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

(0) Obligation:

Clauses:

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

Query: goal()

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="C: goal\n\n", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal, SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(lte(X1, s(s(s(s(0))))), even(X1))\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 2 | ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 3\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 2\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(lte(X1, s(s(s(s(0))))), even(X1)), SCOPE: 2, CLAUSE: 3\n\nX1 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(lte(X22, s(s(s(0)))), even(s(X22)))\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="B: lte(X22, s(s(s(0))))\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: even(s(T1))\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: lte(X22, s(s(s(0)))), SCOPE: 3, CLAUSE: 2 | lte(X22, s(s(s(0)))), SCOPE: 3, CLAUSE: 3\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: lte(X22, s(s(s(0)))), SCOPE: 3, CLAUSE: 2\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: lte(X22, s(s(s(0)))), SCOPE: 3, CLAUSE: 3\n\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: lte(X49, s(s(0)))\n\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: lte(X49, s(s(0))), SCOPE: 4, CLAUSE: 2 | lte(X49, s(s(0))), SCOPE: 4, CLAUSE: 3\n\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: lte(X49, s(s(0))), SCOPE: 4, CLAUSE: 2\n\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: lte(X49, s(s(0))), SCOPE: 4, CLAUSE: 3\n\nX49 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: lte(X73, s(0))\n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: lte(X73, s(0)), SCOPE: 5, CLAUSE: 2 | lte(X73, s(0)), SCOPE: 5, CLAUSE: 3\n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: lte(X73, s(0)), SCOPE: 5, CLAUSE: 2\n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: lte(X73, s(0)), SCOPE: 5, CLAUSE: 3\n\nX73 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: lte(X97, 0)\n\nX97 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: lte(X97, 0), SCOPE: 6, CLAUSE: 2 | lte(X97, 0), SCOPE: 6, CLAUSE: 3\n\nX97 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: lte(X97, 0), SCOPE: 6, CLAUSE: 3\n\nX97 is free", 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];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: even(s(T1)), SCOPE: 7, CLAUSE: 0 | even(s(T1)), SCOPE: 7, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: even(s(T1)), SCOPE: 7, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: even(s(T1)), SCOPE: 7, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="A: even(T7)\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: even(T7), SCOPE: 8, CLAUSE: 0 | even(T7), SCOPE: 8, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: even(T7), SCOPE: 8, CLAUSE: 0\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: even(T7), SCOPE: 8, CLAUSE: 1\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: even(T13)\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \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];
46 [label="46: even(0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: even(0), SCOPE: 9, CLAUSE: 0 | even(0), SCOPE: 9, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: even(0), SCOPE: 9, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 51 [arrowhead = none ];
51 -> 2;

52 [label="ONLY EVAL with clause\ngoal :- ','(lte(X1, s(s(s(s(0))))), even(X1)).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 52 [arrowhead = none ];
52 -> 3;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 53 [arrowhead = none ];
53 -> 4;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 54 [arrowhead = none ];
54 -> 5;

55 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 55 [arrowhead = none ];
55 -> 6;

56 [label="ONLY EVAL with clause\nlte(s(X20), s(X21)) :- lte(X20, X21).\nand substitutionX20 -> X22,\nX1 -> s(X22),\nX21 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 56 [arrowhead = none ];
56 -> 7;

57 [label="ONLY EVAL with clause\nlte(0, X132).\nand substitutionX1 -> 0,\nX132 -> s(s(s(s(0))))", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 57 [arrowhead = none ];
57 -> 46;

58 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 58 [arrowhead = none ];
58 -> 8;

59 [label="SPLIT 2\nnew knowledge:\nT1 is ground\nreplacements:X22 -> T1", fontsize=14, style = filled, fillcolor = violet];
7 -> 59 [arrowhead = none ];
59 -> 9;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 60 [arrowhead = none ];
60 -> 10;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 61 [arrowhead = none ];
61 -> 32;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="ONLY EVAL with clause\nlte(s(X47), s(X48)) :- lte(X47, X48).\nand substitutionX47 -> X49,\nX22 -> s(X49),\nX48 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 64 [arrowhead = none ];
64 -> 13;

