Left Termination of the query pattern bin_tree_in_1(g) w.r.t. the given Prolog program could successfully be proven:

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

(0) Obligation:

Clauses:

bin_tree(void).
bin_tree(tree(X1, Left, Right)) :- ','(bin_tree(Left), bin_tree(Right)).

Queries:

bin_tree(g).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
bin_tree_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

The argument filtering Pi contains the following mapping:
bin_tree_in_g(x1)  =  bin_tree_in_g(x1)
void  =  void
bin_tree_out_g(x1)  =  bin_tree_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

The argument filtering Pi contains the following mapping:
bin_tree_in_g(x1)  =  bin_tree_in_g(x1)
void  =  void
bin_tree_out_g(x1)  =  bin_tree_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)

(3) 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:

BIN_TREE_IN_G(tree(X1, Left, Right)) → U1_G(X1, Left, Right, bin_tree_in_g(Left))
BIN_TREE_IN_G(tree(X1, Left, Right)) → BIN_TREE_IN_G(Left)
U1_G(X1, Left, Right, bin_tree_out_g(Left)) → U2_G(X1, Left, Right, bin_tree_in_g(Right))
U1_G(X1, Left, Right, bin_tree_out_g(Left)) → BIN_TREE_IN_G(Right)

The TRS R consists of the following rules:

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

The argument filtering Pi contains the following mapping:
bin_tree_in_g(x1)  =  bin_tree_in_g(x1)
void  =  void
bin_tree_out_g(x1)  =  bin_tree_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
BIN_TREE_IN_G(x1)  =  BIN_TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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

(4) Obligation:

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

BIN_TREE_IN_G(tree(X1, Left, Right)) → U1_G(X1, Left, Right, bin_tree_in_g(Left))
BIN_TREE_IN_G(tree(X1, Left, Right)) → BIN_TREE_IN_G(Left)
U1_G(X1, Left, Right, bin_tree_out_g(Left)) → U2_G(X1, Left, Right, bin_tree_in_g(Right))
U1_G(X1, Left, Right, bin_tree_out_g(Left)) → BIN_TREE_IN_G(Right)

The TRS R consists of the following rules:

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

The argument filtering Pi contains the following mapping:
bin_tree_in_g(x1)  =  bin_tree_in_g(x1)
void  =  void
bin_tree_out_g(x1)  =  bin_tree_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
BIN_TREE_IN_G(x1)  =  BIN_TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)
U2_G(x1, x2, x3, x4)  =  U2_G(x4)

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:

U1_G(X1, Left, Right, bin_tree_out_g(Left)) → BIN_TREE_IN_G(Right)
BIN_TREE_IN_G(tree(X1, Left, Right)) → U1_G(X1, Left, Right, bin_tree_in_g(Left))
BIN_TREE_IN_G(tree(X1, Left, Right)) → BIN_TREE_IN_G(Left)

The TRS R consists of the following rules:

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

The argument filtering Pi contains the following mapping:
bin_tree_in_g(x1)  =  bin_tree_in_g(x1)
void  =  void
bin_tree_out_g(x1)  =  bin_tree_out_g
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U1_g(x1, x2, x3, x4)  =  U1_g(x3, x4)
U2_g(x1, x2, x3, x4)  =  U2_g(x4)
BIN_TREE_IN_G(x1)  =  BIN_TREE_IN_G(x1)
U1_G(x1, x2, x3, x4)  =  U1_G(x3, x4)

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:

U1_G(Right, bin_tree_out_g) → BIN_TREE_IN_G(Right)
BIN_TREE_IN_G(tree(X1, Left, Right)) → U1_G(Right, bin_tree_in_g(Left))
BIN_TREE_IN_G(tree(X1, Left, Right)) → BIN_TREE_IN_G(Left)

The TRS R consists of the following rules:

bin_tree_in_g(void) → bin_tree_out_g
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(Right, bin_tree_in_g(Left))
U1_g(Right, bin_tree_out_g) → U2_g(bin_tree_in_g(Right))
U2_g(bin_tree_out_g) → bin_tree_out_g

The set Q consists of the following terms:

bin_tree_in_g(x0)
U1_g(x0, x1)
U2_g(x0)

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:

  • BIN_TREE_IN_G(tree(X1, Left, Right)) → U1_G(Right, bin_tree_in_g(Left))
    The graph contains the following edges 1 > 1

  • BIN_TREE_IN_G(tree(X1, Left, Right)) → BIN_TREE_IN_G(Left)
    The graph contains the following edges 1 > 1

  • U1_G(Right, bin_tree_out_g) → BIN_TREE_IN_G(Right)
    The graph contains the following edges 1 >= 1

(10) TRUE

(11) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
bin_tree_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(12) Obligation:

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

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

Pi is empty.

(13) 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:

BIN_TREE_IN_G(tree(X1, Left, Right)) → U1_G(X1, Left, Right, bin_tree_in_g(Left))
BIN_TREE_IN_G(tree(X1, Left, Right)) → BIN_TREE_IN_G(Left)
U1_G(X1, Left, Right, bin_tree_out_g(Left)) → U2_G(X1, Left, Right, bin_tree_in_g(Right))
U1_G(X1, Left, Right, bin_tree_out_g(Left)) → BIN_TREE_IN_G(Right)

The TRS R consists of the following rules:

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

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

(14) Obligation:

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

BIN_TREE_IN_G(tree(X1, Left, Right)) → U1_G(X1, Left, Right, bin_tree_in_g(Left))
BIN_TREE_IN_G(tree(X1, Left, Right)) → BIN_TREE_IN_G(Left)
U1_G(X1, Left, Right, bin_tree_out_g(Left)) → U2_G(X1, Left, Right, bin_tree_in_g(Right))
U1_G(X1, Left, Right, bin_tree_out_g(Left)) → BIN_TREE_IN_G(Right)

The TRS R consists of the following rules:

bin_tree_in_g(void) → bin_tree_out_g(void)
bin_tree_in_g(tree(X1, Left, Right)) → U1_g(X1, Left, Right, bin_tree_in_g(Left))
U1_g(X1, Left, Right, bin_tree_out_g(Left)) → U2_g(X1, Left, Right, bin_tree_in_g(Right))
U2_g(X1, Left, Right, bin_tree_out_g(Right)) → bin_tree_out_g(tree(X1, Left, Right))

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