Left Termination proof of /tmp/tmp3CEBgd/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 PrologToPiTRSViaGraphTransformerProof (⇒, 64 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 PisEmptyProof (⇔, 0 ms)

↳6 YES

(0) Obligation:

Clauses:

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

Query: p(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [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];
9 [label="9: true | p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: 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];
13 [label="13: p(T4, T2), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: p(T7, g(T9))\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: 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];
23 [label="23: p(T7, g(T9)), SCOPE: 2, CLAUSE: 0\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: p(T7, g(T9)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: p(T18, g(T20))\n\nT18 is ground\nX2 is free\np(T18, T2) !=? p(X2, g(X2))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: 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];
41 [label="41: 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];
42 [label="42: 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];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 48 [arrowhead = none ];
48 -> 5;

49 [label="EVAL with clause\np(X2, g(X2)).\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> g(T4)", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 49 [arrowhead = none ];
49 -> 9;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 50 [arrowhead = none ];
50 -> 10;

51 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 51 [arrowhead = none ];
51 -> 13;

52 [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];
10 -> 52 [arrowhead = none ];
52 -> 32;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 53 [arrowhead = none ];
53 -> 35;

54 [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];
13 -> 54 [arrowhead = none ];
54 -> 18;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 55 [arrowhead = none ];
55 -> 19;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 56 [arrowhead = none ];
56 -> 20;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 57 [arrowhead = none ];
57 -> 23;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 58 [arrowhead = none ];
58 -> 24;

59 [label="EVAL with clause\np(X11, g(X11)).\nand substitutionT7 -> T14,\nX11 -> T14,\nT9 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 59 [arrowhead = none ];
59 -> 25;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 60 [arrowhead = none ];
60 -> 26;

61 [label="BACKTRACK\nfor clause: p(X, f(Y)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 61 [arrowhead = none ];
61 -> 28;

62 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 62 [arrowhead = none ];
62 -> 27;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 63 [arrowhead = none ];
63 -> 36;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 64 [arrowhead = none ];
64 -> 41;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 65 [arrowhead = none ];
65 -> 42;

66 [label="EVAL with clause\np(X22, g(X22)).\nand substitutionT18 -> T25,\nX22 -> T25,\nT20 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 66 [arrowhead = none ];
66 -> 43;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 67 [arrowhead = none ];
67 -> 44;

68 [label="BACKTRACK\nfor clause: p(X, f(Y)) :- p(X, g(Y))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 68 [arrowhead = none ];
68 -> 47;

69 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 69 [arrowhead = none ];
69 -> 45;

}

(2) 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(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(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(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(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(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