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

0 Prolog
↳1 CutEliminatorProof (⇒, 0 ms)
↳2 Prolog
↳3 PrologToPiTRSProof (⇒, 0 ms)
↳4 PiTRS
↳5 DependencyPairsProof (⇔, 24 ms)
↳6 PiDP
↳7 DependencyGraphProof (⇔, 0 ms)
↳8 PiDP
↳9 UsableRulesProof (⇔, 0 ms)
↳10 PiDP
↳11 PiDPToQDPProof (⇒, 0 ms)
↳12 QDP
↳13 QDPSizeChangeProof (⇔, 0 ms)
↳14 YES

(0) Obligation:

Clauses:

shuffle(A, [], A) :- !.
shuffle([], B, B) :- !.
shuffle(.(A, RestA), B, .(A, Shuffled)) :- shuffle(RestA, B, Shuffled).
shuffle(A, .(B, RestB), .(B, Shuffled)) :- shuffle(A, RestB, Shuffled).

Query: shuffle(g,g,a)

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

shuffle(A, [], A).
shuffle([], B, B).
shuffle(.(A, RestA), B, .(A, Shuffled)) :- shuffle(RestA, B, Shuffled).
shuffle(A, .(B, RestB), .(B, Shuffled)) :- shuffle(A, RestB, Shuffled).

Query: shuffle(g,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
shuffle_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

shuffle_in_gga(A, [], A) → shuffle_out_gga(A, [], A)
shuffle_in_gga([], B, B) → shuffle_out_gga([], B, B)
shuffle_in_gga(.(A, RestA), B, .(A, Shuffled)) → U1_gga(A, RestA, B, Shuffled, shuffle_in_gga(RestA, B, Shuffled))
shuffle_in_gga(A, .(B, RestB), .(B, Shuffled)) → U2_gga(A, B, RestB, Shuffled, shuffle_in_gga(A, RestB, Shuffled))
U2_gga(A, B, RestB, Shuffled, shuffle_out_gga(A, RestB, Shuffled)) → shuffle_out_gga(A, .(B, RestB), .(B, Shuffled))
U1_gga(A, RestA, B, Shuffled, shuffle_out_gga(RestA, B, Shuffled)) → shuffle_out_gga(.(A, RestA), B, .(A, Shuffled))

The argument filtering Pi contains the following mapping:
shuffle_in_gga(x1, x2, x3)  =  shuffle_in_gga(x1, x2)
[]  =  []
shuffle_out_gga(x1, x2, x3)  =  shuffle_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

shuffle_in_gga(A, [], A) → shuffle_out_gga(A, [], A)
shuffle_in_gga([], B, B) → shuffle_out_gga([], B, B)
shuffle_in_gga(.(A, RestA), B, .(A, Shuffled)) → U1_gga(A, RestA, B, Shuffled, shuffle_in_gga(RestA, B, Shuffled))
shuffle_in_gga(A, .(B, RestB), .(B, Shuffled)) → U2_gga(A, B, RestB, Shuffled, shuffle_in_gga(A, RestB, Shuffled))
U2_gga(A, B, RestB, Shuffled, shuffle_out_gga(A, RestB, Shuffled)) → shuffle_out_gga(A, .(B, RestB), .(B, Shuffled))
U1_gga(A, RestA, B, Shuffled, shuffle_out_gga(RestA, B, Shuffled)) → shuffle_out_gga(.(A, RestA), B, .(A, Shuffled))

The argument filtering Pi contains the following mapping:
shuffle_in_gga(x1, x2, x3)  =  shuffle_in_gga(x1, x2)
[]  =  []
shuffle_out_gga(x1, x2, x3)  =  shuffle_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)

(5) 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:

SHUFFLE_IN_GGA(.(A, RestA), B, .(A, Shuffled)) → U1_GGA(A, RestA, B, Shuffled, shuffle_in_gga(RestA, B, Shuffled))
SHUFFLE_IN_GGA(.(A, RestA), B, .(A, Shuffled)) → SHUFFLE_IN_GGA(RestA, B, Shuffled)
SHUFFLE_IN_GGA(A, .(B, RestB), .(B, Shuffled)) → U2_GGA(A, B, RestB, Shuffled, shuffle_in_gga(A, RestB, Shuffled))
SHUFFLE_IN_GGA(A, .(B, RestB), .(B, Shuffled)) → SHUFFLE_IN_GGA(A, RestB, Shuffled)

The TRS R consists of the following rules:

shuffle_in_gga(A, [], A) → shuffle_out_gga(A, [], A)
shuffle_in_gga([], B, B) → shuffle_out_gga([], B, B)
shuffle_in_gga(.(A, RestA), B, .(A, Shuffled)) → U1_gga(A, RestA, B, Shuffled, shuffle_in_gga(RestA, B, Shuffled))
shuffle_in_gga(A, .(B, RestB), .(B, Shuffled)) → U2_gga(A, B, RestB, Shuffled, shuffle_in_gga(A, RestB, Shuffled))
U2_gga(A, B, RestB, Shuffled, shuffle_out_gga(A, RestB, Shuffled)) → shuffle_out_gga(A, .(B, RestB), .(B, Shuffled))
U1_gga(A, RestA, B, Shuffled, shuffle_out_gga(RestA, B, Shuffled)) → shuffle_out_gga(.(A, RestA), B, .(A, Shuffled))

