Termination w.r.t. Q proof of TRS/TRCSREx1_GM03

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(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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:

LEQ₂(mark₁(X1), X2) -> LEQ₂(X1, X2)
ACTIVE₁(diff₂(X, Y)) -> P₁(X)
P₁(mark₁(X)) -> P₁(X)
LEQ₂(X1, mark₁(X2)) -> LEQ₂(X1, X2)
PROPER₁(leq₂(X1, X2)) -> LEQ₂(proper₁(X1), proper₁(X2))
ACTIVE₁(diff₂(X1, X2)) -> DIFF₂(active₁(X1), X2)
S₁(mark₁(X)) -> S₁(X)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)
ACTIVE₁(leq₂(X1, X2)) -> LEQ₂(active₁(X1), X2)
PROPER₁(diff₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(if₃(X1, X2, X3)) -> IF₃(active₁(X1), X2, X3)
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
IF₃(mark₁(X1), X2, X3) -> IF₃(X1, X2, X3)
ACTIVE₁(diff₂(X, Y)) -> LEQ₂(X, Y)
ACTIVE₁(diff₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(diff₂(X, Y)) -> S₁(diff₂(p₁(X), Y))
ACTIVE₁(leq₂(s₁(X), s₁(Y))) -> LEQ₂(X, Y)
DIFF₂(mark₁(X1), X2) -> DIFF₂(X1, X2)
P₁(ok₁(X)) -> P₁(X)
ACTIVE₁(leq₂(X1, X2)) -> LEQ₂(X1, active₁(X2))
ACTIVE₁(leq₂(X1, X2)) -> ACTIVE₁(X2)
TOP₁(ok₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> S₁(active₁(X))
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(diff₂(X, Y)) -> IF₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y)))
PROPER₁(diff₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(p₁(X)) -> PROPER₁(X)
ACTIVE₁(diff₂(X1, X2)) -> DIFF₂(X1, active₁(X2))
TOP₁(mark₁(X)) -> PROPER₁(X)
PROPER₁(leq₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(leq₂(X1, X2)) -> PROPER₁(X2)
DIFF₂(ok₁(X1), ok₁(X2)) -> DIFF₂(X1, X2)
ACTIVE₁(diff₂(X1, X2)) -> ACTIVE₁(X2)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X1)
IF₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> IF₃(X1, X2, X3)
PROPER₁(diff₂(X1, X2)) -> DIFF₂(proper₁(X1), proper₁(X2))
PROPER₁(if₃(X1, X2, X3)) -> IF₃(proper₁(X1), proper₁(X2), proper₁(X3))
PROPER₁(p₁(X)) -> P₁(proper₁(X))
PROPER₁(s₁(X)) -> S₁(proper₁(X))
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X3)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
S₁(ok₁(X)) -> S₁(X)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X2)
DIFF₂(X1, mark₁(X2)) -> DIFF₂(X1, X2)
ACTIVE₁(diff₂(X, Y)) -> DIFF₂(p₁(X), Y)
ACTIVE₁(leq₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(p₁(X)) -> P₁(active₁(X))
LEQ₂(ok₁(X1), ok₁(X2)) -> LEQ₂(X1, X2)
ACTIVE₁(if₃(X1, X2, X3)) -> ACTIVE₁(X1)
TOP₁(ok₁(X)) -> TOP₁(active₁(X))

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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:

