Termination w.r.t. Q proof of TRS/TRCSRExSec11_1_Luc02a

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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:

FIRST₂(X1, mark₁(X2)) -> FIRST₂(X1, X2)
RECIP₁(mark₁(X)) -> RECIP₁(X)
ACTIVE₁(recip₁(X)) -> RECIP₁(active₁(X))
S₁(mark₁(X)) -> S₁(X)
FIRST₂(mark₁(X1), X2) -> FIRST₂(X1, X2)
RECIP₁(ok₁(X)) -> RECIP₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)
PROPER₁(half₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl₁(s₁(X))) -> DBL₁(X)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(terms₁(N)) -> SQR₁(N)
ACTIVE₁(first₂(X1, X2)) -> FIRST₂(active₁(X1), X2)
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
PROPER₁(sqr₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(active₁(X1), X2)
SQR₁(mark₁(X)) -> SQR₁(X)
TOP₁(ok₁(X)) -> ACTIVE₁(X)
ACTIVE₁(terms₁(N)) -> CONS₂(recip₁(sqr₁(N)), terms₁(s₁(N)))
PROPER₁(sqr₁(X)) -> SQR₁(proper₁(X))
ACTIVE₁(half₁(s₁(s₁(X)))) -> S₁(half₁(X))
ACTIVE₁(s₁(X)) -> S₁(active₁(X))
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(X1, active₁(X2))
PROPER₁(recip₁(X)) -> RECIP₁(proper₁(X))
ACTIVE₁(add₂(s₁(X), Y)) -> ADD₂(X, Y)
ACTIVE₁(sqr₁(s₁(X))) -> ADD₂(sqr₁(X), dbl₁(X))
ACTIVE₁(terms₁(N)) -> RECIP₁(sqr₁(N))
ACTIVE₁(first₂(s₁(X), cons₂(Y, Z))) -> FIRST₂(X, Z)
TOP₁(mark₁(X)) -> PROPER₁(X)
ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
PROPER₁(terms₁(X)) -> PROPER₁(X)
DBL₁(ok₁(X)) -> DBL₁(X)
PROPER₁(half₁(X)) -> HALF₁(proper₁(X))
PROPER₁(first₂(X1, X2)) -> PROPER₁(X1)
HALF₁(mark₁(X)) -> HALF₁(X)
PROPER₁(first₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(sqr₁(s₁(X))) -> DBL₁(X)
ACTIVE₁(first₂(X1, X2)) -> FIRST₂(X1, active₁(X2))
ACTIVE₁(dbl₁(X)) -> DBL₁(active₁(X))
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)
ACTIVE₁(first₂(s₁(X), cons₂(Y, Z))) -> CONS₂(Y, first₂(X, Z))
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
PROPER₁(recip₁(X)) -> PROPER₁(X)
ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)
ACTIVE₁(cons₂(X1, X2)) -> CONS₂(active₁(X1), X2)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(half₁(s₁(s₁(X)))) -> HALF₁(X)
ACTIVE₁(sqr₁(s₁(X))) -> S₁(add₂(sqr₁(X), dbl₁(X)))
TERMS₁(ok₁(X)) -> TERMS₁(X)
ACTIVE₁(terms₁(X)) -> TERMS₁(active₁(X))
ACTIVE₁(sqr₁(X)) -> SQR₁(active₁(X))
ACTIVE₁(terms₁(N)) -> TERMS₁(s₁(N))
DBL₁(mark₁(X)) -> DBL₁(X)
PROPER₁(first₂(X1, X2)) -> FIRST₂(proper₁(X1), proper₁(X2))
ACTIVE₁(dbl₁(s₁(X))) -> S₁(dbl₁(X))
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(half₁(X)) -> HALF₁(active₁(X))
ACTIVE₁(sqr₁(s₁(X))) -> SQR₁(X)
FIRST₂(ok₁(X1), ok₁(X2)) -> FIRST₂(X1, X2)
ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)
PROPER₁(dbl₁(X)) -> DBL₁(proper₁(X))
PROPER₁(cons₂(X1, X2)) -> CONS₂(proper₁(X1), proper₁(X2))
SQR₁(ok₁(X)) -> SQR₁(X)
PROPER₁(terms₁(X)) -> TERMS₁(proper₁(X))
TERMS₁(mark₁(X)) -> TERMS₁(X)
TOP₁(ok₁(X)) -> TOP₁(active₁(X))
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(terms₁(N)) -> S₁(N)
CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)
ACTIVE₁(dbl₁(s₁(X))) -> S₁(s₁(dbl₁(X)))
HALF₁(ok₁(X)) -> HALF₁(X)
PROPER₁(dbl₁(X)) -> PROPER₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(s₁(X), Y)) -> S₁(add₂(X, Y))

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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:

