Left Termination of the query pattern minus_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

p(0, 0).
p(s(X), X).
le(0, Y, true).
le(s(X), 0, false).
le(s(X), s(Y), B) :- le(X, Y, B).
minus(X, Y, Z) :- ','(le(X, Y, B), if(B, X, Y, Z)).
if(true, X, Y, 0).
if(false, X, Y, s(Z)) :- ','(p(X, X1), minus(X1, Y, Z)).

Queries:

minus(g,a,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: minus(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: minus(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(le(T7, T10, X8), if(X8, T7, T10, T11))\n\nT7 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(le(T7, T10, X8), if(X8, T7, T10, T11)), SCOPE: 2, CLAUSE: 2 | ','(le(T7, T10, X8), if(X8, T7, T10, T11)), SCOPE: 2, CLAUSE: 3 | ','(le(T7, T10, X8), if(X8, T7, T10, T11)), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(le(T7, T10, X8), if(X8, T7, T10, T11)), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(le(T7, T10, X8), if(X8, T7, T10, T11)), SCOPE: 2, CLAUSE: 3 | ','(le(T7, T10, X8), if(X8, T7, T10, T11)), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: if(true, 0, T17, T18)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: if(true, 0, T17, T18), SCOPE: 3, CLAUSE: 6 | if(true, 0, T17, T18), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: if(true, 0, T17, T18), SCOPE: 3, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: if(true, 0, T17, T18), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(le(T7, T10, X8), if(X8, T7, T10, T11)), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(le(T7, T10, X8), if(X8, T7, T10, T11)), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: if(false, s(T30), 0, T31)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: if(false, s(T30), 0, T31), SCOPE: 4, CLAUSE: 6 | if(false, s(T30), 0, T31), SCOPE: 4, CLAUSE: 7\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: if(false, s(T30), 0, T31), SCOPE: 4, CLAUSE: 7\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(p(s(T38), X60), minus(X60, 0, T40))\n\nT38 is ground\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(s(T38), X60), minus(X60, 0, T40)), SCOPE: 5, CLAUSE: 0 | ','(p(s(T38), X60), minus(X60, 0, T40)), SCOPE: 5, CLAUSE: 1\n\nT38 is ground\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(s(T38), X60), minus(X60, 0, T40)), SCOPE: 5, CLAUSE: 1\n\nT38 is ground\nX60 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: minus(T46, 0, T40)\n\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: minus(T46, 0, T40), SCOPE: 6, CLAUSE: 5\n\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(le(T54, 0, X92), if(X92, T54, 0, T56))\n\nT54 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: le(T54, 0, X92)\n\nT54 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: if(T57, T54, 0, T56)\n\nT54 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: le(T54, 0, X92), SCOPE: 7, CLAUSE: 2 | le(T54, 0, X92), SCOPE: 7, CLAUSE: 3 | le(T54, 0, X92), SCOPE: 7, CLAUSE: 4\n\nT54 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: le(T54, 0, X92), SCOPE: 7, CLAUSE: 2\n\nT54 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: le(T54, 0, X92), SCOPE: 7, CLAUSE: 3 | le(T54, 0, X92), SCOPE: 7, CLAUSE: 4\n\nT54 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: le(T54, 0, X92), SCOPE: 7, CLAUSE: 3\n\nT54 is ground\nX92 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: le(T54, 0, X92), SCOPE: 7, CLAUSE: 4\n\nT54 is ground\nX92 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: if(T57, T54, 0, T56), SCOPE: 8, CLAUSE: 6 | if(T57, T54, 0, T56), SCOPE: 8, CLAUSE: 7\n\nT54 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: if(T57, T54, 0, T56), SCOPE: 8, CLAUSE: 6\n\nT54 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: if(T57, T54, 0, T56), SCOPE: 8, CLAUSE: 7\n\nT54 is ground\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(p(T75, X138), minus(X138, 0, T77))\n\nT75 is ground\nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(p(T75, X138), minus(X138, 0, T77)), SCOPE: 9, CLAUSE: 0 | ','(p(T75, X138), minus(X138, 0, T77)), SCOPE: 9, CLAUSE: 1\n\nT75 is ground\nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T75, X138), minus(X138, 0, T77)), SCOPE: 9, CLAUSE: 0\n\nT75 is ground\nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T75, X138), minus(X138, 