(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
int(0, 0) → .(0, nil)
int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), 0) → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))
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:
INT(0, s(y)) → INT(s(0), s(y))
INT(s(x), s(y)) → INT_LIST(int(x, y))
INT(s(x), s(y)) → INT(x, y)
INT_LIST(.(x, y)) → INT_LIST(y)
The TRS R consists of the following rules:
int(0, 0) → .(0, nil)
int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), 0) → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))
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 1 less node.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
INT_LIST(.(x, y)) → INT_LIST(y)
The TRS R consists of the following rules:
int(0, 0) → .(0, nil)
int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), 0) → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
.(x1, x2) = .(x2)
From the DPs we obtained the following set of size-change graphs:
- INT_LIST(.(x, y)) → INT_LIST(y) (allowed arguments on rhs = {1})
The graph contains the following edges 1 > 1
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(7) TRUE
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
INT(s(x), s(y)) → INT(x, y)
INT(0, s(y)) → INT(s(0), s(y))
The TRS R consists of the following rules:
int(0, 0) → .(0, nil)
int(0, s(y)) → .(0, int(s(0), s(y)))
int(s(x), 0) → nil
int(s(x), s(y)) → int_list(int(x, y))
int_list(nil) → nil
int_list(.(x, y)) → .(s(x), int_list(y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
0 = 0
s(x1) = s(x1)
From the DPs we obtained the following set of size-change graphs:
- INT(s(x), s(y)) → INT(x, y) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 > 2
- INT(0, s(y)) → INT(s(0), s(y)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 2 >= 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(10) TRUE