Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/TRCSRSLASHExIntrod_Zan97

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

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

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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:

P₁(mark₁(X)) -> P₁(X)
ACTIVE₁(zero₁(X)) -> ZERO₁(active₁(X))
PROPER₁(prod₂(X1, X2)) -> PROPER₁(X2)
S₁(mark₁(X)) -> S₁(X)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(if₃(X1, X2, X3)) -> IF₃(active₁(X1), X2, X3)
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
PROD₂(mark₁(X1), X2) -> PROD₂(X1, X2)
ZERO₁(mark₁(X)) -> ZERO₁(X)
IF₃(mark₁(X1), X2, X3) -> IF₃(X1, X2, X3)
PROD₂(ok₁(X1), ok₁(X2)) -> PROD₂(X1, X2)
ACTIVE₁(prod₂(X1, X2)) -> ACTIVE₁(X2)
FACT₁(mark₁(X)) -> FACT₁(X)
PROPER₁(zero₁(X)) -> PROPER₁(X)
P₁(ok₁(X)) -> P₁(X)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(active₁(X1), X2)
ACTIVE₁(fact₁(X)) -> FACT₁(p₁(X))
TOP₁(ok₁(X)) -> ACTIVE₁(X)
ACTIVE₁(prod₂(s₁(X), Y)) -> ADD₂(Y, prod₂(X, Y))
ACTIVE₁(s₁(X)) -> S₁(active₁(X))
ACTIVE₁(add₂(X1, X2)) -> ADD₂(X1, active₁(X2))
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(fact₁(X)) -> PROD₂(X, fact₁(p₁(X)))
ACTIVE₁(add₂(s₁(X), Y)) -> ADD₂(X, Y)
PROPER₁(prod₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(p₁(X)) -> PROPER₁(X)
PROPER₁(prod₂(X1, X2)) -> PROD₂(proper₁(X1), proper₁(X2))
TOP₁(mark₁(X)) -> PROPER₁(X)
ACTIVE₁(fact₁(X)) -> FACT₁(active₁(X))
ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
ACTIVE₁(fact₁(X)) -> P₁(X)
PROD₂(X1, mark₁(X2)) -> PROD₂(X1, X2)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X1)
IF₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> IF₃(X1, X2, X3)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(add₂(X1, X2)) -> ADD₂(proper₁(X1), proper₁(X2))
PROPER₁(if₃(X1, X2, X3)) -> IF₃(proper₁(X1), proper₁(X2), proper₁(X3))
PROPER₁(fact₁(X)) -> PROPER₁(X)
PROPER₁(p₁(X)) -> P₁(proper₁(X))
ACTIVE₁(prod₂(X1, X2)) -> PROD₂(active₁(X1), X2)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X3)
PROPER₁(s₁(X)) -> S₁(proper₁(X))
FACT₁(ok₁(X)) -> FACT₁(X)
PROPER₁(fact₁(X)) -> FACT₁(proper₁(X))
S₁(ok₁(X)) -> S₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(prod₂(s₁(X), Y)) -> PROD₂(X, Y)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X2)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)
ACTIVE₁(prod₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(fact₁(X)) -> S₁(0)
ACTIVE₁(zero₁(X)) -> ACTIVE₁(X)
ACTIVE₁(fact₁(X)) -> IF₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(p₁(X)) -> P₁(active₁(X))
ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)
PROPER₁(zero₁(X)) -> ZERO₁(proper₁(X))
TOP₁(ok₁(X)) -> TOP₁(active₁(X))
ACTIVE₁(if₃(X1, X2, X3)) -> ACTIVE₁(X1)
ACTIVE₁(fact₁(X)) -> ACTIVE₁(X)
ACTIVE₁(prod₂(X1, X2)) -> PROD₂(X1, active₁(X2))
ACTIVE₁(fact₁(X)) -> ZERO₁(X)
ZERO₁(ok₁(X)) -> ZERO₁(X)
ACTIVE₁(add₂(s₁(X), Y)) -> S₁(add₂(X, Y))

