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) 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: !_1 | p(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: 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];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(=(T12, a), ','(=(T13, T14), p(T13, T14, T12)))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(=(T15, T16), p(T15, T16, a))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T19, T19, a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: p(T19, T19, a), SCOPE: 2, CLAUSE: 0 | p(T19, T19, a), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: !_2 | p(T19, T19, a), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 16 [arrowhead = none ];
16 -> 2;

17 [label="EVAL with clause\np(X2, X2, a) :- !_1.\nand substitutionT1 -> T5,\nX2 -> T5,\nT2 -> T5,\nT3 -> a", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 17 [arrowhead = none ];
17 -> 3;

18 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 18 [arrowhead = none ];
18 -> 4;

19 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 19 [arrowhead = none ];
19 -> 5;

20 [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];
4 -> 20 [arrowhead = none ];
20 -> 7;

21 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 21 [arrowhead = none ];
21 -> 6;

22 [label="UNIFY CASE with substitutionT12 -> a,\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 22 [arrowhead = none ];
22 -> 8;

23 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 23 [arrowhead = none ];
23 -> 9;

24 [label="UNIFY CASE with substitutionT15 -> T19,\nT16 -> T19,\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 24 [arrowhead = none ];
24 -> 10;

25 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 25 [arrowhead = none ];
25 -> 11;

26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 26 [arrowhead = none ];
26 -> 12;

27 [label="ONLY EVAL with clause\np(X11, X11, a) :- !_2.\nand substitutionT19 -> T22,\nX11 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 27 [arrowhead = none ];
27 -> 13;

28 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 28 [arrowhead = none ];
28 -> 14;

29 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 29 [arrowhead = none ];
29 -> 15;

}

(2) Obligation:

Clauses:

pA(T5, T5, a).
pA(T22, T22, a).

Query: pA(a,a,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
Transforming Prolog into the following Term Rewriting System:
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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) 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)

(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, 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

(6) 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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(8) YES