Termination w.r.t. Q proof of TRS/nontermin/AG01#4.35.trs

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

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

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

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

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

Q is empty.

The TRS is overlay and locally confluent. By [15] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

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

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

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

EQ(var(l), var(l')) → EQ(l, l')
REN(x, y, apply(t, s)) → REN(x, y, t)
EQ(apply(t, s), apply(t', s')) → EQ(t, t')
EQ(lambda(x, t), lambda(x', t')) → EQ(x, x')
EQ(cons(t, l), cons(t', l')) → EQ(l, l')
EQ(apply(t, s), apply(t', s')) → AND(eq(t, t'), eq(s, s'))
EQ(cons(t, l), cons(t', l')) → EQ(t, t')
REN(var(l), var(k), var(l')) → EQ(l, l')
EQ(lambda(x, t), lambda(x', t')) → AND(eq(x, x'), eq(t, t'))
EQ(cons(t, l), cons(t', l')) → AND(eq(t, t'), eq(l, l'))
REN(x, y, lambda(z, t)) → REN(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))
EQ(lambda(x, t), lambda(x', t')) → EQ(t, t')
EQ(apply(t, s), apply(t', s')) → EQ(s, s')
REN(var(l), var(k), var(l')) → IF(eq(l, l'), var(k), var(l'))
REN(x, y, lambda(z, t)) → REN(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)
REN(x, y, apply(t, s)) → REN(x, y, s)

The TRS R consists of the following rules:

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

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

EQ(var(l), var(l')) → EQ(l, l')
REN(x, y, apply(t, s)) → REN(x, y, t)
EQ(apply(t, s), apply(t', s')) → EQ(t, t')
EQ(lambda(x, t), lambda(x', t')) → EQ(x, x')
EQ(cons(t, l), cons(t', l')) → EQ(l, l')
EQ(apply(t, s), apply(t', s')) → AND(eq(t, t'), eq(s, s'))
EQ(cons(t, l), cons(t', l')) → EQ(t, t')
REN(var(l), var(k), var(l')) → EQ(l, l')
EQ(lambda(x, t), lambda(x', t')) → AND(eq(x, x'), eq(t, t'))
EQ(cons(t, l), cons(t', l')) → AND(eq(t, t'), eq(l, l'))
REN(x, y, lambda(z, t)) → REN(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))
EQ(lambda(x, t), lambda(x', t')) → EQ(t, t')
EQ(apply(t, s), apply(t', s')) → EQ(s, s')
REN(var(l), var(k), var(l')) → IF(eq(l, l'), var(k), var(l'))
REN(x, y, lambda(z, t)) → REN(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)
REN(x, y, apply(t, s)) → REN(x, y, s)

The TRS R consists of the following rules:

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

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

REN(x, y, apply(t, s)) → REN(x, y, t)
EQ(var(l), var(l')) → EQ(l, l')
EQ(apply(t, s), apply(t', s')) → EQ(t, t')
EQ(lambda(x, t), lambda(x', t')) → EQ(x, x')
EQ(cons(t, l), cons(t', l')) → EQ(l, l')
EQ(apply(t, s), apply(t', s')) → AND(eq(t, t'), eq(s, s'))
EQ(cons(t, l), cons(t', l')) → EQ(t, t')
REN(var(l), var(k), var(l')) → EQ(l, l')
EQ(lambda(x, t), lambda(x', t')) → AND(eq(x, x'), eq(t, t'))
EQ(cons(t, l), cons(t', l')) → AND(eq(t, t'), eq(l, l'))
EQ(lambda(x, t), lambda(x', t')) → EQ(t, t')
REN(x, y, lambda(z, t)) → REN(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))
EQ(apply(t, s), apply(t', s')) → EQ(s, s')
REN(var(l), var(k), var(l')) → IF(eq(l, l'), var(k), var(l'))
REN(x, y, apply(t, s)) → REN(x, y, s)
REN(x, y, lambda(z, t)) → REN(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)

The TRS R consists of the following rules:

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 2 SCCs with 5 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                  ↳ QDP

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

EQ(var(l), var(l')) → EQ(l, l')
EQ(apply(t, s), apply(t', s')) → EQ(t, t')
EQ(lambda(x, t), lambda(x', t')) → EQ(x, x')
EQ(cons(t, l), cons(t', l')) → EQ(l, l')
EQ(lambda(x, t), lambda(x', t')) → EQ(t, t')
EQ(apply(t, s), apply(t', s')) → EQ(s, s')
EQ(cons(t, l), cons(t', l')) → EQ(t, t')

The TRS R consists of the following rules:

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

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.