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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:

P₁(mark₁(X)) -> P₁(X)
ACTIVE₁(zero₁(X)) -> ZERO₁(active₁(X))
PROPER₁(prod₂(X1, X2)) -> PROPER₁(X2)
S₁(mark₁(X)) -> S₁(X)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(if₃(X1, X2, X3)) -> IF₃(active₁(X1), X2, X3)
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
PROD₂(mark₁(X1), X2) -> PROD₂(X1, X2)
ZERO₁(mark₁(X)) -> ZERO₁(X)
IF₃(mark₁(X1), X2, X3) -> IF₃(X1, X2, X3)
PROD₂(ok₁(X1), ok₁(X2)) -> PROD₂(X1, X2)
ACTIVE₁(prod₂(X1, X2)) -> ACTIVE₁(X2)
FACT₁(mark₁(X)) -> FACT₁(X)
PROPER₁(zero₁(X)) -> PROPER₁(X)
P₁(ok₁(X)) -> P₁(X)
ACTIVE₁(add₂(X1, X2)) -> ADD₂(active₁(X1), X2)
ACTIVE₁(fact₁(X)) -> FACT₁(p₁(X))
TOP₁(ok₁(X)) -> ACTIVE₁(X)
ACTIVE₁(prod₂(s₁(X), Y)) -> ADD₂(Y, prod₂(X, Y))
ACTIVE₁(s₁(X)) -> S₁(active₁(X))
ACTIVE₁(add₂(X1, X2)) -> ADD₂(X1, active₁(X2))
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(fact₁(X)) -> PROD₂(X, fact₁(p₁(X)))
ACTIVE₁(add₂(s₁(X), Y)) -> ADD₂(X, Y)
PROPER₁(prod₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(p₁(X)) -> PROPER₁(X)
PROPER₁(prod₂(X1, X2)) -> PROD₂(proper₁(X1), proper₁(X2))
TOP₁(mark₁(X)) -> PROPER₁(X)
ACTIVE₁(fact₁(X)) -> FACT₁(active₁(X))
ADD₂(mark₁(X1), X2) -> ADD₂(X1, X2)
ACTIVE₁(fact₁(X)) -> P₁(X)
PROD₂(X1, mark₁(X2)) -> PROD₂(X1, X2)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X1)
IF₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> IF₃(X1, X2, X3)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(add₂(X1, X2)) -> ADD₂(proper₁(X1), proper₁(X2))
PROPER₁(if₃(X1, X2, X3)) -> IF₃(proper₁(X1), proper₁(X2), proper₁(X3))
PROPER₁(fact₁(X)) -> PROPER₁(X)
PROPER₁(p₁(X)) -> P₁(proper₁(X))
ACTIVE₁(prod₂(X1, X2)) -> PROD₂(active₁(X1), X2)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X3)
PROPER₁(s₁(X)) -> S₁(proper₁(X))
FACT₁(ok₁(X)) -> FACT₁(X)
PROPER₁(fact₁(X)) -> FACT₁(proper₁(X))
S₁(ok₁(X)) -> S₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(prod₂(s₁(X), Y)) -> PROD₂(X, Y)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X2)
ADD₂(ok₁(X1), ok₁(X2)) -> ADD₂(X1, X2)
ACTIVE₁(prod₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(fact₁(X)) -> S₁(0)
ACTIVE₁(zero₁(X)) -> ACTIVE₁(X)
ACTIVE₁(fact₁(X)) -> IF₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X))))
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(p₁(X)) -> P₁(active₁(X))
ADD₂(X1, mark₁(X2)) -> ADD₂(X1, X2)
PROPER₁(zero₁(X)) -> ZERO₁(proper₁(X))
TOP₁(ok₁(X)) -> TOP₁(active₁(X))
ACTIVE₁(if₃(X1, X2, X3)) -> ACTIVE₁(X1)
ACTIVE₁(fact₁(X)) -> ACTIVE₁(X)
ACTIVE₁(prod₂(X1, X2)) -> PROD₂(X1, active₁(X2))
ACTIVE₁(fact₁(X)) -> ZERO₁(X)
ZERO₁(ok₁(X)) -> ZERO₁(X)
ACTIVE₁(add₂(s₁(X), Y)) -> S₁(add₂(X, Y))