LEQ₂(mark₁(X1), X2) -> LEQ₂(X1, X2)
ACTIVE₁(diff₂(X, Y)) -> P₁(X)
P₁(mark₁(X)) -> P₁(X)
LEQ₂(X1, mark₁(X2)) -> LEQ₂(X1, X2)
PROPER₁(leq₂(X1, X2)) -> LEQ₂(proper₁(X1), proper₁(X2))
ACTIVE₁(diff₂(X1, X2)) -> DIFF₂(active₁(X1), X2)
S₁(mark₁(X)) -> S₁(X)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)
ACTIVE₁(leq₂(X1, X2)) -> LEQ₂(active₁(X1), X2)
PROPER₁(diff₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(if₃(X1, X2, X3)) -> IF₃(active₁(X1), X2, X3)
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
IF₃(mark₁(X1), X2, X3) -> IF₃(X1, X2, X3)
ACTIVE₁(diff₂(X, Y)) -> LEQ₂(X, Y)
ACTIVE₁(diff₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(diff₂(X, Y)) -> S₁(diff₂(p₁(X), Y))
ACTIVE₁(leq₂(s₁(X), s₁(Y))) -> LEQ₂(X, Y)
DIFF₂(mark₁(X1), X2) -> DIFF₂(X1, X2)
P₁(ok₁(X)) -> P₁(X)
ACTIVE₁(leq₂(X1, X2)) -> LEQ₂(X1, active₁(X2))
ACTIVE₁(leq₂(X1, X2)) -> ACTIVE₁(X2)
TOP₁(ok₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s₁(X)) -> S₁(active₁(X))
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(diff₂(X, Y)) -> IF₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y)))
PROPER₁(diff₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(p₁(X)) -> PROPER₁(X)
ACTIVE₁(diff₂(X1, X2)) -> DIFF₂(X1, active₁(X2))
TOP₁(mark₁(X)) -> PROPER₁(X)
PROPER₁(leq₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(leq₂(X1, X2)) -> PROPER₁(X2)
DIFF₂(ok₁(X1), ok₁(X2)) -> DIFF₂(X1, X2)
ACTIVE₁(diff₂(X1, X2)) -> ACTIVE₁(X2)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X1)
IF₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> IF₃(X1, X2, X3)
PROPER₁(diff₂(X1, X2)) -> DIFF₂(proper₁(X1), proper₁(X2))
PROPER₁(if₃(X1, X2, X3)) -> IF₃(proper₁(X1), proper₁(X2), proper₁(X3))
PROPER₁(p₁(X)) -> P₁(proper₁(X))
PROPER₁(s₁(X)) -> S₁(proper₁(X))
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X3)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
S₁(ok₁(X)) -> S₁(X)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X2)
DIFF₂(X1, mark₁(X2)) -> DIFF₂(X1, X2)
ACTIVE₁(diff₂(X, Y)) -> DIFF₂(p₁(X), Y)
ACTIVE₁(leq₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(p₁(X)) -> P₁(active₁(X))
LEQ₂(ok₁(X1), ok₁(X2)) -> LEQ₂(X1, X2)
ACTIVE₁(if₃(X1, X2, X3)) -> ACTIVE₁(X1)
TOP₁(ok₁(X)) -> TOP₁(active₁(X))

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 8 SCCs with 20 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

DIFF₂(X1, mark₁(X2)) -> DIFF₂(X1, X2)
DIFF₂(ok₁(X1), ok₁(X2)) -> DIFF₂(X1, X2)
DIFF₂(mark₁(X1), X2) -> DIFF₂(X1, X2)

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

DIFF₂(X1, mark₁(X2)) -> DIFF₂(X1, X2)
DIFF₂(mark₁(X1), X2) -> DIFF₂(X1, X2)

The remaining pairs can at least be oriented weakly.

DIFF₂(ok₁(X1), ok₁(X2)) -> DIFF₂(X1, X2)

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

POL( mark₁(x₁) ) = x₁ + 1

POL( DIFF₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( ok₁(x₁) ) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

DIFF₂(ok₁(X1), ok₁(X2)) -> DIFF₂(X1, X2)

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

DIFF₂(ok₁(X1), ok₁(X2)) -> DIFF₂(X1, X2)

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

POL( ok₁(x₁) ) = x₁ + 1

POL( DIFF₂(x₁, x₂) ) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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

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(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

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

The remaining pairs can at least be oriented weakly.

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

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

POL( IF₃(x₁, ..., x₃) ) = x₁ + x₃ + 1

POL( mark₁(x₁) ) = x₁ + 1

POL( ok₁(x₁) ) = x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

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

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

POL( IF₃(x₁, ..., x₃) ) = max{0, x₁ + x₂ - 1}

POL( ok₁(x₁) ) = x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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

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

LEQ₂(ok₁(X1), ok₁(X2)) -> LEQ₂(X1, X2)
LEQ₂(mark₁(X1), X2) -> LEQ₂(X1, X2)
LEQ₂(X1, mark₁(X2)) -> LEQ₂(X1, X2)

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

LEQ₂(mark₁(X1), X2) -> LEQ₂(X1, X2)
LEQ₂(X1, mark₁(X2)) -> LEQ₂(X1, X2)

The remaining pairs can at least be oriented weakly.

