Left Termination proof of /tmp/tmpQ0gcNU/minus1.pl

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

0 Prolog

↳1 BuiltinConflictTransformerProof (⇔, 0 ms)

↳2 Prolog

↳3 PrologToPrologProblemTransformerProof (⇒, 83 ms)

↳4 Prolog

↳5 PrologToPiTRSProof (⇒, 24 ms)

↳6 PiTRS

↳7 DependencyPairsProof (⇔, 0 ms)

↳8 PiDP

↳9 DependencyGraphProof (⇔, 0 ms)

↳10 PiDP

↳11 UsableRulesProof (⇔, 0 ms)

↳12 PiDP

↳13 PiDPToQDPProof (⇒, 0 ms)

↳14 QDP

↳15 QDPSizeChangeProof (⇔, 0 ms)

↳16 YES

(0) Obligation:

Clauses:

m(X, 0, Z) :- ','(!, =(Z, X)).
m(0, Y, Z) :- ','(!, =(Z, 0)).
m(X, Y, Z) :- ','(p(X, A), ','(p(Y, B), m(A, B, Z))).
p(0, 0).
p(s(X), X).
q(X, Y, Z) :- m(X, Y, Z).
q(X, Y, Z).
=(X, X).

Query: q(g,g,a)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

m(X, 0, Z) :- ','(!, user_defined_=(Z, X)).
m(0, Y, Z) :- ','(!, user_defined_=(Z, 0)).
m(X, Y, Z) :- ','(p(X, A), ','(p(Y, B), m(A, B, Z))).
p(0, 0).
p(s(X), X).
q(X, Y, Z) :- m(X, Y, Z).
q(X, Y, Z).
user_defined_=(X, X).

Query: q(g,g,a)

