Left Termination of the query pattern
less(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).

Query: less(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

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: true | less(0, T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX5 is free\nless(T1, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: less(0, T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11))))\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 5 | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(','(call(zero(0)), ','(!_2, failure(a))), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))) | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11))))) | ?_3 | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(zero(0), ','(','(!_2, failure(a)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11))))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(','(!_2, failure(a)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))) | ?_4 | ?_3 | ','(no(zero(0)), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 2, CLAUSE: 6\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(failure(a), ','(p(0, X10), ','(p(T7, X11), less(X10, X11))))\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(failure(a), ','(p(0, X10), ','(p(T7, X11), less(X10, X11)))), SCOPE: 5, CLAUSE: 7\n\nX10 is free\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(no(zero(T10)), ','(p(T10, X21), ','(p(T12, X22), less(X21, X22))))\n\nT10 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T10, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(zero(T10)), ','(p(T10, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 5 | ','(no(zero(T10)), ','(p(T10, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT10 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T10, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(','(call(zero(T15)), ','(!_6, failure(a))), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))) | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(zero(T15), ','(','(!_6, failure(a)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22))))) | ?_7 | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(zero(T15), ','(','(!_6, failure(a)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22))))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(','(!_6, failure(a)), ','(p(0, X21), ','(p(T12, X22), less(X21, X22)))) | ?_8 | ?_7 | ','(no(zero(0)), ','(p(0, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nX21 is free\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_8 | ?_7 | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))\nzero(T15) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(failure(a), ','(p(0, X21), ','(p(T12, X22), less(X21, X22))))\n\nX21 is free\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(failure(a), ','(p(0, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 9, CLAUSE: 7\n\nX21 is free\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ?_7 | ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))\nzero(T15) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(no(zero(T15)), ','(p(T15, X21), ','(p(T12, X22), less(X21, X22)))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T15, T2) !=? less(0, s(X5))\nzero(T15) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(p(T18, X21), ','(p(T12, X22), less(X21, X22)))\n\nT18 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T18, T2) !=? less(0, s(X5))\nzero(T18) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(p(T18, X21), ','(p(T12, X22), less(X21, X22))), SCOPE: 10, CLAUSE: 2 | ','(p(T18, X21), ','(p(T12, X22), less(X21, X22))), SCOPE: 10, CLAUSE: 3\n\nT18 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T18, T2) !=? less(0, s(X5))\nzero(T18) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(T18, X21), ','(p(T12, X22), less(X21, X22))), SCOPE: 10, CLAUSE: 3\n\nT18 is ground\nX5 is free\nX21 is free\nX22 is free\nless(T18, T2) !=? less(0, s(X5))\nzero(T18) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(T12, X22), less(T21, X22))\n\nT21 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(p(T12, X22), less(T21, X22)), SCOPE: 11, CLAUSE: 2 | ','(p(T12, X22), less(T21, X22)), SCOPE: 11, CLAUSE: 3\n\nT21 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(p(T12, X22), less(T21, X22)), SCOPE: 11, CLAUSE: 2\n\nT21 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T12, X22), less(T21, X22)), SCOPE: 11, CLAUSE: 3\n\nT21 is ground\nX22 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: less(T21, 0)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: less(T21, 0), SCOPE: 12, CLAUSE: 0 | less(T21, 0), SCOPE: 12, CLAUSE: 1\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: less(T21, 0), SCOPE: 12, CLAUSE: 1\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(no(zero(T24)), ','(p(T24, X55), ','(p(0, X56), less(X55, X56))))\n\nT24 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(no(zero(T24)), ','(p(T24, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 5 | ','(no(zero(T24)), ','(p(T24, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT24 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(','(call(zero(T27)), ','(!_13, failure(a))), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))) | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(zero(T27), ','(','(!_13, failure(a)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56))))) | ?_14 | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(zero(T27), ','(','(!_13, failure(a)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56))))), SCOPE: 15, CLAUSE: 4 | ?_15 | ?_14 | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(','(!_13, failure(a)), ','(p(0, X55), ','(p(0, X56), less(X55, X56)))) | ?_15 | ?_14 | ','(no(zero(0)), ','(p(0, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ?_15 | ?_14 | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free\nzero(T27) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(failure(a), ','(p(0, X55), ','(p(0, X56), less(X55, X56))))\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(failure(a), ','(p(0, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 16, CLAUSE: 7\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ?_14 | ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free\nzero(T27) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(no(zero(T27)), ','(p(T27, X55), ','(p(0, X56), less(X55, X56)))), SCOPE: 13, CLAUSE: 6\n\nT27 is ground\nX55 is free\nX56 is free\nzero(T27) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(p(T30, X55), ','(p(0, X56), less(X55, X56)))\n\nT30 is ground\nX55 is free\nX56 is free\nzero(T30) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T30, X55), ','(p(0, X56), less(X55, X56))), SCOPE: 17, CLAUSE: 2 | ','(p(T30, X55), ','(p(0, X56), less(X55, X56))), SCOPE: 17, CLAUSE: 3\n\nT30 is ground\nX55 is free\nX56 is free\nzero(T30) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T30, X55), ','(p(0, X56), less(X55, X56))), SCOPE: 17, CLAUSE: 3\n\nT30 is ground\nX55 is free\nX56 is free\nzero(T30) !=? zero(0)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(0, X56), less(T33, X56))\n\nT33 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(0, X56), less(T33, X56)), SCOPE: 18, CLAUSE: 2 | ','(p(0, X56), less(T33, X56)), SCOPE: 18, CLAUSE: 3\n\nT33 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(p(0, X56), less(T33, X56)), SCOPE: 18, CLAUSE: 2\n\nT33 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(0, X56), less(T33, X56)), SCOPE: 18, CLAUSE: 3\n\nT33 is ground\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: less(T33, 0)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(T21, T37)\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 63 [arrowhead = none ];
63 -> 2;

64 [label="EVAL with clause\nless(0, s(X5)).\nand substitutionT1 -> 0,\nX5 -> T4,\nT2 -> s(T4)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 64 [arrowhead = none ];
64 -> 3;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 65 [arrowhead = none ];
65 -> 4;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 66 [arrowhead = none ];
66 -> 5;

67 [label="ONLY EVAL with clause\nless(X19, X20) :- ','(no(zero(X19)), ','(p(X19, X21), ','(p(X20, X22), less(X21, X22)))).\nand substitutionT1 -> T10,\nX19 -> T10,\nT2 -> T12,\nX20 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 67 [arrowhead = none ];
67 -> 15;

68 [label="ONLY EVAL with clause\nless(X8, X9) :- ','(no(zero(X8)), ','(p(X8, X10), ','(p(X9, X11), less(X10, X11)))).\nand substitutionX8 -> 0,\nT2 -> T7,\nX9 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 68 [arrowhead = none ];
68 -> 6;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 69 [arrowhead = none ];
69 -> 7;

70 [label="ONLY EVAL with clause\nno(X14) :- ','(call(X14), ','(!_2, failure(a))).\nand substitutionX14 -> zero(0)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 70 [arrowhead = none ];
70 -> 8;

71 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 71 [arrowhead = none ];
71 -> 9;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 72 [arrowhead = none ];
72 -> 10;

73 [label="ONLY EVAL with clause\nzero(0).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 73 [arrowhead = none ];
73 -> 11;

74 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 74 [arrowhead = none ];
74 -> 12;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 75 [arrowhead = none ];
75 -> 13;

76 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 76 [arrowhead = none ];
76 -> 14;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 77 [arrowhead = none ];
77 -> 16;

78 [label="ONLY EVAL with clause\nno(X25) :- ','(call(X25), ','(!_6, failure(a))).\nand substitutionT10 -> T15,\nX25 -> zero(T15)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 78 [arrowhead = none ];
78 -> 17;

79 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 79 [arrowhead = none ];
79 -> 18;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 80 [arrowhead = none ];
80 -> 19;

81 [label="EVAL with clause\nzero(0).\nand substitutionT15 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 81 [arrowhead = none ];
81 -> 20;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 82 [arrowhead = none ];
82 -> 21;

