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

Query: p(a,a,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: p(T1, T2, T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: p(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: !_1 | 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\nX2 is free\np(T1, T2, T3) !=? p(X2, X2, a)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(=(T12, a), ','(=(T13, T14), p(T13, T14, T12)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(=(T15, T16), p(T15, T16, a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: p(T19, T19, a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: p(T19, T19, a), SCOPE: 2, CLAUSE: 0 | p(T19, T19, a), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: !_2 | p(T19, T19, a), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 38 [arrowhead = none ];
38 -> 5;

39 [label="EVAL with clause\np(X2, X2, a) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> T5,\nT3 -> a", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 39 [arrowhead = none ];
39 -> 11;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 40 [arrowhead = none ];
40 -> 13;

41 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 41 [arrowhead = none ];
41 -> 15;

42 [label="ONLY EVAL with clause\np(X6, X7, X8) :- ','(=(X8, a), ','(=(X6, X7), p(X6, X7, X8))).\nand substitutionT1 -> T13,\nX6 -> T13,\nT2 -> T14,\nX7 -> T14,\nT3 -> T12,\nX8 -> T12,\nT11 -> T12,\nT9 -> T13,\nT10 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 42 [arrowhead = none ];
42 -> 22;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 43 [arrowhead = none ];
43 -> 16;

44 [label="UNIFY CASE with substitutionT12 -> a,\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 44 [arrowhead = none ];
44 -> 23;

45 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 45 [arrowhead = none ];
45 -> 24;

46 [label="UNIFY CASE with substitutionT15 -> T19,\nT16 -> T19,\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 46 [arrowhead = none ];
46 -> 27;

47 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 47 [arrowhead = none ];
47 -> 28;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 48 [arrowhead = none ];
48 -> 31;

49 [label="ONLY EVAL with clause\np(X11, X11, a) :- !_2.\nand substitutionT19 -> T22,\nX11 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 49 [arrowhead = none ];
49 -> 35;

50 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 50 [arrowhead = none ];
50 -> 36;

51 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 51 [arrowhead = none ];
51 -> 37;

}

(2) Obligation:

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

pA_in_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

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)

(3) 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, a) → pA_out_aaa(T5, T5, a)

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

(4) Obligation:

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

pA_in_aaa(T5, T5, a) → pA_out_aaa(T5, T5, a)

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

(5) PisEmptyProof (EQUIVALENT transformation)

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

(6) YES