(3) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q(T1, T2, T3), SCOPE: 1, CLAUSE: 5 | q(T1, T2, T3), SCOPE: 1, CLAUSE: 6\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: m(T7, T8, T10) | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: m(T7, T8, T10), SCOPE: 2, CLAUSE: 0 | m(T7, T8, T10), SCOPE: 2, CLAUSE: 1 | m(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_2, user_defined_=(T17, T15)) | m(T15, 0, T10), SCOPE: 2, CLAUSE: 1 | m(T15, 0, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(T15, 0, T3), SCOPE: 1, CLAUSE: 6\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: m(T7, T8, T10), SCOPE: 2, CLAUSE: 1 | m(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground\nX11 is free\nX12 is free\nm(T7, T8, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: user_defined_=(T17, T15) | q(T15, 0, T3), SCOPE: 1, CLAUSE: 6\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: user_defined_=(T17, T15)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: q(T15, 0, T3), SCOPE: 1, CLAUSE: 6\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: user_defined_=(T17, T15), SCOPE: 3, CLAUSE: 7\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(!_2, user_defined_=(T33, 0)) | m(0, T31, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(0, T31, T3), SCOPE: 1, CLAUSE: 6\n\nT31 is ground\nX11 is free\nX12 is free\nm(0, T31, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: m(T7, T8, T10), SCOPE: 2, CLAUSE: 2 | ?_2 | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground\nX11 is free\nX12 is free\nX29 is free\nX30 is free\nm(T7, T8, T10) !=? m(X11, 0, X12)\nm(T7, T8, T10) !=? m(0, X29, X30)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: user_defined_=(T33, 0) | q(0, T31, T3), SCOPE: 1, CLAUSE: 6\n\nT31 is ground\nX11 is free\nX12 is free\nm(0, T31, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: user_defined_=(T33, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: q(0, T31, T3), SCOPE: 1, CLAUSE: 6\n\nT31 is ground\nX11 is free\nX12 is free\nm(0, T31, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: user_defined_=(T33, 0), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: m(T7, T8, T10), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nX11 is free\nX12 is free\nX29 is free\nX30 is free\nm(T7, T8, T10) !=? m(X11, 0, X12)\nm(T7, T8, T10) !=? m(0, X29, X30)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ?_2 | q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground\nX11 is free\nX12 is free\nX29 is free\nX30 is free\nm(T7, T8, T10) !=? m(X11, 0, X12)\nm(T7, T8, T10) !=? m(0, X29, X30)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(T59, X68), ','(p(T60, X69), m(X68, X69, T62)))\n\nT59 is ground\nT60 is ground\nX11 is free\nX12 is free\nX29 is free\nX30 is free\nX68 is free\nX69 is free\nm(T59, T60, T10) !=? m(X11, 0, X12)\nm(T59, T60, T10) !=? m(0, X29, X30)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(T59, X68), ','(p(T60, X69), m(X68, X69, T62))), SCOPE: 5, CLAUSE: 3 | ','(p(T59, X68), ','(p(T60, X69), m(X68, X69, T62))), SCOPE: 5, CLAUSE: 4\n\nT59 is ground\nT60 is ground\nX11 is free\nX12 is free\nX29 is free\nX30 is free\nX68 is free\nX69 is free\nm(T59, T60, T10) !=? m(X11, 0, X12)\nm(T59, T60, T10) !=? m(0, X29, X30)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(T59, X68), ','(p(T60, X69), m(X68, X69, T62))), SCOPE: 5, CLAUSE: 4\n\nT59 is ground\nT60 is ground\nX11 is free\nX12 is free\nX29 is free\nX30 is free\nX68 is free\nX69 is free\nm(T59, T60, T10) !=? m(X11, 0, X12)\nm(T59, T60, T10) !=? m(0, X29, X30)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(p(T60, X69), m(T66, X69, T62))\n\nT60 is ground\nT66 is ground\nX11 is free\nX12 is free\nX69 is free\nm(s(T66), T60, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T60, X69), m(T66, X69, T62)), SCOPE: 6, CLAUSE: 3 | ','(p(T60, X69), m(T66, X69, T62)), SCOPE: 6, CLAUSE: 4\n\nT60 is ground\nT66 is ground\nX11 is free\nX12 is free\nX69 is free\nm(s(T66), T60, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T60, X69), m(T66, X69, T62)), SCOPE: 6, CLAUSE: 4\n\nT60 is ground\nT66 is ground\nX11 is free\nX12 is free\nX69 is free\nm(s(T66), T60, T10) !=? m(X11, 0, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: m(T66, T71, T62)\n\nT66 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: m(T66, T71, T62), SCOPE: 7, CLAUSE: 0 | m(T66, T71, T62), SCOPE: 7, CLAUSE: 1 | m(T66, T71, T62), SCOPE: 7, CLAUSE: 2\n\nT66 is ground\nT71 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(!_7, user_defined_=(T80, T78)) | m(T78, 0, T62), SCOPE: 7, CLAUSE: 1 | m(T78, 0, T62), SCOPE: 7, CLAUSE: 2\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: m(T66, T71, T62), SCOPE: 7, CLAUSE: 1 | m(T66, T71, T62), SCOPE: 7, CLAUSE: 2\n\nT66 is ground\nT71 is ground\nX87 is free\nX88 is free\nm(T66, T71, T62) !=? m(X87, 0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: user_defined_=(T80, T78)\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: user_defined_=(T80, T78), SCOPE: 8, CLAUSE: 7\n\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\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: ','(!_7, user_defined_=(T90, 0)) | m(0, T88, T62), SCOPE: 7, CLAUSE: 2\n\nT88 is ground\nX87 is free\nX88 is free\nm(0, T88, T62) !=? m(X87, 0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: m(T66, T71, T62), SCOPE: 7, CLAUSE: 2\n\nT66 is ground\nT71 is ground\nX87 is free\nX88 is free\nX96 is free\nX97 is free\nm(T66, T71, T62) !=? m(X87, 0, X88)\nm(T66, T71, T62) !=? m(0, X96, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: user_defined_=(T90, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: user_defined_=(T90, 0), SCOPE: 9, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T102, X115), ','(p(T103, X116), m(X115, X116, T105)))\n\nT102 is ground\nT103 is ground\nX87 is free\nX88 is free\nX96 is free\nX97 is free\nX115 is free\nX116 is free\nm(T102, T103, T62) !=? m(X87, 0, X88)\nm(T102, T103, T62) !=? m(0, X96, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(T102, X115), ','(p(T103, X116), m(X115, X116, T105))), SCOPE: 10, CLAUSE: 3 | ','(p(T102, X115), ','(p(T103, X116), m(X115, X116, T105))), SCOPE: 10, CLAUSE: 4\n\nT102 is ground\nT103 is ground\nX87 is free\nX88 is free\nX96 is free\nX97 is free\nX115 is free\nX116 is free\nm(T102, T103, T62) !=? m(X87, 0, X88)\nm(T102, T103, T62) !=? m(0, X96, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(p(T102, X115), ','(p(T103, X116), m(X115, X116, T105))), SCOPE: 10, CLAUSE: 4\n\nT102 is ground\nT103 is ground\nX87 is free\nX88 is free\nX96 is free\nX97 is free\nX115 is free\nX116 is free\nm(T102, T103, T62) !=? m(X87, 0, X88)\nm(T102, T103, T62) !=? m(0, X96, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(p(T103, X116), m(T109, X116, T105))\n\nT103 is ground\nT109 is ground\nX87 is free\nX88 is free\nX116 is free\nm(s(T109), T103, T62) !=? m(X87, 0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(T103, X116), m(T109, X116, T105)), SCOPE: 11, CLAUSE: 3 | ','(p(T103, X116), m(T109, X116, T105)), SCOPE: 11, CLAUSE: 4\n\nT103 is ground\nT109 is ground\nX87 is free\nX88 is free\nX116 is free\nm(s(T109), T103, T62) !=? m(X87, 0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(p(T103, X116), m(T109, X116, T105)), SCOPE: 11, CLAUSE: 4\n\nT103 is ground\nT109 is ground\nX87 is free\nX88 is free\nX116 is free\nm(s(T109), T103, T62) !=? m(X87, 0, X88)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: m(T109, T114, T105)\n\nT109 is ground\nT114 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: q(T7, T8, T3), SCOPE: 1, CLAUSE: 6\n\nT7 is ground\nT8 is ground\nX11 is free\nX12 is free\nX29 is free\nX30 is free\nm(T7, T8, T10) !=? m(X11, 0, X12)\nm(T7, T8, T10) !=? m(0, X29, X30)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 65 [arrowhead = none ];
65 -> 2;

