(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).

Queries:

shuffle(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

shuffle1(T4, [], T4).
shuffle1([], T5, T5).
shuffle1(.(T6, T7), T8, .(T6, T10)) :- shuffle1(T7, T8, T10).
shuffle1(T11, .(T12, T13), .(T12, T15)) :- shuffle1(T11, T13, T15).

Queries:

shuffle1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
shuffle1_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_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:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_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:

SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → U1_GGA(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → SHUFFLE1_IN_GGA(T7, T8, T10)
SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → U2_GGA(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → SHUFFLE1_IN_GGA(T11, T13, T15)

The TRS R consists of the following rules:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_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)
SHUFFLE1_IN_GGA(x1, x2, x3)  =  SHUFFLE1_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:

SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → U1_GGA(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → SHUFFLE1_IN_GGA(T7, T8, T10)
SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → U2_GGA(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → SHUFFLE1_IN_GGA(T11, T13, T15)

The TRS R consists of the following rules:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_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)
SHUFFLE1_IN_GGA(x1, x2, x3)  =  SHUFFLE1_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:

SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → SHUFFLE1_IN_GGA(T11, T13, T15)
SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → SHUFFLE1_IN_GGA(T7, T8, T10)

The TRS R consists of the following rules:

shuffle1_in_gga(T4, [], T4) → shuffle1_out_gga(T4, [], T4)
shuffle1_in_gga([], T5, T5) → shuffle1_out_gga([], T5, T5)
shuffle1_in_gga(.(T6, T7), T8, .(T6, T10)) → U1_gga(T6, T7, T8, T10, shuffle1_in_gga(T7, T8, T10))
shuffle1_in_gga(T11, .(T12, T13), .(T12, T15)) → U2_gga(T11, T12, T13, T15, shuffle1_in_gga(T11, T13, T15))
U2_gga(T11, T12, T13, T15, shuffle1_out_gga(T11, T13, T15)) → shuffle1_out_gga(T11, .(T12, T13), .(T12, T15))
U1_gga(T6, T7, T8, T10, shuffle1_out_gga(T7, T8, T10)) → shuffle1_out_gga(.(T6, T7), T8, .(T6, T10))

The argument filtering Pi contains the following mapping:
shuffle1_in_gga(x1, x2, x3)  =  shuffle1_in_gga(x1, x2)
[]  =  []
shuffle1_out_gga(x1, x2, x3)  =  shuffle1_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)
SHUFFLE1_IN_GGA(x1, x2, x3)  =  SHUFFLE1_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:

SHUFFLE1_IN_GGA(T11, .(T12, T13), .(T12, T15)) → SHUFFLE1_IN_GGA(T11, T13, T15)
SHUFFLE1_IN_GGA(.(T6, T7), T8, .(T6, T10)) → SHUFFLE1_IN_GGA(T7, T8, T10)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
SHUFFLE1_IN_GGA(x1, x2, x3)  =  SHUFFLE1_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:

SHUFFLE1_IN_GGA(T11, .(T12, T13)) → SHUFFLE1_IN_GGA(T11, T13)
SHUFFLE1_IN_GGA(.(T6, T7), T8) → SHUFFLE1_IN_GGA(T7, T8)

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:

  • SHUFFLE1_IN_GGA(T11, .(T12, T13)) → SHUFFLE1_IN_GGA(T11, T13)
    The graph contains the following edges 1 >= 1, 2 > 2

  • SHUFFLE1_IN_GGA(.(T6, T7), T8) → SHUFFLE1_IN_GGA(T7, T8)
    The graph contains the following edges 1 > 1, 2 >= 2

(14) TRUE