Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/TRCSRSLASHExSec11_1_Luc02a

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

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

active₁(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 strictly oriented and are deleted.

HALF₁(ok₁(X)) -> HALF₁(X)

The remaining pairs can at least by weakly be oriented.

HALF₁(mark₁(X)) -> HALF₁(X)

Used ordering: Combined order from the following AFS and order.
HALF₁(x1) = HALF₁(x1)
mark₁(x1) = x1
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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)

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 strictly oriented and are deleted.

HALF₁(mark₁(X)) -> HALF₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
HALF₁(x1) = HALF₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

mark₁ > HALF₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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 strictly oriented and are deleted.

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

The remaining pairs can at least by weakly be oriented.

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

Used ordering: Combined order from the following AFS and order.
FIRST₂(x1, x2) = x2
mark₁(x1) = x1
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

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(X1, mark₁(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 strictly oriented and are deleted.

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

The remaining pairs can at least by weakly be oriented.

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

Used ordering: Combined order from the following AFS and order.
FIRST₂(x1, x2) = FIRST₁(x2)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

[FIRST₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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 strictly oriented and are deleted.

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

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
FIRST₂(x1, x2) = FIRST₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

[FIRST₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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 strictly oriented and are deleted.

DBL₁(ok₁(X)) -> DBL₁(X)

The remaining pairs can at least by weakly be oriented.

DBL₁(mark₁(X)) -> DBL₁(X)

Used ordering: Combined order from the following AFS and order.
DBL₁(x1) = DBL₁(x1)
mark₁(x1) = x1
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

DBL₁(mark₁(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 strictly oriented and are deleted.

DBL₁(mark₁(X)) -> DBL₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
DBL₁(x1) = DBL₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

mark₁ > DBL₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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 strictly oriented and are deleted.

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

The remaining pairs can at least by weakly be oriented.

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

Used ordering: Combined order from the following AFS and order.
ADD₂(x1, x2) = x2
mark₁(x1) = x1
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

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(X1, mark₁(X2)) -> ADD₂(X1, X2)
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 strictly oriented and are deleted.

ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)

The remaining pairs can at least by weakly be oriented.

ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)

Used ordering: Combined order from the following AFS and order.
ADD₂(x1, x2) = ADD₁(x2)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

[ADD₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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 strictly oriented and are deleted.

ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
ADD₂(x1, x2) = ADD₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

[ADD₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ 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 strictly oriented and are deleted.

S₁(mark₁(X)) -> S₁(X)

The remaining pairs can at least by weakly be oriented.

S₁(ok₁(X)) -> S₁(X)

Used ordering: Combined order from the following AFS and order.
S₁(x1) = S₁(x1)
ok₁(x1) = x1
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S₁(ok₁(X)) -> S₁(X)

The TRS R consists of the following rules:

active₁(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 strictly oriented and are deleted.

S₁(ok₁(X)) -> S₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
S₁(x1) = S₁(x1)
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

ok₁ > S₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(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 strictly oriented and are deleted.

SQR₁(mark₁(X)) -> SQR₁(X)

The remaining pairs can at least by weakly be oriented.

SQR₁(ok₁(X)) -> SQR₁(X)

Used ordering: Combined order from the following AFS and order.
SQR₁(x1) = SQR₁(x1)
ok₁(x1) = x1
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

SQR₁(ok₁(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 strictly oriented and are deleted.

SQR₁(ok₁(X)) -> SQR₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
SQR₁(x1) = SQR₁(x1)
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

ok₁ > SQR₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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 strictly oriented and are deleted.

RECIP₁(ok₁(X)) -> RECIP₁(X)

The remaining pairs can at least by weakly be oriented.

RECIP₁(mark₁(X)) -> RECIP₁(X)

Used ordering: Combined order from the following AFS and order.
RECIP₁(x1) = RECIP₁(x1)
mark₁(x1) = x1
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

RECIP₁(mark₁(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 strictly oriented and are deleted.

RECIP₁(mark₁(X)) -> RECIP₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
RECIP₁(x1) = RECIP₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

mark₁ > RECIP₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(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 strictly oriented and are deleted.

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

The remaining pairs can at least by weakly be oriented.

CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)

Used ordering: Combined order from the following AFS and order.
CONS₂(x1, x2) = x2
mark₁(x1) = mark
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)

The TRS R consists of the following rules:

active₁(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 strictly oriented and are deleted.

CONS₂(mark₁(X1), X2) -> CONS₂(X1, X2)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
CONS₂(x1, x2) = CONS₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

[CONS₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ 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 strictly oriented and are deleted.

TERMS₁(ok₁(X)) -> TERMS₁(X)

The remaining pairs can at least by weakly be oriented.

TERMS₁(mark₁(X)) -> TERMS₁(X)

Used ordering: Combined order from the following AFS and order.
TERMS₁(x1) = TERMS₁(x1)
mark₁(x1) = x1
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

TERMS₁(mark₁(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 strictly oriented and are deleted.

TERMS₁(mark₁(X)) -> TERMS₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
TERMS₁(x1) = TERMS₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

mark₁ > TERMS₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(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 strictly oriented and are deleted.

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)

The remaining pairs can at least by weakly be oriented.

PROPER₁(half₁(X)) -> PROPER₁(X)
PROPER₁(sqr₁(X)) -> PROPER₁(X)

Used ordering: Combined order from the following AFS and order.
PROPER₁(x1) = PROPER₁(x1)
half₁(x1) = x1
recip₁(x1) = recip₁(x1)
add₂(x1, x2) = add₂(x1, x2)
s₁(x1) = s₁(x1)
cons₂(x1, x2) = cons₂(x1, x2)
terms₁(x1) = terms₁(x1)
first₂(x1, x2) = first₂(x1, x2)
dbl₁(x1) = dbl₁(x1)
sqr₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

cons₂ > PROPER₁
first₂ > PROPER₁

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(half₁(X)) -> PROPER₁(X)
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 strictly oriented and are deleted.

PROPER₁(sqr₁(X)) -> PROPER₁(X)

The remaining pairs can at least by weakly be oriented.

PROPER₁(half₁(X)) -> PROPER₁(X)

Used ordering: Combined order from the following AFS and order.
PROPER₁(x1) = PROPER₁(x1)
half₁(x1) = x1
sqr₁(x1) = sqr₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

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(half₁(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 strictly oriented and are deleted.

PROPER₁(half₁(X)) -> PROPER₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
PROPER₁(x1) = PROPER₁(x1)
half₁(x1) = half₁(x1)

Lexicographic Path Order [19].
Precedence:

half₁ > PROPER₁

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

                        ↳ 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 strictly oriented and are deleted.

ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(first₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The remaining pairs can at least by weakly be oriented.

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

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = x1
half₁(x1) = x1
dbl₁(x1) = x1
first₂(x1, x2) = first₂(x1, x2)
cons₂(x1, x2) = x1
terms₁(x1) = x1
add₂(x1, x2) = add₂(x1, x2)
sqr₁(x1) = x1
s₁(x1) = x1
recip₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

trivial

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(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(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 strictly oriented and are deleted.

ACTIVE₁(cons₂(X1, X2)) -> ACTIVE₁(X1)

The remaining pairs can at least by weakly be oriented.

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = x1
half₁(x1) = x1
dbl₁(x1) = x1
cons₂(x1, x2) = cons₂(x1, x2)
terms₁(x1) = x1
sqr₁(x1) = x1
s₁(x1) = x1
recip₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

trivial

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(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(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 strictly oriented and are deleted.

ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least by weakly be oriented.

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = ACTIVE₁(x1)
half₁(x1) = x1
dbl₁(x1) = dbl₁(x1)
terms₁(x1) = x1
sqr₁(x1) = x1
s₁(x1) = x1
recip₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

trivial

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

                        ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(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 strictly oriented and are deleted.

ACTIVE₁(terms₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least by weakly be oriented.

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = ACTIVE₁(x1)
half₁(x1) = x1
terms₁(x1) = terms₁(x1)
sqr₁(x1) = x1
s₁(x1) = x1
recip₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

trivial

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

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(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 strictly oriented and are deleted.

ACTIVE₁(sqr₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least by weakly be oriented.

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = ACTIVE₁(x1)
half₁(x1) = x1
sqr₁(x1) = sqr₁(x1)
s₁(x1) = x1
recip₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

trivial

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

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(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 strictly oriented and are deleted.

ACTIVE₁(s₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least by weakly be oriented.

