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

(0) Obligation:

Clauses:

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

Queries:

bin_tree(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

(2) Obligation:

Clauses:

p7(void, T7) :- bin_tree1(T7).
p7(tree(T14, T15, T16), T7) :- bin_tree1(T15).
p7(tree(T14, T15, T16), T7) :- ','(bin_tree1(T15), p7(T16, T7)).
bin_tree1(void).
bin_tree1(tree(T5, T6, T7)) :- p7(T6, T7).

Queries:

bin_tree1(g).

(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:
bin_tree1_in: (b)
p7_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

bin_tree1_in_g(void) → bin_tree1_out_g(void)
bin_tree1_in_g(tree(T5, T6, T7)) → U4_g(T5, T6, T7, p7_in_gg(T6, T7))
p7_in_gg(void, T7) → U1_gg(T7, bin_tree1_in_g(T7))
U1_gg(T7, bin_tree1_out_g(T7)) → p7_out_gg(void, T7)
p7_in_gg(tree(T14, T15, T16), T7) → U2_gg(T14, T15, T16, T7, bin_tree1_in_g(T15))
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → p7_out_gg(tree(T14, T15, T16), T7)
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → U3_gg(T14, T15, T16, T7, p7_in_gg(T16, T7))
U3_gg(T14, T15, T16, T7, p7_out_gg(T16, T7)) → p7_out_gg(tree(T14, T15, T16), T7)
U4_g(T5, T6, T7, p7_out_gg(T6, T7)) → bin_tree1_out_g(tree(T5, T6, T7))

Pi is empty.

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

bin_tree1_in_g(void) → bin_tree1_out_g(void)
bin_tree1_in_g(tree(T5, T6, T7)) → U4_g(T5, T6, T7, p7_in_gg(T6, T7))
p7_in_gg(void, T7) → U1_gg(T7, bin_tree1_in_g(T7))
U1_gg(T7, bin_tree1_out_g(T7)) → p7_out_gg(void, T7)
p7_in_gg(tree(T14, T15, T16), T7) → U2_gg(T14, T15, T16, T7, bin_tree1_in_g(T15))
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → p7_out_gg(tree(T14, T15, T16), T7)
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → U3_gg(T14, T15, T16, T7, p7_in_gg(T16, T7))
U3_gg(T14, T15, T16, T7, p7_out_gg(T16, T7)) → p7_out_gg(tree(T14, T15, T16), T7)
U4_g(T5, T6, T7, p7_out_gg(T6, T7)) → bin_tree1_out_g(tree(T5, T6, T7))

Pi is empty.

(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:

BIN_TREE1_IN_G(tree(T5, T6, T7)) → U4_G(T5, T6, T7, p7_in_gg(T6, T7))
BIN_TREE1_IN_G(tree(T5, T6, T7)) → P7_IN_GG(T6, T7)
P7_IN_GG(void, T7) → U1_GG(T7, bin_tree1_in_g(T7))
P7_IN_GG(void, T7) → BIN_TREE1_IN_G(T7)
P7_IN_GG(tree(T14, T15, T16), T7) → U2_GG(T14, T15, T16, T7, bin_tree1_in_g(T15))
P7_IN_GG(tree(T14, T15, T16), T7) → BIN_TREE1_IN_G(T15)
U2_GG(T14, T15, T16, T7, bin_tree1_out_g(T15)) → U3_GG(T14, T15, T16, T7, p7_in_gg(T16, T7))
U2_GG(T14, T15, T16, T7, bin_tree1_out_g(T15)) → P7_IN_GG(T16, T7)

The TRS R consists of the following rules:

bin_tree1_in_g(void) → bin_tree1_out_g(void)
bin_tree1_in_g(tree(T5, T6, T7)) → U4_g(T5, T6, T7, p7_in_gg(T6, T7))
p7_in_gg(void, T7) → U1_gg(T7, bin_tree1_in_g(T7))
U1_gg(T7, bin_tree1_out_g(T7)) → p7_out_gg(void, T7)
p7_in_gg(tree(T14, T15, T16), T7) → U2_gg(T14, T15, T16, T7, bin_tree1_in_g(T15))
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → p7_out_gg(tree(T14, T15, T16), T7)
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → U3_gg(T14, T15, T16, T7, p7_in_gg(T16, T7))
U3_gg(T14, T15, T16, T7, p7_out_gg(T16, T7)) → p7_out_gg(tree(T14, T15, T16), T7)
U4_g(T5, T6, T7, p7_out_gg(T6, T7)) → bin_tree1_out_g(tree(T5, T6, T7))

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

(6) Obligation:

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

BIN_TREE1_IN_G(tree(T5, T6, T7)) → U4_G(T5, T6, T7, p7_in_gg(T6, T7))
BIN_TREE1_IN_G(tree(T5, T6, T7)) → P7_IN_GG(T6, T7)
P7_IN_GG(void, T7) → U1_GG(T7, bin_tree1_in_g(T7))
P7_IN_GG(void, T7) → BIN_TREE1_IN_G(T7)
P7_IN_GG(tree(T14, T15, T16), T7) → U2_GG(T14, T15, T16, T7, bin_tree1_in_g(T15))
P7_IN_GG(tree(T14, T15, T16), T7) → BIN_TREE1_IN_G(T15)
U2_GG(T14, T15, T16, T7, bin_tree1_out_g(T15)) → U3_GG(T14, T15, T16, T7, p7_in_gg(T16, T7))
U2_GG(T14, T15, T16, T7, bin_tree1_out_g(T15)) → P7_IN_GG(T16, T7)

