Left Termination proof of /tmp/tmp5y3Riz/minus3.pl

Left Termination of the query pattern q_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 PiDPToQDPProof (⇐)

        ↳9 QDP

        ↳10 QDPSizeChangeProof (⇔)

        ↳11 YES

    ↳12 PiDP

        ↳13 PiDPToQDPProof (⇐)

        ↳14 QDP

        ↳15 QDPSizeChangeProof (⇔)

        ↳16 YES

(0) Obligation:

Clauses:

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

Queries:

q(g,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: q(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: q(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: m(T5, T6, X5)\n\nT5 is ground\nT6 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: m(T5, T6, X5), SCOPE: 2, CLAUSE: 1 | m(T5, T6, X5), SCOPE: 2, CLAUSE: 2 | m(T5, T6, X5), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | m(T9, 0, X5), SCOPE: 2, CLAUSE: 2 | m(T9, 0, X5), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: m(T5, T6, X5), SCOPE: 2, CLAUSE: 2 | m(T5, T6, X5), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX5 is free; \nm(T5, T6, X5) !=? m(X8, 0, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: m(T9, 0, X5), SCOPE: 2, CLAUSE: 2 | m(T9, 0, X5), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: !_2 | m(0, 0, X5), SCOPE: 2, CLAUSE: 3\n\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: m(T9, 0, X5), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nX5 is free; \nm(T9, 0, X5) !=? m(0, X12, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(p(T12, X29), ','(p(0, X30), m(X29, X30, X31)))\n\nT12 is ground\nX31 is free; \nX29 is free; \nX30 is free; \nm(T12, 0, X5) !=? m(0, X12, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T12, X29), ','(p(0, X30), m(X29, X30, X31))), SCOPE: 3, CLAUSE: 4 | ','(p(T12, X29), ','(p(0, X30), m(X29, X30, X31))), SCOPE: 3, CLAUSE: 5\n\nT12 is ground\nX31 is free; \nX29 is free; \nX30 is free; \nm(T12, 0, X5) !=? m(0, X12, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T12, X29), ','(p(0, X30), m(X29, X30, X31))), SCOPE: 3, CLAUSE: 5\n\nT12 is ground\nX31 is free; \nX29 is free; \nX30 is free; \nm(T12, 0, X5) !=? m(0, X12, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(0, X30), m(T15, X30, X31))\n\nT15 is ground\nX31 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(0, X30), m(T15, X30, X31)), SCOPE: 4, CLAUSE: 4 | ','(p(0, X30), m(T15, X30, X31)), SCOPE: 4, CLAUSE: 5\n\nT15 is ground\nX31 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(0, X30), m(T15, X30, X31)), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nX31 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(0, X30), m(T15, X30, X31)), SCOPE: 4, CLAUSE: 5\n\nT15 is ground\nX31 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: m(T15, 0, X31)\n\nT15 is ground\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: m(T15, 0, X31), SCOPE: 5, CLAUSE: 1 | m(T15, 0, X31), SCOPE: 5, CLAUSE: 2 | m(T15, 0, X31), SCOPE: 5, CLAUSE: 3\n\nT15 is ground\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true | m(T22, 0, X31), SCOPE: 5, CLAUSE: 2 | m(T22, 0, X31), SCOPE: 5, CLAUSE: 3\n\nT22 is ground\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: m(T22, 0, X31), SCOPE: 5, CLAUSE: 2 | m(T22, 0, X31), SCOPE: 5, CLAUSE: 3\n\nT22 is ground\nX31 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: !_5 | m(0, 0, X31), SCOPE: 5, CLAUSE: 3\n\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: m(T22, 0, X31), SCOPE: 5, CLAUSE: 3\n\nT22 is ground\nX31 is free; \nm(T22, 0, X31) !=? m(0, X47, 0)", 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: ','(p(T25, X64), ','(p(0, X65), m(X64, X65, X66)))\n\nT25 is ground\nX66 is free; \nX64 is free; \nX65 is free; \nm(T25, 0, X31) !=? m(0, X47, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(p(T25, X64), ','(p(0, X65), m(X64, X65, X66))), SCOPE: 6, CLAUSE: 4 | ','(p(T25, X64), ','(p(0, X65), m(X64, X65, X66))), SCOPE: 6, CLAUSE: 5\n\nT25 is ground\nX66 is free; \nX64 is free; \nX65 is free; \nm(T25, 0, X31) !=? m(0, X47, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(p(T25, X64), ','(p(0, X65), m(X64, X65, X66))), SCOPE: 6, CLAUSE: 5\n\nT25 is ground\nX66 is free; \nX64 is free; \nX65 is free; \nm(T25, 0, X31) !=? m(0, X47, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(p(0, X65), m(T28, X65, X66))\n\nT28 is ground\nX66 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(p(0, X65), m(T28, X65, X66)), SCOPE: 7, CLAUSE: 4 | ','(p(0, X65), m(T28, X65, X66)), SCOPE: 7, CLAUSE: 5\n\nT28 is ground\nX66 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(0, X65), m(T28, X65, X66)), SCOPE: 7, CLAUSE: 4\n\nT28 is ground\nX66 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(0, X65), m(T28, X65, X66)), SCOPE: 7, CLAUSE: 5\n\nT28 is ground\nX66 is free; \nX65 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: m(T28, 0, X66)\n\nT28 is ground\nX66 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: !_2 | m(0, T35, X5), SCOPE: 2, CLAUSE: 3\n\nT35 is ground\nX5 is free; \nm(0, T35, X5) !=? m(X8, 0, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: m(T5, T6, X5), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX5 is free; \nm(T5, T6, X5) !=? m(X8, 0, X8)\nm(T5, T6, X5) !=? m(0, X81, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(p(T40, X98), ','(p(T41, X99), m(X98, X99, X100)))\n\nT40 is ground\nT41 is ground\nX100 is free; \nX98 is free; \nX99 is free; \nm(T40, T41, X5) !=? m(X8, 0, X8)\nm(T40, T41, X5) !=? m(0, X81, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(p(T40, X98), ','(p(T41, X99), m(X98, X99, X100))), SCOPE: 8, CLAUSE: 4 | ','(p(T40, X98), ','(p(T41, X99), m(X98, X99, X100))), SCOPE: 8, CLAUSE: 5\n\nT40 is ground\nT41 is ground\nX100 is free; \nX98 is free; \nX99 is free; \nm(T40, T41, X5) !=? m(X8, 0, X8)\nm(T40, T41, X5) !=? m(0, X81, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(p(T40, X98), ','(p(T41, X99), m(X98, X99, X100))), SCOPE: 8, CLAUSE: 5\n\nT40 is ground\nT41 is ground\nX100 is free; \nX98 is free; \nX99 is free; \nm(T40, T41, X5) !=? m(X8, 0, X8)\nm(T40, T41, X5) !=? m(0, X81, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(p(T41, X99), m(T44, X99, X100))\n\nT41 is ground\nT44 is ground\nX100 is free; \nX99 is free; \nm(s(T44), T41, X5) !=? m(X8, 0, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(p(T41, X99), m(T44, X99, X100)), SCOPE: 9, CLAUSE: 4 | ','(p(T41, X99), m(T44, X99, X100)), SCOPE: 9, CLAUSE: 5\n\nT41 is ground\nT44 is ground\nX100 is free; \nX99 is free; \nm(s(T44), T41, X5) !=? m(X8, 0, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(p(T41, X99), m(T44, X99, X100)), SCOPE: 9, CLAUSE: 5\n\nT41 is ground\nT44 is ground\nX100 is free; \nX99 is free; \nm(s(T44), T41, X5) !=? m(X8, 0, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: m(T44, T48, X100)\n\nT44 is ground\nT48 is ground\nX100 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 2;

53 [label="ONLY EVAL with clause\nq(X3, X4) :- m(X3, X4, X5).\nand substitution\nT1 -> T5\nX3 -> T5\nT2 -> T6\nX4 -> T6", fontsize=14, style = filled, fillcolor = white];
2 -> 53 [arrowhead = none ];
53 -> 3;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="EVAL with clause\nm(X8, 0, X8).\nand substitution\nT5 -> T9\nX8 -> T9\nT6 -> 0\nX5 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 55 [arrowhead = none ];
55 -> 5;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 56 [arrowhead = none ];
56 -> 6;

57 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 57 [arrowhead = none ];
57 -> 7;

58 [label="EVAL with clause\nm(0, X81, 0) :- !_2.\nand substitution\nT5 -> 0\nT6 -> T35\nX81 -> T35\nX5 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 58 [arrowhead = none ];
58 -> 39;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 59 [arrowhead = none ];
59 -> 40;

60 [label="EVAL with clause\nm(0, X12, 0) :- !_2.\nand substitution\nT9 -> 0\nX12 -> 0\nX5 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 60 [arrowhead = none ];
60 -> 8;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 61 [arrowhead = none ];
61 -> 9;

62 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 62 [arrowhead = none ];
62 -> 10;

63 [label="ONLY EVAL with clause\nm(X26, X27, X28) :- ','(p(X26, X29), ','(p(X27, X30), m(X29, X30, X28))).\nand substitution\nT9 -> T12\nX26 -> T12\nX27 -> 0\nX5 -> X31\nX28 -> X31", fontsize=14, style = filled, fillcolor = white];
9 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 64 [arrowhead = none ];
64 -> 11;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 65 [arrowhead = none ];
65 -> 13;

