Left Termination proof of /tmp/tmpF2puQc/minus2.pl

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

0 Prolog

↳1 BuiltinConflictTransformerProof (⇔, 0 ms)

↳2 Prolog

↳3 PrologToPrologProblemTransformerProof (⇒, 80 ms)

↳4 Prolog

↳5 PrologToPiTRSProof (⇒, 17 ms)

↳6 PiTRS

↳7 DependencyPairsProof (⇔, 156 ms)

↳8 PiDP

↳9 DependencyGraphProof (⇔, 0 ms)

↳10 AND

    ↳11 PiDP

        ↳12 UsableRulesProof (⇔, 0 ms)

        ↳13 PiDP

        ↳14 PiDPToQDPProof (⇔, 0 ms)

        ↳15 QDP

        ↳16 QDPSizeChangeProof (⇔, 0 ms)

        ↳17 YES

    ↳18 PiDP

        ↳19 UsableRulesProof (⇔, 0 ms)

        ↳20 PiDP

        ↳21 PiDPToQDPProof (⇒, 0 ms)

        ↳22 QDP

        ↳23 QDPSizeChangeProof (⇔, 0 ms)

        ↳24 YES

    ↳25 PiDP

        ↳26 UsableRulesProof (⇔, 0 ms)

        ↳27 PiDP

        ↳28 PiDPToQDPProof (⇒, 0 ms)

        ↳29 QDP

        ↳30 MRRProof (⇔, 1 ms)

        ↳31 QDP

        ↳32 PisEmptyProof (⇔, 0 ms)

        ↳33 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(0), 0).
p(s(s(X)), s(Y)) :- p(s(X), Y).
=(X, X).

Query: m(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(0), 0).
p(s(s(X)), s(Y)) :- p(s(X), Y).
user_defined_=(X, X).

Query: m(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: m(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: m(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, user_defined_=(T8, T6)) | m(T6, 0, T3), SCOPE: 1, CLAUSE: 1 | m(T6, 0, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: m(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | m(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nm(T1, T2, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: user_defined_=(T8, T6)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: user_defined_=(T8, T6), SCOPE: 2, CLAUSE: 6\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(!_1, user_defined_=(T16, 0)) | m(0, T14, T3), SCOPE: 1, CLAUSE: 2\n\nT14 is ground\nX3 is free\nX4 is free\nm(0, T14, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: m(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nm(T1, T2, T3) !=? m(X3, 0, X4)\nm(T1, T2, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: user_defined_=(T16, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: user_defined_=(T16, 0), SCOPE: 3, CLAUSE: 6\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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26)))\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 3 | ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 4 | ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 5\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 4 | ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 5\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 4\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(p(T23, X21), ','(p(T24, X22), m(X21, X22, T26))), SCOPE: 4, CLAUSE: 5\n\nT23 is ground\nT24 is ground\nX3 is free\nX4 is free\nX10 is free\nX11 is free\nX21 is free\nX22 is free\nm(T23, T24, T3) !=? m(X3, 0, X4)\nm(T23, T24, T3) !=? m(0, X10, X11)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(T24, X22), m(0, X22, T26))\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: p(T24, X22)\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: m(0, T27, T26)\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: p(T24, X22), SCOPE: 5, CLAUSE: 3 | p(T24, X22), SCOPE: 5, CLAUSE: 4 | p(T24, X22), SCOPE: 5, CLAUSE: 5\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: p(T24, X22), SCOPE: 5, CLAUSE: 4 | p(T24, X22), SCOPE: 5, CLAUSE: 5\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: p(T24, X22), SCOPE: 5, CLAUSE: 4\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: p(T24, X22), SCOPE: 5, CLAUSE: 5\n\nT24 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(0), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: p(s(T30), X31)\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: p(s(T30), X31), SCOPE: 6, CLAUSE: 3 | p(s(T30), X31), SCOPE: 6, CLAUSE: 4 | p(s(T30), X31), SCOPE: 6, CLAUSE: 5\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: p(s(T30), X31), SCOPE: 6, CLAUSE: 4 | p(s(T30), X31), SCOPE: 6, CLAUSE: 5\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: p(s(T30), X31), SCOPE: 6, CLAUSE: 4\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: p(s(T30), X31), SCOPE: 6, CLAUSE: 5\n\nT30 is ground\nX31 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: p(s(T33), X40)\n\nT33 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: m(0, T27, T26), SCOPE: 7, CLAUSE: 0 | m(0, T27, T26), SCOPE: 7, CLAUSE: 1 | m(0, T27, T26), SCOPE: 7, CLAUSE: 2\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(!_7, user_defined_=(T38, 0)) | m(0, 0, T26), SCOPE: 7, CLAUSE: 1 | m(0, 0, T26), SCOPE: 7, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: m(0, T27, T26), SCOPE: 7, CLAUSE: 1 | m(0, T27, T26), SCOPE: 7, CLAUSE: 2\n\nT27 is ground\nX47 is free\nX48 is free\nm(0, T27, T26) !=? m(X47, 0, X48)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: user_defined_=(T38, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: user_defined_=(T38, 0), SCOPE: 8, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(!_7, user_defined_=(T49, 0)) | m(0, T47, T26), SCOPE: 7, CLAUSE: 2\n\nT47 is ground\nX47 is free\nX48 is free\nm(0, T47, T26) !=? m(X47, 0, X48)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: user_defined_=(T49, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: user_defined_=(T49, 0), SCOPE: 9, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(p(s(T55), X69), ','(p(T24, X22), m(s(X69), X22, T26)))\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nX69 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: p(s(T55), X69)\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX69 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(p(T24, X22), m(s(T56), X22, T26))\n\nT24 is ground\nT55 is ground\nT56 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: p(T24, X22)\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: m(s(T56), T57, T26)\n\nT56 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: p(T24, X22), SCOPE: 10, CLAUSE: 3 | p(T24, X22), SCOPE: 10, CLAUSE: 4 | p(T24, X22), SCOPE: 10, CLAUSE: 5\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: p(T24, X22), SCOPE: 10, CLAUSE: 4 | p(T24, X22), SCOPE: 10, CLAUSE: 5\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: p(T24, X22), SCOPE: 10, CLAUSE: 4\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: p(T24, X22), SCOPE: 10, CLAUSE: 5\n\nT24 is ground\nT55 is ground\nX3 is free\nX4 is free\nX22 is free\nm(s(s(T55)), T24, T3) !=? m(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: p(s(T60), X78)\n\nT60 is ground\nX78 is free", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 73 [arrowhead = none ];
73 -> 2;

74 [label="EVAL with clause\nm(X3, 0, X4) :- ','(!_1, user_defined_=(X4, X3)).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T8,\nX4 -> T8,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 74 [arrowhead = none ];
74 -> 3;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 75 [arrowhead = none ];
75 -> 4;