66 [label="ONLY EVAL with clause\nq(X4, X5, X6) :- m(X4, X5, X6).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T8,\nX5 -> T8,\nT3 -> T10,\nX6 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 66 [arrowhead = none ];
66 -> 3;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 67 [arrowhead = none ];
67 -> 4;

68 [label="EVAL with clause\nm(X11, 0, X12) :- ','(!_2, user_defined_=(X12, X11)).\nand substitutionT7 -> T15,\nX11 -> T15,\nT8 -> 0,\nT10 -> T17,\nX12 -> T17,\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 68 [arrowhead = none ];
68 -> 5;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 69 [arrowhead = none ];
69 -> 6;

70 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 70 [arrowhead = none ];
70 -> 7;

71 [label="EVAL with clause\nm(0, X29, X30) :- ','(!_2, user_defined_=(X30, 0)).\nand substitutionT7 -> 0,\nT8 -> T31,\nX29 -> T31,\nT10 -> T33,\nX30 -> T33,\nT32 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 71 [arrowhead = none ];
71 -> 16;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 72 [arrowhead = none ];
72 -> 17;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 73 [arrowhead = none ];
73 -> 8;

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 74 [arrowhead = none ];
74 -> 9;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 75 [arrowhead = none ];
75 -> 10;

76 [label="ONLY EVAL with clause\nq(X22, X23, X24).\nand substitutionT15 -> T25,\nX22 -> T25,\nX23 -> 0,\nT3 -> T26,\nX24 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 76 [arrowhead = none ];
76 -> 14;