FIRST₂(X1, mark₁(X2)) -> FIRST₂(X1, X2)
RECIP₁(mark₁(X)) -> RECIP₁(X)
ACTIVE₁(recip₁(X)) -> RECIP₁(active₁(X))
S₁(mark₁(X)) -> S₁(X)
FIRST₂(mark₁(X1), X2) -> FIRST₂(X1, X2)
RECIP₁(ok₁(X)) -> RECIP₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)
PROPER₁(half₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl₁(s₁(X))) -> DBL₁(X)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(terms₁(N)) -> SQR₁(N)
ACTIVE₁(first₂(X1, X2)) -> FIRST₂(active₁(X1), X2)
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
PROPER₁(sqr₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(active₁(X1), X2)
SQR₁(mark₁(X)) -> SQR₁(X)
TOP₁(ok₁(X)) -> ACTIVE₁(X)
ACTIVE₁(terms₁(N)) -> CONS₂(recip₁(sqr₁(N)), terms₁(s₁(N)))
PROPER₁(sqr₁(X)) -> SQR₁(proper₁(X))
ACTIVE₁(half₁(s₁(s₁(X)))) -> S₁(half₁(X))
ACTIVE₁(s₁(X)) -> S₁(active₁(X))
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(X1, active₁(X2))
PROPER₁(recip₁(X)) -> RECIP₁(proper₁(X))
ACTIVE₁(add₂(s₁(X), Y)) -> ADD₂(X, Y)
ACTIVE₁(sqr₁(s₁(X))) -> ADD₂(sqr₁(X), dbl₁(X))
ACTIVE₁(terms₁(N)) -> RECIP₁(sqr₁(N))
ACTIVE₁(first₂(s₁(X), cons₂(Y, Z))) -> FIRST₂(X, Z)
TOP₁(mark₁(X)) -> PROPER₁(X)
ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
PROPER₁(terms₁(X)) -> PROPER₁(X)
DBL₁(ok₁(X)) -> DBL₁(X)
PROPER₁(half₁(X)) -> HALF₁(proper₁(X))
PROPER₁(first₂(X1, X2)) -> PROPER₁(X1)
HALF₁(mark₁(X)) -> HALF₁(X)
PROPER₁(first₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(sqr₁(s₁(X))) -> DBL₁(X)
ACTIVE₁(first₂(X1, X2)) -> FIRST₂(X1, active₁(X2))
ACTIVE₁(dbl₁(X)) -> DBL₁(active₁(X))
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)
ACTIVE₁(first₂(s₁(X), cons₂(Y, Z))) -> CONS₂(Y, first₂(X, Z))
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
PROPER₁(recip₁(X)) -> PROPER₁(X)
ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)
ACTIVE₁(cons₂(X1, X2)) -> CONS₂(active₁(X1), X2)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(half₁(s₁(s₁(X)))) -> HALF₁(X)
ACTIVE₁(sqr₁(s₁(X))) -> S₁(add₂(sqr₁(X), dbl₁(X)))
TERMS₁(ok₁(X)) -> TERMS₁(X)
ACTIVE₁(terms₁(X)) -> TERMS₁(active₁(X))
ACTIVE₁(sqr₁(X)) -> SQR₁(active₁(X))
ACTIVE₁(terms₁(N)) -> TERMS₁(s₁(N))
DBL₁(mark₁(X)) -> DBL₁(X)
PROPER₁(first₂(X1, X2)) -> FIRST₂(proper₁(X1), proper₁(X2))
ACTIVE₁(dbl₁(s₁(X))) -> S₁(dbl₁(X))
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(half₁(X)) -> HALF₁(active₁(X))
ACTIVE₁(sqr₁(s₁(X))) -> SQR₁(X)
FIRST₂(ok₁(X1), ok₁(X2)) -> FIRST₂(X1, X2)
ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)
PROPER₁(dbl₁(X)) -> DBL₁(proper₁(X))
PROPER₁(cons₂(X1, X2)) -> CONS₂(proper₁(X1), proper₁(X2))
SQR₁(ok₁(X)) -> SQR₁(X)
PROPER₁(terms₁(X)) -> TERMS₁(proper₁(X))
TERMS₁(mark₁(X)) -> TERMS₁(X)
TOP₁(ok₁(X)) -> TOP₁(active₁(X))
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(terms₁(N)) -> S₁(N)
CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)
ACTIVE₁(dbl₁(s₁(X))) -> S₁(s₁(dbl₁(X)))
HALF₁(ok₁(X)) -> HALF₁(X)
PROPER₁(dbl₁(X)) -> PROPER₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(s₁(X), Y)) -> S₁(add₂(X, Y))

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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 12 SCCs with 40 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

HALF₁(mark₁(X)) -> HALF₁(X)
HALF₁(ok₁(X)) -> HALF₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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.

