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

0 Prolog
↳1 PrologToPrologProblemTransformerProof (⇐)
↳2 Prolog
↳3 PrologToPiTRSProof (⇐)
↳4 PiTRS
↳5 DependencyPairsProof (⇔)
↳6 PiDP
↳7 DependencyGraphProof (⇔)
↳8 PiDP
↳9 UsableRulesProof (⇔)
↳10 PiDP
↳11 PiDPToQDPProof (⇐)
↳12 QDP
↳13 QDPSizeChangeProof (⇔)
↳14 YES

(0) Obligation:

Clauses:

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

Queries:

minimum(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

minimum1(tree(T5, void, T6), T5).
minimum1(tree(T18, tree(T31, void, T32), T20), T31).
minimum1(tree(T18, tree(T41, T42, T43), T20), T45) :- minimum1(T42, T45).

Queries:

minimum1(g,a).

(3) PrologToPiTRSProof (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 Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

minimum1_in_ga(tree(T5, void, T6), T5) → minimum1_out_ga(tree(T5, void, T6), T5)
minimum1_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimum1_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimum1_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimum1_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimum1_out_ga(T42, T45)) → minimum1_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimum1_out_ga(x1, x2)  =  minimum1_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

minimum1_in_ga(tree(T5, void, T6), T5) → minimum1_out_ga(tree(T5, void, T6), T5)
minimum1_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimum1_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimum1_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimum1_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimum1_out_ga(T42, T45)) → minimum1_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimum1_out_ga(x1, x2)  =  minimum1_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
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)

The TRS R consists of the following rules:

minimum1_in_ga(tree(T5, void, T6), T5) → minimum1_out_ga(tree(T5, void, T6), T5)
minimum1_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimum1_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimum1_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimum1_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimum1_out_ga(T42, T45)) → minimum1_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimum1_out_ga(x1, x2)  =  minimum1_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)
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

(6) 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)

The TRS R consists of the following rules:

minimum1_in_ga(tree(T5, void, T6), T5) → minimum1_out_ga(tree(T5, void, T6), T5)
minimum1_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimum1_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimum1_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimum1_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimum1_out_ga(T42, T45)) → minimum1_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimum1_out_ga(x1, x2)  =  minimum1_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)
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

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) 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)

The TRS R consists of the following rules:

minimum1_in_ga(tree(T5, void, T6), T5) → minimum1_out_ga(tree(T5, void, T6), T5)
minimum1_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimum1_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimum1_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimum1_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimum1_out_ga(T42, T45)) → minimum1_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimum1_out_ga(x1, x2)  =  minimum1_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)
MINIMUM1_IN_GA(x1, x2)  =  MINIMUM1_IN_GA(x1)

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

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) 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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) 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.

(13) 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

(14) YES