Left Termination of the query pattern minimum_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

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

Queries:

minimum(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

(2) Obligation:

Triples:

minimum1(tree(T18, tree(T41, T42, T43), T20), T45) :- minimum1(T42, T45).

Clauses:

minimumc1(tree(T5, void, T6), T5).
minimumc1(tree(T18, tree(T31, void, T32), T20), T31).
minimumc1(tree(T18, tree(T41, T42, T43), T20), T45) :- minimumc1(T42, T45).

Afs:

minimum1(x1, x2)  =  minimum1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
minimum1_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MINIMUM1_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → U1_GA(T18, T41, T42, T43, T20, T45, minimum1_in_ga(T42, T45))
MINIMUM1_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUM1_IN_GA(T42, T45)

R is empty.
The argument filtering Pi contains the following mapping:
minimum1_in_ga(x1, x2)  =  minimum1_in_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
MINIMUM1_IN_GA(x1, x2)  =  MINIMUM1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GA(x1, x2, x3, x4, x5, x7)

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:

MINIMUM1_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → U1_GA(T18, T41, T42, T43, T20, T45, minimum1_in_ga(T42, T45))
MINIMUM1_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUM1_IN_GA(T42, T45)

R is empty.
The argument filtering Pi contains the following mapping:
minimum1_in_ga(x1, x2)  =  minimum1_in_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
MINIMUM1_IN_GA(x1, x2)  =  MINIMUM1_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6, x7)  =  U1_GA(x1, x2, x3, x4, x5, x7)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(6) Obligation:

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

MINIMUM1_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUM1_IN_GA(T42, T45)

R is empty.
The argument filtering Pi contains the following mapping:
tree(x1, x2, x3)  =  tree(x1, x2, x3)
MINIMUM1_IN_GA(x1, x2)  =  MINIMUM1_IN_GA(x1)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

MINIMUM1_IN_GA(tree(T18, tree(T41, T42, T43), T20)) → MINIMUM1_IN_GA(T42)

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

(9) 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:

• MINIMUM1_IN_GA(tree(T18, tree(T41, T42, T43), T20)) → MINIMUM1_IN_GA(T42)
  The graph contains the following edges 1 > 1

(10) YES