Left Termination proof of /tmp/tmpasDWQm/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 PrologToPrologProblemTransformerProof (⇒, 0 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 14 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 PisEmptyProof (⇔, 0 ms)

↳8 YES

(0) Obligation:

Clauses:

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

Query: p(g,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)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: p(T1, T2), SCOPE: 1, CLAUSE: 0 | p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(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];
4 [label="4: ','(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];
5 [label="5: ','(q(T5, X5), p(X5, T7)), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ?_2 | p(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: p(b, T7)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(b, T7), SCOPE: 3, CLAUSE: 0 | p(b, T7), SCOPE: 3, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(b, T7), SCOPE: 3, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(b, T7), SCOPE: 3, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(q(b, X29), p(X29, T18))\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(q(b, X29), p(X29, T18)), SCOPE: 4, CLAUSE: 2\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \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(T5, T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 22 [arrowhead = none ];
22 -> 2;

23 [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];
2 -> 23 [arrowhead = none ];
23 -> 3;

24 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 24 [arrowhead = none ];
24 -> 4;

25 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 25 [arrowhead = none ];
25 -> 5;

26 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 26 [arrowhead = none ];
26 -> 6;

27 [label="EVAL with clause\nq(a, b).\nand substitutionT5 -> a,\nX5 -> b", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 27 [arrowhead = none ];
27 -> 7;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 28 [arrowhead = none ];
28 -> 8;

29 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 29 [arrowhead = none ];
29 -> 18;

30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 30 [arrowhead = none ];
30 -> 9;

31 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 31 [arrowhead = none ];
31 -> 10;

32 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 32 [arrowhead = none ];
32 -> 11;

33 [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];
10 -> 33 [arrowhead = none ];
33 -> 12;

34 [label="EVAL with clause\np(X36, X36).\nand substitutionX36 -> b,\nT7 -> b", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 34 [arrowhead = none ];
34 -> 15;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 35 [arrowhead = none ];
35 -> 16;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 36 [arrowhead = none ];
36 -> 13;

37 [label="BACKTRACK\nfor clause: q(a, b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 37 [arrowhead = none ];
37 -> 14;

38 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 38 [arrowhead = none ];
38 -> 17;

39 [label="EVAL with clause\np(X39, X39).\nand substitutionT5 -> T22,\nX39 -> T22,\nT2 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 39 [arrowhead = none ];
39 -> 19;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 40 [arrowhead = none ];
40 -> 20;

41 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 41 [arrowhead = none ];
41 -> 21;

}

(2) Obligation:

Clauses:

pA(a, b).
pA(T22, T22).

Query: pA(g,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_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(x2)

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_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(x2)

(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_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(x2)

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_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(x2)

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