83 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 83 [arrowhead = none ];
83 -> 22;

84 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 84 [arrowhead = none ];
84 -> 25;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 85 [arrowhead = none ];
85 -> 23;

86 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 86 [arrowhead = none ];
86 -> 24;

87 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 87 [arrowhead = none ];
87 -> 26;

88 [label="ONLY EVAL with clause\nno(X32).\nand substitutionT15 -> T18,\nX32 -> zero(T18)", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 88 [arrowhead = none ];
88 -> 27;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 89 [arrowhead = none ];
89 -> 28;

90 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T18), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 90 [arrowhead = none ];
90 -> 29;

91 [label="EVAL with clause\np(s(X37), X37).\nand substitutionX37 -> T21,\nT18 -> s(T21),\nX21 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 91 [arrowhead = none ];
91 -> 30;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 92 [arrowhead = none ];
92 -> 31;

93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 93 [arrowhead = none ];
93 -> 32;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 94 [arrowhead = none ];
94 -> 33;

95 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 95 [arrowhead = none ];
95 -> 34;

96 [label="EVAL with clause\np(0, 0).\nand substitutionT12 -> 0,\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 96 [arrowhead = none ];
96 -> 35;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 97 [arrowhead = none ];
97 -> 36;

98 [label="EVAL with clause\np(s(X87), X87).\nand substitutionX87 -> T37,\nT12 -> s(T37),\nX22 -> T37,\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 98 [arrowhead = none ];
98 -> 61;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 99 [arrowhead = none ];
99 -> 62;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 100 [arrowhead = none ];
100 -> 37;

101 [label="BACKTRACK\nfor clause: less(0, s(X2))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 101 [arrowhead = none ];
101 -> 38;

102 [label="ONLY EVAL with clause\nless(X53, X54) :- ','(no(zero(X53)), ','(p(X53, X55), ','(p(X54, X56), less(X55, X56)))).\nand substitutionT21 -> T24,\nX53 -> T24,\nX54 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 102 [arrowhead = none ];
102 -> 39;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 103 [arrowhead = none ];
103 -> 40;

104 [label="ONLY EVAL with clause\nno(X61) :- ','(call(X61), ','(!_13, failure(a))).\nand substitutionT24 -> T27,\nX61 -> zero(T27)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 104 [arrowhead = none ];
104 -> 41;

105 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 105 [arrowhead = none ];
105 -> 42;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 106 [arrowhead = none ];
106 -> 43;

107 [label="EVAL with clause\nzero(0).\nand substitutionT27 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 107 [arrowhead = none ];
107 -> 44;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 108 [arrowhead = none ];
108 -> 45;

109 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 109 [arrowhead = none ];
109 -> 46;

110 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 110 [arrowhead = none ];
110 -> 49;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 111 [arrowhead = none ];
111 -> 47;

112 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 112 [arrowhead = none ];
112 -> 48;

113 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 113 [arrowhead = none ];
113 -> 50;

114 [label="ONLY EVAL with clause\nno(X68).\nand substitutionT27 -> T30,\nX68 -> zero(T30)", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 114 [arrowhead = none ];
114 -> 51;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 115 [arrowhead = none ];
115 -> 52;

116 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (zero(T30), zero(0))", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 116 [arrowhead = none ];
116 -> 53;

117 [label="EVAL with clause\np(s(X73), X73).\nand substitutionX73 -> T33,\nT30 -> s(T33),\nX55 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 117 [arrowhead = none ];
117 -> 54;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 118 [arrowhead = none ];
118 -> 55;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 119 [arrowhead = none ];
119 -> 56;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 120 [arrowhead = none ];
120 -> 57;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 121 [arrowhead = none ];
121 -> 58;

122 [label="ONLY EVAL with clause\np(0, 0).\nand substitutionX56 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 122 [arrowhead = none ];
122 -> 59;

123 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 123 [arrowhead = none ];
123 -> 60;

124 [label="INSTANCE with matching:\nT21 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 124 [arrowhead = none , style=dashed];
124 -> 35[style=dashed];

125 [label="INSTANCE with matching:\nT1 -> T21\nT2 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
61 -> 125 [arrowhead = none , style=dashed];
125 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

lessA(s(T33)) :- lessA(T33).
lessB(0, s(T4)).
lessB(s(T21), 0) :- lessA(T21).
lessB(s(T21), s(T37)) :- lessB(T21, T37).