76 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 76 [arrowhead = none ];
76 -> 5;

77 [label="EVAL with clause\nm(0, X10, X11) :- ','(!_1, user_defined_=(X11, 0)).\nand substitutionT1 -> 0,\nT2 -> T14,\nX10 -> T14,\nT3 -> T16,\nX11 -> T16,\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 77 [arrowhead = none ];
77 -> 10;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 78 [arrowhead = none ];
78 -> 11;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 79 [arrowhead = none ];
79 -> 6;

80 [label="EVAL with clause\nuser_defined_=(X7, X7).\nand substitutionT8 -> T11,\nX7 -> T11,\nT6 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 80 [arrowhead = none ];
80 -> 7;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 81 [arrowhead = none ];
81 -> 8;

82 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 82 [arrowhead = none ];
82 -> 9;

83 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 83 [arrowhead = none ];
83 -> 12;

84 [label="ONLY EVAL with clause\nm(X18, X19, X20) :- ','(p(X18, X21), ','(p(X19, X22), m(X21, X22, X20))).\nand substitutionT1 -> T23,\nX18 -> T23,\nT2 -> T24,\nX19 -> T24,\nT3 -> T26,\nX20 -> T26,\nT25 -> T26", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 84 [arrowhead = none ];
84 -> 17;

85 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 85 [arrowhead = none ];
85 -> 13;

86 [label="EVAL with clause\nuser_defined_=(X14, X14).\nand substitutionT16 -> 0,\nX14 -> 0,\nT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 86 [arrowhead = none ];
86 -> 14;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 87 [arrowhead = none ];
87 -> 15;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 88 [arrowhead = none ];
88 -> 16;

89 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 89 [arrowhead = none ];
89 -> 18;

90 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(T23, T24, T3), m(0, X10, X11))", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 90 [arrowhead = none ];
90 -> 19;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 91 [arrowhead = none ];
91 -> 20;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 92 [arrowhead = none ];
92 -> 21;

93 [label="EVAL with clause\np(s(0), 0).\nand substitutionT23 -> s(0),\nX21 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 93 [arrowhead = none ];
93 -> 22;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 94 [arrowhead = none ];
94 -> 23;

95 [label="EVAL with clause\np(s(s(X67)), s(X68)) :- p(s(X67), X68).\nand substitutionX67 -> T55,\nT23 -> s(s(T55)),\nX68 -> X69,\nX21 -> s(X69)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 95 [arrowhead = none ];
95 -> 58;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 96 [arrowhead = none ];
96 -> 59;

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 97 [arrowhead = none ];
97 -> 24;

98 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT27 is ground\nreplacements:X22 -> T27", fontsize=14, style = filled, fillcolor = violet];
22 -> 98 [arrowhead = none ];
98 -> 25;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 99 [arrowhead = none ];
99 -> 26;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 100 [arrowhead = none ];
100 -> 44;

101 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(0), T24, T3), m(X3, 0, X4))", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 101 [arrowhead = none ];
101 -> 27;

102 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 102 [arrowhead = none ];
102 -> 28;

103 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 103 [arrowhead = none ];
103 -> 29;

104 [label="EVAL with clause\np(s(0), 0).\nand substitutionT24 -> s(0),\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 104 [arrowhead = none ];
104 -> 30;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 105 [arrowhead = none ];
105 -> 31;

106 [label="EVAL with clause\np(s(s(X29)), s(X30)) :- p(s(X29), X30).\nand substitutionX29 -> T30,\nT24 -> s(s(T30)),\nX30 -> X31,\nX22 -> s(X31)", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 106 [arrowhead = none ];
106 -> 33;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 107 [arrowhead = none ];
107 -> 34;

108 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 108 [arrowhead = none ];
108 -> 32;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 109 [arrowhead = none ];
109 -> 35;

110 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 110 [arrowhead = none ];
110 -> 36;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 111 [arrowhead = none ];
111 -> 37;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 112 [arrowhead = none ];
112 -> 38;

113 [label="EVAL with clause\np(s(0), 0).\nand substitutionT30 -> 0,\nX31 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 113 [arrowhead = none ];
113 -> 39;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 114 [arrowhead = none ];
114 -> 40;

115 [label="EVAL with clause\np(s(s(X38)), s(X39)) :- p(s(X38), X39).\nand substitutionX38 -> T33,\nT30 -> s(T33),\nX39 -> X40,\nX31 -> s(X40)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 115 [arrowhead = none ];
115 -> 42;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 116 [arrowhead = none ];
116 -> 43;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 117 [arrowhead = none ];
117 -> 41;

118 [label="INSTANCE with matching:\nT30 -> T33\nX31 -> X40", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 118 [arrowhead = none , style=dashed];
118 -> 33[style=dashed];

119 [label="EVAL with clause\nm(X47, 0, X48) :- ','(!_7, user_defined_=(X48, X47)).\nand substitutionX47 -> 0,\nT27 -> 0,\nT26 -> T38,\nX48 -> T38,\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 119 [arrowhead = none ];
119 -> 45;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 120 [arrowhead = none ];
120 -> 46;

121 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 121 [arrowhead = none ];
121 -> 47;

122 [label="ONLY EVAL with clause\nm(0, X56, X57) :- ','(!_7, user_defined_=(X57, 0)).\nand substitutionT27 -> T47,\nX56 -> T47,\nT26 -> T49,\nX57 -> T49,\nT48 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 122 [arrowhead = none ];
122 -> 52;

123 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 123 [arrowhead = none ];
123 -> 48;

124 [label="EVAL with clause\nuser_defined_=(X51, X51).\nand substitutionT38 -> 0,\nX51 -> 0,\nT41 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 124 [arrowhead = none ];
124 -> 49;

125 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 125 [arrowhead = none ];
125 -> 50;

126 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 126 [arrowhead = none ];
126 -> 51;

127 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 127 [arrowhead = none ];
127 -> 53;

128 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 128 [arrowhead = none ];
128 -> 54;

129 [label="EVAL with clause\nuser_defined_=(X60, X60).\nand substitutionT49 -> 0,\nX60 -> 0,\nT52 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 129 [arrowhead = none ];
129 -> 55;

130 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 130 [arrowhead = none ];
130 -> 56;

131 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 131 [arrowhead = none ];
131 -> 57;

132 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
58 -> 132 [arrowhead = none ];
132 -> 60;

133 [label="SPLIT 2\nnew knowledge:\nT55 is ground\nT56 is ground\nreplacements:X69 -> T56", fontsize=14, style = filled, fillcolor = violet];
58 -> 133 [arrowhead = none ];
133 -> 61;

134 [label="INSTANCE with matching:\nT30 -> T55\nX31 -> X69", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 134 [arrowhead = none , style=dashed];
134 -> 33[style=dashed];

135 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
61 -> 135 [arrowhead = none ];
135 -> 62;

136 [label="SPLIT 2\nnew knowledge:\nT24 is ground\nT57 is ground\nreplacements:X22 -> T57", fontsize=14, style = filled, fillcolor = violet];
61 -> 136 [arrowhead = none ];
136 -> 63;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 137 [arrowhead = none ];
137 -> 64;

138 [label="INSTANCE with matching:\nT1 -> s(T56)\nT2 -> T57\nT3 -> T26", fontsize=14, style = filled, fillcolor = lightgrey];
63 -> 138 [arrowhead = none , style=dashed];
138 -> 1[style=dashed];

139 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (m(s(s(T55)), T24, T3), m(X3, 0, X4))", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 139 [arrowhead = none ];
139 -> 65;

140 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 140 [arrowhead = none ];
140 -> 66;

141 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
65 -> 141 [arrowhead = none ];
141 -> 67;

142 [label="EVAL with clause\np(s(0), 0).\nand substitutionT24 -> s(0),\nX22 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
66 -> 142 [arrowhead = none ];
142 -> 68;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 143 [arrowhead = none ];
143 -> 69;

144 [label="EVAL with clause\np(s(s(X76)), s(X77)) :- p(s(X76), X77).\nand substitutionX76 -> T60,\nT24 -> s(s(T60)),\nX77 -> X78,\nX22 -> s(X78)", fontsize=14, style = filled, fillcolor = lightblue];
67 -> 144 [arrowhead = none ];
144 -> 71;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
67 -> 145 [arrowhead = none ];
145 -> 72;

146 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 146 [arrowhead = none ];
146 -> 70;

147 [label="INSTANCE with matching:\nT30 -> T60\nX31 -> X78", fontsize=14, style = filled, fillcolor = lightgrey];
71 -> 147 [arrowhead = none , style=dashed];
147 -> 33[style=dashed];

}

(4) Obligation:

Clauses:

pA(0, 0).
pA(s(T33), s(X40)) :- pA(T33, X40).
pB(s(0), 0).
pB(s(s(T30)), s(X31)) :- pA(T30, X31).
pC(s(0), 0).
pC(s(s(T60)), s(X78)) :- pA(T60, X78).
mD(T11, 0, T11).
mD(0, T14, 0).
mD(s(0), T24, T26) :- pB(T24, X22).
mD(s(0), T24, 0) :- pB(T24, 0).
mD(s(0), T24, 0) :- pB(T24, T47).
mD(s(s(T55)), T24, T26) :- pA(T55, X69).
mD(s(s(T55)), T24, T26) :- ','(pA(T55, T56), pC(T24, X22)).
mD(s(s(T55)), T24, T26) :- ','(pA(T55, T56), ','(pC(T24, T57), mD(s(T56), T57, T26))).

Query: mD(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:
mD_in: (b,b,f)
pB_in: (b,f) (b,b)
pA_in: (b,f) (b,b)
pC_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

mD_in_gga(T11, 0, T11) → mD_out_gga(T11, 0, T11)
mD_in_gga(0, T14, 0) → mD_out_gga(0, T14, 0)
mD_in_gga(s(0), T24, T26) → U4_gga(T24, T26, pB_in_ga(T24, X22))
pB_in_ga(s(0), 0) → pB_out_ga(s(0), 0)
pB_in_ga(s(s(T30)), s(X31)) → U2_ga(T30, X31, pA_in_ga(T30, X31))
pA_in_ga(0, 0) → pA_out_ga(0, 0)
pA_in_ga(s(T33), s(X40)) → U1_ga(T33, X40, pA_in_ga(T33, X40))
U1_ga(T33, X40, pA_out_ga(T33, X40)) → pA_out_ga(s(T33), s(X40))
U2_ga(T30, X31, pA_out_ga(T30, X31)) → pB_out_ga(s(s(T30)), s(X31))
U4_gga(T24, T26, pB_out_ga(T24, X22)) → mD_out_gga(s(0), T24, T26)
mD_in_gga(s(0), T24, 0) → U5_gga(T24, pB_in_gg(T24, 0))
pB_in_gg(s(0), 0) → pB_out_gg(s(0), 0)
pB_in_gg(s(s(T30)), s(X31)) → U2_gg(T30, X31, pA_in_gg(T30, X31))
pA_in_gg(0, 0) → pA_out_gg(0, 0)
pA_in_gg(s(T33), s(X40)) → U1_gg(T33, X40, pA_in_gg(T33, X40))
U1_gg(T33, X40, pA_out_gg(T33, X40)) → pA_out_gg(s(T33), s(X40))
U2_gg(T30, X31, pA_out_gg(T30, X31)) → pB_out_gg(s(s(T30)), s(X31))
U5_gga(T24, pB_out_gg(T24, 0)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(0), T24, 0) → U6_gga(T24, pB_in_ga(T24, T47))
U6_gga(T24, pB_out_ga(T24, T47)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(s(T55)), T24, T26) → U7_gga(T55, T24, T26, pA_in_ga(T55, X69))
U7_gga(T55, T24, T26, pA_out_ga(T55, X69)) → mD_out_gga(s(s(T55)), T24, T26)
mD_in_gga(s(s(T55)), T24, T26) → U8_gga(T55, T24, T26, pA_in_ga(T55, T56))
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U9_gga(T55, T24, T26, pC_in_ga(T24, X22))
pC_in_ga(s(0), 0) → pC_out_ga(s(0), 0)
pC_in_ga(s(s(T60)), s(X78)) → U3_ga(T60, X78, pA_in_ga(T60, X78))
U3_ga(T60, X78, pA_out_ga(T60, X78)) → pC_out_ga(s(s(T60)), s(X78))
U9_gga(T55, T24, T26, pC_out_ga(T24, X22)) → mD_out_gga(s(s(T55)), T24, T26)
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U10_gga(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_gga(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_gga(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U11_gga(T55, T24, T26, mD_out_gga(s(T56), T57, T26)) → mD_out_gga(s(s(T55)), T24, T26)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

mD_in_gga(T11, 0, T11) → mD_out_gga(T11, 0, T11)
mD_in_gga(0, T14, 0) → mD_out_gga(0, T14, 0)
mD_in_gga(s(0), T24, T26) → U4_gga(T24, T26, pB_in_ga(T24, X22))
pB_in_ga(s(0), 0) → pB_out_ga(s(0), 0)
pB_in_ga(s(s(T30)), s(X31)) → U2_ga(T30, X31, pA_in_ga(T30, X31))
pA_in_ga(0, 0) → pA_out_ga(0, 0)
pA_in_ga(s(T33), s(X40)) → U1_ga(T33, X40, pA_in_ga(T33, X40))
U1_ga(T33, X40, pA_out_ga(T33, X40)) → pA_out_ga(s(T33), s(X40))
U2_ga(T30, X31, pA_out_ga(T30, X31)) → pB_out_ga(s(s(T30)), s(X31))
U4_gga(T24, T26, pB_out_ga(T24, X22)) → mD_out_gga(s(0), T24, T26)
mD_in_gga(s(0), T24, 0) → U5_gga(T24, pB_in_gg(T24, 0))
pB_in_gg(s(0), 0) → pB_out_gg(s(0), 0)
pB_in_gg(s(s(T30)), s(X31)) → U2_gg(T30, X31, pA_in_gg(T30, X31))
pA_in_gg(0, 0) → pA_out_gg(0, 0)
pA_in_gg(s(T33), s(X40)) → U1_gg(T33, X40, pA_in_gg(T33, X40))
U1_gg(T33, X40, pA_out_gg(T33, X40)) → pA_out_gg(s(T33), s(X40))
U2_gg(T30, X31, pA_out_gg(T30, X31)) → pB_out_gg(s(s(T30)), s(X31))
U5_gga(T24, pB_out_gg(T24, 0)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(0), T24, 0) → U6_gga(T24, pB_in_ga(T24, T47))
U6_gga(T24, pB_out_ga(T24, T47)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(s(T55)), T24, T26) → U7_gga(T55, T24, T26, pA_in_ga(T55, X69))
U7_gga(T55, T24, T26, pA_out_ga(T55, X69)) → mD_out_gga(s(s(T55)), T24, T26)
mD_in_gga(s(s(T55)), T24, T26) → U8_gga(T55, T24, T26, pA_in_ga(T55, T56))
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U9_gga(T55, T24, T26, pC_in_ga(T24, X22))
pC_in_ga(s(0), 0) → pC_out_ga(s(0), 0)
pC_in_ga(s(s(T60)), s(X78)) → U3_ga(T60, X78, pA_in_ga(T60, X78))
U3_ga(T60, X78, pA_out_ga(T60, X78)) → pC_out_ga(s(s(T60)), s(X78))
U9_gga(T55, T24, T26, pC_out_ga(T24, X22)) → mD_out_gga(s(s(T55)), T24, T26)
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U10_gga(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_gga(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_gga(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U11_gga(T55, T24, T26, mD_out_gga(s(T56), T57, T26)) → mD_out_gga(s(s(T55)), T24, T26)

(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:

MD_IN_GGA(s(0), T24, T26) → U4_GGA(T24, T26, pB_in_ga(T24, X22))
MD_IN_GGA(s(0), T24, T26) → PB_IN_GA(T24, X22)
PB_IN_GA(s(s(T30)), s(X31)) → U2_GA(T30, X31, pA_in_ga(T30, X31))
PB_IN_GA(s(s(T30)), s(X31)) → PA_IN_GA(T30, X31)
PA_IN_GA(s(T33), s(X40)) → U1_GA(T33, X40, pA_in_ga(T33, X40))
PA_IN_GA(s(T33), s(X40)) → PA_IN_GA(T33, X40)
MD_IN_GGA(s(0), T24, 0) → U5_GGA(T24, pB_in_gg(T24, 0))
MD_IN_GGA(s(0), T24, 0) → PB_IN_GG(T24, 0)
PB_IN_GG(s(s(T30)), s(X31)) → U2_GG(T30, X31, pA_in_gg(T30, X31))
PB_IN_GG(s(s(T30)), s(X31)) → PA_IN_GG(T30, X31)
PA_IN_GG(s(T33), s(X40)) → U1_GG(T33, X40, pA_in_gg(T33, X40))
PA_IN_GG(s(T33), s(X40)) → PA_IN_GG(T33, X40)
MD_IN_GGA(s(0), T24, 0) → U6_GGA(T24, pB_in_ga(T24, T47))
MD_IN_GGA(s(0), T24, 0) → PB_IN_GA(T24, T47)
MD_IN_GGA(s(s(T55)), T24, T26) → U7_GGA(T55, T24, T26, pA_in_ga(T55, X69))
MD_IN_GGA(s(s(T55)), T24, T26) → PA_IN_GA(T55, X69)
MD_IN_GGA(s(s(T55)), T24, T26) → U8_GGA(T55, T24, T26, pA_in_ga(T55, T56))
U8_GGA(T55, T24, T26, pA_out_ga(T55, T56)) → U9_GGA(T55, T24, T26, pC_in_ga(T24, X22))
U8_GGA(T55, T24, T26, pA_out_ga(T55, T56)) → PC_IN_GA(T24, X22)
PC_IN_GA(s(s(T60)), s(X78)) → U3_GA(T60, X78, pA_in_ga(T60, X78))
PC_IN_GA(s(s(T60)), s(X78)) → PA_IN_GA(T60, X78)
U8_GGA(T55, T24, T26, pA_out_ga(T55, T56)) → U10_GGA(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_GGA(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_GGA(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U10_GGA(T55, T24, T26, T56, pC_out_ga(T24, T57)) → MD_IN_GGA(s(T56), T57, T26)

The TRS R consists of the following rules:

mD_in_gga(T11, 0, T11) → mD_out_gga(T11, 0, T11)
mD_in_gga(0, T14, 0) → mD_out_gga(0, T14, 0)
mD_in_gga(s(0), T24, T26) → U4_gga(T24, T26, pB_in_ga(T24, X22))
pB_in_ga(s(0), 0) → pB_out_ga(s(0), 0)
pB_in_ga(s(s(T30)), s(X31)) → U2_ga(T30, X31, pA_in_ga(T30, X31))
pA_in_ga(0, 0) → pA_out_ga(0, 0)
pA_in_ga(s(T33), s(X40)) → U1_ga(T33, X40, pA_in_ga(T33, X40))
U1_ga(T33, X40, pA_out_ga(T33, X40)) → pA_out_ga(s(T33), s(X40))
U2_ga(T30, X31, pA_out_ga(T30, X31)) → pB_out_ga(s(s(T30)), s(X31))
U4_gga(T24, T26, pB_out_ga(T24, X22)) → mD_out_gga(s(0), T24, T26)
mD_in_gga(s(0), T24, 0) → U5_gga(T24, pB_in_gg(T24, 0))
pB_in_gg(s(0), 0) → pB_out_gg(s(0), 0)
pB_in_gg(s(s(T30)), s(X31)) → U2_gg(T30, X31, pA_in_gg(T30, X31))
pA_in_gg(0, 0) → pA_out_gg(0, 0)
pA_in_gg(s(T33), s(X40)) → U1_gg(T33, X40, pA_in_gg(T33, X40))
U1_gg(T33, X40, pA_out_gg(T33, X40)) → pA_out_gg(s(T33), s(X40))
U2_gg(T30, X31, pA_out_gg(T30, X31)) → pB_out_gg(s(s(T30)), s(X31))
U5_gga(T24, pB_out_gg(T24, 0)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(0), T24, 0) → U6_gga(T24, pB_in_ga(T24, T47))
U6_gga(T24, pB_out_ga(T24, T47)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(s(T55)), T24, T26) → U7_gga(T55, T24, T26, pA_in_ga(T55, X69))
U7_gga(T55, T24, T26, pA_out_ga(T55, X69)) → mD_out_gga(s(s(T55)), T24, T26)
mD_in_gga(s(s(T55)), T24, T26) → U8_gga(T55, T24, T26, pA_in_ga(T55, T56))
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U9_gga(T55, T24, T26, pC_in_ga(T24, X22))
pC_in_ga(s(0), 0) → pC_out_ga(s(0), 0)
pC_in_ga(s(s(T60)), s(X78)) → U3_ga(T60, X78, pA_in_ga(T60, X78))
U3_ga(T60, X78, pA_out_ga(T60, X78)) → pC_out_ga(s(s(T60)), s(X78))
U9_gga(T55, T24, T26, pC_out_ga(T24, X22)) → mD_out_gga(s(s(T55)), T24, T26)
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U10_gga(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_gga(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_gga(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U11_gga(T55, T24, T26, mD_out_gga(s(T56), T57, T26)) → mD_out_gga(s(s(T55)), T24, T26)

The argument filtering Pi contains the following mapping:
mD_in_gga(x1, x2, x3) = mD_in_gga(x1, x2)
0 = 0
mD_out_gga(x1, x2, x3) = mD_out_gga
s(x1) = s(x1)
U4_gga(x1, x2, x3) = U4_gga(x3)
pB_in_ga(x1, x2) = pB_in_ga(x1)
pB_out_ga(x1, x2) = pB_out_ga(x2)
U2_ga(x1, x2, x3) = U2_ga(x3)
pA_in_ga(x1, x2) = pA_in_ga(x1)
pA_out_ga(x1, x2) = pA_out_ga(x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
U5_gga(x1, x2) = U5_gga(x2)
pB_in_gg(x1, x2) = pB_in_gg(x1, x2)
pB_out_gg(x1, x2) = pB_out_gg
U2_gg(x1, x2, x3) = U2_gg(x3)
pA_in_gg(x1, x2) = pA_in_gg(x1, x2)
pA_out_gg(x1, x2) = pA_out_gg
U1_gg(x1, x2, x3) = U1_gg(x3)
U6_gga(x1, x2) = U6_gga(x2)
U7_gga(x1, x2, x3, x4) = U7_gga(x4)
U8_gga(x1, x2, x3, x4) = U8_gga(x2, x4)
U9_gga(x1, x2, x3, x4) = U9_gga(x4)
pC_in_ga(x1, x2) = pC_in_ga(x1)
pC_out_ga(x1, x2) = pC_out_ga(x2)
U3_ga(x1, x2, x3) = U3_ga(x3)
U10_gga(x1, x2, x3, x4, x5) = U10_gga(x4, x5)
U11_gga(x1, x2, x3, x4) = U11_gga(x4)
MD_IN_GGA(x1, x2, x3) = MD_IN_GGA(x1, x2)
U4_GGA(x1, x2, x3) = U4_GGA(x3)
PB_IN_GA(x1, x2) = PB_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x3)
PA_IN_GA(x1, x2) = PA_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x3)
U5_GGA(x1, x2) = U5_GGA(x2)
PB_IN_GG(x1, x2) = PB_IN_GG(x1, x2)
U2_GG(x1, x2, x3) = U2_GG(x3)
PA_IN_GG(x1, x2) = PA_IN_GG(x1, x2)
U1_GG(x1, x2, x3) = U1_GG(x3)
U6_GGA(x1, x2) = U6_GGA(x2)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x2, x4)
U9_GGA(x1, x2, x3, x4) = U9_GGA(x4)
PC_IN_GA(x1, x2) = PC_IN_GA(x1)
U3_GA(x1, x2, x3) = U3_GA(x3)
U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x4, x5)
U11_GGA(x1, x2, x3, x4) = U11_GGA(x4)

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

(8) Obligation:

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

MD_IN_GGA(s(0), T24, T26) → U4_GGA(T24, T26, pB_in_ga(T24, X22))
MD_IN_GGA(s(0), T24, T26) → PB_IN_GA(T24, X22)
PB_IN_GA(s(s(T30)), s(X31)) → U2_GA(T30, X31, pA_in_ga(T30, X31))
PB_IN_GA(s(s(T30)), s(X31)) → PA_IN_GA(T30, X31)
PA_IN_GA(s(T33), s(X40)) → U1_GA(T33, X40, pA_in_ga(T33, X40))
PA_IN_GA(s(T33), s(X40)) → PA_IN_GA(T33, X40)
MD_IN_GGA(s(0), T24, 0) → U5_GGA(T24, pB_in_gg(T24, 0))
MD_IN_GGA(s(0), T24, 0) → PB_IN_GG(T24, 0)
PB_IN_GG(s(s(T30)), s(X31)) → U2_GG(T30, X31, pA_in_gg(T30, X31))
PB_IN_GG(s(s(T30)), s(X31)) → PA_IN_GG(T30, X31)
PA_IN_GG(s(T33), s(X40)) → U1_GG(T33, X40, pA_in_gg(T33, X40))
PA_IN_GG(s(T33), s(X40)) → PA_IN_GG(T33, X40)
MD_IN_GGA(s(0), T24, 0) → U6_GGA(T24, pB_in_ga(T24, T47))
MD_IN_GGA(s(0), T24, 0) → PB_IN_GA(T24, T47)
MD_IN_GGA(s(s(T55)), T24, T26) → U7_GGA(T55, T24, T26, pA_in_ga(T55, X69))
MD_IN_GGA(s(s(T55)), T24, T26) → PA_IN_GA(T55, X69)
MD_IN_GGA(s(s(T55)), T24, T26) → U8_GGA(T55, T24, T26, pA_in_ga(T55, T56))
U8_GGA(T55, T24, T26, pA_out_ga(T55, T56)) → U9_GGA(T55, T24, T26, pC_in_ga(T24, X22))
U8_GGA(T55, T24, T26, pA_out_ga(T55, T56)) → PC_IN_GA(T24, X22)
PC_IN_GA(s(s(T60)), s(X78)) → U3_GA(T60, X78, pA_in_ga(T60, X78))
PC_IN_GA(s(s(T60)), s(X78)) → PA_IN_GA(T60, X78)
U8_GGA(T55, T24, T26, pA_out_ga(T55, T56)) → U10_GGA(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_GGA(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_GGA(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U10_GGA(T55, T24, T26, T56, pC_out_ga(T24, T57)) → MD_IN_GGA(s(T56), T57, T26)

The TRS R consists of the following rules:

mD_in_gga(T11, 0, T11) → mD_out_gga(T11, 0, T11)
mD_in_gga(0, T14, 0) → mD_out_gga(0, T14, 0)
mD_in_gga(s(0), T24, T26) → U4_gga(T24, T26, pB_in_ga(T24, X22))
pB_in_ga(s(0), 0) → pB_out_ga(s(0), 0)
pB_in_ga(s(s(T30)), s(X31)) → U2_ga(T30, X31, pA_in_ga(T30, X31))
pA_in_ga(0, 0) → pA_out_ga(0, 0)
pA_in_ga(s(T33), s(X40)) → U1_ga(T33, X40, pA_in_ga(T33, X40))
U1_ga(T33, X40, pA_out_ga(T33, X40)) → pA_out_ga(s(T33), s(X40))
U2_ga(T30, X31, pA_out_ga(T30, X31)) → pB_out_ga(s(s(T30)), s(X31))
U4_gga(T24, T26, pB_out_ga(T24, X22)) → mD_out_gga(s(0), T24, T26)
mD_in_gga(s(0), T24, 0) → U5_gga(T24, pB_in_gg(T24, 0))
pB_in_gg(s(0), 0) → pB_out_gg(s(0), 0)
pB_in_gg(s(s(T30)), s(X31)) → U2_gg(T30, X31, pA_in_gg(T30, X31))
pA_in_gg(0, 0) → pA_out_gg(0, 0)
pA_in_gg(s(T33), s(X40)) → U1_gg(T33, X40, pA_in_gg(T33, X40))
U1_gg(T33, X40, pA_out_gg(T33, X40)) → pA_out_gg(s(T33), s(X40))
U2_gg(T30, X31, pA_out_gg(T30, X31)) → pB_out_gg(s(s(T30)), s(X31))
U5_gga(T24, pB_out_gg(T24, 0)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(0), T24, 0) → U6_gga(T24, pB_in_ga(T24, T47))
U6_gga(T24, pB_out_ga(T24, T47)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(s(T55)), T24, T26) → U7_gga(T55, T24, T26, pA_in_ga(T55, X69))
U7_gga(T55, T24, T26, pA_out_ga(T55, X69)) → mD_out_gga(s(s(T55)), T24, T26)
mD_in_gga(s(s(T55)), T24, T26) → U8_gga(T55, T24, T26, pA_in_ga(T55, T56))
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U9_gga(T55, T24, T26, pC_in_ga(T24, X22))
pC_in_ga(s(0), 0) → pC_out_ga(s(0), 0)
pC_in_ga(s(s(T60)), s(X78)) → U3_ga(T60, X78, pA_in_ga(T60, X78))
U3_ga(T60, X78, pA_out_ga(T60, X78)) → pC_out_ga(s(s(T60)), s(X78))
U9_gga(T55, T24, T26, pC_out_ga(T24, X22)) → mD_out_gga(s(s(T55)), T24, T26)
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U10_gga(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_gga(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_gga(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U11_gga(T55, T24, T26, mD_out_gga(s(T56), T57, T26)) → mD_out_gga(s(s(T55)), T24, T26)

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 19 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

PA_IN_GG(s(T33), s(X40)) → PA_IN_GG(T33, X40)

The TRS R consists of the following rules:

mD_in_gga(T11, 0, T11) → mD_out_gga(T11, 0, T11)
mD_in_gga(0, T14, 0) → mD_out_gga(0, T14, 0)
mD_in_gga(s(0), T24, T26) → U4_gga(T24, T26, pB_in_ga(T24, X22))
pB_in_ga(s(0), 0) → pB_out_ga(s(0), 0)
pB_in_ga(s(s(T30)), s(X31)) → U2_ga(T30, X31, pA_in_ga(T30, X31))
pA_in_ga(0, 0) → pA_out_ga(0, 0)
pA_in_ga(s(T33), s(X40)) → U1_ga(T33, X40, pA_in_ga(T33, X40))
U1_ga(T33, X40, pA_out_ga(T33, X40)) → pA_out_ga(s(T33), s(X40))
U2_ga(T30, X31, pA_out_ga(T30, X31)) → pB_out_ga(s(s(T30)), s(X31))
U4_gga(T24, T26, pB_out_ga(T24, X22)) → mD_out_gga(s(0), T24, T26)
mD_in_gga(s(0), T24, 0) → U5_gga(T24, pB_in_gg(T24, 0))
pB_in_gg(s(0), 0) → pB_out_gg(s(0), 0)
pB_in_gg(s(s(T30)), s(X31)) → U2_gg(T30, X31, pA_in_gg(T30, X31))
pA_in_gg(0, 0) → pA_out_gg(0, 0)
pA_in_gg(s(T33), s(X40)) → U1_gg(T33, X40, pA_in_gg(T33, X40))
U1_gg(T33, X40, pA_out_gg(T33, X40)) → pA_out_gg(s(T33), s(X40))
U2_gg(T30, X31, pA_out_gg(T30, X31)) → pB_out_gg(s(s(T30)), s(X31))
U5_gga(T24, pB_out_gg(T24, 0)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(0), T24, 0) → U6_gga(T24, pB_in_ga(T24, T47))
U6_gga(T24, pB_out_ga(T24, T47)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(s(T55)), T24, T26) → U7_gga(T55, T24, T26, pA_in_ga(T55, X69))
U7_gga(T55, T24, T26, pA_out_ga(T55, X69)) → mD_out_gga(s(s(T55)), T24, T26)
mD_in_gga(s(s(T55)), T24, T26) → U8_gga(T55, T24, T26, pA_in_ga(T55, T56))
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U9_gga(T55, T24, T26, pC_in_ga(T24, X22))
pC_in_ga(s(0), 0) → pC_out_ga(s(0), 0)
pC_in_ga(s(s(T60)), s(X78)) → U3_ga(T60, X78, pA_in_ga(T60, X78))
U3_ga(T60, X78, pA_out_ga(T60, X78)) → pC_out_ga(s(s(T60)), s(X78))
U9_gga(T55, T24, T26, pC_out_ga(T24, X22)) → mD_out_gga(s(s(T55)), T24, T26)
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U10_gga(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_gga(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_gga(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U11_gga(T55, T24, T26, mD_out_gga(s(T56), T57, T26)) → mD_out_gga(s(s(T55)), T24, T26)

