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

(0) Obligation:

Clauses:

goal :- ','(lte(X, s(s(s(s(0))))), even(X)).
lte(0, Y).
lte(X, s(Y)) :- ','(no(zero(X)), ','(p(X, P), lte(P, Y))).
even(0).
even(s(s(X))) :- even(X).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: goal()

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: goal\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(lte(X2, s(s(s(s(0))))), even(X2))\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 1 | ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 2\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 1\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 2\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: even(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: even(0), SCOPE: 3, CLAUSE: 3 | even(0), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: even(0), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: even(0), SCOPE: 3, CLAUSE: 4\n\n", 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: ','(','(no(zero(X20)), ','(p(X20, X19), lte(X19, s(s(s(0)))))), even(X20))\n\nX20 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(','(no(zero(X20)), ','(p(X20, X19), lte(X19, s(s(s(0)))))), even(X20)), SCOPE: 4, CLAUSE: 8 | ','(','(no(zero(X20)), ','(p(X20, X19), lte(X19, s(s(s(0)))))), even(X20)), SCOPE: 4, CLAUSE: 9\n\nX20 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(','(call(zero(X27)), ','(!_4, failure(a))), ','(','(p(X27, X19), lte(X19, s(s(s(0))))), even(X27))) | ','(','(no(zero(X20)), ','(p(X20, X19), lte(X19, s(s(s(0)))))), even(X20)), SCOPE: 4, CLAUSE: 9\n\nX20 is free\nX19 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(zero(X27), ','(','(!_4, failure(a)), ','(','(p(X27, X19), lte(X19, s(s(s(0))))), even(X27)))) | ?_5 | ','(','(no(zero(X20)), ','(p(X20, X19), lte(X19, s(s(s(0)))))), even(X20)), SCOPE: 4, CLAUSE: 9\n\nX20 is free\nX19 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(zero(X27), ','(','(!_4, failure(a)), ','(','(p(X27, X19), lte(X19, s(s(s(0))))), even(X27)))), SCOPE: 6, CLAUSE: 7 | ?_6 | ?_5 | ','(','(no(zero(X20)), ','(p(X20, X19), lte(X19, s(s(s(0)))))), even(X20)), SCOPE: 4, CLAUSE: 9\n\nX20 is free\nX19 is free\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(!_4, failure(a)), ','(','(p(0, X19), lte(X19, s(s(s(0))))), even(0))) | ?_6 | ?_5 | ','(','(no(zero(X20)), ','(p(X20, X19), lte(X19, s(s(s(0)))))), even(X20)), SCOPE: 4, CLAUSE: 9\n\nX20 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(failure(a), ','(','(p(0, X19), lte(X19, s(s(s(0))))), even(0)))\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(failure(a), ','(','(p(0, X19), lte(X19, s(s(s(0))))), even(0))), SCOPE: 7, CLAUSE: 10\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 23 [arrowhead = none ];
23 -> 2;

24 [label="ONLY EVAL with clause\ngoal :- ','(lte(X2, s(s(s(s(0))))), even(X2)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 24 [arrowhead = none ];
24 -> 3;

25 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 25 [arrowhead = none ];
25 -> 4;

26 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 26 [arrowhead = none ];
26 -> 5;

27 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 27 [arrowhead = none ];
27 -> 6;

28 [label="ONLY EVAL with clause\nlte(0, X7).\nand substitutionX2 -> 0,\nX7 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 28 [arrowhead = none ];
28 -> 7;

29 [label="ONLY EVAL with clause\nlte(X17, s(X18)) :- ','(no(zero(X17)), ','(p(X17, X19), lte(X19, X18))).\nand substitutionX2 -> X20,\nX17 -> X20,\nX18 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 29 [arrowhead = none ];
29 -> 14;

30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 30 [arrowhead = none ];
30 -> 8;

31 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 31 [arrowhead = none ];
31 -> 9;

32 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 32 [arrowhead = none ];
32 -> 10;

33 [label="ONLY EVAL with clause\neven(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 33 [arrowhead = none ];
33 -> 11;

34 [label="BACKTRACK\nfor clause: even(s(s(X))) :- even(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 34 [arrowhead = none ];
34 -> 13;

35 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 35 [arrowhead = none ];
35 -> 12;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 36 [arrowhead = none ];
36 -> 15;

37 [label="ONLY EVAL with clause\nno(X26) :- ','(call(X26), ','(!_4, failure(a))).\nand substitutionX20 -> X27,\nX26 -> zero(X27)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 37 [arrowhead = none ];
37 -> 16;

38 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 38 [arrowhead = none ];
38 -> 17;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 39 [arrowhead = none ];
39 -> 18;

40 [label="ONLY EVAL with clause\nzero(0).\nand substitutionX27 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 40 [arrowhead = none ];
40 -> 19;

41 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 41 [arrowhead = none ];
41 -> 20;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 42 [arrowhead = none ];
42 -> 21;

43 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 43 [arrowhead = none ];
43 -> 22;

}

(2) Obligation:

Triples:


Clauses:


Afs:

(3) TPisEmptyProof (EQUIVALENT transformation)

There are no more dependency triples. Hence, the dependency triple problem trivially terminates.

(4) YES