Termination w.r.t. Q proof of TRS/TRCSREx2_Luc03b

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ACTIVE₁(from₁(X)) -> FROM₁(s₁(X))
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X1)
PROPER₁(from₁(X)) -> FROM₁(proper₁(X))
ACTIVE₁(fst₂(X1, X2)) -> FST₂(active₁(X1), X2)
PROPER₁(len₁(X)) -> LEN₁(proper₁(X))
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(from₁(X)) -> FROM₁(active₁(X))
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(fst₂(s₁(X), cons₂(Y, Z))) -> FST₂(X, Z)
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
PROPER₁(len₁(X)) -> PROPER₁(X)
ACTIVE₁(fst₂(s₁(X), cons₂(Y, Z))) -> CONS₂(Y, fst₂(X, Z))
FROM₁(mark₁(X)) -> FROM₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(active₁(X1), X2)
ACTIVE₁(len₁(X)) -> LEN₁(active₁(X))
TOP₁(ok₁(X)) -> ACTIVE₁(X)
LEN₁(ok₁(X)) -> LEN₁(X)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(X1, active₁(X2))
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(fst₂(X1, X2)) -> FST₂(X1, active₁(X2))
ACTIVE₁(add₂(s₁(X), Y)) -> ADD₂(X, Y)
FST₂(X1, mark₁(X2)) -> FST₂(X1, X2)
TOP₁(mark₁(X)) -> PROPER₁(X)
ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
PROPER₁(fst₂(X1, X2)) -> FST₂(proper₁(X1), proper₁(X2))
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(add₂(X1, X2)) -> ADD₂(proper₁(X1), proper₁(X2))
PROPER₁(s₁(X)) -> S₁(proper₁(X))
S₁(ok₁(X)) -> S₁(X)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X1)
ACTIVE₁(len₁(cons₂(X, Z))) -> S₁(len₁(Z))
CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)
ACTIVE₁(from₁(X)) -> CONS₂(X, from₁(s₁(X)))
ACTIVE₁(cons₂(X1, X2)) -> CONS₂(active₁(X1), X2)
LEN₁(mark₁(X)) -> LEN₁(X)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(len₁(cons₂(X, Z))) -> LEN₁(Z)
FST₂(ok₁(X1), ok₁(X2)) -> FST₂(X1, X2)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(len₁(X)) -> ACTIVE₁(X)
ACTIVE₁(from₁(X)) -> S₁(X)
ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)
PROPER₁(cons₂(X1, X2)) -> CONS₂(proper₁(X1), proper₁(X2))
TOP₁(ok₁(X)) -> TOP₁(active₁(X))
PROPER₁(from₁(X)) -> PROPER₁(X)
FST₂(mark₁(X1), X2) -> FST₂(X1, X2)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)
FROM₁(ok₁(X)) -> FROM₁(X)
ACTIVE₁(add₂(s₁(X), Y)) -> S₁(add₂(X, Y))

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

ACTIVE₁(from₁(X)) -> FROM₁(s₁(X))
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X1)
PROPER₁(from₁(X)) -> FROM₁(proper₁(X))
ACTIVE₁(fst₂(X1, X2)) -> FST₂(active₁(X1), X2)
PROPER₁(len₁(X)) -> LEN₁(proper₁(X))
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(from₁(X)) -> FROM₁(active₁(X))
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(fst₂(s₁(X), cons₂(Y, Z))) -> FST₂(X, Z)
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
PROPER₁(len₁(X)) -> PROPER₁(X)
ACTIVE₁(fst₂(s₁(X), cons₂(Y, Z))) -> CONS₂(Y, fst₂(X, Z))
FROM₁(mark₁(X)) -> FROM₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(active₁(X1), X2)
ACTIVE₁(len₁(X)) -> LEN₁(active₁(X))
TOP₁(ok₁(X)) -> ACTIVE₁(X)
LEN₁(ok₁(X)) -> LEN₁(X)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(X1, active₁(X2))
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(fst₂(X1, X2)) -> FST₂(X1, active₁(X2))
ACTIVE₁(add₂(s₁(X), Y)) -> ADD₂(X, Y)
FST₂(X1, mark₁(X2)) -> FST₂(X1, X2)
TOP₁(mark₁(X)) -> PROPER₁(X)
ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
PROPER₁(fst₂(X1, X2)) -> FST₂(proper₁(X1), proper₁(X2))
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(add₂(X1, X2)) -> ADD₂(proper₁(X1), proper₁(X2))
PROPER₁(s₁(X)) -> S₁(proper₁(X))
S₁(ok₁(X)) -> S₁(X)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X1)
ACTIVE₁(len₁(cons₂(X, Z))) -> S₁(len₁(Z))
CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)
ACTIVE₁(from₁(X)) -> CONS₂(X, from₁(s₁(X)))
ACTIVE₁(cons₂(X1, X2)) -> CONS₂(active₁(X1), X2)
LEN₁(mark₁(X)) -> LEN₁(X)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(len₁(cons₂(X, Z))) -> LEN₁(Z)
FST₂(ok₁(X1), ok₁(X2)) -> FST₂(X1, X2)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(len₁(X)) -> ACTIVE₁(X)
ACTIVE₁(from₁(X)) -> S₁(X)
ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)
PROPER₁(cons₂(X1, X2)) -> CONS₂(proper₁(X1), proper₁(X2))
TOP₁(ok₁(X)) -> TOP₁(active₁(X))
PROPER₁(from₁(X)) -> PROPER₁(X)
FST₂(mark₁(X1), X2) -> FST₂(X1, X2)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)
FROM₁(ok₁(X)) -> FROM₁(X)
ACTIVE₁(add₂(s₁(X), Y)) -> S₁(add₂(X, Y))

