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

0 Prolog
↳1 PrologToDTProblemTransformerProof (⇒, 88 ms)
↳2 TRIPLES
↳3 TriplesToPiDPProof (⇒, 27 ms)
↳4 PiDP
↳5 DependencyGraphProof (⇔, 6 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 QDPSizeChangeProof (⇔, 0 ms)
        ↳20 YES
    ↳21 PiDP
        ↳22 UsableRulesProof (⇔, 0 ms)
        ↳23 PiDP
        ↳24 PiDPToQDPProof (⇒, 0 ms)
        ↳25 QDP
        ↳26 QDPSizeChangeProof (⇔, 0 ms)
        ↳27 YES

(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).

Query: perm(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

(2) Obligation:

Triples:

permA(.(X1, X2), X3) :- permA(X2, X4).
permA(.(X1, X2), X3) :- ','(permcA(X2, X4), insertB(X1, X4, X3)).
insertB(X1, .(X2, X3), .(X2, X4)) :- insertB(X1, X3, X4).
insertC(X1, .(X2, X3), .(X2, X4)) :- insertC(X1, X3, X4).
permD(.(X1, .(X2, X3)), X4) :- permA(X3, X5).
permD(.(X1, .(X2, X3)), X4) :- ','(permcA(X3, X5), insertB(X2, X5, X6)).
permD(.(X1, .(X2, X3)), X4) :- ','(permcA(X3, X5), ','(insertcB(X2, X5, X6), insertC(X1, X6, X4))).

Clauses:

permcA([], []).
permcA(.(X1, X2), X3) :- ','(permcA(X2, X4), insertcB(X1, X4, X3)).
insertcB(X1, [], .(X1, [])).
insertcB(X1, X2, .(X1, X2)).
insertcB(X1, .(X2, X3), .(X2, X4)) :- insertcB(X1, X3, X4).
insertcC(X1, [], .(X1, [])).
insertcC(X1, X2, .(X1, X2)).
insertcC(X1, .(X2, X3), .(X2, X4)) :- insertcC(X1, X3, X4).

Afs:

permD(x1, x2)  =  permD(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
permD_in: (b,f)
permA_in: (b,f)
permcA_in: (b,f)
insertcB_in: (b,b,f)
insertB_in: (b,b,f)
insertC_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PERMD_IN_GA(.(X1, .(X2, X3)), X4) → U6_GA(X1, X2, X3, X4, permA_in_ga(X3, X5))
PERMD_IN_GA(.(X1, .(X2, X3)), X4) → PERMA_IN_GA(X3, X5)
PERMA_IN_GA(.(X1, X2), X3) → U1_GA(X1, X2, X3, permA_in_ga(X2, X4))
PERMA_IN_GA(.(X1, X2), X3) → PERMA_IN_GA(X2, X4)
PERMA_IN_GA(.(X1, X2), X3) → U2_GA(X1, X2, X3, permcA_in_ga(X2, X4))
U2_GA(X1, X2, X3, permcA_out_ga(X2, X4)) → U3_GA(X1, X2, X3, insertB_in_gga(X1, X4, X3))
U2_GA(X1, X2, X3, permcA_out_ga(X2, X4)) → INSERTB_IN_GGA(X1, X4, X3)
INSERTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → U4_GGA(X1, X2, X3, X4, insertB_in_gga(X1, X3, X4))
INSERTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → INSERTB_IN_GGA(X1, X3, X4)
PERMD_IN_GA(.(X1, .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, permcA_in_ga(X3, X5))
U7_GA(X1, X2, X3, X4, permcA_out_ga(X3, X5)) → U8_GA(X1, X2, X3, X4, insertB_in_gga(X2, X5, X6))
U7_GA(X1, X2, X3, X4, permcA_out_ga(X3, X5)) → INSERTB_IN_GGA(X2, X5, X6)
U7_GA(X1, X2, X3, X4, permcA_out_ga(X3, X5)) → U9_GA(X1, X2, X3, X4, insertcB_in_gga(X2, X5, X6))
U9_GA(X1, X2, X3, X4, insertcB_out_gga(X2, X5, X6)) → U10_GA(X1, X2, X3, X4, insertC_in_gga(X1, X6, X4))
U9_GA(X1, X2, X3, X4, insertcB_out_gga(X2, X5, X6)) → INSERTC_IN_GGA(X1, X6, X4)
INSERTC_IN_GGA(X1, .(X2, X3), .(X2, X4)) → U5_GGA(X1, X2, X3, X4, insertC_in_gga(X1, X3, X4))
INSERTC_IN_GGA(X1, .(X2, X3), .(X2, X4)) → INSERTC_IN_GGA(X1, X3, X4)

