Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 DependencyGraphProof (⇔)
↳4 AND
    ↳5 QDP
        ↳6 QDPOrderProof (⇔)
        ↳7 QDP
        ↳8 QDPOrderProof (⇔)
        ↳9 QDP
        ↳10 QDPOrderProof (⇔)
        ↳11 QDP
        ↳12 QDPOrderProof (⇔)
        ↳13 QDP
        ↳14 PisEmptyProof (⇔)
        ↳15 TRUE
    ↳16 QDP
        ↳17 QDPOrderProof (⇔)
        ↳18 QDP
        ↳19 QDPOrderProof (⇔)
        ↳20 QDP
        ↳21 QDPOrderProof (⇔)
        ↳22 QDP
        ↳23 PisEmptyProof (⇔)
        ↳24 TRUE

(0) Obligation:

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.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

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

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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

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

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

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


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.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
EQ(x0, x1, x2)  =  EQ(x1, x2)

Tags:
EQ has argument tags [2,0,2] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(EQ(x1, x2)) = 1   
POL(apply(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = x1 + x2   
POL(lambda(x1, x2)) = x1 + x2   
POL(var(x1)) = 1 + x1   

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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

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

(8) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


EQ(lambda(x, t), lambda(x', t')) → EQ(x, x')
EQ(lambda(x, t), lambda(x', t')) → EQ(t, t')
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
EQ(x0, x1, x2)  =  EQ(x1)

Tags:
EQ has argument tags [1,0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(EQ(x1, x2)) = 1   
POL(apply(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = x1 + x2   
POL(lambda(x1, x2)) = 1 + x1 + x2   

The following usable rules [FROCOS05] were oriented: none

(9) Obligation:

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

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

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

(10) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


EQ(cons(t, l), cons(t', l')) → EQ(l, l')
EQ(cons(t, l), cons(t', l')) → EQ(t, t')
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
EQ(x0, x1, x2)  =  EQ(x1)

Tags:
EQ has argument tags [0,0,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(EQ(x1, x2)) = x1   
POL(apply(x1, x2)) = x1 + x2   
POL(cons(x1, x2)) = 1 + x1 + x2   

The following usable rules [FROCOS05] were oriented: none

(11) Obligation:

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

EQ(apply(t, s), apply(t', s')) → EQ(t, t')
EQ(apply(t, s), apply(t', s')) → EQ(s, 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)))

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

(12) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


EQ(apply(t, s), apply(t', s')) → EQ(t, t')
EQ(apply(t, s), apply(t', s')) → EQ(s, s')
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
EQ(x0, x1, x2)  =  EQ(x1)

Tags:
EQ has argument tags [3,2,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(EQ(x1, x2)) = 0   
POL(apply(x1, x2)) = 1 + x1 + x2   

The following usable rules [FROCOS05] were oriented: none

(13) Obligation:

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)))

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

(14) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(15) TRUE

(16) Obligation:

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

REN(x, y, apply(t, s)) → REN(x, y, s)
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)

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

(17) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


REN(x, y, apply(t, s)) → REN(x, y, s)
REN(x, y, apply(t, s)) → REN(x, y, t)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
REN(x0, x1, x2, x3)  =  REN(x2, x3)

Tags:
REN has argument tags [3,1,2,0] and root tag 0

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(REN(x1, x2, x3)) = 0   
POL(and(x1, x2)) = 1 + x1   
POL(apply(x1, x2)) = 1 + x1 + x2   
POL(cons(x1, x2)) = 0   
POL(eq(x1, x2)) = 0   
POL(false) = 1   
POL(if(x1, x2, x3)) = 0   
POL(lambda(x1, x2)) = x2   
POL(nil) = 0   
POL(ren(x1, x2, x3)) = x3   
POL(true) = 1   
POL(var(x1)) = 0   

The following usable rules [FROCOS05] were oriented: none

(18) Obligation:

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

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)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


REN(x, y, lambda(z, t)) → REN(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
REN(x0, x1, x2, x3)  =  REN(x1, x2, x3)

Tags:
REN has argument tags [0,2,3,0] and root tag 0

Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(REN(x1, x2, x3)) = x1 + x2   
POL(and(x1, x2)) = 1   
POL(apply(x1, x2)) = 0   
POL(cons(x1, x2)) = 0   
POL(eq(x1, x2)) = 0   
POL(false) = 1   
POL(if(x1, x2, x3)) = 0   
POL(lambda(x1, x2)) = 1 + x1 + x2   
POL(nil) = 0   
POL(ren(x1, x2, x3)) = x3   
POL(true) = 1   
POL(var(x1)) = 0   

The following usable rules [FROCOS05] were oriented: none

(20) Obligation:

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

REN(x, y, lambda(z, t)) → REN(x, y, 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)))

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


REN(x, y, lambda(z, t)) → REN(x, y, ren(z, var(cons(x, cons(y, cons(lambda(z, t), nil)))), t))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
REN(x0, x1, x2, x3)  =  REN(x0, x1, x2, x3)

Tags:
REN has argument tags [0,0,1,3] and root tag 0

Comparison: MS
Underlying order for the size change arcs and the rules of R:
Polynomial interpretation [POLO]:

POL(REN(x1, x2, x3)) = 1   
POL(and(x1, x2)) = 1 + x1   
POL(apply(x1, x2)) = 0   
POL(cons(x1, x2)) = 0   
POL(eq(x1, x2)) = 0   
POL(false) = 1   
POL(if(x1, x2, x3)) = 0   
POL(lambda(x1, x2)) = 1 + x2   
POL(nil) = 0   
POL(ren(x1, x2, x3)) = x3   
POL(true) = 1   
POL(var(x1)) = 0   

The following usable rules [FROCOS05] were oriented: none

(22) Obligation:

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)))

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

(23) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(24) TRUE