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



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

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

Queries:

minimum(g,a).

We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
minimum_in: (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(X1, Left, X2), X) → U1_ga(X1, Left, X2, X, minimum_in_ga(Left, X))
U1_ga(X1, Left, X2, X, minimum_out_ga(Left, X)) → minimum_out_ga(tree(X1, Left, X2), 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



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

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(X1, Left, X2), X) → U1_ga(X1, Left, X2, X, minimum_in_ga(Left, X))
U1_ga(X1, Left, X2, X, minimum_out_ga(Left, X)) → minimum_out_ga(tree(X1, Left, X2), 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)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

MINIMUM_IN_GA(tree(X1, Left, X2), X) → U1_GA(X1, Left, X2, X, minimum_in_ga(Left, X))
MINIMUM_IN_GA(tree(X1, Left, X2), 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(X1, Left, X2), X) → U1_ga(X1, Left, X2, X, minimum_in_ga(Left, X))
U1_ga(X1, Left, X2, X, minimum_out_ga(Left, X)) → minimum_out_ga(tree(X1, Left, X2), 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

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

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

MINIMUM_IN_GA(tree(X1, Left, X2), X) → U1_GA(X1, Left, X2, X, minimum_in_ga(Left, X))
MINIMUM_IN_GA(tree(X1, Left, X2), 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(X1, Left, X2), X) → U1_ga(X1, Left, X2, X, minimum_in_ga(Left, X))
U1_ga(X1, Left, X2, X, minimum_out_ga(Left, X)) → minimum_out_ga(tree(X1, Left, X2), 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
The approximation of the Dependency Graph [30] contains 1 SCC with 1 less node.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

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

MINIMUM_IN_GA(tree(X1, Left, X2), 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(X1, Left, X2), X) → U1_ga(X1, Left, X2, X, minimum_in_ga(Left, X))
U1_ga(X1, Left, X2, X, minimum_out_ga(Left, X)) → minimum_out_ga(tree(X1, Left, X2), 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
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

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

MINIMUM_IN_GA(tree(X1, Left, X2), 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
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ QDPSizeChangeProof

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

MINIMUM_IN_GA(tree(X1, Left, X2)) → MINIMUM_IN_GA(Left)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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: