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

(0) Obligation:

Clauses:

p(X, Y, Z) :- ','(=(X, Y), ','(=(Z, 1), !)).
p(X, Y, Z) :- ','(=(Z, 1), ','(=(Y, X), p(X, Y, Z))).
=(X, X).

Query: p(a,a,a)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

p(X, Y, Z) :- ','(user_defined_=(X, Y), ','(user_defined_=(Z, 1), !)).
p(X, Y, Z) :- ','(user_defined_=(Z, 1), ','(user_defined_=(Y, X), p(X, Y, Z))).
user_defined_=(X, X).

Query: p(a,a,a)

(3) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: p(T1, T2, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(user_defined_=(T10, T11), ','(user_defined_=(T12, 1), !_1)) | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(user_defined_=(T10, T11), ','(user_defined_=(T12, 1), !_1)), SCOPE: 2, CLAUSE: 2 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(user_defined_=(T16, 1), !_1) | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(user_defined_=(T16, 1), !_1), SCOPE: 3, CLAUSE: 2 | ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: !_1 | ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ?_3 | ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ?_2 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(user_defined_=(T35, 1), ','(user_defined_=(T36, T37), p(T37, T36, T35)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(user_defined_=(T35, 1), ','(user_defined_=(T36, T37), p(T37, T36, T35))), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(user_defined_=(T44, T45), p(T45, T44, 1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(user_defined_=(T44, T45), p(T45, T44, 1)), SCOPE: 5, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(T51, T51, 1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: p(T51, T51, 1), SCOPE: 6, CLAUSE: 0 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(user_defined_=(T57, T57), ','(user_defined_=(1, 1), !_6)) | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(user_defined_=(T57, T57), ','(user_defined_=(1, 1), !_6)), SCOPE: 7, CLAUSE: 2 | ?_7 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(user_defined_=(1, 1), !_6) | ?_7 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(user_defined_=(1, 1), !_6), SCOPE: 8, CLAUSE: 2 | ?_8 | ?_7 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: !_6 | ?_8 | ?_7 | p(T51, T51, 1), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(user_defined_=(T76, 1), ','(user_defined_=(T77, T78), p(T78, T77, T76)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(user_defined_=(T76, 1), ','(user_defined_=(T77, T78), p(T78, T77, T76))), SCOPE: 9, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(user_defined_=(T85, T86), p(T86, T85, 1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(user_defined_=(T85, T86), p(T86, T85, 1)), SCOPE: 10, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: p(T92, T92, 1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 37 [arrowhead = none ];
37 -> 2;

38 [label="ONLY EVAL with clause\np(X4, X5, X6) :- ','(user_defined_=(X4, X5), ','(user_defined_=(X6, 1), !_1)).\nand substitutionT1 -> T10,\nX4 -> T10,\nT2 -> T11,\nX5 -> T11,\nT3 -> T12,\nX6 -> T12,\nT7 -> T10,\nT8 -> T11,\nT9 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 38 [arrowhead = none ];
38 -> 3;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 39 [arrowhead = none ];
39 -> 4;

40 [label="EVAL with clause\nuser_defined_=(X9, X9).\nand substitutionT10 -> T15,\nX9 -> T15,\nT11 -> T15,\nT12 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 40 [arrowhead = none ];
40 -> 5;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 41 [arrowhead = none ];
41 -> 6;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 42 [arrowhead = none ];
42 -> 7;

43 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 43 [arrowhead = none ];
43 -> 29;

44 [label="EVAL with clause\nuser_defined_=(X12, X12).\nand substitutionT16 -> 1,\nX12 -> 1,\nT19 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 44 [arrowhead = none ];
44 -> 8;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 45 [arrowhead = none ];
45 -> 9;

46 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 46 [arrowhead = none ];
46 -> 10;

47 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 47 [arrowhead = none ];
47 -> 12;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 48 [arrowhead = none ];
48 -> 11;

49 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 49 [arrowhead = none ];
49 -> 13;