0, T77)), SCOPE: 9, CLAUSE: 1\n\nT75 is ground\nX138 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: minus(0, 0, T77)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: minus(0, 0, T77), SCOPE: 10, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(le(0, 0, X166), if(X166, 0, 0, T84))\n\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: le(0, 0, X166)\n\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: if(T85, 0, 0, T84)\n\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: le(0, 0, X166), SCOPE: 11, CLAUSE: 2 | le(0, 0, X166), SCOPE: 11, CLAUSE: 3 | le(0, 0, X166), SCOPE: 11, CLAUSE: 4\n\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: le(0, 0, X166), SCOPE: 11, CLAUSE: 2\n\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: le(0, 0, X166), SCOPE: 11, CLAUSE: 3 | le(0, 0, X166), SCOPE: 11, CLAUSE: 4\n\nX166 is free", 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="65: le(0, 0, X166), SCOPE: 11, CLAUSE: 4\n\nX166 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: if(T85, 0, 0, T84), SCOPE: 12, CLAUSE: 6 | if(T85, 0, 0, T84), SCOPE: 12, CLAUSE: 7\n\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: if(T85, 0, 0, T84), SCOPE: 12, CLAUSE: 6\n\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: if(T85, 0, 0, T84), SCOPE: 12, CLAUSE: 7\n\nT85 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(p(0, X208), minus(X208, 0, T89))\n\nX208 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(p(0, X208), minus(X208, 0, T89)), SCOPE: 13, CLAUSE: 0 | ','(p(0, X208), minus(X208, 0, T89)), SCOPE: 13, CLAUSE: 1\n\nX208 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(p(0, X208), minus(X208, 0, T89)), SCOPE: 13, CLAUSE: 0\n\nX208 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(p(0, X208), minus(X208, 0, T89)), SCOPE: 13, CLAUSE: 1\n\nX208 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: minus(0, 0, T89)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: minus(T96, 0, T77)\n\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(le(T103, T105, X249), if(X249, s(T103), s(T105), T106))\n\nT103 is ground\nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: le(T103, T105, X249)\n\nT103 is ground\nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: if(T109, s(T103), s(T110), T111)\n\nT103 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: le(T103, T105, X249), SCOPE: 14, CLAUSE: 2 | le(T103, T105, X249), SCOPE: 14, CLAUSE: 3 | le(T103, T105, X249), SCOPE: 14, CLAUSE: 4\n\nT103 is ground\nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: le(T103, T105, X249), SCOPE: 14, CLAUSE: 2\n\nT103 is ground\nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: le(T103, T105, X249), SCOPE: 14, CLAUSE: 3 | le(T103, T105, X249), SCOPE: 14, CLAUSE: 4\n\nT103 is ground\nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: le(T103, T105, X249), SCOPE: 14, CLAUSE: 3\n\nT103 is ground\nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: le(T103, T105, X249), SCOPE: 14, CLAUSE: 4\n\nT103 is ground\nX249 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: le(T128, T130, X280)\n\nT128 is ground\nX280 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: if(T109, s(T103), s(T110), T111), SCOPE: 15, CLAUSE: 6 | if(T109, s(T103), s(T110), T111), SCOPE: 15, CLAUSE: 7\n\nT103 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: if(T109, s(T103), s(T110), T111), SCOPE: 15, CLAUSE: 6\n\nT103 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: if(T109, s(T103), s(T110), T111), SCOPE: 15, CLAUSE: 7\n\nT103 is ground\nT109 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: ','(p(s(T153), X312), minus(X312, s(T157), T156))\n\nT153 is ground\nX312 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: ','(p(s(T153), X312), minus(X312, s(T157), T156)), SCOPE: 16, CLAUSE: 0 | ','(p(s(T153), X312), minus(X312, s(T157), T156)), SCOPE: 16, CLAUSE: 1\n\nT153 is ground\nX312 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(p(s(T153), X312), minus(X312, s(T157), T156)), SCOPE: 16, CLAUSE: 1\n\nT153 is ground\nX312 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: minus(T165, s(T157), T156)\n\nT165 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 110 [arrowhead = none ];
110 -> 2;