77 [label="EVAL with clause\nuser_defined_=(X15, X15).\nand substitutionT17 -> T20,\nX15 -> T20,\nT15 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 77 [arrowhead = none ];
77 -> 11;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 78 [arrowhead = none ];
78 -> 12;

79 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 79 [arrowhead = none ];
79 -> 13;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 80 [arrowhead = none ];
80 -> 15;

81 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 81 [arrowhead = none ];
81 -> 18;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 82 [arrowhead = none ];
82 -> 27;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 83 [arrowhead = none ];
83 -> 28;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 84 [arrowhead = none ];
84 -> 19;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 85 [arrowhead = none ];
85 -> 20;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 86 [arrowhead = none ];
86 -> 21;

87 [label="ONLY EVAL with clause\nq(X40, X41, X42).\nand substitutionX40 -> 0,\nT31 -> T41,\nX41 -> T41,\nT3 -> T42,\nX42 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 87 [arrowhead = none ];
87 -> 25;

88 [label="EVAL with clause\nuser_defined_=(X33, X33).\nand substitutionT33 -> 0,\nX33 -> 0,\nT36 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 88 [arrowhead = none ];
88 -> 22;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 89 [arrowhead = none ];
89 -> 23;

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 90 [arrowhead = none ];
90 -> 24;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 91 [arrowhead = none ];
91 -> 26;

92 [label="ONLY EVAL with clause\nm(X65, X66, X67) :- ','(p(X65, X68), ','(p(X66, X69), m(X68, X69, X67))).\nand substitutionT7 -> T59,\nX65 -> T59,\nT8 -> T60,\nX66 -> T60,\nT10 -> T62,\nX67 -> T62,\nT61 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 92 [arrowhead = none ];
92 -> 29;

93 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 93 [arrowhead = none ];
93 -> 62;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 94 [arrowhead = none ];
94 -> 30;

95 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T59, T60, T10), m(0, X29, X30))", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 95 [arrowhead = none ];
95 -> 31;

96 [label="EVAL with clause\np(s(X75), X75).\nand substitutionX75 -> T66,\nT59 -> s(T66),\nX68 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 96 [arrowhead = none ];
96 -> 32;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 97 [arrowhead = none ];
97 -> 33;

98 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 98 [arrowhead = none ];
98 -> 34;

99 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(T66), T60, T10), m(X11, 0, X12))", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 99 [arrowhead = none ];
99 -> 35;

100 [label="EVAL with clause\np(s(X80), X80).\nand substitutionX80 -> T71,\nT60 -> s(T71),\nX69 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 100 [arrowhead = none ];
100 -> 36;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 101 [arrowhead = none ];
101 -> 37;

102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 102 [arrowhead = none ];
102 -> 38;

103 [label="EVAL with clause\nm(X87, 0, X88) :- ','(!_7, user_defined_=(X88, X87)).\nand substitutionT66 -> T78,\nX87 -> T78,\nT71 -> 0,\nT62 -> T80,\nX88 -> T80,\nT79 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 103 [arrowhead = none ];
103 -> 39;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 104 [arrowhead = none ];
104 -> 40;

105 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 105 [arrowhead = none ];
105 -> 41;

106 [label="EVAL with clause\nm(0, X96, X97) :- ','(!_7, user_defined_=(X97, 0)).\nand substitutionT66 -> 0,\nT71 -> T88,\nX96 -> T88,\nT62 -> T90,\nX97 -> T90,\nT89 -> T90", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 106 [arrowhead = none ];
106 -> 46;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 107 [arrowhead = none ];
107 -> 47;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 108 [arrowhead = none ];
108 -> 42;

109 [label="EVAL with clause\nuser_defined_=(X91, X91).\nand substitutionT80 -> T83,\nX91 -> T83,\nT78 -> T83", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 109 [arrowhead = none ];
109 -> 43;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 110 [arrowhead = none ];
110 -> 44;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 111 [arrowhead = none ];
111 -> 45;

112 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 112 [arrowhead = none ];
112 -> 48;