50 [label="ONLY EVAL with clause\np(X22, X23, X24) :- ','(user_defined_=(X24, 1), ','(user_defined_=(X23, X22), p(X22, X23, X24))).\nand substitutionT1 -> T37,\nX22 -> T37,\nT2 -> T36,\nX23 -> T36,\nT3 -> T35,\nX24 -> T35,\nT34 -> T35,\nT33 -> T36,\nT32 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 50 [arrowhead = none ];
50 -> 14;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 51 [arrowhead = none ];
51 -> 15;

52 [label="EVAL with clause\nuser_defined_=(X30, X30).\nand substitutionT35 -> 1,\nX30 -> 1,\nT43 -> 1,\nT36 -> T44,\nT37 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 52 [arrowhead = none ];
52 -> 16;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 53 [arrowhead = none ];
53 -> 17;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 54 [arrowhead = none ];
54 -> 18;

55 [label="EVAL with clause\nuser_defined_=(X36, X36).\nand substitutionT44 -> T51,\nX36 -> T51,\nT45 -> T51,\nT50 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 55 [arrowhead = none ];
55 -> 19;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 56 [arrowhead = none ];
56 -> 20;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 57 [arrowhead = none ];
57 -> 21;

58 [label="ONLY EVAL with clause\np(X46, X47, X48) :- ','(user_defined_=(X46, X47), ','(user_defined_=(X48, 1), !_6)).\nand substitutionT51 -> T57,\nX46 -> T57,\nX47 -> T57,\nX48 -> 1,\nT56 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 58 [arrowhead = none ];
58 -> 22;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 59 [arrowhead = none ];
59 -> 23;

60 [label="ONLY EVAL with clause\nuser_defined_=(X51, X51).\nand substitutionT57 -> T60,\nX51 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 60 [arrowhead = none ];
60 -> 24;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 61 [arrowhead = none ];
61 -> 25;

62 [label="ONLY EVAL with clause\nuser_defined_=(X54, X54).\nand substitutionX54 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 62 [arrowhead = none ];
62 -> 26;

63 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 63 [arrowhead = none ];
63 -> 27;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 64 [arrowhead = none ];
64 -> 28;

65 [label="ONLY EVAL with clause\np(X64, X65, X66) :- ','(user_defined_=(X66, 1), ','(user_defined_=(X65, X64), p(X64, X65, X66))).\nand substitutionT1 -> T78,\nX64 -> T78,\nT2 -> T77,\nX65 -> T77,\nT3 -> T76,\nX66 -> T76,\nT75 -> T76,\nT74 -> T77,\nT73 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 66 [arrowhead = none ];
66 -> 31;

67 [label="EVAL with clause\nuser_defined_=(X72, X72).\nand substitutionT76 -> 1,\nX72 -> 1,\nT84 -> 1,\nT77 -> T85,\nT78 -> T86", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 67 [arrowhead = none ];
67 -> 32;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 69 [arrowhead = none ];
69 -> 34;

70 [label="EVAL with clause\nuser_defined_=(X78, X78).\nand substitutionT85 -> T92,\nX78 -> T92,\nT86 -> T92,\nT91 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 70 [arrowhead = none ];
70 -> 35;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 71 [arrowhead = none ];
71 -> 36;

72 [label="INSTANCE with matching:\nT51 -> T92", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 72 [arrowhead = none , style=dashed];
72 -> 19[style=dashed];

}

(4) Obligation:

Clauses:

pA(T60).
pB(T15, T15, 1).
pB(T51, T51, 1) :- pA(T51).
pB(T92, T92, 1) :- pA(T92).

Query: pB(a,a,a)

(5) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
pB_in: (f,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

pB_in_aaa(T15, T15, 1) → pB_out_aaa(T15, T15, 1)
pB_in_aaa(T51, T51, 1) → U1_aaa(T51, pA_in_a(T51))
pA_in_a(T60) → pA_out_a(T60)
U1_aaa(T51, pA_out_a(T51)) → pB_out_aaa(T51, T51, 1)
pB_in_aaa(T92, T92, 1) → U2_aaa(T92, pA_in_a(T92))
U2_aaa(T92, pA_out_a(T92)) → pB_out_aaa(T92, T92, 1)

The argument filtering Pi contains the following mapping:
pB_in_aaa(x1, x2, x3)  =  pB_in_aaa
pB_out_aaa(x1, x2, x3)  =  pB_out_aaa(x3)
U1_aaa(x1, x2)  =  U1_aaa(x2)
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a
U2_aaa(x1, x2)  =  U2_aaa(x2)
1  =  1