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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 10 SCCs with 28 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ 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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ 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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

          ↳ 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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

            ↳ QDPOrderProof

              ↳ 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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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

P₁(mark₁(X)) -> P₁(X)
P₁(ok₁(X)) -> P₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

P₁(ok₁(X)) -> P₁(X)

The remaining pairs can at least by weakly be oriented.

P₁(mark₁(X)) -> P₁(X)

Used ordering: Combined order from the following AFS and order.
P₁(x1) = P₁(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

P₁(mark₁(X)) -> P₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

P₁(mark₁(X)) -> P₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
P₁(x1) = P₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

mark₁ > P₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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

PROD₂(ok₁(X1), ok₁(X2)) -> PROD₂(X1, X2)
PROD₂(X1, mark₁(X2)) -> PROD₂(X1, X2)
PROD₂(mark₁(X1), X2) -> PROD₂(X1, X2)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

PROD₂(ok₁(X1), ok₁(X2)) -> PROD₂(X1, X2)

The remaining pairs can at least by weakly be oriented.

PROD₂(X1, mark₁(X2)) -> PROD₂(X1, X2)
PROD₂(mark₁(X1), X2) -> PROD₂(X1, X2)

Used ordering: Combined order from the following AFS and order.
PROD₂(x1, x2) = PROD₁(x2)
ok₁(x1) = ok₁(x1)
mark₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

ok₁ > PROD₁

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

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

PROD₂(X1, mark₁(X2)) -> PROD₂(X1, X2)
PROD₂(mark₁(X1), X2) -> PROD₂(X1, X2)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

PROD₂(X1, mark₁(X2)) -> PROD₂(X1, X2)

The remaining pairs can at least by weakly be oriented.

PROD₂(mark₁(X1), X2) -> PROD₂(X1, X2)

Used ordering: Combined order from the following AFS and order.
PROD₂(x1, x2) = PROD₁(x2)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

[PROD₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

PROD₂(mark₁(X1), X2) -> PROD₂(X1, X2)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

PROD₂(mark₁(X1), X2) -> PROD₂(X1, X2)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
PROD₂(x1, x2) = PROD₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

[PROD₁, mark_1]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ 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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

            ↳ 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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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

ZERO₁(mark₁(X)) -> ZERO₁(X)
ZERO₁(ok₁(X)) -> ZERO₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

ZERO₁(ok₁(X)) -> ZERO₁(X)

The remaining pairs can at least by weakly be oriented.

ZERO₁(mark₁(X)) -> ZERO₁(X)

Used ordering: Combined order from the following AFS and order.
ZERO₁(x1) = ZERO₁(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ZERO₁(mark₁(X)) -> ZERO₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

ZERO₁(mark₁(X)) -> ZERO₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
ZERO₁(x1) = ZERO₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

mark₁ > ZERO₁

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

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

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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

IF₃(mark₁(X1), X2, X3) -> IF₃(X1, X2, X3)
IF₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> IF₃(X1, X2, X3)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

IF₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> IF₃(X1, X2, X3)

The remaining pairs can at least by weakly be oriented.

