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


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


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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Left Termination of the query pattern

overlap(g,g)

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 [EQUIVALENT, 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:

overlap(Xs, Ys) :- ','(member2(X, Xs), member1(X, Ys)).
has_a_or_b(Xs) :- overlap(Xs, .(a, .(b, []))).
member1(X, .(Y, Xs)) :- member1(X, Xs).
member1(X, .(X, Xs)).
member2(X, .(Y, Xs)) :- member2(X, Xs).
member2(X, .(X, Xs)).


Query: overlap(g,g)
----------------------------------------

(1) PrologToPiTRSProof (SOUND)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:

overlap_in_2: (b,b)

member2_in_2: (f,b)

member1_in_2: (b,b)

Transforming Prolog into the following Term Rewriting System:

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

   overlap_in_gg(Xs, Ys) -> U1_gg(Xs, Ys, member2_in_ag(X, Xs))
   member2_in_ag(X, .(Y, Xs)) -> U5_ag(X, Y, Xs, member2_in_ag(X, Xs))
   member2_in_ag(X, .(X, Xs)) -> member2_out_ag(X, .(X, Xs))
   U5_ag(X, Y, Xs, member2_out_ag(X, Xs)) -> member2_out_ag(X, .(Y, Xs))
   U1_gg(Xs, Ys, member2_out_ag(X, Xs)) -> U2_gg(Xs, Ys, member1_in_gg(X, Ys))
   member1_in_gg(X, .(Y, Xs)) -> U4_gg(X, Y, Xs, member1_in_gg(X, Xs))
   member1_in_gg(X, .(X, Xs)) -> member1_out_gg(X, .(X, Xs))
   U4_gg(X, Y, Xs, member1_out_gg(X, Xs)) -> member1_out_gg(X, .(Y, Xs))
   U2_gg(Xs, Ys, member1_out_gg(X, Ys)) -> overlap_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
overlap_in_gg(x1, x2)  =  overlap_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)

member2_in_ag(x1, x2)  =  member2_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)

member2_out_ag(x1, x2)  =  member2_out_ag(x1)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

member1_in_gg(x1, x2)  =  member1_in_gg(x1, x2)

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

member1_out_gg(x1, x2)  =  member1_out_gg

overlap_out_gg(x1, x2)  =  overlap_out_gg





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

(2)
Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:

   overlap_in_gg(Xs, Ys) -> U1_gg(Xs, Ys, member2_in_ag(X, Xs))
   member2_in_ag(X, .(Y, Xs)) -> U5_ag(X, Y, Xs, member2_in_ag(X, Xs))
   member2_in_ag(X, .(X, Xs)) -> member2_out_ag(X, .(X, Xs))
   U5_ag(X, Y, Xs, member2_out_ag(X, Xs)) -> member2_out_ag(X, .(Y, Xs))
   U1_gg(Xs, Ys, member2_out_ag(X, Xs)) -> U2_gg(Xs, Ys, member1_in_gg(X, Ys))
   member1_in_gg(X, .(Y, Xs)) -> U4_gg(X, Y, Xs, member1_in_gg(X, Xs))
   member1_in_gg(X, .(X, Xs)) -> member1_out_gg(X, .(X, Xs))
   U4_gg(X, Y, Xs, member1_out_gg(X, Xs)) -> member1_out_gg(X, .(Y, Xs))
   U2_gg(Xs, Ys, member1_out_gg(X, Ys)) -> overlap_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
overlap_in_gg(x1, x2)  =  overlap_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)

member2_in_ag(x1, x2)  =  member2_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)

member2_out_ag(x1, x2)  =  member2_out_ag(x1)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

member1_in_gg(x1, x2)  =  member1_in_gg(x1, x2)

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

member1_out_gg(x1, x2)  =  member1_out_gg

overlap_out_gg(x1, x2)  =  overlap_out_gg



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

(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:

   OVERLAP_IN_GG(Xs, Ys) -> U1_GG(Xs, Ys, member2_in_ag(X, Xs))
   OVERLAP_IN_GG(Xs, Ys) -> MEMBER2_IN_AG(X, Xs)
   MEMBER2_IN_AG(X, .(Y, Xs)) -> U5_AG(X, Y, Xs, member2_in_ag(X, Xs))
   MEMBER2_IN_AG(X, .(Y, Xs)) -> MEMBER2_IN_AG(X, Xs)
   U1_GG(Xs, Ys, member2_out_ag(X, Xs)) -> U2_GG(Xs, Ys, member1_in_gg(X, Ys))
   U1_GG(Xs, Ys, member2_out_ag(X, Xs)) -> MEMBER1_IN_GG(X, Ys)
   MEMBER1_IN_GG(X, .(Y, Xs)) -> U4_GG(X, Y, Xs, member1_in_gg(X, Xs))
   MEMBER1_IN_GG(X, .(Y, Xs)) -> MEMBER1_IN_GG(X, Xs)

