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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: goal\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: goal, SCOPE: 1, CLAUSE: 0\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(lte(X2, s(s(s(s(0))))), even(X2))\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(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];
12 [label="12: ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 1\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(lte(X2, s(s(s(s(0))))), even(X2)), SCOPE: 2, CLAUSE: 2\n\nX2 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: even(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: even(0), SCOPE: 3, CLAUSE: 3 | even(0), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: even(0), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: even(0), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(','(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];
39 [label="39: ','(','(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];
41 [label="41: ','(','(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];
43 [label="43: ','(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];
44 [label="44: ','(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];
46 [label="46: ','(','(!_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];
47 [label="47: ','(failure(a), ','(','(p(0, X19), lte(X19, s(s(s(0))))), even(0)))\n\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(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];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 4;

53 [label="ONLY EVAL with clause\ngoal :- ','(lte(X2, s(s(s(s(0))))), even(X2)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 53 [arrowhead = none ];
53 -> 7;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 54 [arrowhead = none ];
54 -> 10;

55 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 55 [arrowhead = none ];
55 -> 12;

56 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 56 [arrowhead = none ];
56 -> 14;

57 [label="ONLY EVAL with clause\nlte(0, X7).\nand substitutionX2 -> 0,\nX7 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 57 [arrowhead = none ];
57 -> 16;

58 [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];
14 -> 58 [arrowhead = none ];
58 -> 35;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 59 [arrowhead = none ];
59 -> 18;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 60 [arrowhead = none ];
60 -> 20;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 61 [arrowhead = none ];
61 -> 21;

62 [label="ONLY EVAL with clause\neven(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 62 [arrowhead = none ];
62 -> 24;

63 [label="BACKTRACK\nfor clause: even(s(s(X))) :- even(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 63 [arrowhead = none ];
63 -> 30;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 64 [arrowhead = none ];
64 -> 25;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 65 [arrowhead = none ];
65 -> 39;

66 [label="ONLY EVAL with clause\nno(X26) :- ','(call(X26), ','(!_4, failure(a))).\nand substitutionX20 -> X27,\nX26 -> zero(X27)", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 66 [arrowhead = none ];
66 -> 41;

67 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 67 [arrowhead = none ];
67 -> 43;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 68 [arrowhead = none ];
68 -> 44;

69 [label="ONLY EVAL with clause\nzero(0).\nand substitutionX27 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 69 [arrowhead = none ];
69 -> 46;

70 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 70 [arrowhead = none ];
70 -> 47;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 71 [arrowhead = none ];
71 -> 50;

72 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 72 [arrowhead = none ];
72 -> 51;

}

(2) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

goalA_in_ → goalA_out_

Pi is empty.

(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:

goalA_in_ → goalA_out_

Pi is empty.
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:

goalA_in_ → goalA_out_

Pi is empty.
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.

(6) YES