HALF₁(mark₁(X)) -> HALF₁(X)
HALF₁(ok₁(X)) -> HALF₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( HALF₁(x₁) ) = 2x₁

POL( mark₁(x₁) ) = 3x₁ + 1

POL( ok₁(x₁) ) = 2x₁ + 2

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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

FIRST₂(X1, mark₁(X2)) -> FIRST₂(X1, X2)
FIRST₂(ok₁(X1), ok₁(X2)) -> FIRST₂(X1, X2)
FIRST₂(mark₁(X1), X2) -> FIRST₂(X1, X2)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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.

FIRST₂(X1, mark₁(X2)) -> FIRST₂(X1, X2)
FIRST₂(ok₁(X1), ok₁(X2)) -> FIRST₂(X1, X2)

The remaining pairs can at least be oriented weakly.

FIRST₂(mark₁(X1), X2) -> FIRST₂(X1, X2)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FIRST₂(x₁, x₂) ) = 3x₂ + 2

POL( mark₁(x₁) ) = 2x₁ + 1

POL( ok₁(x₁) ) = x₁ + 3

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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

FIRST₂(mark₁(X1), X2) -> FIRST₂(X1, X2)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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.

FIRST₂(mark₁(X1), X2) -> FIRST₂(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FIRST₂(x₁, x₂) ) = 2x₁ + 3x₂ + 2

POL( mark₁(x₁) ) = x₁ + 1

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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

DBL₁(mark₁(X)) -> DBL₁(X)
DBL₁(ok₁(X)) -> DBL₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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.

DBL₁(mark₁(X)) -> DBL₁(X)
DBL₁(ok₁(X)) -> DBL₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( mark₁(x₁) ) = 3x₁ + 1

POL( DBL₁(x₁) ) = 2x₁

POL( ok₁(x₁) ) = 2x₁ + 2

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ 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(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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)
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 Order [17,21] with Interpretation:

POL( ADD₂(x₁, x₂) ) = 3x₂ + 2

POL( mark₁(x₁) ) = 2x₁ + 1

POL( ok₁(x₁) ) = x₁ + 3

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

          ↳ 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(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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 Order [17,21] with Interpretation:

POL( mark₁(x₁) ) = x₁ + 1

POL( ADD₂(x₁, x₂) ) = 2x₁ + 3x₂ + 2

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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S₁(ok₁(X)) -> S₁(X)
S₁(mark₁(X)) -> S₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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)
S₁(mark₁(X)) -> S₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( S₁(x₁) ) = 2x₁

POL( mark₁(x₁) ) = 2x₁ + 2

POL( ok₁(x₁) ) = 3x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

SQR₁(ok₁(X)) -> SQR₁(X)
SQR₁(mark₁(X)) -> SQR₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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.

SQR₁(ok₁(X)) -> SQR₁(X)
SQR₁(mark₁(X)) -> SQR₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( SQR₁(x₁) ) = 2x₁

POL( mark₁(x₁) ) = 2x₁ + 2

POL( ok₁(x₁) ) = 3x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

RECIP₁(mark₁(X)) -> RECIP₁(X)
RECIP₁(ok₁(X)) -> RECIP₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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.

RECIP₁(mark₁(X)) -> RECIP₁(X)
RECIP₁(ok₁(X)) -> RECIP₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( mark₁(x₁) ) = 3x₁ + 1

POL( ok₁(x₁) ) = 2x₁ + 2

POL( RECIP₁(x₁) ) = 2x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ 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(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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)
CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( CONS₂(x₁, x₂) ) = max{0, 2x₁ + 2x₂ - 1}

POL( mark₁(x₁) ) = 2x₁ + 1

POL( ok₁(x₁) ) = x₁ + 3

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

TERMS₁(mark₁(X)) -> TERMS₁(X)
TERMS₁(ok₁(X)) -> TERMS₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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.

TERMS₁(mark₁(X)) -> TERMS₁(X)
TERMS₁(ok₁(X)) -> TERMS₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( TERMS₁(x₁) ) = 2x₁

POL( mark₁(x₁) ) = 3x₁ + 1

POL( ok₁(x₁) ) = 2x₁ + 2

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

PROPER₁(half₁(X)) -> PROPER₁(X)
PROPER₁(recip₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(terms₁(X)) -> PROPER₁(X)
PROPER₁(first₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(first₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbl₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(sqr₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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(half₁(X)) -> PROPER₁(X)
PROPER₁(first₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(first₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(sqr₁(X)) -> PROPER₁(X)

The remaining pairs can at least be oriented weakly.