66 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T12, 0, X5), m(0, X12, 0))", fontsize=14, style = filled, fillcolor = white];
13 -> 66 [arrowhead = none ];
66 -> 14;

67 [label="EVAL with clause\np(s(X36), X36).\nand substitution\nX36 -> T15\nT12 -> s(T15)\nX29 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 67 [arrowhead = none ];
67 -> 15;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 68 [arrowhead = none ];
68 -> 16;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 69 [arrowhead = none ];
69 -> 17;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 70 [arrowhead = none ];
70 -> 18;

71 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 71 [arrowhead = none ];
71 -> 19;

72 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX30 -> 0", fontsize=14, style = filled, fillcolor = white];
18 -> 72 [arrowhead = none ];
72 -> 20;

73 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
19 -> 73 [arrowhead = none ];
73 -> 38;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 74 [arrowhead = none ];
74 -> 21;

75 [label="ONLY EVAL with clause\nm(X44, 0, X44).\nand substitution\nT15 -> T22\nX44 -> T22\nX31 -> T22", fontsize=14, style = filled, fillcolor = white];
21 -> 75 [arrowhead = none ];
75 -> 22;

76 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 76 [arrowhead = none ];
76 -> 23;

77 [label="EVAL with clause\nm(0, X47, 0) :- !_5.\nand substitution\nT22 -> 0\nX47 -> 0\nX31 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 77 [arrowhead = none ];
77 -> 24;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 78 [arrowhead = none ];
78 -> 25;

79 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="ONLY EVAL with clause\nm(X61, X62, X63) :- ','(p(X61, X64), ','(p(X62, X65), m(X64, X65, X63))).\nand substitution\nT22 -> T25\nX61 -> T25\nX62 -> 0\nX31 -> X66\nX63 -> X66", fontsize=14, style = filled, fillcolor = white];
25 -> 80 [arrowhead = none ];
80 -> 28;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 81 [arrowhead = none ];
81 -> 27;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 82 [arrowhead = none ];
82 -> 29;

83 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T25, 0, X31), m(0, X47, 0))", fontsize=14, style = filled, fillcolor = white];
29 -> 83 [arrowhead = none ];
83 -> 30;

84 [label="EVAL with clause\np(s(X71), X71).\nand substitution\nX71 -> T28\nT25 -> s(T28)\nX64 -> T28", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 84 [arrowhead = none ];
84 -> 31;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 85 [arrowhead = none ];
85 -> 32;

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 86 [arrowhead = none ];
86 -> 33;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 87 [arrowhead = none ];
87 -> 34;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 88 [arrowhead = none ];
88 -> 35;

89 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX65 -> 0", fontsize=14, style = filled, fillcolor = white];
34 -> 89 [arrowhead = none ];
89 -> 36;

90 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
35 -> 90 [arrowhead = none ];
90 -> 37;

91 [label="INSTANCE with matching:\nT15 -> T28\nX31 -> X66", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 91 [arrowhead = none , style=dashed];
91 -> 20[style=dashed];

92 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 92 [arrowhead = none ];
92 -> 41;

93 [label="ONLY EVAL with clause\nm(X95, X96, X97) :- ','(p(X95, X98), ','(p(X96, X99), m(X98, X99, X97))).\nand substitution\nT5 -> T40\nX95 -> T40\nT6 -> T41\nX96 -> T41\nX5 -> X100\nX97 -> X100", fontsize=14, style = filled, fillcolor = white];
40 -> 93 [arrowhead = none ];
93 -> 43;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 94 [arrowhead = none ];
94 -> 42;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 95 [arrowhead = none ];
95 -> 44;

96 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T40, T41, X5), m(0, X81, 0))", fontsize=14, style = filled, fillcolor = white];
44 -> 96 [arrowhead = none ];
96 -> 45;

97 [label="EVAL with clause\np(s(X105), X105).\nand substitution\nX105 -> T44\nT40 -> s(T44)\nX98 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 97 [arrowhead = none ];
97 -> 46;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 98 [arrowhead = none ];
98 -> 47;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 99 [arrowhead = none ];
99 -> 48;

100 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(T44), T41, X5), m(X8, 0, X8))", fontsize=14, style = filled, fillcolor = white];
48 -> 100 [arrowhead = none ];
100 -> 49;

101 [label="EVAL with clause\np(s(X110), X110).\nand substitution\nX110 -> T48\nT41 -> s(T48)\nX99 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 101 [arrowhead = none ];
101 -> 50;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 102 [arrowhead = none ];
102 -> 51;

103 [label="INSTANCE with matching:\nT5 -> T44\nT6 -> T48\nX5 -> X100", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 103 [arrowhead = none , style=dashed];
103 -> 3[style=dashed];

}

(2) Obligation:

Triples:

m20(s(T28), X66) :- m20(T28, X66).
m3(s(T15), 0, X31) :- m20(T15, X31).
m3(s(T44), s(T48), X100) :- m3(T44, T48, X100).
q1(T5, T6) :- m3(T5, T6, X5).