EQ(apply(t, s), apply(t', s')) → EQ(t, t')
EQ(lambda(x, t), lambda(x', t')) → EQ(x, x')
EQ(cons(t, l), cons(t', l')) → EQ(l, l')
EQ(lambda(x, t), lambda(x', t')) → EQ(t, t')
EQ(apply(t, s), apply(t', s')) → EQ(s, s')
EQ(cons(t, l), cons(t', l')) → EQ(t, t')

The remaining pairs can at least be oriented weakly.

EQ(var(l), var(l')) → EQ(l, l')

Used ordering: Combined order from the following AFS and order.
EQ(x1, x2) = EQ(x1, x2)
var(x1) = x1
apply(x1, x2) = apply(x1, x2)
lambda(x1, x2) = lambda(x1, x2)
cons(x1, x2) = cons(x1, x2)

Recursive Path Order [2].
Precedence:

[EQ₂, apply₂, lambda_2]

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                  ↳ QDP

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

EQ(var(l), var(l')) → EQ(l, l')

The TRS R consists of the following rules:

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

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.

EQ(var(l), var(l')) → EQ(l, l')

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

Recursive Path Order [2].
Precedence:

var₁ > EQ₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ PisEmptyProof

                  ↳ QDP

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

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

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

REN(x, y, apply(t, s)) → REN(x, y, t)
REN(x, y, lambda(z, t)) → REN(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))
REN(x, y, lambda(z, t)) → REN(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)
REN(x, y, apply(t, s)) → REN(x, y, s)

The TRS R consists of the following rules:

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

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.

REN(x, y, apply(t, s)) → REN(x, y, t)
REN(x, y, lambda(z, t)) → REN(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))
REN(x, y, lambda(z, t)) → REN(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)
REN(x, y, apply(t, s)) → REN(x, y, s)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
REN(x1, x2, x3) = REN(x3)
apply(x1, x2) = apply(x1, x2)
lambda(x1, x2) = lambda(x1, x2)
ren(x1, x2, x3) = x3
var(x1) = var
cons(x1, x2) = cons
nil = nil
eq(x1, x2) = eq
and(x1, x2) = and(x1, x2)
false = false
true = true
if(x1, x2, x3) = if

Recursive Path Order [2].
Precedence:

REN₁ > lambda₂ > [var, cons, if]
REN₁ > [nil, true] > [var, cons, if]
apply₂ > [var, cons, if]
eq > [var, cons, if]
and₂ > [var, cons, if]
false > [var, cons, if]

The following usable rules [14] were oriented:

ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
if(false, var(k), var(l')) → var(l')
if(true, var(k), var(l')) → var(k)
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ EdgeDeletionProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

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

and(true, y) → y
and(false, y) → false
eq(nil, nil) → true
eq(cons(t, l), nil) → false
eq(nil, cons(t, l)) → false
eq(cons(t, l), cons(t', l')) → and(eq(t, t'), eq(l, l'))
eq(var(l), var(l')) → eq(l, l')
eq(var(l), apply(t, s)) → false
eq(var(l), lambda(x, t)) → false
eq(apply(t, s), var(l)) → false
eq(apply(t, s), apply(t', s')) → and(eq(t, t'), eq(s, s'))
eq(apply(t, s), lambda(x, t)) → false
eq(lambda(x, t), var(l)) → false
eq(lambda(x, t), apply(t, s)) → false
eq(lambda(x, t), lambda(x', t')) → and(eq(x, x'), eq(t, t'))
if(true, var(k), var(l')) → var(k)
if(false, var(k), var(l')) → var(l')
ren(var(l), var(k), var(l')) → if(eq(l, l'), var(k), var(l'))
ren(x, y, apply(t, s)) → apply(ren(x, y, t), ren(x, y, s))
ren(x, y, lambda(z, t)) → lambda(var(cons(x, cons(y, cons(lambda(z, t), nil)))), ren(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)))

The set Q consists of the following terms:

and(true, x0)
and(false, x0)
eq(nil, nil)
eq(cons(x0, x1), nil)
eq(nil, cons(x0, x1))
eq(cons(x0, x1), cons(x2, x3))
eq(var(x0), var(x1))
eq(var(x0), apply(x1, x2))
eq(var(x0), lambda(x1, x2))
eq(apply(x0, x1), var(x2))
eq(apply(x0, x1), apply(x2, x3))
eq(apply(x0, x1), lambda(x2, x0))
eq(lambda(x0, x1), var(x2))
eq(lambda(x0, x1), apply(x1, x2))
eq(lambda(x0, x1), lambda(x2, x3))
if(true, var(x0), var(x1))
if(false, var(x0), var(x1))
ren(var(x0), var(x1), var(x2))
ren(x0, x1, apply(x2, x3))
ren(x0, x1, lambda(x2, x3))

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