The argument filtering Pi contains the following mapping:
shuffle_in_gga(x1, x2, x3)  =  shuffle_in_gga(x1, x2)
[]  =  []
shuffle_out_gga(x1, x2, x3)  =  shuffle_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)
SHUFFLE_IN_GGA(x1, x2, x3)  =  SHUFFLE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x2, x5)

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

(6) Obligation:

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

SHUFFLE_IN_GGA(.(A, RestA), B, .(A, Shuffled)) → U1_GGA(A, RestA, B, Shuffled, shuffle_in_gga(RestA, B, Shuffled))
SHUFFLE_IN_GGA(.(A, RestA), B, .(A, Shuffled)) → SHUFFLE_IN_GGA(RestA, B, Shuffled)
SHUFFLE_IN_GGA(A, .(B, RestB), .(B, Shuffled)) → U2_GGA(A, B, RestB, Shuffled, shuffle_in_gga(A, RestB, Shuffled))
SHUFFLE_IN_GGA(A, .(B, RestB), .(B, Shuffled)) → SHUFFLE_IN_GGA(A, RestB, Shuffled)

The TRS R consists of the following rules:

shuffle_in_gga(A, [], A) → shuffle_out_gga(A, [], A)
shuffle_in_gga([], B, B) → shuffle_out_gga([], B, B)
shuffle_in_gga(.(A, RestA), B, .(A, Shuffled)) → U1_gga(A, RestA, B, Shuffled, shuffle_in_gga(RestA, B, Shuffled))
shuffle_in_gga(A, .(B, RestB), .(B, Shuffled)) → U2_gga(A, B, RestB, Shuffled, shuffle_in_gga(A, RestB, Shuffled))
U2_gga(A, B, RestB, Shuffled, shuffle_out_gga(A, RestB, Shuffled)) → shuffle_out_gga(A, .(B, RestB), .(B, Shuffled))
U1_gga(A, RestA, B, Shuffled, shuffle_out_gga(RestA, B, Shuffled)) → shuffle_out_gga(.(A, RestA), B, .(A, Shuffled))

The argument filtering Pi contains the following mapping:
shuffle_in_gga(x1, x2, x3)  =  shuffle_in_gga(x1, x2)
[]  =  []
shuffle_out_gga(x1, x2, x3)  =  shuffle_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)
SHUFFLE_IN_GGA(x1, x2, x3)  =  SHUFFLE_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5)  =  U1_GGA(x1, x5)
U2_GGA(x1, x2, x3, x4, x5)  =  U2_GGA(x2, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

SHUFFLE_IN_GGA(A, .(B, RestB), .(B, Shuffled)) → SHUFFLE_IN_GGA(A, RestB, Shuffled)
SHUFFLE_IN_GGA(.(A, RestA), B, .(A, Shuffled)) → SHUFFLE_IN_GGA(RestA, B, Shuffled)

The TRS R consists of the following rules:

shuffle_in_gga(A, [], A) → shuffle_out_gga(A, [], A)
shuffle_in_gga([], B, B) → shuffle_out_gga([], B, B)
shuffle_in_gga(.(A, RestA), B, .(A, Shuffled)) → U1_gga(A, RestA, B, Shuffled, shuffle_in_gga(RestA, B, Shuffled))
shuffle_in_gga(A, .(B, RestB), .(B, Shuffled)) → U2_gga(A, B, RestB, Shuffled, shuffle_in_gga(A, RestB, Shuffled))
U2_gga(A, B, RestB, Shuffled, shuffle_out_gga(A, RestB, Shuffled)) → shuffle_out_gga(A, .(B, RestB), .(B, Shuffled))
U1_gga(A, RestA, B, Shuffled, shuffle_out_gga(RestA, B, Shuffled)) → shuffle_out_gga(.(A, RestA), B, .(A, Shuffled))

The argument filtering Pi contains the following mapping:
shuffle_in_gga(x1, x2, x3)  =  shuffle_in_gga(x1, x2)
[]  =  []
shuffle_out_gga(x1, x2, x3)  =  shuffle_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U1_gga(x1, x2, x3, x4, x5)  =  U1_gga(x1, x5)
U2_gga(x1, x2, x3, x4, x5)  =  U2_gga(x2, x5)
SHUFFLE_IN_GGA(x1, x2, x3)  =  SHUFFLE_IN_GGA(x1, x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

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

SHUFFLE_IN_GGA(A, .(B, RestB), .(B, Shuffled)) → SHUFFLE_IN_GGA(A, RestB, Shuffled)
SHUFFLE_IN_GGA(.(A, RestA), B, .(A, Shuffled)) → SHUFFLE_IN_GGA(RestA, B, Shuffled)

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

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

(11) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) Obligation:

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

SHUFFLE_IN_GGA(A, .(B, RestB)) → SHUFFLE_IN_GGA(A, RestB)
SHUFFLE_IN_GGA(.(A, RestA), B) → SHUFFLE_IN_GGA(RestA, B)

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

(13) 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:

  • SHUFFLE_IN_GGA(A, .(B, RestB)) → SHUFFLE_IN_GGA(A, RestB)
    The graph contains the following edges 1 >= 1, 2 > 2

  • SHUFFLE_IN_GGA(.(A, RestA), B) → SHUFFLE_IN_GGA(RestA, B)
    The graph contains the following edges 1 > 1, 2 >= 2

(14) YES