Left Termination proof of /tmp/tmpRMVnKt/pl2.3.1.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 47 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 PisEmptyProof (⇔, 0 ms)

↳6 YES

(0) Obligation:

Clauses:

p(X, Z) :- ','(q(X, Y), p(Y, Z)).
p(X, X).
q(a, b).

Query: p(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: p(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(q(T5, X5), p(X5, T7)) | p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(q(T5, X5), p(X5, T7)), SCOPE: 2, CLAUSE: 2 | ?_2 | p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(q(T5, X5), p(X5, T7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ?_2 | p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(b, T7)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: p(b, T7), SCOPE: 3, CLAUSE: 0 | p(b, T7), SCOPE: 3, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: p(b, T7), SCOPE: 3, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(b, T7), SCOPE: 3, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(q(b, X29), p(X29, T18))\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(q(b, X29), p(X29, T18)), SCOPE: 4, CLAUSE: 2\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 54 [arrowhead = none ];
54 -> 5;

55 [label="ONLY EVAL with clause\np(X3, X4) :- ','(q(X3, X5), p(X5, X4)).\nand substitutionT1 -> T5,\nX3 -> T5,\nT2 -> T7,\nX4 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 55 [arrowhead = none ];
55 -> 7;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 56 [arrowhead = none ];
56 -> 10;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 57 [arrowhead = none ];
57 -> 14;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 58 [arrowhead = none ];
58 -> 16;

59 [label="EVAL with clause\nq(a, b).\nand substitutionT5 -> a,\nX5 -> b", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 59 [arrowhead = none ];
59 -> 18;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 60 [arrowhead = none ];
60 -> 19;

61 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 61 [arrowhead = none ];
61 -> 50;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 62 [arrowhead = none ];
62 -> 25;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 63 [arrowhead = none ];
63 -> 27;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 64 [arrowhead = none ];
64 -> 29;

65 [label="ONLY EVAL with clause\np(X27, X28) :- ','(q(X27, X29), p(X29, X28)).\nand substitutionX27 -> b,\nT7 -> T18,\nX28 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 65 [arrowhead = none ];
65 -> 42;

66 [label="EVAL with clause\np(X36, X36).\nand substitutionX36 -> b,\nT7 -> b", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 66 [arrowhead = none ];
66 -> 47;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 67 [arrowhead = none ];
67 -> 48;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 68 [arrowhead = none ];
68 -> 45;

69 [label="BACKTRACK\nfor clause: q(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 69 [arrowhead = none ];
69 -> 46;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 70 [arrowhead = none ];
70 -> 49;

71 [label="EVAL with clause\np(X39, X39).\nand substitutionT5 -> T22,\nX39 -> T22,\nT2 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 71 [arrowhead = none ];
71 -> 51;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 72 [arrowhead = none ];
72 -> 52;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 73 [arrowhead = none ];
73 -> 53;

}

(2) Obligation:

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

pA_in_ga(a, b) → pA_out_ga(a, b)
pA_in_ga(T22, T22) → pA_out_ga(T22, T22)

The argument filtering Pi contains the following mapping:
pA_in_ga(x1, x2) = pA_in_ga(x1)
a = a
pA_out_ga(x1, x2) = pA_out_ga(x1, x2)

(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_ga(a, b) → pA_out_ga(a, b)
pA_in_ga(T22, T22) → pA_out_ga(T22, T22)

The argument filtering Pi contains the following mapping:
pA_in_ga(x1, x2) = pA_in_ga(x1)
a = a
pA_out_ga(x1, x2) = pA_out_ga(x1, x2)

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_ga(a, b) → pA_out_ga(a, b)
pA_in_ga(T22, T22) → pA_out_ga(T22, T22)

The argument filtering Pi contains the following mapping:
pA_in_ga(x1, x2) = pA_in_ga(x1)
a = a
pA_out_ga(x1, x2) = pA_out_ga(x1, x2)

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.

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) PisEmptyProof (EQUIVALENT transformation)

(6) YES