The TRS R consists of the following rules:

   overlap_in_gg(Xs, Ys) -> U1_gg(Xs, Ys, member2_in_ag(X, Xs))
   member2_in_ag(X, .(Y, Xs)) -> U5_ag(X, Y, Xs, member2_in_ag(X, Xs))
   member2_in_ag(X, .(X, Xs)) -> member2_out_ag(X, .(X, Xs))
   U5_ag(X, Y, Xs, member2_out_ag(X, Xs)) -> member2_out_ag(X, .(Y, Xs))
   U1_gg(Xs, Ys, member2_out_ag(X, Xs)) -> U2_gg(Xs, Ys, member1_in_gg(X, Ys))
   member1_in_gg(X, .(Y, Xs)) -> U4_gg(X, Y, Xs, member1_in_gg(X, Xs))
   member1_in_gg(X, .(X, Xs)) -> member1_out_gg(X, .(X, Xs))
   U4_gg(X, Y, Xs, member1_out_gg(X, Xs)) -> member1_out_gg(X, .(Y, Xs))
   U2_gg(Xs, Ys, member1_out_gg(X, Ys)) -> overlap_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
overlap_in_gg(x1, x2)  =  overlap_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)

member2_in_ag(x1, x2)  =  member2_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)

member2_out_ag(x1, x2)  =  member2_out_ag(x1)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

member1_in_gg(x1, x2)  =  member1_in_gg(x1, x2)

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

member1_out_gg(x1, x2)  =  member1_out_gg

overlap_out_gg(x1, x2)  =  overlap_out_gg

OVERLAP_IN_GG(x1, x2)  =  OVERLAP_IN_GG(x1, x2)

U1_GG(x1, x2, x3)  =  U1_GG(x2, x3)

MEMBER2_IN_AG(x1, x2)  =  MEMBER2_IN_AG(x2)

U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)

U2_GG(x1, x2, x3)  =  U2_GG(x3)

MEMBER1_IN_GG(x1, x2)  =  MEMBER1_IN_GG(x1, x2)

U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)


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

(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   OVERLAP_IN_GG(Xs, Ys) -> U1_GG(Xs, Ys, member2_in_ag(X, Xs))
   OVERLAP_IN_GG(Xs, Ys) -> MEMBER2_IN_AG(X, Xs)
   MEMBER2_IN_AG(X, .(Y, Xs)) -> U5_AG(X, Y, Xs, member2_in_ag(X, Xs))
   MEMBER2_IN_AG(X, .(Y, Xs)) -> MEMBER2_IN_AG(X, Xs)
   U1_GG(Xs, Ys, member2_out_ag(X, Xs)) -> U2_GG(Xs, Ys, member1_in_gg(X, Ys))
   U1_GG(Xs, Ys, member2_out_ag(X, Xs)) -> MEMBER1_IN_GG(X, Ys)
   MEMBER1_IN_GG(X, .(Y, Xs)) -> U4_GG(X, Y, Xs, member1_in_gg(X, Xs))
   MEMBER1_IN_GG(X, .(Y, Xs)) -> MEMBER1_IN_GG(X, Xs)

The TRS R consists of the following rules:

   overlap_in_gg(Xs, Ys) -> U1_gg(Xs, Ys, member2_in_ag(X, Xs))
   member2_in_ag(X, .(Y, Xs)) -> U5_ag(X, Y, Xs, member2_in_ag(X, Xs))
   member2_in_ag(X, .(X, Xs)) -> member2_out_ag(X, .(X, Xs))
   U5_ag(X, Y, Xs, member2_out_ag(X, Xs)) -> member2_out_ag(X, .(Y, Xs))
   U1_gg(Xs, Ys, member2_out_ag(X, Xs)) -> U2_gg(Xs, Ys, member1_in_gg(X, Ys))
   member1_in_gg(X, .(Y, Xs)) -> U4_gg(X, Y, Xs, member1_in_gg(X, Xs))
   member1_in_gg(X, .(X, Xs)) -> member1_out_gg(X, .(X, Xs))
   U4_gg(X, Y, Xs, member1_out_gg(X, Xs)) -> member1_out_gg(X, .(Y, Xs))
   U2_gg(Xs, Ys, member1_out_gg(X, Ys)) -> overlap_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
overlap_in_gg(x1, x2)  =  overlap_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)

member2_in_ag(x1, x2)  =  member2_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)

member2_out_ag(x1, x2)  =  member2_out_ag(x1)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

member1_in_gg(x1, x2)  =  member1_in_gg(x1, x2)

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

member1_out_gg(x1, x2)  =  member1_out_gg

overlap_out_gg(x1, x2)  =  overlap_out_gg

OVERLAP_IN_GG(x1, x2)  =  OVERLAP_IN_GG(x1, x2)

U1_GG(x1, x2, x3)  =  U1_GG(x2, x3)

MEMBER2_IN_AG(x1, x2)  =  MEMBER2_IN_AG(x2)

U5_AG(x1, x2, x3, x4)  =  U5_AG(x4)

U2_GG(x1, x2, x3)  =  U2_GG(x3)

MEMBER1_IN_GG(x1, x2)  =  MEMBER1_IN_GG(x1, x2)

