Left Termination proof of /tmp/tmpx2hnXk/plumer453b.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 0 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 8 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 PisEmptyProof (⇔, 0 ms)

↳8 YES

(0) Obligation:

Clauses:

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

Query: f(a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: f(T1)\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: f(T1), SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(p(T4), q(T4))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 1 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: q(a) | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\np(T4) !=? p(a)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: q(a), SCOPE: 3, CLAUSE: 3 | ?_3 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ?_3 | ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(p(T4), q(T4)), SCOPE: 2, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(','(p(a), !_2), q(T9))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(','(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];
12 [label="12: ','(!_2, q(T9)) | ','(','(p(a), !_2), q(T9)), SCOPE: 4, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: q(T9)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: q(T9), SCOPE: 5, CLAUSE: 3\n\n", 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];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(','(p(a), !_2), q(T14))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(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];
20 [label="20: ','(!_2, q(T14)) | ','(','(p(a), !_2), q(T14)), SCOPE: 6, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: q(T14)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: q(T14), SCOPE: 7, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 26 [arrowhead = none ];
26 -> 2;

27 [label="ONLY EVAL with clause\nf(X2) :- ','(p(X2), q(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 27 [arrowhead = none ];
27 -> 3;

28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 28 [arrowhead = none ];
28 -> 4;

29 [label="EVAL with clause\np(a).\nand substitutionT4 -> a", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 29 [arrowhead = none ];
29 -> 5;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 30 [arrowhead = none ];
30 -> 6;

31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 31 [arrowhead = none ];
31 -> 7;

32 [label="ONLY EVAL with clause\np(X8) :- ','(p(a), !_2).\nand substitutionT4 -> T14,\nX8 -> T14,\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 32 [arrowhead = none ];
32 -> 18;

33 [label="BACKTRACK\nfor clause: q(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="ONLY EVAL with clause\np(X5) :- ','(p(a), !_2).\nand substitutionT4 -> T9,\nX5 -> T9,\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 35 [arrowhead = none ];
35 -> 10;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 36 [arrowhead = none ];
36 -> 11;

37 [label="ONLY EVAL with clause\np(a).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 37 [arrowhead = none ];
37 -> 12;

38 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 38 [arrowhead = none ];
38 -> 13;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 39 [arrowhead = none ];
39 -> 14;

40 [label="EVAL with clause\nq(b).\nand substitutionT9 -> b", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 40 [arrowhead = none ];
40 -> 15;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 41 [arrowhead = none ];
41 -> 16;

42 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 42 [arrowhead = none ];
42 -> 17;

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 43 [arrowhead = none ];
43 -> 19;

44 [label="ONLY EVAL with clause\np(a).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 44 [arrowhead = none ];
44 -> 20;

45 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 45 [arrowhead = none ];
45 -> 21;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 46 [arrowhead = none ];
46 -> 22;

47 [label="EVAL with clause\nq(b).\nand substitutionT14 -> b", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 47 [arrowhead = none ];
47 -> 23;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 48 [arrowhead = none ];
48 -> 24;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 49 [arrowhead = none ];
49 -> 25;

}

(2) Obligation:

Clauses:

fA(b).
fA(b).

Query: fA(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:

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)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) 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)

(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:

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

(6) 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

(7) PisEmptyProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) PisEmptyProof (EQUIVALENT transformation)

(8) YES