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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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

minimum(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) 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:

minimum(tree(X, void, X1), X).
minimum(tree(X2, Left, X3), X) :- minimum(Left, X).


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

minimum_in_2: (b,f)

Transforming Prolog into the following Term Rewriting System:

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

   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)

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

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

void  =  void

minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)

U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(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:

   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)

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

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

void  =  void

minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)

U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(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:

   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> U1_GA(X2, Left, X3, X, minimum_in_ga(Left, X))
   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> MINIMUM_IN_GA(Left, X)

The TRS R consists of the following rules:

   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)

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

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

void  =  void

minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)

U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)

MINIMUM_IN_GA(x1, x2)  =  MINIMUM_IN_GA(x1)

U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(x5)


We have to consider all (P,R,Pi)-chains
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(4)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> U1_GA(X2, Left, X3, X, minimum_in_ga(Left, X))
   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> MINIMUM_IN_GA(Left, X)

The TRS R consists of the following rules:

   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)

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

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

void  =  void

minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)

U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)

MINIMUM_IN_GA(x1, x2)  =  MINIMUM_IN_GA(x1)

U1_GA(x1, x2, x3, x4, x5)  =  U1_GA(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 1 less node.
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(6)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> MINIMUM_IN_GA(Left, X)

The TRS R consists of the following rules:

   minimum_in_ga(tree(X, void, X1), X) -> minimum_out_ga(tree(X, void, X1), X)
   minimum_in_ga(tree(X2, Left, X3), X) -> U1_ga(X2, Left, X3, X, minimum_in_ga(Left, X))
   U1_ga(X2, Left, X3, X, minimum_out_ga(Left, X)) -> minimum_out_ga(tree(X2, Left, X3), X)

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

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

void  =  void

minimum_out_ga(x1, x2)  =  minimum_out_ga(x2)

U1_ga(x1, x2, x3, x4, x5)  =  U1_ga(x5)

MINIMUM_IN_GA(x1, x2)  =  MINIMUM_IN_GA(x1)


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.
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(8)
Obligation:
Pi DP problem:
The TRS P consists of the following rules:

   MINIMUM_IN_GA(tree(X2, Left, X3), X) -> MINIMUM_IN_GA(Left, X)

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

MINIMUM_IN_GA(x1, x2)  =  MINIMUM_IN_GA(x1)


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.
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(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MINIMUM_IN_GA(tree(X2, Left, X3)) -> MINIMUM_IN_GA(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:
*MINIMUM_IN_GA(tree(X2, Left, X3)) -> MINIMUM_IN_GA(Left)
The graph contains the following edges 1 > 1


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(12)
YES