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, X, 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, X, 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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: p(T1, T2, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: !_1 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nX2 is free\np(T1, T2, T3) !=? p(X2, X2, 1)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(user_defined_=(T12, 1), ','(user_defined_=(T13, T14), p(T14, T13, T12)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(user_defined_=(T12, 1), ','(user_defined_=(T13, T14), p(T14, T13, T12))), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(user_defined_=(T18, T19), p(T19, T18, 1))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(user_defined_=(T18, T19), p(T19, T18, 1)), SCOPE: 3, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: p(T24, T24, 1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: p(T24, T24, 1), SCOPE: 4, CLAUSE: 0 | p(T24, T24, 1), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: !_4 | p(T24, T24, 1), SCOPE: 4, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 6;

53 [label="EVAL with clause\np(X2, X2, 1) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> T5,\nT3 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 53 [arrowhead = none ];
53 -> 7;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 54 [arrowhead = none ];
54 -> 11;

55 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 55 [arrowhead = none ];
55 -> 14;

56 [label="ONLY EVAL with clause\np(X6, X7, X8) :- ','(user_defined_=(X8, 1), ','(user_defined_=(X7, X6), p(X6, X7, X8))).\nand substitutionT1 -> T14,\nX6 -> T14,\nT2 -> T13,\nX7 -> T13,\nT3 -> T12,\nX8 -> T12,\nT11 -> T12,\nT10 -> T13,\nT9 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 56 [arrowhead = none ];
56 -> 20;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 57 [arrowhead = none ];
57 -> 18;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 58 [arrowhead = none ];
58 -> 22;

59 [label="EVAL with clause\nuser_defined_=(X11, X11).\nand substitutionT12 -> 1,\nX11 -> 1,\nT17 -> 1,\nT13 -> T18,\nT14 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 59 [arrowhead = none ];
59 -> 25;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 60 [arrowhead = none ];
60 -> 28;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 61 [arrowhead = none ];
61 -> 30;

62 [label="EVAL with clause\nuser_defined_=(X15, X15).\nand substitutionT18 -> T24,\nX15 -> T24,\nT19 -> T24,\nT23 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 62 [arrowhead = none ];
62 -> 34;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 63 [arrowhead = none ];
63 -> 36;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 64 [arrowhead = none ];
64 -> 48;

65 [label="ONLY EVAL with clause\np(X19, X19, 1) :- !_4.\nand substitutionT24 -> T28,\nX19 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 65 [arrowhead = none ];
65 -> 49;

66 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 66 [arrowhead = none ];
66 -> 50;

67 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 67 [arrowhead = none ];
67 -> 51;

}

(4) Obligation:

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

pA_in_aaa(T5, T5, 1) → pA_out_aaa(T5, T5, 1)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(x3)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
P is empty.
The TRS R consists of the following rules:

pA_in_aaa(T5, T5, 1) → pA_out_aaa(T5, T5, 1)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(x3)

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

(6) Obligation:

Pi DP problem:
P is empty.
The TRS R consists of the following rules:

pA_in_aaa(T5, T5, 1) → pA_out_aaa(T5, T5, 1)

The argument filtering Pi contains the following mapping:
pA_in_aaa(x1, x2, x3)  =  pA_in_aaa
pA_out_aaa(x1, x2, x3)  =  pA_out_aaa(x3)

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

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,R,Pi) chain.

(8) YES