111 [label="ONLY EVAL with clause\nminus(X5, X6, X7) :- ','(le(X5, X6, X8), if(X8, X5, X6, X7)).\nand substitution\nT1 -> T7\nX5 -> T7\nT2 -> T10\nX6 -> T10\nT3 -> T11\nX7 -> T11\nT8 -> T10\nT9 -> T11", fontsize=14, style = filled, fillcolor = white];
2 -> 111 [arrowhead = none ];
111 -> 3;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 112 [arrowhead = none ];
112 -> 4;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 113 [arrowhead = none ];
113 -> 5;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 114 [arrowhead = none ];
114 -> 6;

115 [label="EVAL with clause\nle(0, X13, true).\nand substitution\nT7 -> 0\nT10 -> T17\nX13 -> T17\nX8 -> true\nT16 -> T17\nT11 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 115 [arrowhead = none ];
115 -> 7;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 116 [arrowhead = none ];
116 -> 8;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 117 [arrowhead = none ];
117 -> 16;

118 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 118 [arrowhead = none ];
118 -> 17;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 119 [arrowhead = none ];
119 -> 9;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 120 [arrowhead = none ];
120 -> 10;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 121 [arrowhead = none ];
121 -> 11;

122 [label="EVAL with clause\nif(true, X27, X28, 0).\nand substitution\nX27 -> 0\nT17 -> T25\nX28 -> T25\nT18 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 122 [arrowhead = none ];
122 -> 12;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 123 [arrowhead = none ];
123 -> 13;

124 [label="BACKTRACK\nfor clause: if(false, X, Y, s(Z)) :- ','(p(X, X1), minus(X1, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 124 [arrowhead = none ];
124 -> 15;

125 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 125 [arrowhead = none ];
125 -> 14;

126 [label="EVAL with clause\nle(s(X38), 0, false).\nand substitution\nX38 -> T30\nT7 -> s(T30)\nT10 -> 0\nX8 -> false\nT11 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 126 [arrowhead = none ];
126 -> 18;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 127 [arrowhead = none ];
127 -> 19;

128 [label="EVAL with clause\nle(s(X246), s(X247), X248) :- le(X246, X247, X248).\nand substitution\nX246 -> T103\nT7 -> s(T103)\nX247 -> T105\nT10 -> s(T105)\nX8 -> X249\nX248 -> X249\nT104 -> T105\nT11 -> T106", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 128 [arrowhead = none ];
128 -> 82;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 129 [arrowhead = none ];
129 -> 83;

130 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 130 [arrowhead = none ];
130 -> 20;

131 [label="BACKTRACK\nfor clause: if(true, X, Y, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
20 -> 131 [arrowhead = none ];
131 -> 21;

132 [label="EVAL with clause\nif(false, X57, X58, s(X59)) :- ','(p(X57, X60), minus(X60, X58, X59)).\nand substitution\nT30 -> T38\nX57 -> s(T38)\nX58 -> 0\nX59 -> T40\nT31 -> s(T40)\nT39 -> T40", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 132 [arrowhead = none ];
132 -> 22;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 133 [arrowhead = none ];
133 -> 23;

134 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 134 [arrowhead = none ];
134 -> 24;

135 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
24 -> 135 [arrowhead = none ];
135 -> 25;

