Left Termination of the query pattern perm_in_2(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 AND
      ↳7 PiDP
        ↳8 UsableRulesProof (⇔)
        ↳9 PiDP
        ↳10 PiDPToQDPProof (⇐)
        ↳11 QDP
        ↳12 QDPSizeChangeProof (⇔)
        ↳13 TRUE
      ↳14 PiDP
        ↳15 UsableRulesProof (⇔)
        ↳16 PiDP
        ↳17 PiDPToQDPProof (⇐)
        ↳18 QDP
        ↳19 QDPSizeChangeProof (⇔)
        ↳20 TRUE
  ↳21 PrologToPiTRSProof (⇐)
    ↳22 PiTRS
    ↳23 DependencyPairsProof (⇔)
    ↳24 PiDP
    ↳25 DependencyGraphProof (⇔)
    ↳26 AND
      ↳27 PiDP
        ↳28 UsableRulesProof (⇔)
        ↳29 PiDP
        ↳30 PiDPToQDPProof (⇐)
        ↳31 QDP
        ↳32 QDPSizeChangeProof (⇔)
        ↳33 TRUE
      ↳34 PiDP
        ↳35 UsableRulesProof (⇔)
        ↳36 PiDP
        ↳37 PiDPToQDPProof (⇐)
        ↳38 QDP

(0) Obligation:

Clauses:

perm([], []).
perm(.(X, L), Z) :- ','(perm(L, Y), insert(X, Y, Z)).
insert(X, [], .(X, [])).
insert(X, L, .(X, L)).
insert(X, .(H, L1), .(H, L2)) :- insert(X, L1, L2).

Queries:

perm(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:
perm_in: (b,f)
insert_in: (b,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, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, 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, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, 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, L), Z) → U1_GA(X, L, Z, perm_in_ga(L, Y))
PERM_IN_GA(.(X, L), Z) → PERM_IN_GA(L, Y)
U1_GA(X, L, Z, perm_out_ga(L, Y)) → U2_GA(X, L, Z, insert_in_gga(X, Y, Z))
U1_GA(X, L, Z, perm_out_ga(L, Y)) → INSERT_IN_GGA(X, Y, Z)
INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → U3_GGA(X, H, L1, L2, insert_in_gga(X, L1, L2))
INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → INSERT_IN_GGA(X, L1, L2)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
INSERT_IN_GGA(x1, x2, x3)  =  INSERT_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x2, 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, L), Z) → U1_GA(X, L, Z, perm_in_ga(L, Y))
PERM_IN_GA(.(X, L), Z) → PERM_IN_GA(L, Y)
U1_GA(X, L, Z, perm_out_ga(L, Y)) → U2_GA(X, L, Z, insert_in_gga(X, Y, Z))
U1_GA(X, L, Z, perm_out_ga(L, Y)) → INSERT_IN_GGA(X, Y, Z)
INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → U3_GGA(X, H, L1, L2, insert_in_gga(X, L1, L2))
INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → INSERT_IN_GGA(X, L1, L2)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x4)
INSERT_IN_GGA(x1, x2, x3)  =  INSERT_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x2, 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 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → INSERT_IN_GGA(X, L1, L2)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
INSERT_IN_GGA(x1, x2, x3)  =  INSERT_IN_GGA(x1, 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:

INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → INSERT_IN_GGA(X, L1, L2)

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

INSERT_IN_GGA(X, .(H, L1)) → INSERT_IN_GGA(X, L1)

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:

  • INSERT_IN_GGA(X, .(H, L1)) → INSERT_IN_GGA(X, L1)
    The graph contains the following edges 1 >= 1, 2 > 2

(13) TRUE

(14) Obligation:

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

PERM_IN_GA(.(X, L), Z) → PERM_IN_GA(L, Y)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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)  =  U1_ga(x1, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x2, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)

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:

PERM_IN_GA(.(X, L), Z) → PERM_IN_GA(L, Y)

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

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:

PERM_IN_GA(.(X, L)) → PERM_IN_GA(L)

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

(19) 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:

  • PERM_IN_GA(.(X, L)) → PERM_IN_GA(L)
    The graph contains the following edges 1 > 1

(20) TRUE

(21) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm_in: (b,f)
insert_in: (b,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, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(22) Obligation:

