Left Termination of the query pattern less_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

less(0, s(X2)).
less(X, Y) :- ','(no(zero(X)), ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1)))).
p(0, 0).
p(s(X), X).
zero(0).
no(X) :- ','(X, ','(!, failure(a))).
no(X3).
failure(b).

Queries:

less(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: less(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: less(T1, T2), SCOPE: 1, CLAUSE: 0 | less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: less(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(no(zero(T4)), ','(p(T4, X11), ','(p(T6, X12), less(X11, X12))))\n\nT4 is ground\nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(no(zero(T4)), ','(p(T4, X11), ','(p(T6, X12), less(X11, X12)))), SCOPE: 2, CLAUSE: 5 | ','(no(zero(T4)), ','(p(T4, X11), ','(p(T6, X12), less(X11, X12)))), SCOPE: 2, CLAUSE: 6 | ?_2\n\nT4 is ground\nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(zero(T7), ','(!_2, ','(failure(a), ','(p(T7, X11), ','(p(T6, X12), less(X11, X12)))))) | ','(no(zero(T7)), ','(p(T7, X11), ','(p(T6, X12), less(X11, X12)))), SCOPE: 2, CLAUSE: 6 | ?_2\n\nT7 is ground\nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(zero(T7), ','(!_2, ','(failure(a), ','(p(T7, X11), ','(p(T6, X12), less(X11, X12)))))), SCOPE: 3, CLAUSE: 4 | ','(no(zero(T7)), ','(p(T7, X11), ','(p(T6, X12), less(X11, X12)))), SCOPE: 2, CLAUSE: 6 | ?_2\n\nT7 is ground\nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(!_2, ','(failure(a), ','(p(0, X11), ','(p(T6, X12), less(X11, X12))))) | ','(no(zero(0)), ','(p(0, X11), ','(p(T6, X12), less(X11, X12)))), SCOPE: 2, CLAUSE: 6 | ?_2\n\nX11 is free\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(no(zero(T7)), ','(p(T7, X11), ','(p(T6, X12), less(X11, X12)))), SCOPE: 2, CLAUSE: 6\n\nT7 is ground\nX11 is free; \nX12 is free; \nzero(T7) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(failure(a), ','(p(0, X11), ','(p(T6, X12), less(X11, X12))))\n\nX11 is free\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(failure(a), ','(p(0, X11), ','(p(T6, X12), less(X11, X12)))), SCOPE: 4, CLAUSE: 7\n\nX11 is free\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T8, X11), ','(p(T6, X12), less(X11, X12)))\n\nT8 is ground\nX11 is free; \nX12 is free; \nzero(T8) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(T8, X11), ','(p(T6, X12), less(X11, X12))), SCOPE: 5, CLAUSE: 2 | ','(p(T8, X11), ','(p(T6, X12), less(X11, X12))), SCOPE: 5, CLAUSE: 3\n\nT8 is ground\nX11 is free; \nX12 is free; \nzero(T8) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T8, X11), ','(p(T6, X12), less(X11, X12))), SCOPE: 5, CLAUSE: 3\n\nT8 is ground\nX11 is free; \nX12 is free; \nzero(T8) !=? zero(0)\np(T8, X11) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(p(T6, X12), less(T9, X12))\n\nT9 is ground\nX11 is free; \nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\nX11 is free\nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(T6, X12), less(T9, X12)), SCOPE: 6, CLAUSE: 2 | ','(p(T6, X12), less(T9, X12)), SCOPE: 6, CLAUSE: 3\n\nT9 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: less(T9, 0) | ','(p(T6, X12), less(T9, X12)), SCOPE: 6, CLAUSE: 3\n\nT9 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T6, X12), less(T9, X12)), SCOPE: 6, CLAUSE: 3\n\nT9 is ground\nX12 is free; \np(T6, X12) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: less(T9, 0)\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(p(T6, X12), less(T9, X12)), SCOPE: 6, CLAUSE: 3\n\nT9 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(T9, 0), SCOPE: 7, CLAUSE: 0 | less(T9, 0), SCOPE: 7, CLAUSE: 1\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T9, 0), SCOPE: 7, CLAUSE: 1\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(no(zero(T10)), ','(p(T10, X24), ','(p(0, X25), less(X24, X25))))\n\nT10 is ground\nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(no(zero(T10)), ','(p(T10, X24), ','(p(0, X25), less(X24, X25)))), SCOPE: 8, CLAUSE: 5 | ','(no(zero(T10)), ','(p(T10, X24), ','(p(0, X25), less(X24, X25)))), SCOPE: 8, CLAUSE: 6 | ?_8\n\nT10 is ground\nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(zero(T11), ','(!_8, ','(failure(a), ','(p(T11, X24), ','(p(0, X25), less(X24, X25)))))) | ','(no(zero(T11)), ','(p(T11, X24), ','(p(0, X25), less(X24, X25)))), SCOPE: 8, CLAUSE: 6 | ?_8\n\nT11 is ground\nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(zero(T11), ','(!_8, ','(failure(a), ','(p(T11, X24), ','(p(0, X25), less(X24, X25)))))), SCOPE: 9, CLAUSE: 4 | ','(no(zero(T11)), ','(p(T11, X24), ','(p(0, X25), less(X24, X25)))), SCOPE: 8, CLAUSE: 6 | ?_8\n\nT11 is ground\nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(!_8, ','(failure(a), ','(p(0, X24), ','(p(0, X25), less(X24, X25))))) | ','(no(zero(0)), ','(p(0, X24), ','(p(0, X25), less(X24, X25)))), SCOPE: 8, CLAUSE: 6 | ?_8\n\nX24 is free\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(no(zero(T11)), ','(p(T11, X24), ','(p(0, X25), less(X24, X25)))), SCOPE: 8, CLAUSE: 6\n\nT11 is ground\nX24 is free; \nX25 is free; \nzero(T11) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(failure(a), ','(p(0, X24), ','(p(0, X25), less(X24, X25))))\n\nX24 is free\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(failure(a), ','(p(0, X24), ','(p(0, X25), less(X24, X25)))), SCOPE: 10, CLAUSE: 7\n\nX24 is free\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(p(T12, X24), ','(p(0, X25), less(X24, X25)))\n\nT12 is ground\nX24 is free; \nX25 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(p(T12, X24), ','(p(0, X25), less(X24, X25))), SCOPE: 11, CLAUSE: 2 | ','(p(T12, X24), ','(p(0, X25), less(X24, X25))), SCOPE: 11, CLAUSE: 3\n\nT12 is ground\nX24 is free; \nX25 is free; \nzero(T12) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(p(T12, X24), ','(p(0, X25), less(X24, X25))), SCOPE: 11, CLAUSE: 3\n\nT12 is ground\nX24 is free; \nX25 is free; \nzero(T12) !=? zero(0)\np(T12, X24) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(p(0, X25), less(T13, X25))\n\nT13 is ground\nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\nX24 is free\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(p(0, X25), less(T13, X25)), SCOPE: 12, CLAUSE: 2 | ','(p(0, X25), less(T13, X25)), SCOPE: 12, CLAUSE: 3\n\nT13 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: less(T13, 0) | ','(p(0, X25), less(T13, X25)), SCOPE: 12, CLAUSE: 3\n\nT13 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T13, 0)\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(p(0, X25), less(T13, X25)), SCOPE: 12, CLAUSE: 3\n\nT13 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: less(T9, T15)\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\nX12 is free\nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: less(T9, T17)\n\nT9 is ground\nX12 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\nX12 is free\nX36 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 2;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 53 [arrowhead = none ];
53 -> 3;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
2 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="EVAL with clause\nless(0, s(X6)).\nand substitution\nT1 -> 0\nX6 -> T3\nT2 -> s(T3)", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 55 [arrowhead = none ];
55 -> 5;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 56 [arrowhead = none ];
56 -> 6;

57 [label="EVAL with clause\nless(X9, X10) :- ','(no(zero(X9)), ','(p(X9, X11), ','(p(X10, X12), less(X11, X12)))).\nand substitution\nT1 -> T4\nX9 -> T4\nT2 -> T6\nX10 -> T6\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 57 [arrowhead = none ];
57 -> 8;

58 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 58 [arrowhead = none ];
58 -> 7;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 59 [arrowhead = none ];
59 -> 9;

60 [label="EVAL with clause\nno(X14) :- ','(X14, ','(!, failure(a))).\nand substitution\nT4 -> T7\nX14 -> zero(T7)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 60 [arrowhead = none ];
60 -> 10;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 61 [arrowhead = none ];
61 -> 11;

62 [label="EVAL with clause\nzero(0).\nand substitution\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 62 [arrowhead = none ];
62 -> 12;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 63 [arrowhead = none ];
63 -> 13;

64 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 64 [arrowhead = none ];
64 -> 14;

65 [label="EVAL with clause\nno(X16).\nand substitution\nT7 -> T8\nX16 -> zero(T8)", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 65 [arrowhead = none ];
65 -> 17;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 66 [arrowhead = none ];
66 -> 15;

67 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 67 [arrowhead = none ];
67 -> 16;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 68 [arrowhead = none ];
68 -> 18;

69 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T8), zero(0))", fontsize=14, style = filled, fillcolor = white];
18 -> 69 [arrowhead = none ];
69 -> 19;

70 [label="EVAL with clause\np(s(X18), X18).\nand substitution\nX18 -> T9\nT8 -> s(T9)\nX11 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 70 [arrowhead = none ];
70 -> 20;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 71 [arrowhead = none ];
71 -> 21;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 72 [arrowhead = none ];
72 -> 22;

73 [label="EVAL with clause\np(0, 0).\nand substitution\nT6 -> 0\nX12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 73 [arrowhead = none ];
73 -> 23;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 74 [arrowhead = none ];
74 -> 24;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 75 [arrowhead = none ];
75 -> 25;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
23 -> 76 [arrowhead = none ];
76 -> 26;

77 [label="EVAL with clause\np(s(X36), X36).\nand substitution\nX36 -> T17\nT6 -> s(T17)\nX12 -> T17\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 77 [arrowhead = none ];
77 -> 50;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 78 [arrowhead = none ];
78 -> 51;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 79 [arrowhead = none ];
79 -> 27;

80 [label="EVAL with clause\np(s(X34), X34).\nand substitution\nX34 -> T15\nT6 -> s(T15)\nX12 -> T15\nT14 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 80 [arrowhead = none ];
80 -> 48;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 81 [arrowhead = none ];
81 -> 49;

82 [label="BACKTRACK\nfor clause: less(0, s(X2))because of non-unification", fontsize=14, style = filled, fillcolor = white];
27 -> 82 [arrowhead = none ];
82 -> 28;

83 [label="EVAL with clause\nless(X22, X23) :- ','(no(zero(X22)), ','(p(X22, X24), ','(p(X23, X25), less(X24, X25)))).\nand substitution\nT9 -> T10\nX22 -> T10\nX23 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 83 [arrowhead = none ];
83 -> 29;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 84 [arrowhead = none ];
84 -> 30;

85 [label="EVAL with clause\nno(X27) :- ','(X27, ','(!, failure(a))).\nand substitution\nT10 -> T11\nX27 -> zero(T11)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 85 [arrowhead = none ];
85 -> 31;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 86 [arrowhead = none ];
86 -> 32;

87 [label="EVAL with clause\nzero(0).\nand substitution\nT11 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 87 [arrowhead = none ];
87 -> 33;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 88 [arrowhead = none ];
88 -> 34;

89 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 89 [arrowhead = none ];
89 -> 35;

90 [label="EVAL with clause\nno(X29).\nand substitution\nT11 -> T12\nX29 -> zero(T12)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 90 [arrowhead = none ];
90 -> 38;

91 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 91 [arrowhead = none ];
91 -> 36;

92 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
36 -> 92 [arrowhead = none ];
92 -> 37;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 93 [arrowhead = none ];
93 -> 39;

94 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T12), zero(0))", fontsize=14, style = filled, fillcolor = white];
39 -> 94 [arrowhead = none ];
94 -> 40;

95 [label="EVAL with clause\np(s(X31), X31).\nand substitution\nX31 -> T13\nT12 -> s(T13)\nX24 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 95 [arrowhead = none ];
95 -> 41;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 96 [arrowhead = none ];
96 -> 42;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 97 [arrowhead = none ];
97 -> 43;

98 [label="EVAL with clause\np(0, 0).\nand substitution\nX25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 98 [arrowhead = none ];
98 -> 44;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 99 [arrowhead = none ];
99 -> 45;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 100 [arrowhead = none ];
100 -> 46;

101 [label="INSTANCE with matching:\nT9 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 101 [arrowhead = none , style=dashed];
101 -> 25[style=dashed];

102 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
46 -> 102 [arrowhead = none ];
102 -> 47;

103 [label="INSTANCE with matching:\nT1 -> T9\nT2 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 103 [arrowhead = none , style=dashed];
103 -> 1[style=dashed];

104 [label="INSTANCE with matching:\nT1 -> T9\nT2 -> T17", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 104 [arrowhead = none , style=dashed];
104 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

less25(s(T13)) :- less25(T13).
less1(0, s(T3)).
less1(s(s(T13)), 0) :- less25(T13).
less1(s(T9), s(T15)) :- less1(T9, T15).
less1(s(T9), s(T17)) :- less1(T9, T17).

Queries:

less1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
less1_in: (b,f)
less25_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

LESS1_IN_GA(s(s(T13)), 0) → U2_GA(T13, less25_in_g(T13))
LESS1_IN_GA(s(s(T13)), 0) → LESS25_IN_G(T13)
LESS25_IN_G(s(T13)) → U1_G(T13, less25_in_g(T13))
LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)
LESS1_IN_GA(s(T9), s(T15)) → U3_GA(T9, T15, less1_in_ga(T9, T15))
LESS1_IN_GA(s(T9), s(T15)) → LESS1_IN_GA(T9, T15)
LESS1_IN_GA(s(T9), s(T17)) → U4_GA(T9, T17, less1_in_ga(T9, T17))

The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS1_IN_GA(x1, x2)  =  LESS1_IN_GA(x1)
U2_GA(x1, x2)  =  U2_GA(x2)
LESS25_IN_G(x1)  =  LESS25_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U4_GA(x1, x2, x3)  =  U4_GA(x3)

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

(6) Obligation:

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

LESS1_IN_GA(s(s(T13)), 0) → U2_GA(T13, less25_in_g(T13))
LESS1_IN_GA(s(s(T13)), 0) → LESS25_IN_G(T13)
LESS25_IN_G(s(T13)) → U1_G(T13, less25_in_g(T13))
LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)
LESS1_IN_GA(s(T9), s(T15)) → U3_GA(T9, T15, less1_in_ga(T9, T15))
LESS1_IN_GA(s(T9), s(T15)) → LESS1_IN_GA(T9, T15)
LESS1_IN_GA(s(T9), s(T17)) → U4_GA(T9, T17, less1_in_ga(T9, T17))

The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS1_IN_GA(x1, x2)  =  LESS1_IN_GA(x1)
U2_GA(x1, x2)  =  U2_GA(x2)
LESS25_IN_G(x1)  =  LESS25_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GA(x1, x2, x3)  =  U3_GA(x3)
U4_GA(x1, x2, x3)  =  U4_GA(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)

The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS25_IN_G(x1)  =  LESS25_IN_G(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)

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

(14) 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:

• LESS25_IN_G(s(T13)) → LESS25_IN_G(T13)
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

LESS1_IN_GA(s(T9), s(T15)) → LESS1_IN_GA(T9, T15)

The TRS R consists of the following rules:

less1_in_ga(0, s(T3)) → less1_out_ga(0, s(T3))
less1_in_ga(s(s(T13)), 0) → U2_ga(T13, less25_in_g(T13))
less25_in_g(s(T13)) → U1_g(T13, less25_in_g(T13))
U1_g(T13, less25_out_g(T13)) → less25_out_g(s(T13))
U2_ga(T13, less25_out_g(T13)) → less1_out_ga(s(s(T13)), 0)
less1_in_ga(s(T9), s(T15)) → U3_ga(T9, T15, less1_in_ga(T9, T15))
less1_in_ga(s(T9), s(T17)) → U4_ga(T9, T17, less1_in_ga(T9, T17))
U4_ga(T9, T17, less1_out_ga(T9, T17)) → less1_out_ga(s(T9), s(T17))
U3_ga(T9, T15, less1_out_ga(T9, T15)) → less1_out_ga(s(T9), s(T15))

The argument filtering Pi contains the following mapping:
less1_in_ga(x1, x2)  =  less1_in_ga(x1)
0  =  0
less1_out_ga(x1, x2)  =  less1_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
less25_in_g(x1)  =  less25_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
less25_out_g(x1)  =  less25_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
U4_ga(x1, x2, x3)  =  U4_ga(x3)
LESS1_IN_GA(x1, x2)  =  LESS1_IN_GA(x1)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

LESS1_IN_GA(s(T9), s(T15)) → LESS1_IN_GA(T9, T15)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LESS1_IN_GA(x1, x2)  =  LESS1_IN_GA(x1)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

LESS1_IN_GA(s(T9)) → LESS1_IN_GA(T9)

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

(21) 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:

• LESS1_IN_GA(s(T9)) → LESS1_IN_GA(T9)
  The graph contains the following edges 1 > 1

(22) TRUE