Left Termination of the query pattern
perm1(g,a)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog
↳1 PrologToPiTRSProof (⇒, 0 ms)
↳2 PiTRS
↳3 DependencyPairsProof (⇔, 22 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 0 ms)
↳6 AND
    ↳7 PiDP
        ↳8 UsableRulesProof (⇔, 0 ms)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇒, 0 ms)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔, 0 ms)
        ↳13 YES
    ↳14 PiDP
        ↳15 UsableRulesProof (⇔, 0 ms)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇒, 0 ms)
        ↳18 QDP
        ↳19 MRRProof (⇔, 8 ms)
        ↳20 QDP
        ↳21 PisEmptyProof (⇔, 0 ms)
        ↳22 YES

(0) Obligation:

Clauses:

perm1([], []).
perm1(Xs, .(X, Ys)) :- ','(select(X, Xs, Zs), perm1(Zs, Ys)).
select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).

Query: perm1(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:
perm1_in: (b,f)
select_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs))
select_in_aga(X, .(X, Xs), Xs) → select_out_aga(X, .(X, Xs), Xs)
select_in_aga(X, .(Y, Xs), .(Y, Zs)) → U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) → select_out_aga(X, .(Y, Xs), .(Y, Zs))
U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys))
U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) → perm1_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs))
select_in_aga(X, .(X, Xs), Xs) → select_out_aga(X, .(X, Xs), Xs)
select_in_aga(X, .(Y, Xs), .(Y, Zs)) → U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) → select_out_aga(X, .(Y, Xs), .(Y, Zs))
U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys))
U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) → perm1_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)

(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:

PERM1_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, select_in_aga(X, Xs, Zs))
PERM1_IN_GA(Xs, .(X, Ys)) → SELECT_IN_AGA(X, Xs, Zs)
SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) → U3_AGA(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) → SELECT_IN_AGA(X, Xs, Zs)
U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → U2_GA(Xs, X, Ys, perm1_in_ga(Zs, Ys))
U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → PERM1_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs))
select_in_aga(X, .(X, Xs), Xs) → select_out_aga(X, .(X, Xs), Xs)
select_in_aga(X, .(Y, Xs), .(Y, Zs)) → U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) → select_out_aga(X, .(Y, Xs), .(Y, Zs))
U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys))
U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) → perm1_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)

We have to consider all (P,R,Pi)-chains

(4) Obligation:

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

PERM1_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, select_in_aga(X, Xs, Zs))
PERM1_IN_GA(Xs, .(X, Ys)) → SELECT_IN_AGA(X, Xs, Zs)
SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) → U3_AGA(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) → SELECT_IN_AGA(X, Xs, Zs)
U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → U2_GA(Xs, X, Ys, perm1_in_ga(Zs, Ys))
U1_GA(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → PERM1_IN_GA(Zs, Ys)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs))
select_in_aga(X, .(X, Xs), Xs) → select_out_aga(X, .(X, Xs), Xs)
select_in_aga(X, .(Y, Xs), .(Y, Zs)) → U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) → select_out_aga(X, .(Y, Xs), .(Y, Zs))
U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys))
U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) → perm1_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
SELECT_IN_AGA(x1, x2, x3)  =  SELECT_IN_AGA(x2)
U3_AGA(x1, x2, x3, x4, x5)  =  U3_AGA(x2, x5)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x2, x4)

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:

SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) → SELECT_IN_AGA(X, Xs, Zs)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs))
select_in_aga(X, .(X, Xs), Xs) → select_out_aga(X, .(X, Xs), Xs)
select_in_aga(X, .(Y, Xs), .(Y, Zs)) → U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) → select_out_aga(X, .(Y, Xs), .(Y, Zs))
U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys))
U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) → perm1_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
SELECT_IN_AGA(x1, x2, x3)  =  SELECT_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:

SELECT_IN_AGA(X, .(Y, Xs), .(Y, Zs)) → SELECT_IN_AGA(X, Xs, Zs)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
SELECT_IN_AGA(x1, x2, x3)  =  SELECT_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:

SELECT_IN_AGA(.(Y, Xs)) → SELECT_IN_AGA(Xs)

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:

  • SELECT_IN_AGA(.(Y, Xs)) → SELECT_IN_AGA(Xs)
    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(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → PERM1_IN_GA(Zs, Ys)
PERM1_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, select_in_aga(X, Xs, Zs))

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(Xs, .(X, Ys)) → U1_ga(Xs, X, Ys, select_in_aga(X, Xs, Zs))
select_in_aga(X, .(X, Xs), Xs) → select_out_aga(X, .(X, Xs), Xs)
select_in_aga(X, .(Y, Xs), .(Y, Zs)) → U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) → select_out_aga(X, .(Y, Xs), .(Y, Zs))
U1_ga(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → U2_ga(Xs, X, Ys, perm1_in_ga(Zs, Ys))
U2_ga(Xs, X, Ys, perm1_out_ga(Zs, Ys)) → perm1_out_ga(Xs, .(X, Ys))

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2)  =  perm1_in_ga(x1)
[]  =  []
perm1_out_ga(x1, x2)  =  perm1_out_ga(x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x2, x4)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)

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(Xs, X, Ys, select_out_aga(X, Xs, Zs)) → PERM1_IN_GA(Zs, Ys)
PERM1_IN_GA(Xs, .(X, Ys)) → U1_GA(Xs, X, Ys, select_in_aga(X, Xs, Zs))

The TRS R consists of the following rules:

select_in_aga(X, .(X, Xs), Xs) → select_out_aga(X, .(X, Xs), Xs)
select_in_aga(X, .(Y, Xs), .(Y, Zs)) → U3_aga(X, Y, Xs, Zs, select_in_aga(X, Xs, Zs))
U3_aga(X, Y, Xs, Zs, select_out_aga(X, Xs, Zs)) → select_out_aga(X, .(Y, Xs), .(Y, Zs))

The argument filtering Pi contains the following mapping:
select_in_aga(x1, x2, x3)  =  select_in_aga(x2)
.(x1, x2)  =  .(x1, x2)
select_out_aga(x1, x2, x3)  =  select_out_aga(x1, x3)
U3_aga(x1, x2, x3, x4, x5)  =  U3_aga(x2, x5)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)

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(select_out_aga(X, Zs)) → PERM1_IN_GA(Zs)
PERM1_IN_GA(Xs) → U1_GA(select_in_aga(Xs))

The TRS R consists of the following rules:

select_in_aga(.(X, Xs)) → select_out_aga(X, Xs)
select_in_aga(.(Y, Xs)) → U3_aga(Y, select_in_aga(Xs))
U3_aga(Y, select_out_aga(X, Zs)) → select_out_aga(X, .(Y, Zs))

The set Q consists of the following terms:

select_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(select_out_aga(X, Zs)) → PERM1_IN_GA(Zs)
PERM1_IN_GA(Xs) → U1_GA(select_in_aga(Xs))

Strictly oriented rules of the TRS R:

select_in_aga(.(X, Xs)) → select_out_aga(X, Xs)
select_in_aga(.(Y, Xs)) → U3_aga(Y, select_in_aga(Xs))
U3_aga(Y, select_out_aga(X, Zs)) → select_out_aga(X, .(Y, Zs))

Used ordering: Knuth-Bendix order [KBO] with precedence:
U1GA1 > .2 > selectinaga1 > U3aga2 > PERM1INGA1 > selectoutaga2

and weight map:

select_in_aga_1=1
U1_GA_1=1
PERM1_IN_GA_1=3
._2=0
select_out_aga_2=1
U3_aga_2=0

The variable weight is 2

(20) Obligation:

Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

select_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