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

(0) Obligation:

Clauses:

even(0) :- !.
even(N) :- ','(p(N, P), odd(P)).
odd(s(0)) :- !.
odd(s(N)) :- even(P).
p(0, 0).
p(s(X), X).

Query: even(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: even(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: even(T1), SCOPE: 1, CLAUSE: 0 | even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: !_1 | even(0), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: even(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\neven(T1) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(T3, X3), odd(X3))\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 4 | ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T3, X3), odd(X3)), SCOPE: 2, CLAUSE: 5\n\nT3 is ground\nX3 is free\neven(T3) !=? even(0)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: odd(T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: odd(T6), SCOPE: 3, CLAUSE: 2 | odd(T6), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: !_3 | odd(s(0)), SCOPE: 3, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: odd(T6), SCOPE: 3, CLAUSE: 3\n\nT6 is ground\nodd(T6) !=? odd(s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: even(X11)\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: even(X11), SCOPE: 4, CLAUSE: 0 | even(X11), SCOPE: 4, CLAUSE: 1\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: !_4 | even(X11), SCOPE: 4, CLAUSE: 1\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 67 [arrowhead = none ];
67 -> 5;

68 [label="EVAL with clause\neven(0) :- !_1.\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 68 [arrowhead = none ];
68 -> 9;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 69 [arrowhead = none ];
69 -> 12;

70 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 70 [arrowhead = none ];
70 -> 13;

71 [label="ONLY EVAL with clause\neven(X2) :- ','(p(X2, X3), odd(X3)).\nand substitutionT1 -> T3,\nX2 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 71 [arrowhead = none ];
71 -> 22;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 72 [arrowhead = none ];
72 -> 17;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 73 [arrowhead = none ];
73 -> 23;

74 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (even(T3), even(0))", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 74 [arrowhead = none ];
74 -> 24;

75 [label="EVAL with clause\np(s(X6), X6).\nand substitutionX6 -> T6,\nT3 -> s(T6),\nX3 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 75 [arrowhead = none ];
75 -> 27;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 76 [arrowhead = none ];
76 -> 28;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 77 [arrowhead = none ];
77 -> 50;

78 [label="EVAL with clause\nodd(s(0)) :- !_3.\nand substitutionT6 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 78 [arrowhead = none ];
78 -> 54;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 79 [arrowhead = none ];
79 -> 56;

80 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 80 [arrowhead = none ];
80 -> 59;

81 [label="EVAL with clause\nodd(s(X10)) :- even(X11).\nand substitutionX10 -> T9,\nT6 -> s(T9)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 81 [arrowhead = none ];
81 -> 61;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 82 [arrowhead = none ];
82 -> 62;

83 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 83 [arrowhead = none ];
83 -> 60;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 84 [arrowhead = none ];
84 -> 63;

85 [label="ONLY EVAL with clause\neven(0) :- !_4.\nand substitutionX11 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
63 -> 85 [arrowhead = none ];
85 -> 64;

86 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 86 [arrowhead = none ];
86 -> 65;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 87 [arrowhead = none ];
87 -> 66;

}

(2) Obligation:

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

evenA_in_g(0) → evenA_out_g(0)
evenA_in_g(s(s(0))) → evenA_out_g(s(s(0)))
evenA_in_g(s(s(T9))) → evenA_out_g(s(s(T9)))

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:

evenA_in_g(0) → evenA_out_g(0)
evenA_in_g(s(s(0))) → evenA_out_g(s(s(0)))
evenA_in_g(s(s(T9))) → evenA_out_g(s(s(T9)))

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:

evenA_in_g(0) → evenA_out_g(0)
evenA_in_g(s(s(0))) → evenA_out_g(s(s(0)))
evenA_in_g(s(s(T9))) → evenA_out_g(s(s(T9)))

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