136 [label="ONLY EVAL with clause\np(s(X75), X75).\nand substitution\nT38 -> T46\nX75 -> T46\nX60 -> T46", fontsize=14, style = filled, fillcolor = white];
25 -> 136 [arrowhead = none ];
136 -> 26;

137 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 137 [arrowhead = none ];
137 -> 27;

138 [label="ONLY EVAL with clause\nminus(X89, X90, X91) :- ','(le(X89, X90, X92), if(X92, X89, X90, X91)).\nand substitution\nT46 -> T54\nX89 -> T54\nX90 -> 0\nT40 -> T56\nX91 -> T56\nT55 -> T56", fontsize=14, style = filled, fillcolor = white];
27 -> 138 [arrowhead = none ];
138 -> 28;

139 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
28 -> 139 [arrowhead = none ];
139 -> 29;

140 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nreplacements:\nX92 -> T57", fontsize=14, style = filled, fillcolor = violet];
28 -> 140 [arrowhead = none ];
140 -> 30;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 141 [arrowhead = none ];
141 -> 31;

142 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 142 [arrowhead = none ];
142 -> 43;

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 143 [arrowhead = none ];
143 -> 32;

144 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 144 [arrowhead = none ];
144 -> 33;

145 [label="EVAL with clause\nle(0, X101, true).\nand substitution\nT54 -> 0\nX101 -> 0\nX92 -> true", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 145 [arrowhead = none ];
145 -> 34;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 146 [arrowhead = none ];
146 -> 35;

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 147 [arrowhead = none ];
147 -> 37;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
33 -> 148 [arrowhead = none ];
148 -> 38;

149 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 149 [arrowhead = none ];
149 -> 36;

150 [label="EVAL with clause\nle(s(X107), 0, false).\nand substitution\nX107 -> T62\nT54 -> s(T62)\nX92 -> false", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 150 [arrowhead = none ];
150 -> 39;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 151 [arrowhead = none ];
151 -> 40;

152 [label="BACKTRACK\nfor clause: le(s(X), s(Y), B) :- le(X, Y, B)because of non-unification", fontsize=14, style = filled, fillcolor = white];
38 -> 152 [arrowhead = none ];
152 -> 42;

153 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 153 [arrowhead = none ];
153 -> 41;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 154 [arrowhead = none ];
154 -> 44;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 155 [arrowhead = none ];
155 -> 45;

156 [label="EVAL with clause\nif(true, X124, X125, 0).\nand substitution\nT57 -> true\nT54 -> T70\nX124 -> T70\nX125 -> 0\nT56 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 156 [arrowhead = none ];
156 -> 46;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 157 [arrowhead = none ];
157 -> 47;

158 [label="EVAL with clause\nif(false, X135, X136, s(X137)) :- ','(p(X135, X138), minus(X138, X136, X137)).\nand substitution\nT57 -> false\nT54 -> T75\nX135 -> T75\nX136 -> 0\nX137 -> T77\nT56 -> s(T77)\nT76 -> T77", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 158 [arrowhead = none ];
158 -> 49;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 159 [arrowhead = none ];
159 -> 50;

160 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 160 [arrowhead = none ];
160 -> 48;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 161 [arrowhead = none ];
161 -> 51;

162 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 162 [arrowhead = none ];
162 -> 52;

163 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
51 -> 163 [arrowhead = none ];
163 -> 53;

164 [label="EVAL with clause\np(0, 0).\nand substitution\nT75 -> 0\nX138 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 164 [arrowhead = none ];
164 -> 54;

165 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 165 [arrowhead = none ];
165 -> 55;

166 [label="EVAL with clause\np(s(X232), X232).\nand substitution\nX232 -> T96\nT75 -> s(T96)\nX138 -> T96", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 166 [arrowhead = none ];
166 -> 80;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 167 [arrowhead = none ];
167 -> 81;

