Left Termination proof of /tmp/tmpg8ovEp/example1.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 (⇒, 9 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 PisEmptyProof (⇔, 0 ms)

↳8 YES

(0) Obligation:

Clauses:

p(X, g(X)).
p(X, f(Y)) :- p(X, g(Y)).

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: true | p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: p(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX2 is free\np(T1, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: p(T7, g(T9))\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: p(T7, g(T9)), SCOPE: 2, CLAUSE: 0 | p(T7, g(T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: p(T7, g(T9)), SCOPE: 2, CLAUSE: 0\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: p(T7, g(T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: p(T18, g(T20))\n\nT18 is ground\nX2 is free\np(T18, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: p(T18, g(T20)), SCOPE: 3, CLAUSE: 0 | p(T18, g(T20)), SCOPE: 3, CLAUSE: 1\n\nT18 is ground\nX2 is free\np(T18, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T18, g(T20)), SCOPE: 3, CLAUSE: 0\n\nT18 is ground\nX2 is free\np(T18, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: p(T18, g(T20)), SCOPE: 3, CLAUSE: 1\n\nT18 is ground\nX2 is free\np(T18, T2) !=? p(X2, g(X2))", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 24 [arrowhead = none ];
24 -> 2;

25 [label="EVAL with clause\np(X2, g(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> g(T4)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 25 [arrowhead = none ];
25 -> 3;

26 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 26 [arrowhead = none ];
26 -> 4;

27 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 27 [arrowhead = none ];
27 -> 5;

28 [label="EVAL with clause\np(X16, f(X17)) :- p(X16, g(X17)).\nand substitutionT1 -> T18,\nX16 -> T18,\nX17 -> T20,\nT2 -> f(T20),\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 28 [arrowhead = none ];
28 -> 15;

29 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 29 [arrowhead = none ];
29 -> 16;

30 [label="EVAL with clause\np(X5, f(X6)) :- p(X5, g(X6)).\nand substitutionT4 -> T7,\nX5 -> T7,\nX6 -> T9,\nT2 -> f(T9),\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 30 [arrowhead = none ];
30 -> 6;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 31 [arrowhead = none ];
31 -> 7;

32 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 32 [arrowhead = none ];
32 -> 8;

33 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 33 [arrowhead = none ];
33 -> 9;

34 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 34 [arrowhead = none ];
34 -> 10;

35 [label="EVAL with clause\np(X11, g(X11)).\nand substitutionT7 -> T14,\nX11 -> T14,\nT9 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 35 [arrowhead = none ];
35 -> 11;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="BACKTRACK\nfor clause: p(X, f(Y)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 37 [arrowhead = none ];
37 -> 14;

38 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 38 [arrowhead = none ];
38 -> 13;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 39 [arrowhead = none ];
39 -> 17;

40 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 40 [arrowhead = none ];
40 -> 18;

41 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 41 [arrowhead = none ];
41 -> 19;

42 [label="EVAL with clause\np(X22, g(X22)).\nand substitutionT18 -> T25,\nX22 -> T25,\nT20 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 42 [arrowhead = none ];
42 -> 20;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 43 [arrowhead = none ];
43 -> 21;

44 [label="BACKTRACK\nfor clause: p(X, f(Y)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 44 [arrowhead = none ];
44 -> 23;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 45 [arrowhead = none ];
45 -> 22;

}

(2) Obligation:

Clauses:

pA(T4, g(T4)).
pA(T14, f(T14)).
pA(T25, f(T25)).

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(T4, g(T4)) → pA_out_ga(T4, g(T4))
pA_in_ga(T14, f(T14)) → pA_out_ga(T14, f(T14))

The argument filtering Pi contains the following mapping:
pA_in_ga(x1, x2) = pA_in_ga(x1)
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(T4, g(T4)) → pA_out_ga(T4, g(T4))
pA_in_ga(T14, f(T14)) → pA_out_ga(T14, f(T14))

The argument filtering Pi contains the following mapping:
pA_in_ga(x1, x2) = pA_in_ga(x1)
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(T4, g(T4)) → pA_out_ga(T4, g(T4))
pA_in_ga(T14, f(T14)) → pA_out_ga(T14, f(T14))

The argument filtering Pi contains the following mapping:
pA_in_ga(x1, x2) = pA_in_ga(x1)
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(T4, g(T4)) → pA_out_ga(T4, g(T4))
pA_in_ga(T14, f(T14)) → pA_out_ga(T14, f(T14))

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