113 [label="ONLY EVAL with clause\nm(X112, X113, X114) :- ','(p(X112, X115), ','(p(X113, X116), m(X115, X116, X114))).\nand substitutionT66 -> T102,\nX112 -> T102,\nT71 -> T103,\nX113 -> T103,\nT62 -> T105,\nX114 -> T105,\nT104 -> T105", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 113 [arrowhead = none ];
113 -> 53;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 114 [arrowhead = none ];
114 -> 49;

115 [label="EVAL with clause\nuser_defined_=(X100, X100).\nand substitutionT90 -> 0,\nX100 -> 0,\nT93 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 115 [arrowhead = none ];
115 -> 50;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 116 [arrowhead = none ];
116 -> 51;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 117 [arrowhead = none ];
117 -> 52;

118 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 118 [arrowhead = none ];
118 -> 54;

119 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T102, T103, T62), m(0, X96, X97))", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 119 [arrowhead = none ];
119 -> 55;

120 [label="EVAL with clause\np(s(X122), X122).\nand substitutionX122 -> T109,\nT102 -> s(T109),\nX115 -> T109", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 120 [arrowhead = none ];
120 -> 56;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 121 [arrowhead = none ];
121 -> 57;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 122 [arrowhead = none ];
122 -> 58;

123 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(T109), T103, T62), m(X87, 0, X88))", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 123 [arrowhead = none ];
123 -> 59;

124 [label="EVAL with clause\np(s(X127), X127).\nand substitutionX127 -> T114,\nT103 -> s(T114),\nX116 -> T114", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 124 [arrowhead = none ];
124 -> 60;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 125 [arrowhead = none ];
125 -> 61;

126 [label="INSTANCE with matching:\nT66 -> T109\nT71 -> T114\nT62 -> T105", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 126 [arrowhead = none , style=dashed];
126 -> 36[style=dashed];

127 [label="ONLY EVAL with clause\nq(X136, X137, X138).\nand substitutionT7 -> T123,\nX136 -> T123,\nT8 -> T124,\nX137 -> T124,\nT3 -> T125,\nX138 -> T125", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 127 [arrowhead = none ];
127 -> 63;

128 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 128 [arrowhead = none ];
128 -> 64;

}

(4) Obligation:

Clauses:

mA(T83, 0, T83).
mA(0, T88, 0).
mA(s(T109), s(T114), T105) :- mA(T109, T114, T105).
qB(T20, 0, T20).
qB(T25, 0, T26).
qB(0, T31, 0).
qB(0, T41, T42).
qB(s(T66), s(T71), T62) :- mA(T66, T71, T62).
qB(T123, T124, T125).

Query: qB(g,g,a)

(5) PrologToPiTRSProof (SOUND transformation)

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

qB_in_gga(T20, 0, T20) → qB_out_gga(T20, 0, T20)
qB_in_gga(T25, 0, T26) → qB_out_gga(T25, 0, T26)
qB_in_gga(0, T31, 0) → qB_out_gga(0, T31, 0)
qB_in_gga(0, T41, T42) → qB_out_gga(0, T41, T42)
qB_in_gga(s(T66), s(T71), T62) → U2_gga(T66, T71, T62, mA_in_gga(T66, T71, T62))
mA_in_gga(T83, 0, T83) → mA_out_gga(T83, 0, T83)
mA_in_gga(0, T88, 0) → mA_out_gga(0, T88, 0)
mA_in_gga(s(T109), s(T114), T105) → U1_gga(T109, T114, T105, mA_in_gga(T109, T114, T105))
U1_gga(T109, T114, T105, mA_out_gga(T109, T114, T105)) → mA_out_gga(s(T109), s(T114), T105)
U2_gga(T66, T71, T62, mA_out_gga(T66, T71, T62)) → qB_out_gga(s(T66), s(T71), T62)
qB_in_gga(T123, T124, T125) → qB_out_gga(T123, T124, T125)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

