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

(0) Obligation:

Clauses:

p(X) :- ','(q(X), r(X)).
p(X) :- =(X, 2).
q(X) :- ','(=(X, 1), !).
q(X) :- =(X, 3).
r(X) :- ','(=(X, 1), !).
r(X) :- =(X, 4).
=(X, X).

Query: p(a)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

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

(2) Obligation:

Clauses:

p(X) :- ','(q(X), r(X)).
p(X) :- user_defined_=(X, 2).
q(X) :- ','(user_defined_=(X, 1), !).
q(X) :- user_defined_=(X, 3).
r(X) :- ','(user_defined_=(X, 1), !).
r(X) :- user_defined_=(X, 4).
user_defined_=(X, X).

Query: p(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)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(q(T4), r(T4)) | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(q(T4), r(T4)), SCOPE: 2, CLAUSE: 2 | ','(q(T4), r(T4)), SCOPE: 2, CLAUSE: 3 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(','(user_defined_=(T9, 1), !_2), r(T9)) | ','(q(T4), r(T4)), SCOPE: 2, CLAUSE: 3 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(','(user_defined_=(T9, 1), !_2), r(T9)), SCOPE: 3, CLAUSE: 6 | ?_3 | ','(q(T4), r(T4)), SCOPE: 2, CLAUSE: 3 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, r(1)) | ?_3 | ','(q(T4), r(T4)), SCOPE: 2, CLAUSE: 3 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ?_3 | ','(q(T4), r(T4)), SCOPE: 2, CLAUSE: 3 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: r(1) | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: r(1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: r(1), SCOPE: 4, CLAUSE: 4 | r(1), SCOPE: 4, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(user_defined_=(1, 1), !_4) | r(1), SCOPE: 4, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(user_defined_=(1, 1), !_4), SCOPE: 5, CLAUSE: 6 | ?_5 | r(1), SCOPE: 4, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: !_4 | ?_5 | r(1), SCOPE: 4, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: user_defined_=(T17, 2)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: user_defined_=(T17, 2), SCOPE: 6, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(q(T4), r(T4)), SCOPE: 2, CLAUSE: 3 | ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(q(T4), r(T4)), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ?_2 | p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(user_defined_=(T28, 3), r(T28))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(user_defined_=(T28, 3), r(T28)), SCOPE: 7, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: r(3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: r(3), SCOPE: 8, CLAUSE: 4 | r(3), SCOPE: 8, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(user_defined_=(3, 1), !_8) | r(3), SCOPE: 8, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(user_defined_=(3, 1), !_8), SCOPE: 9, CLAUSE: 6 | ?_9 | r(3), SCOPE: 8, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ?_9 | r(3), SCOPE: 8, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: r(3), SCOPE: 8, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: user_defined_=(3, 4)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: user_defined_=(3, 4), SCOPE: 10, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(T1), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: user_defined_=(T36, 2)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: user_defined_=(T36, 2), SCOPE: 11, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 44 [arrowhead = none ];
44 -> 2;

45 [label="ONLY EVAL with clause\np(X2) :- ','(q(X2), r(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 45 [arrowhead = none ];
45 -> 3;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 46 [arrowhead = none ];
46 -> 4;

47 [label="ONLY EVAL with clause\nq(X5) :- ','(user_defined_=(X5, 1), !_2).\nand substitutionT4 -> T9,\nX5 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 47 [arrowhead = none ];
47 -> 5;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 48 [arrowhead = none ];
48 -> 6;

49 [label="EVAL with clause\nuser_defined_=(X8, X8).\nand substitutionT9 -> 1,\nX8 -> 1,\nT12 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 49 [arrowhead = none ];
49 -> 7;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 50 [arrowhead = none ];
50 -> 8;

51 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 51 [arrowhead = none ];
51 -> 9;

52 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 52 [arrowhead = none ];
52 -> 23;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 53 [arrowhead = none ];
53 -> 10;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 54 [arrowhead = none ];
54 -> 11;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 55 [arrowhead = none ];
55 -> 12;