65 [label="ONLY EVAL with clause\nlte(0, X115).\nand substitutionX22 -> 0,\nX115 -> s(s(s(0)))", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 66 [arrowhead = none ];
66 -> 14;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 67 [arrowhead = none ];
67 -> 15;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 68 [arrowhead = none ];
68 -> 16;

69 [label="ONLY EVAL with clause\nlte(s(X71), s(X72)) :- lte(X71, X72).\nand substitutionX71 -> X73,\nX49 -> s(X73),\nX72 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 69 [arrowhead = none ];
69 -> 17;

70 [label="ONLY EVAL with clause\nlte(0, X112).\nand substitutionX49 -> 0,\nX112 -> s(s(0))", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 70 [arrowhead = none ];
70 -> 28;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 72 [arrowhead = none ];
72 -> 19;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 73 [arrowhead = none ];
73 -> 20;

74 [label="ONLY EVAL with clause\nlte(s(X95), s(X96)) :- lte(X95, X96).\nand substitutionX95 -> X97,\nX73 -> s(X97),\nX96 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 74 [arrowhead = none ];
74 -> 21;

75 [label="ONLY EVAL with clause\nlte(0, X109).\nand substitutionX73 -> 0,\nX109 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 75 [arrowhead = none ];
75 -> 26;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 76 [arrowhead = none ];
76 -> 22;

77 [label="BACKTRACK\nfor clause: lte(s(X), s(Y)) :- lte(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 77 [arrowhead = none ];
77 -> 23;

78 [label="ONLY EVAL with clause\nlte(0, X106).\nand substitutionX97 -> 0,\nX106 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 78 [arrowhead = none ];
78 -> 24;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 79 [arrowhead = none ];
79 -> 25;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 80 [arrowhead = none ];
80 -> 27;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 81 [arrowhead = none ];
81 -> 29;

82 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 82 [arrowhead = none ];
82 -> 31;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 83 [arrowhead = none ];
83 -> 33;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 84 [arrowhead = none ];
84 -> 34;

85 [label="EVAL with clause\neven(s(s(X121))) :- even(X121).\nand substitutionX121 -> T7,\nT1 -> s(T7)", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 85 [arrowhead = none ];
85 -> 35;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 86 [arrowhead = none ];
86 -> 36;

87 [label="BACKTRACK\nfor clause: even(0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 87 [arrowhead = none ];
87 -> 45;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 88 [arrowhead = none ];
88 -> 37;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 89 [arrowhead = none ];
89 -> 38;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
37 -> 90 [arrowhead = none ];
90 -> 39;

91 [label="EVAL with clause\neven(s(s(X127))) :- even(X127).\nand substitutionX127 -> T13,\nT7 -> s(s(T13))", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 91 [arrowhead = none ];
91 -> 40;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 92 [arrowhead = none ];
92 -> 41;

93 [label="EVAL with clause\neven(0).\nand substitutionT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 93 [arrowhead = none ];
93 -> 42;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 94 [arrowhead = none ];
94 -> 43;

95 [label="INSTANCE with matching:\nT7 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 95 [arrowhead = none , style=dashed];
95 -> 35[style=dashed];

96 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 96 [arrowhead = none ];
96 -> 44;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 97 [arrowhead = none ];
97 -> 47;

98 [label="BACKTRACK\nfor clause: even(s(s(X))) :- even(X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 98 [arrowhead = none ];
98 -> 48;

99 [label="ONLY EVAL with clause\neven(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 99 [arrowhead = none ];
99 -> 49;

100 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 100 [arrowhead = none ];
100 -> 50;

}

(2) Obligation:

Triples:

evenA(s(s(X1))) :- evenA(X1).
goalC :- ','(ltecB(s(X1)), evenA(X1)).

Clauses:

evencA(s(s(X1))) :- evencA(X1).
evencA(0).
ltecB(s(s(s(0)))).
ltecB(s(s(0))).
ltecB(s(0)).
ltecB(0).