168 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 168 [arrowhead = none ];
168 -> 56;

169 [label="ONLY EVAL with clause\nminus(X163, X164, X165) :- ','(le(X163, X164, X166), if(X166, X163, X164, X165)).\nand substitution\nX163 -> 0\nX164 -> 0\nT77 -> T84\nX165 -> T84\nT83 -> T84", fontsize=14, style = filled, fillcolor = white];
56 -> 169 [arrowhead = none ];
169 -> 57;

170 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
57 -> 170 [arrowhead = none ];
170 -> 58;

171 [label="SPLIT 2\nnew knowledge:\nT85 is ground\nreplacements:\nX166 -> T85", fontsize=14, style = filled, fillcolor = violet];
57 -> 171 [arrowhead = none ];
171 -> 59;

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 172 [arrowhead = none ];
172 -> 60;

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
59 -> 173 [arrowhead = none ];
173 -> 67;

174 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 174 [arrowhead = none ];
174 -> 61;

175 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 175 [arrowhead = none ];
175 -> 62;

176 [label="ONLY EVAL with clause\nle(0, X177, true).\nand substitution\nX177 -> 0\nX166 -> true", fontsize=14, style = filled, fillcolor = white];
61 -> 176 [arrowhead = none ];
176 -> 63;

177 [label="BACKTRACK\nfor clause: le(s(X), 0, false)because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 177 [arrowhead = none ];
177 -> 65;

178 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 178 [arrowhead = none ];
178 -> 64;

179 [label="BACKTRACK\nfor clause: le(s(X), s(Y), B) :- le(X, Y, B)because of non-unification", fontsize=14, style = filled, fillcolor = white];
65 -> 179 [arrowhead = none ];
179 -> 66;

180 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
67 -> 180 [arrowhead = none ];
180 -> 68;

181 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
67 -> 181 [arrowhead = none ];
181 -> 69;

182 [label="EVAL with clause\nif(true, X194, X195, 0).\nand substitution\nT85 -> true\nX194 -> 0\nX195 -> 0\nT84 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 182 [arrowhead = none ];
182 -> 70;

183 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 183 [arrowhead = none ];
183 -> 71;

184 [label="EVAL with clause\nif(false, X205, X206, s(X207)) :- ','(p(X205, X208), minus(X208, X206, X207)).\nand substitution\nT85 -> false\nX205 -> 0\nX206 -> 0\nX207 -> T89\nT84 -> s(T89)\nT88 -> T89", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 184 [arrowhead = none ];
184 -> 73;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 185 [arrowhead = none ];
185 -> 74;

186 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 186 [arrowhead = none ];
186 -> 72;

187 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 187 [arrowhead = none ];
187 -> 75;

188 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
75 -> 188 [arrowhead = none ];
188 -> 76;

189 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
75 -> 189 [arrowhead = none ];
189 -> 77;

190 [label="ONLY EVAL with clause\np(0, 0).\nand substitution\nX208 -> 0", fontsize=14, style = filled, fillcolor = white];
76 -> 190 [arrowhead = none ];
190 -> 78;

191 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = white];
77 -> 191 [arrowhead = none ];
191 -> 79;

192 [label="INSTANCE with matching:\nT77 -> T89", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 192 [arrowhead = none , style=dashed];
192 -> 54[style=dashed];

193 [label="INSTANCE with matching:\nT46 -> T96\nT40 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
80 -> 193 [arrowhead = none , style=dashed];
193 -> 26[style=dashed];

194 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
82 -> 194 [arrowhead = none ];
194 -> 84;

195 [label="SPLIT 2\nnew knowledge:\nT109 is ground\nreplacements:\nX249 -> T109\nT105 -> T110\nT106 -> T111", fontsize=14, style = filled, fillcolor = violet];
82 -> 195 [arrowhead = none ];
195 -> 85;

196 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
84 -> 196 [arrowhead = none ];
196 -> 86;