Query: lessB(g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
lessB_in: (b,f)
lessA_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_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:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_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:

LESSB_IN_GA(s(T21), 0) → U2_GA(T21, lessA_in_g(T21))
LESSB_IN_GA(s(T21), 0) → LESSA_IN_G(T21)
LESSA_IN_G(s(T33)) → U1_G(T33, lessA_in_g(T33))
LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)
LESSB_IN_GA(s(T21), s(T37)) → U3_GA(T21, T37, lessB_in_ga(T21, T37))
LESSB_IN_GA(s(T21), s(T37)) → LESSB_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESSB_IN_GA(x1, x2)  =  LESSB_IN_GA(x1)
U2_GA(x1, x2)  =  U2_GA(x2)
LESSA_IN_G(x1)  =  LESSA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GA(x1, x2, x3)  =  U3_GA(x3)

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

(6) Obligation:

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

LESSB_IN_GA(s(T21), 0) → U2_GA(T21, lessA_in_g(T21))
LESSB_IN_GA(s(T21), 0) → LESSA_IN_G(T21)
LESSA_IN_G(s(T33)) → U1_G(T33, lessA_in_g(T33))
LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)
LESSB_IN_GA(s(T21), s(T37)) → U3_GA(T21, T37, lessB_in_ga(T21, T37))
LESSB_IN_GA(s(T21), s(T37)) → LESSB_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESSB_IN_GA(x1, x2)  =  LESSB_IN_GA(x1)
U2_GA(x1, x2)  =  U2_GA(x2)
LESSA_IN_G(x1)  =  LESSA_IN_G(x1)
U1_G(x1, x2)  =  U1_G(x2)
U3_GA(x1, x2, x3)  =  U3_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 4 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESSA_IN_G(x1)  =  LESSA_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:

LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)

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:

LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)

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:

• LESSA_IN_G(s(T33)) → LESSA_IN_G(T33)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

LESSB_IN_GA(s(T21), s(T37)) → LESSB_IN_GA(T21, T37)

The TRS R consists of the following rules:

lessB_in_ga(0, s(T4)) → lessB_out_ga(0, s(T4))
lessB_in_ga(s(T21), 0) → U2_ga(T21, lessA_in_g(T21))
lessA_in_g(s(T33)) → U1_g(T33, lessA_in_g(T33))
U1_g(T33, lessA_out_g(T33)) → lessA_out_g(s(T33))
U2_ga(T21, lessA_out_g(T21)) → lessB_out_ga(s(T21), 0)
lessB_in_ga(s(T21), s(T37)) → U3_ga(T21, T37, lessB_in_ga(T21, T37))
U3_ga(T21, T37, lessB_out_ga(T21, T37)) → lessB_out_ga(s(T21), s(T37))

The argument filtering Pi contains the following mapping:
lessB_in_ga(x1, x2)  =  lessB_in_ga(x1)
0  =  0
lessB_out_ga(x1, x2)  =  lessB_out_ga
s(x1)  =  s(x1)
U2_ga(x1, x2)  =  U2_ga(x2)
lessA_in_g(x1)  =  lessA_in_g(x1)
U1_g(x1, x2)  =  U1_g(x2)
lessA_out_g(x1)  =  lessA_out_g
U3_ga(x1, x2, x3)  =  U3_ga(x3)
LESSB_IN_GA(x1, x2)  =  LESSB_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:

LESSB_IN_GA(s(T21), s(T37)) → LESSB_IN_GA(T21, T37)

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

LESSB_IN_GA(s(T21)) → LESSB_IN_GA(T21)

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:

• LESSB_IN_GA(s(T21)) → LESSB_IN_GA(T21)
  The graph contains the following edges 1 > 1

(22) YES