56 [label="ONLY EVAL with clause\np(X17) :- user_defined_=(X17, 2).\nand substitutionT1 -> T17,\nX17 -> T17,\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="ONLY EVAL with clause\nr(X11) :- ','(user_defined_=(X11, 1), !_4).\nand substitutionX11 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 57 [arrowhead = none ];
57 -> 13;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 58 [arrowhead = none ];
58 -> 14;

59 [label="ONLY EVAL with clause\nuser_defined_=(X14, X14).\nand substitutionX14 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 59 [arrowhead = none ];
59 -> 15;

60 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 60 [arrowhead = none ];
60 -> 16;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 61 [arrowhead = none ];
61 -> 17;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 62 [arrowhead = none ];
62 -> 19;

63 [label="EVAL with clause\nuser_defined_=(X20, X20).\nand substitutionT17 -> 2,\nX20 -> 2,\nT20 -> 2", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 63 [arrowhead = none ];
63 -> 20;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 64 [arrowhead = none ];
64 -> 21;

65 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 65 [arrowhead = none ];
65 -> 22;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 66 [arrowhead = none ];
66 -> 24;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 67 [arrowhead = none ];
67 -> 25;

68 [label="ONLY EVAL with clause\nq(X25) :- user_defined_=(X25, 3).\nand substitutionT4 -> T28,\nX25 -> T28,\nT27 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 68 [arrowhead = none ];
68 -> 26;

69 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 69 [arrowhead = none ];
69 -> 38;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 70 [arrowhead = none ];
70 -> 27;

71 [label="EVAL with clause\nuser_defined_=(X28, X28).\nand substitutionT28 -> 3,\nX28 -> 3,\nT31 -> 3", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 71 [arrowhead = none ];
71 -> 28;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 72 [arrowhead = none ];
72 -> 29;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 73 [arrowhead = none ];
73 -> 30;

74 [label="ONLY EVAL with clause\nr(X31) :- ','(user_defined_=(X31, 1), !_8).\nand substitutionX31 -> 3", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 74 [arrowhead = none ];
74 -> 31;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 75 [arrowhead = none ];
75 -> 32;

76 [label="BACKTRACK\nfor clause: user_defined_=(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 76 [arrowhead = none ];
76 -> 33;

77 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 77 [arrowhead = none ];
77 -> 34;

78 [label="ONLY EVAL with clause\nr(X35) :- user_defined_=(X35, 4).\nand substitutionX35 -> 3", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 78 [arrowhead = none ];
78 -> 35;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 79 [arrowhead = none ];
79 -> 36;

80 [label="BACKTRACK\nfor clause: user_defined_=(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 80 [arrowhead = none ];
80 -> 37;

81 [label="ONLY EVAL with clause\np(X39) :- user_defined_=(X39, 2).\nand substitutionT1 -> T36,\nX39 -> T36,\nT35 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 81 [arrowhead = none ];
81 -> 39;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 82 [arrowhead = none ];
82 -> 40;

83 [label="EVAL with clause\nuser_defined_=(X42, X42).\nand substitutionT36 -> 2,\nX42 -> 2,\nT39 -> 2", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 83 [arrowhead = none ];
83 -> 41;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 84 [arrowhead = none ];
84 -> 42;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 85 [arrowhead = none ];
85 -> 43;

}

(4) Obligation:

Clauses:

pA(1).
pA(2).
pA(2).

Query: pA(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:
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

pA_in_a(1) → pA_out_a(1)
pA_in_a(2) → pA_out_a(2)

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

pA_in_a(1) → pA_out_a(1)
pA_in_a(2) → pA_out_a(2)

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a(x1)

(7) 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_a(1) → pA_out_a(1)
pA_in_a(2) → pA_out_a(2)

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a(x1)

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

(8) Obligation:

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

pA_in_a(1) → pA_out_a(1)
pA_in_a(2) → pA_out_a(2)

The argument filtering Pi contains the following mapping:
pA_in_a(x1)  =  pA_in_a
pA_out_a(x1)  =  pA_out_a(x1)

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

(9) PisEmptyProof (EQUIVALENT transformation)

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

(10) YES