Clauses:

mc20(T22, T22).
mc20(0, 0).
mc20(s(T28), X66) :- mc20(T28, X66).
mc3(T9, 0, T9).
mc3(0, 0, 0).
mc3(s(T15), 0, X31) :- mc20(T15, X31).
mc3(0, T35, 0).
mc3(s(T44), s(T48), X100) :- mc3(T44, T48, X100).

Afs:

q1(x1, x2) = q1(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
q1_in: (b,b)
m3_in: (b,b,f)
m20_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

Q1_IN_GG(T5, T6) → U4_GG(T5, T6, m3_in_gga(T5, T6, X5))
Q1_IN_GG(T5, T6) → M3_IN_GGA(T5, T6, X5)
M3_IN_GGA(s(T15), 0, X31) → U2_GGA(T15, X31, m20_in_ga(T15, X31))
M3_IN_GGA(s(T15), 0, X31) → M20_IN_GA(T15, X31)
M20_IN_GA(s(T28), X66) → U1_GA(T28, X66, m20_in_ga(T28, X66))
M20_IN_GA(s(T28), X66) → M20_IN_GA(T28, X66)
M3_IN_GGA(s(T44), s(T48), X100) → U3_GGA(T44, T48, X100, m3_in_gga(T44, T48, X100))
M3_IN_GGA(s(T44), s(T48), X100) → M3_IN_GGA(T44, T48, X100)

R is empty.
The argument filtering Pi contains the following mapping:
m3_in_gga(x1, x2, x3) = m3_in_gga(x1, x2)
s(x1) = s(x1)
0 = 0
m20_in_ga(x1, x2) = m20_in_ga(x1)
Q1_IN_GG(x1, x2) = Q1_IN_GG(x1, x2)
U4_GG(x1, x2, x3) = U4_GG(x1, x2, x3)
M3_IN_GGA(x1, x2, x3) = M3_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3) = U2_GGA(x1, x3)
M20_IN_GA(x1, x2) = M20_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)

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:

Q1_IN_GG(T5, T6) → U4_GG(T5, T6, m3_in_gga(T5, T6, X5))
Q1_IN_GG(T5, T6) → M3_IN_GGA(T5, T6, X5)
M3_IN_GGA(s(T15), 0, X31) → U2_GGA(T15, X31, m20_in_ga(T15, X31))
M3_IN_GGA(s(T15), 0, X31) → M20_IN_GA(T15, X31)
M20_IN_GA(s(T28), X66) → U1_GA(T28, X66, m20_in_ga(T28, X66))
M20_IN_GA(s(T28), X66) → M20_IN_GA(T28, X66)
M3_IN_GGA(s(T44), s(T48), X100) → U3_GGA(T44, T48, X100, m3_in_gga(T44, T48, X100))
M3_IN_GGA(s(T44), s(T48), X100) → M3_IN_GGA(T44, T48, X100)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

M20_IN_GA(s(T28), X66) → M20_IN_GA(T28, X66)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
M20_IN_GA(x1, x2) = M20_IN_GA(x1)

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

(8) PiDPToQDPProof (SOUND transformation)

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

(9) Obligation:

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

M20_IN_GA(s(T28)) → M20_IN_GA(T28)

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

(10) 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:

M20_IN_GA(s(T28)) → M20_IN_GA(T28)
The graph contains the following edges 1 > 1

(11) YES

(12) Obligation:

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

M3_IN_GGA(s(T44), s(T48), X100) → M3_IN_GGA(T44, T48, X100)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
M3_IN_GGA(x1, x2, x3) = M3_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:

M3_IN_GGA(s(T44), s(T48)) → M3_IN_GGA(T44, T48)

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:

M3_IN_GGA(s(T44), s(T48)) → M3_IN_GGA(T44, T48)
The graph contains the following edges 1 > 1, 2 > 2

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) PiDPToQDPProof (SOUND transformation)

(9) Obligation:

(10) QDPSizeChangeProof (EQUIVALENT transformation)

(11) YES

(12) Obligation:

(13) PiDPToQDPProof (SOUND transformation)

(14) Obligation:

(15) QDPSizeChangeProof (EQUIVALENT transformation)

(16) YES