(0) Obligation:
Clauses:
perm([], []).
perm(.(X, Y), .(U, V)) :- ','(delete(U, .(X, Y), W), perm(W, V)).
delete(X, .(X, Y), Y).
delete(U, .(X, Y), .(X, Z)) :- delete(U, Y, Z).
Query: perm(g,a)
(1) PrologToPiTRSProof (SOUND transformation)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f)
delete_in: (f,b,f)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, perm_in_ga(W, V))
U2_ga(X, Y, U, V, perm_out_ga(W, V)) → perm_out_ga(.(X, Y), .(U, V))
The argument filtering Pi contains the following mapping:
perm_in_ga(
x1,
x2) =
perm_in_ga(
x1)
[] =
[]
perm_out_ga(
x1,
x2) =
perm_out_ga(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x5)
delete_in_aga(
x1,
x2,
x3) =
delete_in_aga(
x2)
delete_out_aga(
x1,
x2,
x3) =
delete_out_aga(
x1,
x3)
U3_aga(
x1,
x2,
x3,
x4,
x5) =
U3_aga(
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x3,
x5)
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(2) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, perm_in_ga(W, V))
U2_ga(X, Y, U, V, perm_out_ga(W, V)) → perm_out_ga(.(X, Y), .(U, V))
The argument filtering Pi contains the following mapping:
perm_in_ga(
x1,
x2) =
perm_in_ga(
x1)
[] =
[]
perm_out_ga(
x1,
x2) =
perm_out_ga(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x5)
delete_in_aga(
x1,
x2,
x3) =
delete_in_aga(
x2)
delete_out_aga(
x1,
x2,
x3) =
delete_out_aga(
x1,
x3)
U3_aga(
x1,
x2,
x3,
x4,
x5) =
U3_aga(
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x3,
x5)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
PERM_IN_GA(.(X, Y), .(U, V)) → U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
PERM_IN_GA(.(X, Y), .(U, V)) → DELETE_IN_AGA(U, .(X, Y), W)
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z))
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → DELETE_IN_AGA(U, Y, Z)
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_GA(X, Y, U, V, perm_in_ga(W, V))
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → PERM_IN_GA(W, V)
The TRS R consists of the following rules:
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, perm_in_ga(W, V))
U2_ga(X, Y, U, V, perm_out_ga(W, V)) → perm_out_ga(.(X, Y), .(U, V))
The argument filtering Pi contains the following mapping:
perm_in_ga(
x1,
x2) =
perm_in_ga(
x1)
[] =
[]
perm_out_ga(
x1,
x2) =
perm_out_ga(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x5)
delete_in_aga(
x1,
x2,
x3) =
delete_in_aga(
x2)
delete_out_aga(
x1,
x2,
x3) =
delete_out_aga(
x1,
x3)
U3_aga(
x1,
x2,
x3,
x4,
x5) =
U3_aga(
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x3,
x5)
PERM_IN_GA(
x1,
x2) =
PERM_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x5)
DELETE_IN_AGA(
x1,
x2,
x3) =
DELETE_IN_AGA(
x2)
U3_AGA(
x1,
x2,
x3,
x4,
x5) =
U3_AGA(
x2,
x5)
U2_GA(
x1,
x2,
x3,
x4,
x5) =
U2_GA(
x3,
x5)
We have to consider all (P,R,Pi)-chains
(4) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
PERM_IN_GA(.(X, Y), .(U, V)) → U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
PERM_IN_GA(.(X, Y), .(U, V)) → DELETE_IN_AGA(U, .(X, Y), W)
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → U3_AGA(U, X, Y, Z, delete_in_aga(U, Y, Z))
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → DELETE_IN_AGA(U, Y, Z)
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_GA(X, Y, U, V, perm_in_ga(W, V))
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → PERM_IN_GA(W, V)
The TRS R consists of the following rules:
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, perm_in_ga(W, V))
U2_ga(X, Y, U, V, perm_out_ga(W, V)) → perm_out_ga(.(X, Y), .(U, V))
The argument filtering Pi contains the following mapping:
perm_in_ga(
x1,
x2) =
perm_in_ga(
x1)
[] =
[]
perm_out_ga(
x1,
x2) =
perm_out_ga(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x5)
delete_in_aga(
x1,
x2,
x3) =
delete_in_aga(
x2)
delete_out_aga(
x1,
x2,
x3) =
delete_out_aga(
x1,
x3)
U3_aga(
x1,
x2,
x3,
x4,
x5) =
U3_aga(
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x3,
x5)
PERM_IN_GA(
x1,
x2) =
PERM_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x5)
DELETE_IN_AGA(
x1,
x2,
x3) =
DELETE_IN_AGA(
x2)
U3_AGA(
x1,
x2,
x3,
x4,
x5) =
U3_AGA(
x2,
x5)
U2_GA(
x1,
x2,
x3,
x4,
x5) =
U2_GA(
x3,
x5)
We have to consider all (P,R,Pi)-chains
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 3 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → DELETE_IN_AGA(U, Y, Z)
The TRS R consists of the following rules:
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, perm_in_ga(W, V))
U2_ga(X, Y, U, V, perm_out_ga(W, V)) → perm_out_ga(.(X, Y), .(U, V))
The argument filtering Pi contains the following mapping:
perm_in_ga(
x1,
x2) =
perm_in_ga(
x1)
[] =
[]
perm_out_ga(
x1,
x2) =
perm_out_ga(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x5)
delete_in_aga(
x1,
x2,
x3) =
delete_in_aga(
x2)
delete_out_aga(
x1,
x2,
x3) =
delete_out_aga(
x1,
x3)
U3_aga(
x1,
x2,
x3,
x4,
x5) =
U3_aga(
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x3,
x5)
DELETE_IN_AGA(
x1,
x2,
x3) =
DELETE_IN_AGA(
x2)
We have to consider all (P,R,Pi)-chains
(8) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(9) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
DELETE_IN_AGA(U, .(X, Y), .(X, Z)) → DELETE_IN_AGA(U, Y, Z)
R is empty.