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

pB_in_aaa(T15, T15, 1) → pB_out_aaa(T15, T15, 1)
pB_in_aaa(T51, T51, 1) → U1_aaa(T51, pA_in_a(T51))
pA_in_a(T60) → pA_out_a(T60)
U1_aaa(T51, pA_out_a(T51)) → pB_out_aaa(T51, T51, 1)
pB_in_aaa(T92, T92, 1) → U2_aaa(T92, pA_in_a(T92))
U2_aaa(T92, pA_out_a(T92)) → pB_out_aaa(T92, T92, 1)

The argument filtering Pi contains the following mapping:
pB_in_aaa(x1, x2, x3)  =  pB_in_aaa
pB_out_aaa(x1, x2, x3)  =  pB_out_aaa(x3)
U1_aaa(x1, x2)  =  U1_aaa(x2)
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a
U2_aaa(x1, x2)  =  U2_aaa(x2)
1  =  1

(7) 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:

PB_IN_AAA(T51, T51, 1) → U1_AAA(T51, pA_in_a(T51))
PB_IN_AAA(T51, T51, 1) → PA_IN_A(T51)
PB_IN_AAA(T92, T92, 1) → U2_AAA(T92, pA_in_a(T92))

The TRS R consists of the following rules:

pB_in_aaa(T15, T15, 1) → pB_out_aaa(T15, T15, 1)
pB_in_aaa(T51, T51, 1) → U1_aaa(T51, pA_in_a(T51))
pA_in_a(T60) → pA_out_a(T60)
U1_aaa(T51, pA_out_a(T51)) → pB_out_aaa(T51, T51, 1)
pB_in_aaa(T92, T92, 1) → U2_aaa(T92, pA_in_a(T92))
U2_aaa(T92, pA_out_a(T92)) → pB_out_aaa(T92, T92, 1)

The argument filtering Pi contains the following mapping:
pB_in_aaa(x1, x2, x3)  =  pB_in_aaa
pB_out_aaa(x1, x2, x3)  =  pB_out_aaa(x3)
U1_aaa(x1, x2)  =  U1_aaa(x2)
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a
U2_aaa(x1, x2)  =  U2_aaa(x2)
1  =  1
PB_IN_AAA(x1, x2, x3)  =  PB_IN_AAA
U1_AAA(x1, x2)  =  U1_AAA(x2)
PA_IN_A(x1)  =  PA_IN_A
U2_AAA(x1, x2)  =  U2_AAA(x2)

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

(8) Obligation:

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

PB_IN_AAA(T51, T51, 1) → U1_AAA(T51, pA_in_a(T51))
PB_IN_AAA(T51, T51, 1) → PA_IN_A(T51)
PB_IN_AAA(T92, T92, 1) → U2_AAA(T92, pA_in_a(T92))

The TRS R consists of the following rules:

pB_in_aaa(T15, T15, 1) → pB_out_aaa(T15, T15, 1)
pB_in_aaa(T51, T51, 1) → U1_aaa(T51, pA_in_a(T51))
pA_in_a(T60) → pA_out_a(T60)
U1_aaa(T51, pA_out_a(T51)) → pB_out_aaa(T51, T51, 1)
pB_in_aaa(T92, T92, 1) → U2_aaa(T92, pA_in_a(T92))
U2_aaa(T92, pA_out_a(T92)) → pB_out_aaa(T92, T92, 1)

The argument filtering Pi contains the following mapping:
pB_in_aaa(x1, x2, x3)  =  pB_in_aaa
pB_out_aaa(x1, x2, x3)  =  pB_out_aaa(x3)
U1_aaa(x1, x2)  =  U1_aaa(x2)
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a
U2_aaa(x1, x2)  =  U2_aaa(x2)
1  =  1
PB_IN_AAA(x1, x2, x3)  =  PB_IN_AAA
U1_AAA(x1, x2)  =  U1_AAA(x2)
PA_IN_A(x1)  =  PA_IN_A
U2_AAA(x1, x2)  =  U2_AAA(x2)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 0 SCCs with 3 less nodes.

(10) TRUE