/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

tree_member(a,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) PiDP
(7) UsableRulesProof [EQUIVALENT, 0 ms]
(8) PiDP
(9) PiDPToQDPProof [SOUND, 0 ms]
(10) QDP
(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Clauses:

tree_member(X, tree(X, X1, X2)).
tree_member(X, tree(X3, Left, X4)) :- tree_member(X, Left).
tree_member(X, tree(X5, X6, Right)) :- tree_member(X, Right).


Query: tree_member(a,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:

tree_member_in_2: (f,b)

Transforming Prolog into the following Term Rewriting System:

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

   tree_member_in_ag(X, tree(X, X1, X2)) -> tree_member_out_ag(X, tree(X, X1, X2))
   tree_member_in_ag(X, tree(X3, Left, X4)) -> U1_ag(X, X3, Left, X4, tree_member_in_ag(X, Left))
   tree_member_in_ag(X, tree(X5, X6, Right)) -> U2_ag(X, X5, X6, Right, tree_member_in_ag(X, Right))
   U2_ag(X, X5, X6, Right, tree_member_out_ag(X, Right)) -> tree_member_out_ag(X, tree(X5, X6, Right))
   U1_ag(X, X3, Left, X4, tree_member_out_ag(X, Left)) -> tree_member_out_ag(X, tree(X3, Left, X4))

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

tree(x1, x2, x3)  =  tree(x1, x2, x3)

tree_member_out_ag(x1, x2)  =  tree_member_out_ag(x1)

U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)





Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



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

   tree_member_in_ag(X, tree(X, X1, X2)) -> tree_member_out_ag(X, tree(X, X1, X2))
   tree_member_in_ag(X, tree(X3, Left, X4)) -> U1_ag(X, X3, Left, X4, tree_member_in_ag(X, Left))
   tree_member_in_ag(X, tree(X5, X6, Right)) -> U2_ag(X, X5, X6, Right, tree_member_in_ag(X, Right))
   U2_ag(X, X5, X6, Right, tree_member_out_ag(X, Right)) -> tree_member_out_ag(X, tree(X5, X6, Right))
   U1_ag(X, X3, Left, X4, tree_member_out_ag(X, Left)) -> tree_member_out_ag(X, tree(X3, Left, X4))

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

tree(x1, x2, x3)  =  tree(x1, x2, x3)

tree_member_out_ag(x1, x2)  =  tree_member_out_ag(x1)

U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)



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(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:

   TREE_MEMBER_IN_AG(X, tree(X3, Left, X4)) -> U1_AG(X, X3, Left, X4, tree_member_in_ag(X, Left))
   TREE_MEMBER_IN_AG(X, tree(X3, Left, X4)) -> TREE_MEMBER_IN_AG(X, Left)
   TREE_MEMBER_IN_AG(X, tree(X5, X6, Right)) -> U2_AG(X, X5, X6, Right, tree_member_in_ag(X, Right))
   TREE_MEMBER_IN_AG(X, tree(X5, X6, Right)) -> TREE_MEMBER_IN_AG(X, Right)

The TRS R consists of the following rules:

   tree_member_in_ag(X, tree(X, X1, X2)) -> tree_member_out_ag(X, tree(X, X1, X2))
   tree_member_in_ag(X, tree(X3, Left, X4)) -> U1_ag(X, X3, Left, X4, tree_member_in_ag(X, Left))
   tree_member_in_ag(X, tree(X5, X6, Right)) -> U2_ag(X, X5, X6, Right, tree_member_in_ag(X, Right))
   U2_ag(X, X5, X6, Right, tree_member_out_ag(X, Right)) -> tree_member_out_ag(X, tree(X5, X6, Right))
   U1_ag(X, X3, Left, X4, tree_member_out_ag(X, Left)) -> tree_member_out_ag(X, tree(X3, Left, X4))

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

tree(x1, x2, x3)  =  tree(x1, x2, x3)

tree_member_out_ag(x1, x2)  =  tree_member_out_ag(x1)

U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)

TREE_MEMBER_IN_AG(x1, x2)  =  TREE_MEMBER_IN_AG(x2)

U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x5)

U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x5)


