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)) :- ','(p(X, P), lte(P, Y)).
even(0).
even(s(s(X))) :- even(X).
p(0, 0).
p(s(X), X).

Query: goal()

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [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(X1, s(s(s(s(0))))), even(X1))\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 1 | ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 2\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(!_2, even(0)) | ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 2\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: even(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: even(0), SCOPE: 3, CLAUSE: 3 | even(0), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: even(0), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: 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];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 29 [arrowhead = none ];
29 -> 4;

30 [label="ONLY EVAL with clause\ngoal :- ','(lte(X1, s(s(s(s(0))))), even(X1)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 30 [arrowhead = none ];
30 -> 7;

31 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 31 [arrowhead = none ];
31 -> 11;

32 [label="ONLY EVAL with clause\nlte(0, X4) :- !_2.\nand substitutionX1 -> 0,\nX4 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 32 [arrowhead = none ];
32 -> 12;

33 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 33 [arrowhead = none ];
33 -> 14;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 34 [arrowhead = none ];
34 -> 16;

35 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 35 [arrowhead = none ];
35 -> 19;

36 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 36 [arrowhead = none ];
36 -> 22;

37 [label="ONLY EVAL with clause\neven(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 37 [arrowhead = none ];
37 -> 24;

38 [label="BACKTRACK\nfor clause: even(s(s(X))) :- even(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 38 [arrowhead = none ];
38 -> 28;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 39 [arrowhead = none ];
39 -> 26;

}

(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