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

PA_IN_GG(s(T33), s(X40)) → PA_IN_GG(T33, X40)

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

(14) PiDPToQDPProof (EQUIVALENT transformation)

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

(15) Obligation:

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

PA_IN_GG(s(T33), s(X40)) → PA_IN_GG(T33, X40)

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

(16) 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:

PA_IN_GG(s(T33), s(X40)) → PA_IN_GG(T33, X40)
The graph contains the following edges 1 > 1, 2 > 2

(17) YES

(18) Obligation:

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

PA_IN_GA(s(T33), s(X40)) → PA_IN_GA(T33, X40)

The TRS R consists of the following rules:

mD_in_gga(T11, 0, T11) → mD_out_gga(T11, 0, T11)
mD_in_gga(0, T14, 0) → mD_out_gga(0, T14, 0)
mD_in_gga(s(0), T24, T26) → U4_gga(T24, T26, pB_in_ga(T24, X22))
pB_in_ga(s(0), 0) → pB_out_ga(s(0), 0)
pB_in_ga(s(s(T30)), s(X31)) → U2_ga(T30, X31, pA_in_ga(T30, X31))
pA_in_ga(0, 0) → pA_out_ga(0, 0)
pA_in_ga(s(T33), s(X40)) → U1_ga(T33, X40, pA_in_ga(T33, X40))
U1_ga(T33, X40, pA_out_ga(T33, X40)) → pA_out_ga(s(T33), s(X40))
U2_ga(T30, X31, pA_out_ga(T30, X31)) → pB_out_ga(s(s(T30)), s(X31))
U4_gga(T24, T26, pB_out_ga(T24, X22)) → mD_out_gga(s(0), T24, T26)
mD_in_gga(s(0), T24, 0) → U5_gga(T24, pB_in_gg(T24, 0))
pB_in_gg(s(0), 0) → pB_out_gg(s(0), 0)
pB_in_gg(s(s(T30)), s(X31)) → U2_gg(T30, X31, pA_in_gg(T30, X31))
pA_in_gg(0, 0) → pA_out_gg(0, 0)
pA_in_gg(s(T33), s(X40)) → U1_gg(T33, X40, pA_in_gg(T33, X40))
U1_gg(T33, X40, pA_out_gg(T33, X40)) → pA_out_gg(s(T33), s(X40))
U2_gg(T30, X31, pA_out_gg(T30, X31)) → pB_out_gg(s(s(T30)), s(X31))
U5_gga(T24, pB_out_gg(T24, 0)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(0), T24, 0) → U6_gga(T24, pB_in_ga(T24, T47))
U6_gga(T24, pB_out_ga(T24, T47)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(s(T55)), T24, T26) → U7_gga(T55, T24, T26, pA_in_ga(T55, X69))
U7_gga(T55, T24, T26, pA_out_ga(T55, X69)) → mD_out_gga(s(s(T55)), T24, T26)
mD_in_gga(s(s(T55)), T24, T26) → U8_gga(T55, T24, T26, pA_in_ga(T55, T56))
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U9_gga(T55, T24, T26, pC_in_ga(T24, X22))
pC_in_ga(s(0), 0) → pC_out_ga(s(0), 0)
pC_in_ga(s(s(T60)), s(X78)) → U3_ga(T60, X78, pA_in_ga(T60, X78))
U3_ga(T60, X78, pA_out_ga(T60, X78)) → pC_out_ga(s(s(T60)), s(X78))
U9_gga(T55, T24, T26, pC_out_ga(T24, X22)) → mD_out_gga(s(s(T55)), T24, T26)
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U10_gga(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_gga(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_gga(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U11_gga(T55, T24, T26, mD_out_gga(s(T56), T57, T26)) → mD_out_gga(s(s(T55)), T24, T26)

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

PA_IN_GA(s(T33), s(X40)) → PA_IN_GA(T33, X40)

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

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

PA_IN_GA(s(T33)) → PA_IN_GA(T33)

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

(23) 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:

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

(24) YES

(25) Obligation:

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

MD_IN_GGA(s(s(T55)), T24, T26) → U8_GGA(T55, T24, T26, pA_in_ga(T55, T56))
U8_GGA(T55, T24, T26, pA_out_ga(T55, T56)) → U10_GGA(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_GGA(T55, T24, T26, T56, pC_out_ga(T24, T57)) → MD_IN_GGA(s(T56), T57, T26)

The TRS R consists of the following rules:

mD_in_gga(T11, 0, T11) → mD_out_gga(T11, 0, T11)
mD_in_gga(0, T14, 0) → mD_out_gga(0, T14, 0)
mD_in_gga(s(0), T24, T26) → U4_gga(T24, T26, pB_in_ga(T24, X22))
pB_in_ga(s(0), 0) → pB_out_ga(s(0), 0)
pB_in_ga(s(s(T30)), s(X31)) → U2_ga(T30, X31, pA_in_ga(T30, X31))
pA_in_ga(0, 0) → pA_out_ga(0, 0)
pA_in_ga(s(T33), s(X40)) → U1_ga(T33, X40, pA_in_ga(T33, X40))
U1_ga(T33, X40, pA_out_ga(T33, X40)) → pA_out_ga(s(T33), s(X40))
U2_ga(T30, X31, pA_out_ga(T30, X31)) → pB_out_ga(s(s(T30)), s(X31))
U4_gga(T24, T26, pB_out_ga(T24, X22)) → mD_out_gga(s(0), T24, T26)
mD_in_gga(s(0), T24, 0) → U5_gga(T24, pB_in_gg(T24, 0))
pB_in_gg(s(0), 0) → pB_out_gg(s(0), 0)
pB_in_gg(s(s(T30)), s(X31)) → U2_gg(T30, X31, pA_in_gg(T30, X31))
pA_in_gg(0, 0) → pA_out_gg(0, 0)
pA_in_gg(s(T33), s(X40)) → U1_gg(T33, X40, pA_in_gg(T33, X40))
U1_gg(T33, X40, pA_out_gg(T33, X40)) → pA_out_gg(s(T33), s(X40))
U2_gg(T30, X31, pA_out_gg(T30, X31)) → pB_out_gg(s(s(T30)), s(X31))
U5_gga(T24, pB_out_gg(T24, 0)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(0), T24, 0) → U6_gga(T24, pB_in_ga(T24, T47))
U6_gga(T24, pB_out_ga(T24, T47)) → mD_out_gga(s(0), T24, 0)
mD_in_gga(s(s(T55)), T24, T26) → U7_gga(T55, T24, T26, pA_in_ga(T55, X69))
U7_gga(T55, T24, T26, pA_out_ga(T55, X69)) → mD_out_gga(s(s(T55)), T24, T26)
mD_in_gga(s(s(T55)), T24, T26) → U8_gga(T55, T24, T26, pA_in_ga(T55, T56))
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U9_gga(T55, T24, T26, pC_in_ga(T24, X22))
pC_in_ga(s(0), 0) → pC_out_ga(s(0), 0)
pC_in_ga(s(s(T60)), s(X78)) → U3_ga(T60, X78, pA_in_ga(T60, X78))
U3_ga(T60, X78, pA_out_ga(T60, X78)) → pC_out_ga(s(s(T60)), s(X78))
U9_gga(T55, T24, T26, pC_out_ga(T24, X22)) → mD_out_gga(s(s(T55)), T24, T26)
U8_gga(T55, T24, T26, pA_out_ga(T55, T56)) → U10_gga(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_gga(T55, T24, T26, T56, pC_out_ga(T24, T57)) → U11_gga(T55, T24, T26, mD_in_gga(s(T56), T57, T26))
U11_gga(T55, T24, T26, mD_out_gga(s(T56), T57, T26)) → mD_out_gga(s(s(T55)), T24, T26)