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
DELETE_IN_AGA(
x1,
x2,
x3) =
DELETE_IN_AGA(
x2)
We have to consider all (P,R,Pi)-chains
(10) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
DELETE_IN_AGA(.(X, Y)) → DELETE_IN_AGA(Y)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(12) 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:
- DELETE_IN_AGA(.(X, Y)) → DELETE_IN_AGA(Y)
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → PERM_IN_GA(W, V)
PERM_IN_GA(.(X, Y), .(U, V)) → U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
The TRS R consists of the following rules:
perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, Y), .(U, V)) → U1_ga(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
U1_ga(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → U2_ga(X, Y, U, V, perm_in_ga(W, V))
U2_ga(X, Y, U, V, perm_out_ga(W, V)) → perm_out_ga(.(X, Y), .(U, V))
The argument filtering Pi contains the following mapping:
perm_in_ga(
x1,
x2) =
perm_in_ga(
x1)
[] =
[]
perm_out_ga(
x1,
x2) =
perm_out_ga(
x2)
.(
x1,
x2) =
.(
x1,
x2)
U1_ga(
x1,
x2,
x3,
x4,
x5) =
U1_ga(
x5)
delete_in_aga(
x1,
x2,
x3) =
delete_in_aga(
x2)
delete_out_aga(
x1,
x2,
x3) =
delete_out_aga(
x1,
x3)
U3_aga(
x1,
x2,
x3,
x4,
x5) =
U3_aga(
x2,
x5)
U2_ga(
x1,
x2,
x3,
x4,
x5) =
U2_ga(
x3,
x5)
PERM_IN_GA(
x1,
x2) =
PERM_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x5)
We have to consider all (P,R,Pi)-chains
(15) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(16) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
U1_GA(X, Y, U, V, delete_out_aga(U, .(X, Y), W)) → PERM_IN_GA(W, V)
PERM_IN_GA(.(X, Y), .(U, V)) → U1_GA(X, Y, U, V, delete_in_aga(U, .(X, Y), W))
The TRS R consists of the following rules:
delete_in_aga(X, .(X, Y), Y) → delete_out_aga(X, .(X, Y), Y)
delete_in_aga(U, .(X, Y), .(X, Z)) → U3_aga(U, X, Y, Z, delete_in_aga(U, Y, Z))
U3_aga(U, X, Y, Z, delete_out_aga(U, Y, Z)) → delete_out_aga(U, .(X, Y), .(X, Z))
The argument filtering Pi contains the following mapping:
.(
x1,
x2) =
.(
x1,
x2)
delete_in_aga(
x1,
x2,
x3) =
delete_in_aga(
x2)
delete_out_aga(
x1,
x2,
x3) =
delete_out_aga(
x1,
x3)
U3_aga(
x1,
x2,
x3,
x4,
x5) =
U3_aga(
x2,
x5)
PERM_IN_GA(
x1,
x2) =
PERM_IN_GA(
x1)
U1_GA(
x1,
x2,
x3,
x4,
x5) =
U1_GA(
x5)
We have to consider all (P,R,Pi)-chains
(17) PiDPToQDPProof (SOUND transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
U1_GA(delete_out_aga(U, W)) → PERM_IN_GA(W)
PERM_IN_GA(.(X, Y)) → U1_GA(delete_in_aga(.(X, Y)))
The TRS R consists of the following rules:
delete_in_aga(.(X, Y)) → delete_out_aga(X, Y)
delete_in_aga(.(X, Y)) → U3_aga(X, delete_in_aga(Y))
U3_aga(X, delete_out_aga(U, Z)) → delete_out_aga(U, .(X, Z))
The set Q consists of the following terms:
delete_in_aga(x0)
U3_aga(x0, x1)
We have to consider all (P,Q,R)-chains.
(19) 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:
U1_GA(delete_out_aga(U, W)) → PERM_IN_GA(W)
PERM_IN_GA(.(X, Y)) → U1_GA(delete_in_aga(.(X, Y)))
Strictly oriented rules of the TRS R:
delete_in_aga(.(X, Y)) → delete_out_aga(X, Y)
delete_in_aga(.(X, Y)) → U3_aga(X, delete_in_aga(Y))
U3_aga(X, delete_out_aga(U, Z)) → delete_out_aga(U, .(X, Z))
Used ordering: Knuth-Bendix order [KBO] with precedence:
.2 > deleteinaga1 > U3aga2 > U1GA1 > PERMINGA1 > deleteoutaga2
and weight map:
delete_in_aga_1=1
U1_GA_1=1
PERM_IN_GA_1=3
._2=0
delete_out_aga_2=1
U3_aga_2=0
The variable weight is 1
(20) Obligation:
Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:
delete_in_aga(x0)
U3_aga(x0, x1)
We have to consider all (P,Q,R)-chains.
(21) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(22) YES