IF₃(mark₁(X1), X2, X3) -> IF₃(X1, X2, X3)

Used ordering: Combined order from the following AFS and order.
IF₃(x1, x2, x3) = IF₃(x1, x2, x3)
mark₁(x1) = x1
ok₁(x1) = ok₁(x1)

Lexicographic Path Order [19].
Precedence:

ok₁ > IF₃

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

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

IF₃(mark₁(X1), X2, X3) -> IF₃(X1, X2, X3)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

IF₃(mark₁(X1), X2, X3) -> IF₃(X1, X2, X3)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
IF₃(x1, x2, x3) = IF₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

[IF₁, mark_1]

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

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

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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

FACT₁(mark₁(X)) -> FACT₁(X)
FACT₁(ok₁(X)) -> FACT₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

FACT₁(ok₁(X)) -> FACT₁(X)

The remaining pairs can at least by weakly be oriented.

FACT₁(mark₁(X)) -> FACT₁(X)

Used ordering: Combined order from the following AFS and order.
FACT₁(x1) = FACT₁(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

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

FACT₁(mark₁(X)) -> FACT₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.

FACT₁(mark₁(X)) -> FACT₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
FACT₁(x1) = FACT₁(x1)
mark₁(x1) = mark₁(x1)

Lexicographic Path Order [19].
Precedence:

mark₁ > FACT₁

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

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

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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

PROPER₁(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(zero₁(X)) -> PROPER₁(X)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(fact₁(X)) -> PROPER₁(X)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X3)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X1)
PROPER₁(prod₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(prod₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(p₁(X)) -> PROPER₁(X)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X2)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(add₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X3)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X1)
PROPER₁(prod₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(prod₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(add₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X2)

The remaining pairs can at least by weakly be oriented.

PROPER₁(zero₁(X)) -> PROPER₁(X)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(fact₁(X)) -> PROPER₁(X)
PROPER₁(p₁(X)) -> PROPER₁(X)

Used ordering: Combined order from the following AFS and order.
PROPER₁(x1) = PROPER₁(x1)
add₂(x1, x2) = add₂(x1, x2)
zero₁(x1) = x1
s₁(x1) = x1
fact₁(x1) = x1
if₃(x1, x2, x3) = if₃(x1, x2, x3)
prod₂(x1, x2) = prod₂(x1, x2)
p₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

add₂ > PROPER₁
if₃ > PROPER₁
prod₂ > PROPER₁

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

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

PROPER₁(zero₁(X)) -> PROPER₁(X)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(fact₁(X)) -> PROPER₁(X)
PROPER₁(p₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(s₁(X)) -> PROPER₁(X)

The remaining pairs can at least by weakly be oriented.

PROPER₁(zero₁(X)) -> PROPER₁(X)
PROPER₁(fact₁(X)) -> PROPER₁(X)
PROPER₁(p₁(X)) -> PROPER₁(X)

Used ordering: Combined order from the following AFS and order.
PROPER₁(x1) = PROPER₁(x1)
zero₁(x1) = x1
s₁(x1) = s₁(x1)
fact₁(x1) = x1
p₁(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

PROPER₁(zero₁(X)) -> PROPER₁(X)
PROPER₁(fact₁(X)) -> PROPER₁(X)
PROPER₁(p₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(fact₁(X)) -> PROPER₁(X)

The remaining pairs can at least by weakly be oriented.

PROPER₁(zero₁(X)) -> PROPER₁(X)
PROPER₁(p₁(X)) -> PROPER₁(X)

Used ordering: Combined order from the following AFS and order.
PROPER₁(x1) = PROPER₁(x1)
zero₁(x1) = x1
fact₁(x1) = fact₁(x1)
p₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

[PROPER₁, fact_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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

PROPER₁(zero₁(X)) -> PROPER₁(X)
PROPER₁(p₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(p₁(X)) -> PROPER₁(X)

The remaining pairs can at least by weakly be oriented.