The TRS R consists of the following rules:

permcA_in_ga([], []) → permcA_out_ga([], [])
permcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, permcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, permcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, insertcB_in_gga(X1, X4, X3))
insertcB_in_gga(X1, [], .(X1, [])) → insertcB_out_gga(X1, [], .(X1, []))
insertcB_in_gga(X1, X2, .(X1, X2)) → insertcB_out_gga(X1, X2, .(X1, X2))
insertcB_in_gga(X1, .(X2, X3), .(X2, X4)) → U14_gga(X1, X2, X3, X4, insertcB_in_gga(X1, X3, X4))
U14_gga(X1, X2, X3, X4, insertcB_out_gga(X1, X3, X4)) → insertcB_out_gga(X1, .(X2, X3), .(X2, X4))
U13_ga(X1, X2, X3, insertcB_out_gga(X1, X4, X3)) → permcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
permA_in_ga(x1, x2)  =  permA_in_ga(x1)
permcA_in_ga(x1, x2)  =  permcA_in_ga(x1)
[]  =  []
permcA_out_ga(x1, x2)  =  permcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
insertcB_in_gga(x1, x2, x3)  =  insertcB_in_gga(x1, x2)
insertcB_out_gga(x1, x2, x3)  =  insertcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
insertB_in_gga(x1, x2, x3)  =  insertB_in_gga(x1, x2)
insertC_in_gga(x1, x2, x3)  =  insertC_in_gga(x1, x2)
PERMD_IN_GA(x1, x2)  =  PERMD_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
PERMA_IN_GA(x1, x2)  =  PERMA_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)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
INSERTB_IN_GGA(x1, x2, x3)  =  INSERTB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
INSERTC_IN_GGA(x1, x2, x3)  =  INSERTC_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

PERMD_IN_GA(.(X1, .(X2, X3)), X4) → U6_GA(X1, X2, X3, X4, permA_in_ga(X3, X5))
PERMD_IN_GA(.(X1, .(X2, X3)), X4) → PERMA_IN_GA(X3, X5)
PERMA_IN_GA(.(X1, X2), X3) → U1_GA(X1, X2, X3, permA_in_ga(X2, X4))
PERMA_IN_GA(.(X1, X2), X3) → PERMA_IN_GA(X2, X4)
PERMA_IN_GA(.(X1, X2), X3) → U2_GA(X1, X2, X3, permcA_in_ga(X2, X4))
U2_GA(X1, X2, X3, permcA_out_ga(X2, X4)) → U3_GA(X1, X2, X3, insertB_in_gga(X1, X4, X3))
U2_GA(X1, X2, X3, permcA_out_ga(X2, X4)) → INSERTB_IN_GGA(X1, X4, X3)
INSERTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → U4_GGA(X1, X2, X3, X4, insertB_in_gga(X1, X3, X4))
INSERTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → INSERTB_IN_GGA(X1, X3, X4)
PERMD_IN_GA(.(X1, .(X2, X3)), X4) → U7_GA(X1, X2, X3, X4, permcA_in_ga(X3, X5))
U7_GA(X1, X2, X3, X4, permcA_out_ga(X3, X5)) → U8_GA(X1, X2, X3, X4, insertB_in_gga(X2, X5, X6))
U7_GA(X1, X2, X3, X4, permcA_out_ga(X3, X5)) → INSERTB_IN_GGA(X2, X5, X6)
U7_GA(X1, X2, X3, X4, permcA_out_ga(X3, X5)) → U9_GA(X1, X2, X3, X4, insertcB_in_gga(X2, X5, X6))
U9_GA(X1, X2, X3, X4, insertcB_out_gga(X2, X5, X6)) → U10_GA(X1, X2, X3, X4, insertC_in_gga(X1, X6, X4))
U9_GA(X1, X2, X3, X4, insertcB_out_gga(X2, X5, X6)) → INSERTC_IN_GGA(X1, X6, X4)
INSERTC_IN_GGA(X1, .(X2, X3), .(X2, X4)) → U5_GGA(X1, X2, X3, X4, insertC_in_gga(X1, X3, X4))
INSERTC_IN_GGA(X1, .(X2, X3), .(X2, X4)) → INSERTC_IN_GGA(X1, X3, X4)