The TRS R consists of the following rules:

bin_tree1_in_g(void) → bin_tree1_out_g(void)
bin_tree1_in_g(tree(T5, T6, T7)) → U4_g(T5, T6, T7, p7_in_gg(T6, T7))
p7_in_gg(void, T7) → U1_gg(T7, bin_tree1_in_g(T7))
U1_gg(T7, bin_tree1_out_g(T7)) → p7_out_gg(void, T7)
p7_in_gg(tree(T14, T15, T16), T7) → U2_gg(T14, T15, T16, T7, bin_tree1_in_g(T15))
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → p7_out_gg(tree(T14, T15, T16), T7)
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → U3_gg(T14, T15, T16, T7, p7_in_gg(T16, T7))
U3_gg(T14, T15, T16, T7, p7_out_gg(T16, T7)) → p7_out_gg(tree(T14, T15, T16), T7)
U4_g(T5, T6, T7, p7_out_gg(T6, T7)) → bin_tree1_out_g(tree(T5, T6, T7))

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

BIN_TREE1_IN_G(tree(T5, T6, T7)) → P7_IN_GG(T6, T7)
P7_IN_GG(void, T7) → BIN_TREE1_IN_G(T7)
P7_IN_GG(tree(T14, T15, T16), T7) → U2_GG(T14, T15, T16, T7, bin_tree1_in_g(T15))
U2_GG(T14, T15, T16, T7, bin_tree1_out_g(T15)) → P7_IN_GG(T16, T7)
P7_IN_GG(tree(T14, T15, T16), T7) → BIN_TREE1_IN_G(T15)

The TRS R consists of the following rules:

bin_tree1_in_g(void) → bin_tree1_out_g(void)
bin_tree1_in_g(tree(T5, T6, T7)) → U4_g(T5, T6, T7, p7_in_gg(T6, T7))
p7_in_gg(void, T7) → U1_gg(T7, bin_tree1_in_g(T7))
U1_gg(T7, bin_tree1_out_g(T7)) → p7_out_gg(void, T7)
p7_in_gg(tree(T14, T15, T16), T7) → U2_gg(T14, T15, T16, T7, bin_tree1_in_g(T15))
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → p7_out_gg(tree(T14, T15, T16), T7)
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → U3_gg(T14, T15, T16, T7, p7_in_gg(T16, T7))
U3_gg(T14, T15, T16, T7, p7_out_gg(T16, T7)) → p7_out_gg(tree(T14, T15, T16), T7)
U4_g(T5, T6, T7, p7_out_gg(T6, T7)) → bin_tree1_out_g(tree(T5, T6, T7))

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

(9) PiDPToQDPProof (EQUIVALENT transformation)

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

(10) Obligation:

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

BIN_TREE1_IN_G(tree(T5, T6, T7)) → P7_IN_GG(T6, T7)
P7_IN_GG(void, T7) → BIN_TREE1_IN_G(T7)
P7_IN_GG(tree(T14, T15, T16), T7) → U2_GG(T14, T15, T16, T7, bin_tree1_in_g(T15))
U2_GG(T14, T15, T16, T7, bin_tree1_out_g(T15)) → P7_IN_GG(T16, T7)
P7_IN_GG(tree(T14, T15, T16), T7) → BIN_TREE1_IN_G(T15)

The TRS R consists of the following rules:

bin_tree1_in_g(void) → bin_tree1_out_g(void)
bin_tree1_in_g(tree(T5, T6, T7)) → U4_g(T5, T6, T7, p7_in_gg(T6, T7))
p7_in_gg(void, T7) → U1_gg(T7, bin_tree1_in_g(T7))
U1_gg(T7, bin_tree1_out_g(T7)) → p7_out_gg(void, T7)
p7_in_gg(tree(T14, T15, T16), T7) → U2_gg(T14, T15, T16, T7, bin_tree1_in_g(T15))
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → p7_out_gg(tree(T14, T15, T16), T7)
U2_gg(T14, T15, T16, T7, bin_tree1_out_g(T15)) → U3_gg(T14, T15, T16, T7, p7_in_gg(T16, T7))
U3_gg(T14, T15, T16, T7, p7_out_gg(T16, T7)) → p7_out_gg(tree(T14, T15, T16), T7)
U4_g(T5, T6, T7, p7_out_gg(T6, T7)) → bin_tree1_out_g(tree(T5, T6, T7))

The set Q consists of the following terms:

bin_tree1_in_g(x0)
p7_in_gg(x0, x1)
U1_gg(x0, x1)
U2_gg(x0, x1, x2, x3, x4)
U3_gg(x0, x1, x2, x3, x4)
U4_g(x0, x1, x2, x3)

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

(11) 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:

  • P7_IN_GG(tree(T14, T15, T16), T7) → U2_GG(T14, T15, T16, T7, bin_tree1_in_g(T15))
    The graph contains the following edges 1 > 1, 1 > 2, 1 > 3, 2 >= 4

  • BIN_TREE1_IN_G(tree(T5, T6, T7)) → P7_IN_GG(T6, T7)
    The graph contains the following edges 1 > 1, 1 > 2

  • U2_GG(T14, T15, T16, T7, bin_tree1_out_g(T15)) → P7_IN_GG(T16, T7)
    The graph contains the following edges 3 >= 1, 4 >= 2

  • P7_IN_GG(void, T7) → BIN_TREE1_IN_G(T7)
    The graph contains the following edges 2 >= 1

  • P7_IN_GG(tree(T14, T15, T16), T7) → BIN_TREE1_IN_G(T15)
    The graph contains the following edges 1 > 1

(12) YES