(26) UsableRulesProof (EQUIVALENT transformation)

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

(27) Obligation:

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

MD_IN_GGA(s(s(T55)), T24, T26) → U8_GGA(T55, T24, T26, pA_in_ga(T55, T56))
U8_GGA(T55, T24, T26, pA_out_ga(T55, T56)) → U10_GGA(T55, T24, T26, T56, pC_in_ga(T24, T57))
U10_GGA(T55, T24, T26, T56, pC_out_ga(T24, T57)) → MD_IN_GGA(s(T56), T57, T26)

The TRS R consists of the following rules:

pA_in_ga(0, 0) → pA_out_ga(0, 0)
pA_in_ga(s(T33), s(X40)) → U1_ga(T33, X40, pA_in_ga(T33, X40))
pC_in_ga(s(0), 0) → pC_out_ga(s(0), 0)
pC_in_ga(s(s(T60)), s(X78)) → U3_ga(T60, X78, pA_in_ga(T60, X78))
U1_ga(T33, X40, pA_out_ga(T33, X40)) → pA_out_ga(s(T33), s(X40))
U3_ga(T60, X78, pA_out_ga(T60, X78)) → pC_out_ga(s(s(T60)), s(X78))

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
pA_in_ga(x1, x2) = pA_in_ga(x1)
pA_out_ga(x1, x2) = pA_out_ga(x2)
U1_ga(x1, x2, x3) = U1_ga(x3)
pC_in_ga(x1, x2) = pC_in_ga(x1)
pC_out_ga(x1, x2) = pC_out_ga(x2)
U3_ga(x1, x2, x3) = U3_ga(x3)
MD_IN_GGA(x1, x2, x3) = MD_IN_GGA(x1, x2)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x2, x4)
U10_GGA(x1, x2, x3, x4, x5) = U10_GGA(x4, x5)

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

(28) PiDPToQDPProof (SOUND transformation)

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

(29) Obligation:

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

MD_IN_GGA(s(s(T55)), T24) → U8_GGA(T24, pA_in_ga(T55))
U8_GGA(T24, pA_out_ga(T56)) → U10_GGA(T56, pC_in_ga(T24))
U10_GGA(T56, pC_out_ga(T57)) → MD_IN_GGA(s(T56), T57)

The TRS R consists of the following rules:

pA_in_ga(0) → pA_out_ga(0)
pA_in_ga(s(T33)) → U1_ga(pA_in_ga(T33))
pC_in_ga(s(0)) → pC_out_ga(0)
pC_in_ga(s(s(T60))) → U3_ga(pA_in_ga(T60))
U1_ga(pA_out_ga(X40)) → pA_out_ga(s(X40))
U3_ga(pA_out_ga(X78)) → pC_out_ga(s(X78))

The set Q consists of the following terms:

pA_in_ga(x0)
pC_in_ga(x0)
U1_ga(x0)
U3_ga(x0)

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

(30) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MD_IN_GGA(s(s(T55)), T24) → U8_GGA(T24, pA_in_ga(T55))
U8_GGA(T24, pA_out_ga(T56)) → U10_GGA(T56, pC_in_ga(T24))
U10_GGA(T56, pC_out_ga(T57)) → MD_IN_GGA(s(T56), T57)

Strictly oriented rules of the TRS R:

pA_in_ga(0) → pA_out_ga(0)
pA_in_ga(s(T33)) → U1_ga(pA_in_ga(T33))
pC_in_ga(s(0)) → pC_out_ga(0)
pC_in_ga(s(s(T60))) → U3_ga(pA_in_ga(T60))
U1_ga(pA_out_ga(X40)) → pA_out_ga(s(X40))
U3_ga(pA_out_ga(X78)) → pC_out_ga(s(X78))

Used ordering: Knuth-Bendix order [KBO] with precedence:

0 > MD_{IN_GGA₂} > U3_ga₁ > pC_{out_ga₁} > U10_GGA₂ > pA_{in_ga₁} > U1_ga₁ > pA_{out_ga₁} > pC_{in_ga₁} > U8_GGA₂ > s₁

and weight map:

0=1
pA_in_ga_1=2
pA_out_ga_1=1
s_1=3
U1_ga_1=3
pC_in_ga_1=1
pC_out_ga_1=2
U3_ga_1=4
MD_IN_GGA_2=0
U8_GGA_2=3
U10_GGA_2=2

The variable weight is 1

(31) Obligation:

Q DP problem:
P is empty.
R is empty.
The set Q consists of the following terms:

pA_in_ga(x0)
pC_in_ga(x0)
U1_ga(x0)
U3_ga(x0)

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

(32) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(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) Complex Obligation (AND)

(11) Obligation:

(12) UsableRulesProof (EQUIVALENT transformation)

(13) Obligation:

(14) PiDPToQDPProof (EQUIVALENT transformation)

(15) Obligation:

(16) QDPSizeChangeProof (EQUIVALENT transformation)

(17) YES

(18) Obligation:

(19) UsableRulesProof (EQUIVALENT transformation)

(20) Obligation:

(21) PiDPToQDPProof (SOUND transformation)

(22) Obligation:

(23) QDPSizeChangeProof (EQUIVALENT transformation)

(24) YES

(25) Obligation:

(26) UsableRulesProof (EQUIVALENT transformation)

(27) Obligation:

(28) PiDPToQDPProof (SOUND transformation)

(29) Obligation:

(30) MRRProof (EQUIVALENT transformation)

(31) Obligation:

(32) PisEmptyProof (EQUIVALENT transformation)

(33) YES