LEQ₂(ok₁(X1), ok₁(X2)) -> LEQ₂(X1, X2)

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

POL( mark₁(x₁) ) = x₁ + 1

POL( ok₁(x₁) ) = x₁

POL( LEQ₂(x₁, x₂) ) = x₁ + x₂ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

LEQ₂(ok₁(X1), ok₁(X2)) -> LEQ₂(X1, X2)

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

LEQ₂(ok₁(X1), ok₁(X2)) -> LEQ₂(X1, X2)

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

POL( LEQ₂(x₁, x₂) ) = x₁

POL( ok₁(x₁) ) = x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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

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(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

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

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

POL( S₁(x₁) ) = x₁ + 1

POL( mark₁(x₁) ) = x₁ + 1

POL( ok₁(x₁) ) = x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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

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(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

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

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

POL( P₁(x₁) ) = x₁ + 1

POL( mark₁(x₁) ) = x₁ + 1

POL( ok₁(x₁) ) = x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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

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

PROPER₁(diff₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(leq₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(leq₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X3)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X1)
PROPER₁(diff₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X2)
PROPER₁(p₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

PROPER₁(diff₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(leq₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(leq₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X3)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X1)
PROPER₁(diff₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(if₃(X1, X2, X3)) -> PROPER₁(X2)
PROPER₁(p₁(X)) -> PROPER₁(X)

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

POL( leq₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( s₁(x₁) ) = x₁ + 1

POL( if₃(x₁, ..., x₃) ) = x₁ + x₂ + x₃ + 1

POL( PROPER₁(x₁) ) = x₁ + 1

POL( diff₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( p₁(x₁) ) = x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

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

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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

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

ACTIVE₁(diff₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(if₃(X1, X2, X3)) -> ACTIVE₁(X1)
ACTIVE₁(diff₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(leq₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(leq₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)

The TRS R consists of the following rules:

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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 oriented strictly and are deleted.

ACTIVE₁(diff₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(if₃(X1, X2, X3)) -> ACTIVE₁(X1)
ACTIVE₁(diff₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(leq₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(s₁(X)) -> ACTIVE₁(X)
ACTIVE₁(leq₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(p₁(X)) -> ACTIVE₁(X)

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

POL( leq₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( s₁(x₁) ) = x₁ + 1

POL( if₃(x₁, ..., x₃) ) = x₁ + x₂ + x₃ + 1

POL( ACTIVE₁(x₁) ) = x₁ + 1

POL( diff₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( p₁(x₁) ) = x₁ + 1

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

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

active₁(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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

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(p₁(0)) -> mark₁(0)
active₁(p₁(s₁(X))) -> mark₁(X)
active₁(leq₂(0, Y)) -> mark₁(true)
active₁(leq₂(s₁(X), 0)) -> mark₁(false)
active₁(leq₂(s₁(X), s₁(Y))) -> mark₁(leq₂(X, Y))
active₁(if₃(true, X, Y)) -> mark₁(X)
active₁(if₃(false, X, Y)) -> mark₁(Y)
active₁(diff₂(X, Y)) -> mark₁(if₃(leq₂(X, Y), 0, s₁(diff₂(p₁(X), Y))))
active₁(p₁(X)) -> p₁(active₁(X))
active₁(s₁(X)) -> s₁(active₁(X))
active₁(leq₂(X1, X2)) -> leq₂(active₁(X1), X2)
active₁(leq₂(X1, X2)) -> leq₂(X1, active₁(X2))
active₁(if₃(X1, X2, X3)) -> if₃(active₁(X1), X2, X3)
active₁(diff₂(X1, X2)) -> diff₂(active₁(X1), X2)
active₁(diff₂(X1, X2)) -> diff₂(X1, active₁(X2))
p₁(mark₁(X)) -> mark₁(p₁(X))
s₁(mark₁(X)) -> mark₁(s₁(X))
leq₂(mark₁(X1), X2) -> mark₁(leq₂(X1, X2))
leq₂(X1, mark₁(X2)) -> mark₁(leq₂(X1, X2))
if₃(mark₁(X1), X2, X3) -> mark₁(if₃(X1, X2, X3))
diff₂(mark₁(X1), X2) -> mark₁(diff₂(X1, X2))
diff₂(X1, mark₁(X2)) -> mark₁(diff₂(X1, X2))
proper₁(p₁(X)) -> p₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(leq₂(X1, X2)) -> leq₂(proper₁(X1), proper₁(X2))
proper₁(true) -> ok₁(true)
proper₁(false) -> ok₁(false)
proper₁(if₃(X1, X2, X3)) -> if₃(proper₁(X1), proper₁(X2), proper₁(X3))
proper₁(diff₂(X1, X2)) -> diff₂(proper₁(X1), proper₁(X2))
p₁(ok₁(X)) -> ok₁(p₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
leq₂(ok₁(X1), ok₁(X2)) -> ok₁(leq₂(X1, X2))
if₃(ok₁(X1), ok₁(X2), ok₁(X3)) -> ok₁(if₃(X1, X2, X3))
diff₂(ok₁(X1), ok₁(X2)) -> ok₁(diff₂(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.