qB_in_gga(T20, 0, T20) → qB_out_gga(T20, 0, T20)
qB_in_gga(T25, 0, T26) → qB_out_gga(T25, 0, T26)
qB_in_gga(0, T31, 0) → qB_out_gga(0, T31, 0)
qB_in_gga(0, T41, T42) → qB_out_gga(0, T41, T42)
qB_in_gga(s(T66), s(T71), T62) → U2_gga(T66, T71, T62, mA_in_gga(T66, T71, T62))
mA_in_gga(T83, 0, T83) → mA_out_gga(T83, 0, T83)
mA_in_gga(0, T88, 0) → mA_out_gga(0, T88, 0)
mA_in_gga(s(T109), s(T114), T105) → U1_gga(T109, T114, T105, mA_in_gga(T109, T114, T105))
U1_gga(T109, T114, T105, mA_out_gga(T109, T114, T105)) → mA_out_gga(s(T109), s(T114), T105)
U2_gga(T66, T71, T62, mA_out_gga(T66, T71, T62)) → qB_out_gga(s(T66), s(T71), T62)
qB_in_gga(T123, T124, T125) → qB_out_gga(T123, T124, T125)

(7) 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:

QB_IN_GGA(s(T66), s(T71), T62) → U2_GGA(T66, T71, T62, mA_in_gga(T66, T71, T62))
QB_IN_GGA(s(T66), s(T71), T62) → MA_IN_GGA(T66, T71, T62)
MA_IN_GGA(s(T109), s(T114), T105) → U1_GGA(T109, T114, T105, mA_in_gga(T109, T114, T105))
MA_IN_GGA(s(T109), s(T114), T105) → MA_IN_GGA(T109, T114, T105)

The TRS R consists of the following rules:

qB_in_gga(T20, 0, T20) → qB_out_gga(T20, 0, T20)
qB_in_gga(T25, 0, T26) → qB_out_gga(T25, 0, T26)
qB_in_gga(0, T31, 0) → qB_out_gga(0, T31, 0)
qB_in_gga(0, T41, T42) → qB_out_gga(0, T41, T42)
qB_in_gga(s(T66), s(T71), T62) → U2_gga(T66, T71, T62, mA_in_gga(T66, T71, T62))
mA_in_gga(T83, 0, T83) → mA_out_gga(T83, 0, T83)
mA_in_gga(0, T88, 0) → mA_out_gga(0, T88, 0)
mA_in_gga(s(T109), s(T114), T105) → U1_gga(T109, T114, T105, mA_in_gga(T109, T114, T105))
U1_gga(T109, T114, T105, mA_out_gga(T109, T114, T105)) → mA_out_gga(s(T109), s(T114), T105)
U2_gga(T66, T71, T62, mA_out_gga(T66, T71, T62)) → qB_out_gga(s(T66), s(T71), T62)
qB_in_gga(T123, T124, T125) → qB_out_gga(T123, T124, T125)

The argument filtering Pi contains the following mapping:
qB_in_gga(x1, x2, x3) = qB_in_gga(x1, x2)
0 = 0
qB_out_gga(x1, x2, x3) = qB_out_gga
s(x1) = s(x1)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
mA_in_gga(x1, x2, x3) = mA_in_gga(x1, x2)
mA_out_gga(x1, x2, x3) = mA_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
QB_IN_GGA(x1, x2, x3) = QB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x4)
MA_IN_GGA(x1, x2, x3) = MA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)

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

(8) Obligation:

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

QB_IN_GGA(s(T66), s(T71), T62) → U2_GGA(T66, T71, T62, mA_in_gga(T66, T71, T62))
QB_IN_GGA(s(T66), s(T71), T62) → MA_IN_GGA(T66, T71, T62)
MA_IN_GGA(s(T109), s(T114), T105) → U1_GGA(T109, T114, T105, mA_in_gga(T109, T114, T105))
MA_IN_GGA(s(T109), s(T114), T105) → MA_IN_GGA(T109, T114, T105)