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

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)

(23) 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, L), Z) → U1_GA(X, L, Z, perm_in_ga(L, Y))
PERM_IN_GA(.(X, L), Z) → PERM_IN_GA(L, Y)
U1_GA(X, L, Z, perm_out_ga(L, Y)) → U2_GA(X, L, Z, insert_in_gga(X, Y, Z))
U1_GA(X, L, Z, perm_out_ga(L, Y)) → INSERT_IN_GGA(X, Y, Z)
INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → U3_GGA(X, H, L1, L2, insert_in_gga(X, L1, L2))
INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → INSERT_IN_GGA(X, L1, L2)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
INSERT_IN_GGA(x1, x2, x3)  =  INSERT_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)

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

(24) Obligation:

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

PERM_IN_GA(.(X, L), Z) → U1_GA(X, L, Z, perm_in_ga(L, Y))
PERM_IN_GA(.(X, L), Z) → PERM_IN_GA(L, Y)
U1_GA(X, L, Z, perm_out_ga(L, Y)) → U2_GA(X, L, Z, insert_in_gga(X, Y, Z))
U1_GA(X, L, Z, perm_out_ga(L, Y)) → INSERT_IN_GGA(X, Y, Z)
INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → U3_GGA(X, H, L1, L2, insert_in_gga(X, L1, L2))
INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → INSERT_IN_GGA(X, L1, L2)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x2, x4)
INSERT_IN_GGA(x1, x2, x3)  =  INSERT_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4, x5)  =  U3_GGA(x1, x2, x3, x5)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(26) Complex Obligation (AND)

(27) Obligation:

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

INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → INSERT_IN_GGA(X, L1, L2)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
INSERT_IN_GGA(x1, x2, x3)  =  INSERT_IN_GGA(x1, x2)

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

(28) UsableRulesProof (EQUIVALENT transformation)

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

(29) Obligation:

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

INSERT_IN_GGA(X, .(H, L1), .(H, L2)) → INSERT_IN_GGA(X, L1, L2)

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

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

(30) PiDPToQDPProof (SOUND transformation)

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

(31) Obligation:

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

INSERT_IN_GGA(X, .(H, L1)) → INSERT_IN_GGA(X, L1)

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

(32) 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:

  • INSERT_IN_GGA(X, .(H, L1)) → INSERT_IN_GGA(X, L1)
    The graph contains the following edges 1 >= 1, 2 > 2

(33) TRUE

(34) Obligation:

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

PERM_IN_GA(.(X, L), Z) → PERM_IN_GA(L, Y)

The TRS R consists of the following rules:

perm_in_ga([], []) → perm_out_ga([], [])
perm_in_ga(.(X, L), Z) → U1_ga(X, L, Z, perm_in_ga(L, Y))
U1_ga(X, L, Z, perm_out_ga(L, Y)) → U2_ga(X, L, Z, insert_in_gga(X, Y, Z))
insert_in_gga(X, [], .(X, [])) → insert_out_gga(X, [], .(X, []))
insert_in_gga(X, L, .(X, L)) → insert_out_gga(X, L, .(X, L))
insert_in_gga(X, .(H, L1), .(H, L2)) → U3_gga(X, H, L1, L2, insert_in_gga(X, L1, L2))
U3_gga(X, H, L1, L2, insert_out_gga(X, L1, L2)) → insert_out_gga(X, .(H, L1), .(H, L2))
U2_ga(X, L, Z, insert_out_gga(X, Y, Z)) → perm_out_ga(.(X, L), Z)

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(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4)  =  U2_ga(x1, x2, x4)
insert_in_gga(x1, x2, x3)  =  insert_in_gga(x1, x2)
insert_out_gga(x1, x2, x3)  =  insert_out_gga(x1, x2, x3)
U3_gga(x1, x2, x3, x4, x5)  =  U3_gga(x1, x2, x3, x5)
PERM_IN_GA(x1, x2)  =  PERM_IN_GA(x1)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

PERM_IN_GA(.(X, L), Z) → PERM_IN_GA(L, Y)

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

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

(37) PiDPToQDPProof (SOUND transformation)

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

(38) Obligation:

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

PERM_IN_GA(.(X, L)) → PERM_IN_GA(L)

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