Left Termination proof of /tmp/tmpetDfjJ/plumer453b.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 57 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 PisEmptyProof (⇔, 0 ms)

↳6 YES

(0) Obligation:

Clauses:

f(X) :- ','(p(X), q(X)).
p(a).
p(X) :- ','(p(a), !).
q(b).

Query: f(a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: f(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: f(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(p(T4), q(T4))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 1 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: q(a) | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\np(T4) !=? p(a)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: q(a), SCOPE: 3, CLAUSE: 3 | ?_3 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ?_3 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(','(p(a), !_2), q(T9))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(','(p(a), !_2), q(T9)), SCOPE: 4, CLAUSE: 1 | ','(','(p(a), !_2), q(T9)), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(!_2, q(T9)) | ','(','(p(a), !_2), q(T9)), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: q(T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: q(T9), SCOPE: 5, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(','(p(a), !_2), q(T14))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(','(p(a), !_2), q(T14)), SCOPE: 6, CLAUSE: 1 | ','(','(p(a), !_2), q(T14)), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(!_2, q(T14)) | ','(','(p(a), !_2), q(T14)), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: q(T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: q(T14), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 65 [arrowhead = none ];
65 -> 6;

66 [label="ONLY EVAL with clause\nf(X2) :- ','(p(X2), q(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 66 [arrowhead = none ];
66 -> 8;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 67 [arrowhead = none ];
67 -> 11;

68 [label="EVAL with clause\np(a).\nand substitutionT4 -> a", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 68 [arrowhead = none ];
68 -> 12;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 69 [arrowhead = none ];
69 -> 15;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 70 [arrowhead = none ];
70 -> 16;

71 [label="ONLY EVAL with clause\np(X8) :- ','(p(a), !_2).\nand substitutionT4 -> T14,\nX8 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 71 [arrowhead = none ];
71 -> 44;

72 [label="BACKTRACK\nfor clause: q(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 72 [arrowhead = none ];
72 -> 19;

73 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 73 [arrowhead = none ];
73 -> 21;

74 [label="ONLY EVAL with clause\np(X5) :- ','(p(a), !_2).\nand substitutionT4 -> T9,\nX5 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 74 [arrowhead = none ];
74 -> 25;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 75 [arrowhead = none ];
75 -> 27;

76 [label="ONLY EVAL with clause\np(a).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 76 [arrowhead = none ];
76 -> 29;

77 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 77 [arrowhead = none ];
77 -> 31;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 78 [arrowhead = none ];
78 -> 34;

79 [label="EVAL with clause\nq(b).\nand substitutionT9 -> b", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 79 [arrowhead = none ];
79 -> 38;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 80 [arrowhead = none ];
80 -> 39;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 81 [arrowhead = none ];
81 -> 40;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 82 [arrowhead = none ];
82 -> 47;

83 [label="ONLY EVAL with clause\np(a).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 83 [arrowhead = none ];
83 -> 52;

84 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 84 [arrowhead = none ];
84 -> 54;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 85 [arrowhead = none ];
85 -> 58;

86 [label="EVAL with clause\nq(b).\nand substitutionT14 -> b", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 86 [arrowhead = none ];
86 -> 60;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 87 [arrowhead = none ];
87 -> 62;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 88 [arrowhead = none ];
88 -> 64;

}

(2) Obligation:

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

fA_in_a(b) → fA_out_a(b)

The argument filtering Pi contains the following mapping:
fA_in_a(x1) = fA_in_a
fA_out_a(x1) = fA_out_a(x1)

(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:

fA_in_a(b) → fA_out_a(b)

The argument filtering Pi contains the following mapping:
fA_in_a(x1) = fA_in_a
fA_out_a(x1) = fA_out_a(x1)

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:

fA_in_a(b) → fA_out_a(b)

The argument filtering Pi contains the following mapping:
fA_in_a(x1) = fA_in_a
fA_out_a(x1) = fA_out_a(x1)

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