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 9 SCCs with 24 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

S₁(ok₁(X)) -> S₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

S₁(ok₁(X)) -> S₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(S₁(x₁)) = x₁
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LEN₁(mark₁(X)) -> LEN₁(X)
LEN₁(ok₁(X)) -> LEN₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

LEN₁(mark₁(X)) -> LEN₁(X)

The remaining pairs can at least be oriented weakly.

LEN₁(ok₁(X)) -> LEN₁(X)

Used ordering: Polynomial interpretation [21]:

POL(LEN₁(x₁)) = x₁
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

LEN₁(ok₁(X)) -> LEN₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

LEN₁(ok₁(X)) -> LEN₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(LEN₁(x₁)) = x₁
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)
ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)

The remaining pairs can at least be oriented weakly.

ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)

Used ordering: Polynomial interpretation [21]:

POL(ADD₂(x₁, x₂)) = x₂
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)

The remaining pairs can at least be oriented weakly.

ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)

Used ordering: Polynomial interpretation [21]:

POL(ADD₂(x₁, x₂)) = x₂
POL(mark₁(x₁)) = 0
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ADD₂(x₁, x₂)) = x₁
POL(mark₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FROM₁(mark₁(X)) -> FROM₁(X)
FROM₁(ok₁(X)) -> FROM₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FROM₁(mark₁(X)) -> FROM₁(X)

The remaining pairs can at least be oriented weakly.

FROM₁(ok₁(X)) -> FROM₁(X)

Used ordering: Polynomial interpretation [21]:

POL(FROM₁(x₁)) = x₁
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FROM₁(ok₁(X)) -> FROM₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FROM₁(ok₁(X)) -> FROM₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FROM₁(x₁)) = x₁
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FST₂(ok₁(X1), ok₁(X2)) -> FST₂(X1, X2)
FST₂(mark₁(X1), X2) -> FST₂(X1, X2)
FST₂(X1, mark₁(X2)) -> FST₂(X1, X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FST₂(ok₁(X1), ok₁(X2)) -> FST₂(X1, X2)

The remaining pairs can at least be oriented weakly.

FST₂(mark₁(X1), X2) -> FST₂(X1, X2)
FST₂(X1, mark₁(X2)) -> FST₂(X1, X2)

Used ordering: Polynomial interpretation [21]:

POL(FST₂(x₁, x₂)) = x₁
POL(mark₁(x₁)) = x₁
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FST₂(mark₁(X1), X2) -> FST₂(X1, X2)
FST₂(X1, mark₁(X2)) -> FST₂(X1, X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FST₂(mark₁(X1), X2) -> FST₂(X1, X2)

The remaining pairs can at least be oriented weakly.

FST₂(X1, mark₁(X2)) -> FST₂(X1, X2)

Used ordering: Polynomial interpretation [21]:

POL(FST₂(x₁, x₂)) = x₁
POL(mark₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

FST₂(X1, mark₁(X2)) -> FST₂(X1, X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

FST₂(X1, mark₁(X2)) -> FST₂(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FST₂(x₁, x₂)) = x₂
POL(mark₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)
CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)

The remaining pairs can at least be oriented weakly.

CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)

Used ordering: Polynomial interpretation [21]:

POL(CONS₂(x₁, x₂)) = x₂
POL(mark₁(x₁)) = 0
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(CONS₂(x₁, x₂)) = x₁
POL(mark₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(len₁(X)) -> PROPER₁(X)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)

The remaining pairs can at least be oriented weakly.

PROPER₁(fst₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(len₁(X)) -> PROPER₁(X)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X1)

Used ordering: Polynomial interpretation [21]:

POL(PROPER₁(x₁)) = x₁
POL(add₂(x₁, x₂)) = 1 + x₁ + x₂
POL(cons₂(x₁, x₂)) = x₁ + x₂
POL(from₁(x₁)) = x₁
POL(fst₂(x₁, x₂)) = x₁ + x₂
POL(len₁(x₁)) = x₁
POL(s₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER₁(fst₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(len₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER₁(fst₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(fst₂(X1, X2)) -> PROPER₁(X1)

The remaining pairs can at least be oriented weakly.

PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(len₁(X)) -> PROPER₁(X)

Used ordering: Polynomial interpretation [21]:

POL(PROPER₁(x₁)) = x₁
POL(cons₂(x₁, x₂)) = x₁ + x₂
POL(from₁(x₁)) = x₁
POL(fst₂(x₁, x₂)) = 1 + x₁ + x₂
POL(len₁(x₁)) = x₁
POL(s₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(len₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER₁(s₁(X)) -> PROPER₁(X)

The remaining pairs can at least be oriented weakly.

PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(len₁(X)) -> PROPER₁(X)

Used ordering: Polynomial interpretation [21]:

POL(PROPER₁(x₁)) = x₁
POL(cons₂(x₁, x₂)) = x₁ + x₂
POL(from₁(x₁)) = x₁
POL(len₁(x₁)) = x₁
POL(s₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(len₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)

The remaining pairs can at least be oriented weakly.

PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(len₁(X)) -> PROPER₁(X)

Used ordering: Polynomial interpretation [21]:

POL(PROPER₁(x₁)) = x₁
POL(cons₂(x₁, x₂)) = 1 + x₁ + x₂
POL(from₁(x₁)) = x₁
POL(len₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(len₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER₁(from₁(X)) -> PROPER₁(X)

The remaining pairs can at least be oriented weakly.

PROPER₁(len₁(X)) -> PROPER₁(X)

Used ordering: Polynomial interpretation [21]:

POL(PROPER₁(x₁)) = x₁
POL(from₁(x₁)) = 1 + x₁
POL(len₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

PROPER₁(len₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

PROPER₁(len₁(X)) -> PROPER₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROPER₁(x₁)) = x₁
POL(len₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE₁(len₁(X)) -> ACTIVE₁(X)
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE₁(len₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least be oriented weakly.

ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

Used ordering: Polynomial interpretation [21]:

POL(ACTIVE₁(x₁)) = x₁
POL(add₂(x₁, x₂)) = x₁ + x₂
POL(cons₂(x₁, x₂)) = x₁
POL(from₁(x₁)) = x₁
POL(fst₂(x₁, x₂)) = x₁ + x₂
POL(len₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(fst₂(X1, X2)) -> ACTIVE₁(X2)

The remaining pairs can at least be oriented weakly.

ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

Used ordering: Polynomial interpretation [21]:

POL(ACTIVE₁(x₁)) = x₁
POL(add₂(x₁, x₂)) = x₁ + x₂
POL(cons₂(x₁, x₂)) = x₁
POL(from₁(x₁)) = x₁
POL(fst₂(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)

The remaining pairs can at least be oriented weakly.

ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

Used ordering: Polynomial interpretation [21]:

POL(ACTIVE₁(x₁)) = x₁
POL(add₂(x₁, x₂)) = x₁ + x₂
POL(cons₂(x₁, x₂)) = 1 + x₁
POL(from₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(from₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The remaining pairs can at least be oriented weakly.