Afs:

goalC  =  goalC

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
evenA_in: (b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

GOALC_IN_ → U2_¹(ltecB_in_a(s(X1)))
U2_¹(ltecB_out_a(s(X1))) → U3_¹(evenA_in_g(X1))
U2_¹(ltecB_out_a(s(X1))) → EVENA_IN_G(X1)
EVENA_IN_G(s(s(X1))) → U1_G(X1, evenA_in_g(X1))
EVENA_IN_G(s(s(X1))) → EVENA_IN_G(X1)

The TRS R consists of the following rules:

ltecB_in_a(s(s(s(0)))) → ltecB_out_a(s(s(s(0))))
ltecB_in_a(s(s(0))) → ltecB_out_a(s(s(0)))
ltecB_in_a(s(0)) → ltecB_out_a(s(0))
ltecB_in_a(0) → ltecB_out_a(0)

The argument filtering Pi contains the following mapping:
ltecB_in_a(x1)  =  ltecB_in_a
ltecB_out_a(x1)  =  ltecB_out_a(x1)
s(x1)  =  s(x1)
evenA_in_g(x1)  =  evenA_in_g(x1)
GOALC_IN_  =  GOALC_IN_
U2_¹(x1)  =  U2_¹(x1)
U3_¹(x1)  =  U3_¹(x1)
EVENA_IN_G(x1)  =  EVENA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)

We have to consider all (P,R,Pi)-chains

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

GOALC_IN_ → U2_¹(ltecB_in_a(s(X1)))
U2_¹(ltecB_out_a(s(X1))) → U3_¹(evenA_in_g(X1))
U2_¹(ltecB_out_a(s(X1))) → EVENA_IN_G(X1)
EVENA_IN_G(s(s(X1))) → U1_G(X1, evenA_in_g(X1))
EVENA_IN_G(s(s(X1))) → EVENA_IN_G(X1)

The TRS R consists of the following rules:

ltecB_in_a(s(s(s(0)))) → ltecB_out_a(s(s(s(0))))
ltecB_in_a(s(s(0))) → ltecB_out_a(s(s(0)))
ltecB_in_a(s(0)) → ltecB_out_a(s(0))
ltecB_in_a(0) → ltecB_out_a(0)

The argument filtering Pi contains the following mapping:
ltecB_in_a(x1)  =  ltecB_in_a
ltecB_out_a(x1)  =  ltecB_out_a(x1)
s(x1)  =  s(x1)
evenA_in_g(x1)  =  evenA_in_g(x1)
GOALC_IN_  =  GOALC_IN_
U2_¹(x1)  =  U2_¹(x1)
U3_¹(x1)  =  U3_¹(x1)
EVENA_IN_G(x1)  =  EVENA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x1, x2)

We have to consider all (P,R,Pi)-chains

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

EVENA_IN_G(s(s(X1))) → EVENA_IN_G(X1)

The TRS R consists of the following rules:

ltecB_in_a(s(s(s(0)))) → ltecB_out_a(s(s(s(0))))
ltecB_in_a(s(s(0))) → ltecB_out_a(s(s(0)))
ltecB_in_a(s(0)) → ltecB_out_a(s(0))
ltecB_in_a(0) → ltecB_out_a(0)

The argument filtering Pi contains the following mapping:
ltecB_in_a(x1)  =  ltecB_in_a
ltecB_out_a(x1)  =  ltecB_out_a(x1)
s(x1)  =  s(x1)
EVENA_IN_G(x1)  =  EVENA_IN_G(x1)

We have to consider all (P,R,Pi)-chains

(7) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

EVENA_IN_G(s(s(X1))) → EVENA_IN_G(X1)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains

(9) PiDPToQDPProof (EQUIVALENT transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(10) Obligation:

Q DP problem:
The TRS P consists of the following rules:

EVENA_IN_G(s(s(X1))) → EVENA_IN_G(X1)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(11) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• EVENA_IN_G(s(s(X1))) → EVENA_IN_G(X1)
  The graph contains the following edges 1 > 1

(12) YES