(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(lambda(x), y) → lambda(a(x, 1))
a(lambda(x), y) → lambda(a(x, a(y, t)))
a(a(x, y), z) → a(x, a(y, z))
lambda(x) → x
a(x, y) → x
a(x, y) → y
Q is empty.
(1) QTRSRRRProof (EQUIVALENT transformation)
Used ordering:
Polynomial interpretation [POLO]:
POL(1) = 0
POL(a(x1, x2)) = x1 + x2
POL(lambda(x1)) = 1 + x1
POL(t) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
lambda(x) → x
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(lambda(x), y) → lambda(a(x, 1))
a(lambda(x), y) → lambda(a(x, a(y, t)))
a(a(x, y), z) → a(x, a(y, z))
a(x, y) → x
a(x, y) → y
Q is empty.
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(lambda(x), y) → A(x, 1)
A(lambda(x), y) → A(x, a(y, t))
A(lambda(x), y) → A(y, t)
A(a(x, y), z) → A(x, a(y, z))
A(a(x, y), z) → A(y, z)
The TRS R consists of the following rules:
a(lambda(x), y) → lambda(a(x, 1))
a(lambda(x), y) → lambda(a(x, a(y, t)))
a(a(x, y), z) → a(x, a(y, z))
a(x, y) → x
a(x, y) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
A(lambda(x), y) → A(x, 1)
A(lambda(x), y) → A(x, a(y, t))
A(lambda(x), y) → A(y, t)
Used ordering: Polynomial interpretation [POLO]:
POL(1) = 0
POL(A(x1, x2)) = x1 + x2
POL(a(x1, x2)) = x1 + x2
POL(lambda(x1)) = 1 + x1
POL(t) = 0
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(x, y), z) → A(x, a(y, z))
A(a(x, y), z) → A(y, z)
The TRS R consists of the following rules:
a(lambda(x), y) → lambda(a(x, 1))
a(lambda(x), y) → lambda(a(x, a(y, t)))
a(a(x, y), z) → a(x, a(y, z))
a(x, y) → x
a(x, y) → y
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- A(a(x, y), z) → A(x, a(y, z))
The graph contains the following edges 1 > 1
- A(a(x, y), z) → A(y, z)
The graph contains the following edges 1 > 1, 2 >= 2
(8) TRUE