The TRS R consists of the following rules:

qB_in_gga(T20, 0, T20) → qB_out_gga(T20, 0, T20)
qB_in_gga(T25, 0, T26) → qB_out_gga(T25, 0, T26)
qB_in_gga(0, T31, 0) → qB_out_gga(0, T31, 0)
qB_in_gga(0, T41, T42) → qB_out_gga(0, T41, T42)
qB_in_gga(s(T66), s(T71), T62) → U2_gga(T66, T71, T62, mA_in_gga(T66, T71, T62))
mA_in_gga(T83, 0, T83) → mA_out_gga(T83, 0, T83)
mA_in_gga(0, T88, 0) → mA_out_gga(0, T88, 0)
mA_in_gga(s(T109), s(T114), T105) → U1_gga(T109, T114, T105, mA_in_gga(T109, T114, T105))
U1_gga(T109, T114, T105, mA_out_gga(T109, T114, T105)) → mA_out_gga(s(T109), s(T114), T105)
U2_gga(T66, T71, T62, mA_out_gga(T66, T71, T62)) → qB_out_gga(s(T66), s(T71), T62)
qB_in_gga(T123, T124, T125) → qB_out_gga(T123, T124, T125)

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(10) Obligation:

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

MA_IN_GGA(s(T109), s(T114), T105) → MA_IN_GGA(T109, T114, T105)

The TRS R consists of the following rules:

qB_in_gga(T20, 0, T20) → qB_out_gga(T20, 0, T20)
qB_in_gga(T25, 0, T26) → qB_out_gga(T25, 0, T26)
qB_in_gga(0, T31, 0) → qB_out_gga(0, T31, 0)
qB_in_gga(0, T41, T42) → qB_out_gga(0, T41, T42)
qB_in_gga(s(T66), s(T71), T62) → U2_gga(T66, T71, T62, mA_in_gga(T66, T71, T62))
mA_in_gga(T83, 0, T83) → mA_out_gga(T83, 0, T83)
mA_in_gga(0, T88, 0) → mA_out_gga(0, T88, 0)
mA_in_gga(s(T109), s(T114), T105) → U1_gga(T109, T114, T105, mA_in_gga(T109, T114, T105))
U1_gga(T109, T114, T105, mA_out_gga(T109, T114, T105)) → mA_out_gga(s(T109), s(T114), T105)
U2_gga(T66, T71, T62, mA_out_gga(T66, T71, T62)) → qB_out_gga(s(T66), s(T71), T62)
qB_in_gga(T123, T124, T125) → qB_out_gga(T123, T124, T125)

The argument filtering Pi contains the following mapping:
qB_in_gga(x1, x2, x3) = qB_in_gga(x1, x2)
0 = 0
qB_out_gga(x1, x2, x3) = qB_out_gga
s(x1) = s(x1)
U2_gga(x1, x2, x3, x4) = U2_gga(x4)
mA_in_gga(x1, x2, x3) = mA_in_gga(x1, x2)
mA_out_gga(x1, x2, x3) = mA_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
MA_IN_GGA(x1, x2, x3) = MA_IN_GGA(x1, x2)

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

(11) UsableRulesProof (EQUIVALENT transformation)

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

(12) Obligation:

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

MA_IN_GGA(s(T109), s(T114), T105) → MA_IN_GGA(T109, T114, T105)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
MA_IN_GGA(x1, x2, x3) = MA_IN_GGA(x1, x2)

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

(13) PiDPToQDPProof (SOUND transformation)

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

(14) Obligation:

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

MA_IN_GGA(s(T109), s(T114)) → MA_IN_GGA(T109, T114)

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

(15) 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:

MA_IN_GGA(s(T109), s(T114)) → MA_IN_GGA(T109, T114)
The graph contains the following edges 1 > 1, 2 > 2

(0) Obligation:

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

(2) Obligation:

(3) PrologToPrologProblemTransformerProof (SOUND transformation)

(4) Obligation:

(5) PrologToPiTRSProof (SOUND transformation)

(6) Obligation:

(7) DependencyPairsProof (EQUIVALENT transformation)

(8) Obligation:

(9) DependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) UsableRulesProof (EQUIVALENT transformation)

(12) Obligation:

(13) PiDPToQDPProof (SOUND transformation)

(14) Obligation:

(15) QDPSizeChangeProof (EQUIVALENT transformation)

(16) YES