PROPER₁(zero₁(X)) -> PROPER₁(X)

Used ordering: Combined order from the following AFS and order.
PROPER₁(x1) = PROPER₁(x1)
zero₁(x1) = x1
p₁(x1) = p₁(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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

PROPER₁(zero₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(zero₁(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)
zero₁(x1) = zero₁(x1)

Lexicographic Path Order [19].
Precedence:

zero₁ > PROPER₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

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

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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

ACTIVE₁(prod₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(prod₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(fact₁(X)) -> ACTIVE₁(X)
ACTIVE₁(if₃(X1, X2, X3)) -> ACTIVE₁(X1)
ACTIVE₁(zero₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(prod₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(prod₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(fact₁(X)) -> ACTIVE₁(X)
ACTIVE₁(if₃(X1, X2, X3)) -> ACTIVE₁(X1)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(add₂(X1, X2)) -> ACTIVE₁(X2)

The remaining pairs can at least by weakly be oriented.

ACTIVE₁(zero₁(X)) -> ACTIVE₁(X)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = ACTIVE₁(x1)
prod₂(x1, x2) = prod₂(x1, x2)
add₂(x1, x2) = add₂(x1, x2)
fact₁(x1) = fact₁(x1)
if₃(x1, x2, x3) = if₂(x1, x3)
zero₁(x1) = x1
s₁(x1) = s₁(x1)
p₁(x1) = x1

Lexicographic Path Order [19].
Precedence:

[ACTIVE₁, prod₂, fact₁, s_1]

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

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

ACTIVE₁(zero₁(X)) -> ACTIVE₁(X)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(p₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least by weakly be oriented.

ACTIVE₁(zero₁(X)) -> ACTIVE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVE₁(x1) = ACTIVE₁(x1)
zero₁(x1) = x1
p₁(x1) = p₁(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

                    ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(zero₁(X)) -> ACTIVE₁(X)

The TRS R consists of the following rules:

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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(zero₁(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)
zero₁(x1) = zero₁(x1)

Lexicographic Path Order [19].
Precedence:

zero₁ > ACTIVE₁

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

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

          ↳ QDP

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

active₁(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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

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(fact₁(X)) -> mark₁(if₃(zero₁(X), s₁(0), prod₂(X, fact₁(p₁(X)))))
active₁(add₂(0, X)) -> mark₁(X)
active₁(add₂(s₁(X), Y)) -> mark₁(s₁(add₂(X, Y)))
active₁(prod₂(0, X)) -> mark₁(0)
active₁(prod₂(s₁(X), Y)) -> mark₁(add₂(Y, prod₂(X, Y)))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(zero₁(0)) -> mark₁(true)
active₁(zero₁(s₁(X))) -> mark₁(false)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(fact₁(X)) -> fact₁(active₁(X))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(zero₁(X)) -> zero₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(prod₂(X1, X2)) -> prod₂(active₁(X1), X2)
active₁(prod₂(X1, X2)) -> prod₂(X1, active₁(X2))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(add₂(X1, X2)) -> add₂(active₁(X1), X2)
active₁(add₂(X1, X2)) -> add₂(X1, active₁(X2))
fact₁(mark₁(X)) -> mark₁(fact₁(X))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
zero₁(mark₁(X)) -> mark₁(zero₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
prod₂(mark₁(X1), X2) -> mark₁(prod₂(X1, X2))
prod₂(X1, mark₁(X2)) -> mark₁(prod₂(X1, X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
add₂(mark₁(X1), X2) -> mark₁(add₂(X1, X2))
add₂(X1, mark₁(X2)) -> mark₁(add₂(X1, X2))
proper₁(fact₁(X)) -> fact₁(proper₁(X))
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(zero₁(X)) -> zero₁(proper₁(X))
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(prod₂(X1, X2)) -> prod₂(proper₁(X1), proper₁(X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(add₂(X1, X2)) -> add₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
fact₁(ok₁(X)) -> ok₁(fact₁(X))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
zero₁(ok₁(X)) -> ok₁(zero₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
prod₂(ok₁(X1), ok₁(X2)) -> ok₁(prod₂(X1, X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
add₂(ok₁(X1), ok₁(X2)) -> ok₁(add₂(X1, X2))
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.