/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.pl /export/starexec/sandbox2/output/output_files


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YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.pl
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

perm(g,a)

w.r.t. the given Prolog program could successfully be proven:

(0) Prolog
(1) PrologToPiTRSProof [SOUND, 0 ms]
(2) PiTRS
(3) DependencyPairsProof [EQUIVALENT, 0 ms]
(4) PiDP
(5) DependencyGraphProof [EQUIVALENT, 0 ms]
(6) AND
    (7) PiDP
        (8) UsableRulesProof [EQUIVALENT, 0 ms]
        (9) PiDP
        (10) PiDPToQDPProof [SOUND, 0 ms]
        (11) QDP
        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) PiDP
        (15) UsableRulesProof [EQUIVALENT, 0 ms]
        (16) PiDP
        (17) PiDPToQDPProof [SOUND, 0 ms]
        (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) 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) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

perm_in_2: (b,f)

insert_in_3: (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)
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)
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)
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)
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)
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)
YES

----------------------------------------

(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)
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)
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)
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)
YES