PROPER₁(recip₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(terms₁(X)) -> PROPER₁(X)
PROPER₁(dbl₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( half₁(x₁) ) = 3x₁ + 1

POL( dbl₁(x₁) ) = 2x₁

POL( PROPER₁(x₁) ) = 3x₁

POL( first₂(x₁, x₂) ) = 2x₁ + 2x₂ + 1

POL( recip₁(x₁) ) = x₁

POL( terms₁(x₁) ) = 2x₁

POL( sqr₁(x₁) ) = 2x₁ + 2

POL( add₂(x₁, x₂) ) = 2x₁ + 2x₂

POL( s₁(x₁) ) = x₁

POL( cons₂(x₁, x₂) ) = 2x₁ + 2x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

PROPER₁(recip₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(terms₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbl₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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)
PROPER₁(terms₁(X)) -> PROPER₁(X)

The remaining pairs can at least be oriented weakly.

PROPER₁(recip₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbl₁(X)) -> PROPER₁(X)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( dbl₁(x₁) ) = 2x₁

POL( PROPER₁(x₁) ) = 3x₁

POL( terms₁(x₁) ) = 2x₁ + 1

POL( recip₁(x₁) ) = x₁

POL( add₂(x₁, x₂) ) = 3x₁ + 3x₂

POL( s₁(x₁) ) = x₁ + 2

POL( cons₂(x₁, x₂) ) = x₁ + 2x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(recip₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbl₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbl₁(X)) -> PROPER₁(X)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)

The remaining pairs can at least be oriented weakly.

PROPER₁(recip₁(X)) -> PROPER₁(X)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( dbl₁(x₁) ) = 2x₁ + 2

POL( cons₂(x₁, x₂) ) = 3x₁ + 3x₂ + 1

POL( PROPER₁(x₁) ) = 3x₁

POL( recip₁(x₁) ) = 3x₁

POL( add₂(x₁, x₂) ) = 2x₁ + 2x₂ + 3

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(recip₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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(recip₁(X)) -> PROPER₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER₁(x₁) ) = 2x₁ + 3

POL( recip₁(x₁) ) = 3x₁ + 3

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The remaining pairs can at least be oriented weakly.

ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE₁(x₁) ) = 3x₁

POL( half₁(x₁) ) = x₁ + 3

POL( dbl₁(x₁) ) = x₁ + 2

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( terms₁(x₁) ) = 3x₁

POL( sqr₁(x₁) ) = 3x₁ + 3

POL( recip₁(x₁) ) = 2x₁

POL( add₂(x₁, x₂) ) = 3x₁ + 2x₂ + 1

POL( s₁(x₁) ) = 2x₁ + 1

POL( cons₂(x₁, x₂) ) = 2x₁ + x₂

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least be oriented weakly.

ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE₁(x₁) ) = 2x₁

POL( cons₂(x₁, x₂) ) = 2x₁ + 3x₂ + 2

POL( first₂(x₁, x₂) ) = 3x₁ + 3x₂

POL( terms₁(x₁) ) = x₁ + 2

POL( recip₁(x₁) ) = 2x₁ + 2

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)

The TRS R consists of the following rules:

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE₁(x₁) ) = x₁ + 3

POL( first₂(x₁, x₂) ) = 2x₁ + 2x₂ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

          ↳ QDP

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

active₁(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

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(terms₁(N)) -> mark₁(cons₂(recip₁(sqr₁(N)), terms₁(s₁(N))))
active₁(sqr₁(0)) -> mark₁(0)
active₁(sqr₁(s₁(X))) -> mark₁(s₁(add₂(sqr₁(X), dbl₁(X))))
active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(first₂(0, X)) -> mark₁(nil)
active₁(first₂(s₁(X), cons₂(Y, Z))) -> mark₁(cons₂(Y, first₂(X, Z)))
active₁(half₁(0)) -> mark₁(0)
active₁(half₁(s₁(0))) -> mark₁(0)
active₁(half₁(s₁(s₁(X)))) -> mark₁(s₁(half₁(X)))
active₁(half₁(dbl₁(X))) -> mark₁(X)
active₁(terms₁(X)) -> terms₁(active₁(X))
active₁(cons₂(X1, X2)) -> cons₂(active₁(X1), X2)
active₁(recip₁(X)) -> recip₁(active₁(X))
active₁(sqr₁(X)) -> sqr₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(first₂(X1, X2)) -> first₂(active₁(X1), X2)
active₁(first₂(X1, X2)) -> first₂(X1, active₁(X2))
active₁(half₁(X)) -> half₁(active₁(X))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
half₁(mark₁(X)) -> mark₁(half₁(X))
proper₁(terms₁(X)) -> terms₁(proper₁(X))
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(recip₁(X)) -> recip₁(proper₁(X))
proper₁(sqr₁(X)) -> sqr₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(first₂(X1, X2)) -> first₂(proper₁(X1), proper₁(X2))
proper₁(nil) -> ok₁(nil)
proper₁(half₁(X)) -> half₁(proper₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
half₁(ok₁(X)) -> ok₁(half₁(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.