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

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

   TREE_MEMBER_IN_AG(X, tree(X3, Left, X4)) -> U1_AG(X, X3, Left, X4, tree_member_in_ag(X, Left))
   TREE_MEMBER_IN_AG(X, tree(X3, Left, X4)) -> TREE_MEMBER_IN_AG(X, Left)
   TREE_MEMBER_IN_AG(X, tree(X5, X6, Right)) -> U2_AG(X, X5, X6, Right, tree_member_in_ag(X, Right))
   TREE_MEMBER_IN_AG(X, tree(X5, X6, Right)) -> TREE_MEMBER_IN_AG(X, Right)

The TRS R consists of the following rules:

   tree_member_in_ag(X, tree(X, X1, X2)) -> tree_member_out_ag(X, tree(X, X1, X2))
   tree_member_in_ag(X, tree(X3, Left, X4)) -> U1_ag(X, X3, Left, X4, tree_member_in_ag(X, Left))
   tree_member_in_ag(X, tree(X5, X6, Right)) -> U2_ag(X, X5, X6, Right, tree_member_in_ag(X, Right))
   U2_ag(X, X5, X6, Right, tree_member_out_ag(X, Right)) -> tree_member_out_ag(X, tree(X5, X6, Right))
   U1_ag(X, X3, Left, X4, tree_member_out_ag(X, Left)) -> tree_member_out_ag(X, tree(X3, Left, X4))

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

tree(x1, x2, x3)  =  tree(x1, x2, x3)

tree_member_out_ag(x1, x2)  =  tree_member_out_ag(x1)

U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)

TREE_MEMBER_IN_AG(x1, x2)  =  TREE_MEMBER_IN_AG(x2)

U1_AG(x1, x2, x3, x4, x5)  =  U1_AG(x5)

U2_AG(x1, x2, x3, x4, x5)  =  U2_AG(x5)


We have to consider all (P,R,Pi)-chains
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(5) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
----------------------------------------

(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   TREE_MEMBER_IN_AG(X, tree(X5, X6, Right)) -> TREE_MEMBER_IN_AG(X, Right)
   TREE_MEMBER_IN_AG(X, tree(X3, Left, X4)) -> TREE_MEMBER_IN_AG(X, Left)

The TRS R consists of the following rules:

   tree_member_in_ag(X, tree(X, X1, X2)) -> tree_member_out_ag(X, tree(X, X1, X2))
   tree_member_in_ag(X, tree(X3, Left, X4)) -> U1_ag(X, X3, Left, X4, tree_member_in_ag(X, Left))
   tree_member_in_ag(X, tree(X5, X6, Right)) -> U2_ag(X, X5, X6, Right, tree_member_in_ag(X, Right))
   U2_ag(X, X5, X6, Right, tree_member_out_ag(X, Right)) -> tree_member_out_ag(X, tree(X5, X6, Right))
   U1_ag(X, X3, Left, X4, tree_member_out_ag(X, Left)) -> tree_member_out_ag(X, tree(X3, Left, X4))

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

tree(x1, x2, x3)  =  tree(x1, x2, x3)

tree_member_out_ag(x1, x2)  =  tree_member_out_ag(x1)

U1_ag(x1, x2, x3, x4, x5)  =  U1_ag(x5)

U2_ag(x1, x2, x3, x4, x5)  =  U2_ag(x5)

TREE_MEMBER_IN_AG(x1, x2)  =  TREE_MEMBER_IN_AG(x2)


We have to consider all (P,R,Pi)-chains
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(7) UsableRulesProof (EQUIVALENT)
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:

   TREE_MEMBER_IN_AG(X, tree(X5, X6, Right)) -> TREE_MEMBER_IN_AG(X, Right)
   TREE_MEMBER_IN_AG(X, tree(X3, Left, X4)) -> TREE_MEMBER_IN_AG(X, Left)

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

TREE_MEMBER_IN_AG(x1, x2)  =  TREE_MEMBER_IN_AG(x2)


We have to consider all (P,R,Pi)-chains
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(9) PiDPToQDPProof (SOUND)
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:

   TREE_MEMBER_IN_AG(tree(X5, X6, Right)) -> TREE_MEMBER_IN_AG(Right)
   TREE_MEMBER_IN_AG(tree(X3, Left, X4)) -> TREE_MEMBER_IN_AG(Left)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
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(11) 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:
*TREE_MEMBER_IN_AG(tree(X5, X6, Right)) -> TREE_MEMBER_IN_AG(Right)
The graph contains the following edges 1 > 1


*TREE_MEMBER_IN_AG(tree(X3, Left, X4)) -> TREE_MEMBER_IN_AG(Left)
The graph contains the following edges 1 > 1


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(12)
YES