Left Termination of the query pattern reverse_concatenate_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

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

(0) Obligation:

Clauses:

concatenate([], L, L).
concatenate(.(X, L1), L2, .(X, L3)) :- concatenate(L1, L2, L3).
member(X, .(X, L)).
member(X, .(Y, L)) :- member(X, L).
reverse(L, L1) :- reverse_concatenate(L, [], L1).
reverse_concatenate([], L, L).
reverse_concatenate(.(X, L1), L2, L3) :- reverse_concatenate(L1, .(X, L2), L3).

Queries:

reverse_concatenate(g,g,a).

(1) PrologToPiTRSProof (SOUND transformation)

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

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(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:

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(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:

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → U4_GGA(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)

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

(4) Obligation:

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

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → U4_GGA(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x5)
REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2))

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

(11) 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:

  • REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2))
    The graph contains the following edges 1 > 1

(12) TRUE

(13) PrologToPiTRSProof (SOUND transformation)

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

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)

(15) 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:

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → U4_GGA(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)

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

(16) Obligation:

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

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → U4_GGA(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

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

(18) Obligation:

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

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)

The TRS R consists of the following rules:

reverse_concatenate_in_gga([], L, L) → reverse_concatenate_out_gga([], L, L)
reverse_concatenate_in_gga(.(X, L1), L2, L3) → U4_gga(X, L1, L2, L3, reverse_concatenate_in_gga(L1, .(X, L2), L3))
U4_gga(X, L1, L2, L3, reverse_concatenate_out_gga(L1, .(X, L2), L3)) → reverse_concatenate_out_gga(.(X, L1), L2, L3)

The argument filtering Pi contains the following mapping:
reverse_concatenate_in_gga(x1, x2, x3)  =  reverse_concatenate_in_gga(x1, x2)
[]  =  []
reverse_concatenate_out_gga(x1, x2, x3)  =  reverse_concatenate_out_gga(x1, x2, x3)
.(x1, x2)  =  .(x1, x2)
U4_gga(x1, x2, x3, x4, x5)  =  U4_gga(x1, x2, x3, x5)
REVERSE_CONCATENATE_IN_GGA(x1, x2, x3)  =  REVERSE_CONCATENATE_IN_GGA(x1, x2)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2, L3) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2), L3)

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

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

REVERSE_CONCATENATE_IN_GGA(.(X, L1), L2) → REVERSE_CONCATENATE_IN_GGA(L1, .(X, L2))

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