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)
ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = ACTIVE₁(x1)
half₁(x1) = x1
s₁(x1) = s₁(x1)
recip₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

[ACTIVE₁, s_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

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(half₁(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 strictly oriented and are deleted.

ACTIVE₁(recip₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least by weakly be oriented.

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = ACTIVE₁(x1)
half₁(x1) = x1
recip₁(x1) = recip₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

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

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(half₁(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 strictly oriented and are deleted.

ACTIVE₁(half₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = ACTIVE₁(x1)
half₁(x1) = half₁(x1)

Lexicographic Path Order [19].
Precedence:

half₁ > ACTIVE₁

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

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ PisEmptyProof

          ↳ QDP

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

active₁(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

            ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

active₁(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 strictly oriented and are deleted.

TOP₁(mark₁(X)) -> TOP₁(proper₁(X))

The remaining pairs can at least by weakly be oriented.

TOP₁(ok₁(X)) -> TOP₁(active₁(X))

Used ordering: Combined order from the following AFS and order.
TOP₁(x1) = x1
ok₁(x1) = x1
active₁(x1) = x1
mark₁(x1) = mark₁(x1)
proper₁(x1) = x1
terms₁(x1) = terms₁(x1)
cons₂(x1, x2) = x1
recip₁(x1) = recip₁(x1)
sqr₁(x1) = sqr₁(x1)
s₁(x1) = s₁(x1)
0 = 0
add₂(x1, x2) = add₂(x1, x2)
dbl₁(x1) = dbl₁(x1)
first₂(x1, x2) = first₂(x1, x2)
nil = nil
half₁(x1) = half₁(x1)

Lexicographic Path Order [19].
Precedence:

terms₁ > [sqr₁, dbl_1] > add₂ > [mark₁, recip₁, s_1] > [0, nil]
first₂ > [mark₁, recip₁, s_1] > [0, nil]
half₁ > [mark₁, recip₁, s_1] > [0, nil]

The following usable rules [14] were oriented:

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))
half₁(mark₁(X)) -> mark₁(half₁(X))
half₁(ok₁(X)) -> ok₁(half₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
s₁(mark₁(X)) -> mark₁(s₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))
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))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

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

TOP₁(ok₁(X)) -> TOP₁(active₁(X))

The TRS R consists of the following rules:

active₁(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 strictly oriented and are deleted.

TOP₁(ok₁(X)) -> TOP₁(active₁(X))

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
TOP₁(x1) = TOP₁(x1)
ok₁(x1) = ok₁(x1)
active₁(x1) = x1
terms₁(x1) = terms₁(x1)
mark₁(x1) = mark
cons₂(x1, x2) = cons₁(x2)
recip₁(x1) = recip₁(x1)
sqr₁(x1) = x1
s₁(x1) = x1
0 = 0
add₂(x1, x2) = x2
dbl₁(x1) = dbl₁(x1)
first₂(x1, x2) = first₁(x1)
nil = nil
half₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

TOP₁ > mark
[terms₁, cons_1] > first₁ > [ok₁, recip_1] > mark
[terms₁, cons_1] > first₁ > nil > mark
dbl₁ > [ok₁, recip_1] > mark
dbl₁ > 0 > mark

The following usable rules [14] were oriented:

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))
half₁(mark₁(X)) -> mark₁(half₁(X))
half₁(ok₁(X)) -> ok₁(half₁(X))
first₂(mark₁(X1), X2) -> mark₁(first₂(X1, X2))
first₂(X1, mark₁(X2)) -> mark₁(first₂(X1, X2))
first₂(ok₁(X1), ok₁(X2)) -> ok₁(first₂(X1, X2))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
s₁(mark₁(X)) -> mark₁(s₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
sqr₁(mark₁(X)) -> mark₁(sqr₁(X))
sqr₁(ok₁(X)) -> ok₁(sqr₁(X))
recip₁(mark₁(X)) -> mark₁(recip₁(X))
recip₁(ok₁(X)) -> ok₁(recip₁(X))
cons₂(mark₁(X1), X2) -> mark₁(cons₂(X1, X2))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
terms₁(mark₁(X)) -> mark₁(terms₁(X))
terms₁(ok₁(X)) -> ok₁(terms₁(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

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

active₁(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.