U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)


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

(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.
----------------------------------------

(6)
Complex Obligation (AND)

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

(7)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MEMBER1_IN_GG(X, .(Y, Xs)) -> MEMBER1_IN_GG(X, Xs)

The TRS R consists of the following rules:

   overlap_in_gg(Xs, Ys) -> U1_gg(Xs, Ys, member2_in_ag(X, Xs))
   member2_in_ag(X, .(Y, Xs)) -> U5_ag(X, Y, Xs, member2_in_ag(X, Xs))
   member2_in_ag(X, .(X, Xs)) -> member2_out_ag(X, .(X, Xs))
   U5_ag(X, Y, Xs, member2_out_ag(X, Xs)) -> member2_out_ag(X, .(Y, Xs))
   U1_gg(Xs, Ys, member2_out_ag(X, Xs)) -> U2_gg(Xs, Ys, member1_in_gg(X, Ys))
   member1_in_gg(X, .(Y, Xs)) -> U4_gg(X, Y, Xs, member1_in_gg(X, Xs))
   member1_in_gg(X, .(X, Xs)) -> member1_out_gg(X, .(X, Xs))
   U4_gg(X, Y, Xs, member1_out_gg(X, Xs)) -> member1_out_gg(X, .(Y, Xs))
   U2_gg(Xs, Ys, member1_out_gg(X, Ys)) -> overlap_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
overlap_in_gg(x1, x2)  =  overlap_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)

member2_in_ag(x1, x2)  =  member2_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)

member2_out_ag(x1, x2)  =  member2_out_ag(x1)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

member1_in_gg(x1, x2)  =  member1_in_gg(x1, x2)

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

member1_out_gg(x1, x2)  =  member1_out_gg

overlap_out_gg(x1, x2)  =  overlap_out_gg

MEMBER1_IN_GG(x1, x2)  =  MEMBER1_IN_GG(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:

   MEMBER1_IN_GG(X, .(Y, Xs)) -> MEMBER1_IN_GG(X, Xs)

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

(10) PiDPToQDPProof (EQUIVALENT)
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:

   MEMBER1_IN_GG(X, .(Y, Xs)) -> MEMBER1_IN_GG(X, Xs)

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:
*MEMBER1_IN_GG(X, .(Y, Xs)) -> MEMBER1_IN_GG(X, Xs)
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:

   MEMBER2_IN_AG(X, .(Y, Xs)) -> MEMBER2_IN_AG(X, Xs)

The TRS R consists of the following rules:

   overlap_in_gg(Xs, Ys) -> U1_gg(Xs, Ys, member2_in_ag(X, Xs))
   member2_in_ag(X, .(Y, Xs)) -> U5_ag(X, Y, Xs, member2_in_ag(X, Xs))
   member2_in_ag(X, .(X, Xs)) -> member2_out_ag(X, .(X, Xs))
   U5_ag(X, Y, Xs, member2_out_ag(X, Xs)) -> member2_out_ag(X, .(Y, Xs))
   U1_gg(Xs, Ys, member2_out_ag(X, Xs)) -> U2_gg(Xs, Ys, member1_in_gg(X, Ys))
   member1_in_gg(X, .(Y, Xs)) -> U4_gg(X, Y, Xs, member1_in_gg(X, Xs))
   member1_in_gg(X, .(X, Xs)) -> member1_out_gg(X, .(X, Xs))
   U4_gg(X, Y, Xs, member1_out_gg(X, Xs)) -> member1_out_gg(X, .(Y, Xs))
   U2_gg(Xs, Ys, member1_out_gg(X, Ys)) -> overlap_out_gg(Xs, Ys)

The argument filtering Pi contains the following mapping:
overlap_in_gg(x1, x2)  =  overlap_in_gg(x1, x2)

U1_gg(x1, x2, x3)  =  U1_gg(x2, x3)

member2_in_ag(x1, x2)  =  member2_in_ag(x2)

.(x1, x2)  =  .(x1, x2)

U5_ag(x1, x2, x3, x4)  =  U5_ag(x4)

member2_out_ag(x1, x2)  =  member2_out_ag(x1)

U2_gg(x1, x2, x3)  =  U2_gg(x3)

member1_in_gg(x1, x2)  =  member1_in_gg(x1, x2)

U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

member1_out_gg(x1, x2)  =  member1_out_gg

overlap_out_gg(x1, x2)  =  overlap_out_gg

MEMBER2_IN_AG(x1, x2)  =  MEMBER2_IN_AG(x2)


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:

   MEMBER2_IN_AG(X, .(Y, Xs)) -> MEMBER2_IN_AG(X, Xs)

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

MEMBER2_IN_AG(x1, x2)  =  MEMBER2_IN_AG(x2)


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:

   MEMBER2_IN_AG(.(Y, Xs)) -> MEMBER2_IN_AG(Xs)

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:
*MEMBER2_IN_AG(.(Y, Xs)) -> MEMBER2_IN_AG(Xs)
The graph contains the following edges 1 > 1


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

(20)
YES