ACTIVE₁(from₁(X)) -> ACTIVE₁(X)

Used ordering: Polynomial interpretation [21]:

POL(ACTIVE₁(x₁)) = x₁
POL(add₂(x₁, x₂)) = 1 + x₁ + x₂
POL(from₁(x₁)) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

          ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

ACTIVE₁(from₁(X)) -> ACTIVE₁(X)

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

ACTIVE₁(from₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE₁(x₁)) = x₁
POL(from₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ PisEmptyProof

          ↳ QDP

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP₁(ok₁(X)) -> TOP₁(active₁(X))
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

TOP₁(mark₁(X)) -> TOP₁(proper₁(X))

The remaining pairs can at least be oriented weakly.

TOP₁(ok₁(X)) -> TOP₁(active₁(X))

Used ordering: Polynomial interpretation [21]:

POL(0) = 0
POL(TOP₁(x₁)) = x₁
POL(active₁(x₁)) = x₁
POL(add₂(x₁, x₂)) = 1 + x₁ + x₂
POL(cons₂(x₁, x₂)) = x₁
POL(from₁(x₁)) = 1 + x₁
POL(fst₂(x₁, x₂)) = 1 + x₁ + x₂
POL(len₁(x₁)) = 1 + x₁
POL(mark₁(x₁)) = 1 + x₁
POL(nil) = 0
POL(ok₁(x₁)) = x₁
POL(proper₁(x₁)) = x₁
POL(s₁(x₁)) = 0

The following usable rules [14] were oriented:

add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
active₁(len₁(nil)) -> mark₁(0)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
proper₁(0) -> ok₁(0)
s₁(ok₁(X)) -> ok₁(s₁(X))
from₁(ok₁(X)) -> ok₁(from₁(X))
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(add₂(0, X)) -> mark₁(X)
proper₁(s₁(X)) -> s₁(proper₁(X))
len₁(mark₁(X)) -> mark₁(len₁(X))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(len₁(X)) -> len₁(active₁(X))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(nil) -> ok₁(nil)
active₁(from₁(X)) -> from₁(active₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(fst₂(0, Z)) -> mark₁(nil)
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
from₁(mark₁(X)) -> mark₁(from₁(X))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
len₁(ok₁(X)) -> ok₁(len₁(X))
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

TOP₁(ok₁(X)) -> TOP₁(active₁(X))

The TRS R consists of the following rules:

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

TOP₁(ok₁(X)) -> TOP₁(active₁(X))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(0) = 0
POL(TOP₁(x₁)) = x₁
POL(active₁(x₁)) = x₁
POL(add₂(x₁, x₂)) = x₂
POL(cons₂(x₁, x₂)) = x₁
POL(from₁(x₁)) = x₁
POL(fst₂(x₁, x₂)) = x₁
POL(len₁(x₁)) = x₁
POL(mark₁(x₁)) = 0
POL(nil) = 0
POL(ok₁(x₁)) = 1 + x₁
POL(s₁(x₁)) = 0

The following usable rules [14] were oriented:

active₁(len₁(nil)) -> mark₁(0)
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
from₁(ok₁(X)) -> ok₁(from₁(X))
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(add₂(0, X)) -> mark₁(X)
len₁(mark₁(X)) -> mark₁(len₁(X))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(len₁(X)) -> len₁(active₁(X))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
active₁(from₁(X)) -> from₁(active₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
active₁(fst₂(0, Z)) -> mark₁(nil)
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
from₁(mark₁(X)) -> mark₁(from₁(X))
len₁(ok₁(X)) -> ok₁(len₁(X))
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

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

active₁(fst₂(0, Z)) -> mark₁(nil)
active₁(fst₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, fst₂(X, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(len₁(nil)) -> mark₁(0)
active₁(len₁(cons₂(X, Z))) -> mark₁(s₁(len₁(Z)))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(active₁(X1), X2)
active₁(fst₂(X1, X2)) -> fst₂(X1, active₁(X2))
active₁(from₁(X)) -> from₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(len₁(X)) -> len₁(active₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
fst₂(mark₁(X1), X2) -> mark₁(fst₂(X1, X2))
fst₂(X1, mark₁(X2)) -> mark₁(fst₂(X1, X2))
from₁(mark₁(X)) -> mark₁(from₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
len₁(mark₁(X)) -> mark₁(len₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(fst₂(X1, X2)) -> fst₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(len₁(X)) -> len₁(proper₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
fst₂(ok₁(X1), ok₁(X2)) -> ok₁(fst₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
len₁(ok₁(X)) -> ok₁(len₁(X))
top₁(mark₁(X)) -> top₁(proper₁(X))
top₁(ok₁(X)) -> top₁(active₁(X))

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