The TRS R consists of the following rules:

permcA_in_ga([], []) → permcA_out_ga([], [])
permcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, permcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, permcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, insertcB_in_gga(X1, X4, X3))
insertcB_in_gga(X1, [], .(X1, [])) → insertcB_out_gga(X1, [], .(X1, []))
insertcB_in_gga(X1, X2, .(X1, X2)) → insertcB_out_gga(X1, X2, .(X1, X2))
insertcB_in_gga(X1, .(X2, X3), .(X2, X4)) → U14_gga(X1, X2, X3, X4, insertcB_in_gga(X1, X3, X4))
U14_gga(X1, X2, X3, X4, insertcB_out_gga(X1, X3, X4)) → insertcB_out_gga(X1, .(X2, X3), .(X2, X4))
U13_ga(X1, X2, X3, insertcB_out_gga(X1, X4, X3)) → permcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
permA_in_ga(x1, x2)  =  permA_in_ga(x1)
permcA_in_ga(x1, x2)  =  permcA_in_ga(x1)
[]  =  []
permcA_out_ga(x1, x2)  =  permcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
insertcB_in_gga(x1, x2, x3)  =  insertcB_in_gga(x1, x2)
insertcB_out_gga(x1, x2, x3)  =  insertcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
insertB_in_gga(x1, x2, x3)  =  insertB_in_gga(x1, x2)
insertC_in_gga(x1, x2, x3)  =  insertC_in_gga(x1, x2)
PERMD_IN_GA(x1, x2)  =  PERMD_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x1, x2, x3, x5)
PERMA_IN_GA(x1, x2)  =  PERMA_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)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x2, x4)
INSERTB_IN_GGA(x1, x2, x3)  =  INSERTB_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3, x4, x5)  =  U4_GGA(x1, x2, x3, x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5)  =  U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5)  =  U10_GA(x1, x2, x3, x5)
INSERTC_IN_GGA(x1, x2, x3)  =  INSERTC_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

INSERTC_IN_GGA(X1, .(X2, X3), .(X2, X4)) → INSERTC_IN_GGA(X1, X3, X4)

The TRS R consists of the following rules:

permcA_in_ga([], []) → permcA_out_ga([], [])
permcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, permcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, permcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, insertcB_in_gga(X1, X4, X3))
insertcB_in_gga(X1, [], .(X1, [])) → insertcB_out_gga(X1, [], .(X1, []))
insertcB_in_gga(X1, X2, .(X1, X2)) → insertcB_out_gga(X1, X2, .(X1, X2))
insertcB_in_gga(X1, .(X2, X3), .(X2, X4)) → U14_gga(X1, X2, X3, X4, insertcB_in_gga(X1, X3, X4))
U14_gga(X1, X2, X3, X4, insertcB_out_gga(X1, X3, X4)) → insertcB_out_gga(X1, .(X2, X3), .(X2, X4))
U13_ga(X1, X2, X3, insertcB_out_gga(X1, X4, X3)) → permcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
permcA_in_ga(x1, x2)  =  permcA_in_ga(x1)
[]  =  []
permcA_out_ga(x1, x2)  =  permcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
insertcB_in_gga(x1, x2, x3)  =  insertcB_in_gga(x1, x2)
insertcB_out_gga(x1, x2, x3)  =  insertcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
INSERTC_IN_GGA(x1, x2, x3)  =  INSERTC_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:

INSERTC_IN_GGA(X1, .(X2, X3), .(X2, X4)) → INSERTC_IN_GGA(X1, X3, X4)

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

INSERTC_IN_GGA(X1, .(X2, X3)) → INSERTC_IN_GGA(X1, X3)

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:

  • INSERTC_IN_GGA(X1, .(X2, X3)) → INSERTC_IN_GGA(X1, X3)
    The graph contains the following edges 1 >= 1, 2 > 2

(13) YES

(14) Obligation:

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

INSERTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → INSERTB_IN_GGA(X1, X3, X4)

The TRS R consists of the following rules:

permcA_in_ga([], []) → permcA_out_ga([], [])
permcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, permcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, permcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, insertcB_in_gga(X1, X4, X3))
insertcB_in_gga(X1, [], .(X1, [])) → insertcB_out_gga(X1, [], .(X1, []))
insertcB_in_gga(X1, X2, .(X1, X2)) → insertcB_out_gga(X1, X2, .(X1, X2))
insertcB_in_gga(X1, .(X2, X3), .(X2, X4)) → U14_gga(X1, X2, X3, X4, insertcB_in_gga(X1, X3, X4))
U14_gga(X1, X2, X3, X4, insertcB_out_gga(X1, X3, X4)) → insertcB_out_gga(X1, .(X2, X3), .(X2, X4))
U13_ga(X1, X2, X3, insertcB_out_gga(X1, X4, X3)) → permcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
permcA_in_ga(x1, x2)  =  permcA_in_ga(x1)
[]  =  []
permcA_out_ga(x1, x2)  =  permcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
insertcB_in_gga(x1, x2, x3)  =  insertcB_in_gga(x1, x2)
insertcB_out_gga(x1, x2, x3)  =  insertcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
INSERTB_IN_GGA(x1, x2, x3)  =  INSERTB_IN_GGA(x1, x2)

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:

INSERTB_IN_GGA(X1, .(X2, X3), .(X2, X4)) → INSERTB_IN_GGA(X1, X3, X4)

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

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:

INSERTB_IN_GGA(X1, .(X2, X3)) → INSERTB_IN_GGA(X1, X3)

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:

  • INSERTB_IN_GGA(X1, .(X2, X3)) → INSERTB_IN_GGA(X1, X3)
    The graph contains the following edges 1 >= 1, 2 > 2

(20) YES

(21) Obligation:

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

PERMA_IN_GA(.(X1, X2), X3) → PERMA_IN_GA(X2, X4)

The TRS R consists of the following rules:

permcA_in_ga([], []) → permcA_out_ga([], [])
permcA_in_ga(.(X1, X2), X3) → U12_ga(X1, X2, X3, permcA_in_ga(X2, X4))
U12_ga(X1, X2, X3, permcA_out_ga(X2, X4)) → U13_ga(X1, X2, X3, insertcB_in_gga(X1, X4, X3))
insertcB_in_gga(X1, [], .(X1, [])) → insertcB_out_gga(X1, [], .(X1, []))
insertcB_in_gga(X1, X2, .(X1, X2)) → insertcB_out_gga(X1, X2, .(X1, X2))
insertcB_in_gga(X1, .(X2, X3), .(X2, X4)) → U14_gga(X1, X2, X3, X4, insertcB_in_gga(X1, X3, X4))
U14_gga(X1, X2, X3, X4, insertcB_out_gga(X1, X3, X4)) → insertcB_out_gga(X1, .(X2, X3), .(X2, X4))
U13_ga(X1, X2, X3, insertcB_out_gga(X1, X4, X3)) → permcA_out_ga(.(X1, X2), X3)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
permcA_in_ga(x1, x2)  =  permcA_in_ga(x1)
[]  =  []
permcA_out_ga(x1, x2)  =  permcA_out_ga(x1, x2)
U12_ga(x1, x2, x3, x4)  =  U12_ga(x1, x2, x4)
U13_ga(x1, x2, x3, x4)  =  U13_ga(x1, x2, x4)
insertcB_in_gga(x1, x2, x3)  =  insertcB_in_gga(x1, x2)
insertcB_out_gga(x1, x2, x3)  =  insertcB_out_gga(x1, x2, x3)
U14_gga(x1, x2, x3, x4, x5)  =  U14_gga(x1, x2, x3, x5)
PERMA_IN_GA(x1, x2)  =  PERMA_IN_GA(x1)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

PERMA_IN_GA(.(X1, X2), X3) → PERMA_IN_GA(X2, X4)

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

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

PERMA_IN_GA(.(X1, X2)) → PERMA_IN_GA(X2)

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

(26) 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:

  • PERMA_IN_GA(.(X1, X2)) → PERMA_IN_GA(X2)
    The graph contains the following edges 1 > 1

(27) YES