197 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 197 [arrowhead = none ];
197 -> 99;

198 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 198 [arrowhead = none ];
198 -> 87;

199 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
86 -> 199 [arrowhead = none ];
199 -> 88;

200 [label="EVAL with clause\nle(0, X261, true).\nand substitution\nT103 -> 0\nT105 -> T118\nX261 -> T118\nX249 -> true", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 200 [arrowhead = none ];
200 -> 89;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 201 [arrowhead = none ];
201 -> 90;

202 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
88 -> 202 [arrowhead = none ];
202 -> 92;

203 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
88 -> 203 [arrowhead = none ];
203 -> 93;

204 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 204 [arrowhead = none ];
204 -> 91;

205 [label="EVAL with clause\nle(s(X267), 0, false).\nand substitution\nX267 -> T123\nT103 -> s(T123)\nT105 -> 0\nX249 -> false", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 205 [arrowhead = none ];
205 -> 94;

206 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 206 [arrowhead = none ];
206 -> 95;

207 [label="EVAL with clause\nle(s(X277), s(X278), X279) :- le(X277, X278, X279).\nand substitution\nX277 -> T128\nT103 -> s(T128)\nX278 -> T130\nT105 -> s(T130)\nX249 -> X280\nX279 -> X280\nT129 -> T130", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 207 [arrowhead = none ];
207 -> 97;

208 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 208 [arrowhead = none ];
208 -> 98;

209 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
94 -> 209 [arrowhead = none ];
209 -> 96;

210 [label="INSTANCE with matching:\nT103 -> T128\nT105 -> T130\nX249 -> X280", fontsize=14, style = filled, fillcolor = lightgrey];
97 -> 210 [arrowhead = none , style=dashed];
210 -> 84[style=dashed];

211 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
99 -> 211 [arrowhead = none ];
211 -> 100;

212 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
99 -> 212 [arrowhead = none ];
212 -> 101;

213 [label="EVAL with clause\nif(true, X298, X299, 0).\nand substitution\nT109 -> true\nT103 -> T145\nX298 -> s(T145)\nT110 -> T146\nX299 -> s(T146)\nT111 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 213 [arrowhead = none ];
213 -> 102;

214 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 214 [arrowhead = none ];
214 -> 103;

215 [label="EVAL with clause\nif(false, X309, X310, s(X311)) :- ','(p(X309, X312), minus(X312, X310, X311)).\nand substitution\nT109 -> false\nT103 -> T153\nX309 -> s(T153)\nT110 -> T157\nX310 -> s(T157)\nX311 -> T156\nT111 -> s(T156)\nT155 -> T156\nT154 -> T157", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 215 [arrowhead = none ];
215 -> 105;

216 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 216 [arrowhead = none ];
216 -> 106;

217 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
102 -> 217 [arrowhead = none ];
217 -> 104;

218 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
105 -> 218 [arrowhead = none ];
218 -> 107;

219 [label="BACKTRACK\nfor clause: p(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = white];
107 -> 219 [arrowhead = none ];
219 -> 108;

220 [label="ONLY EVAL with clause\np(s(X327), X327).\nand substitution\nT153 -> T165\nX327 -> T165\nX312 -> T165", fontsize=14, style = filled, fillcolor = white];
108 -> 220 [arrowhead = none ];
220 -> 109;

221 [label="INSTANCE with matching:\nT1 -> T165\nT2 -> s(T157)\nT3 -> T156", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 221 [arrowhead = none , style=dashed];
221 -> 1[style=dashed];

}

(2) Obligation:

Triples:

minus54(s(T89)) :- ','(lec58(false), minus54(T89)).
minus26(0, s(T77)) :- ','(lec29(0, false), minus54(T77)).
minus26(s(T96), s(T77)) :- ','(lec29(s(T96), false), minus26(T96, T77)).
le84(s(T128), s(T130), X280) :- le84(T128, T130, X280).
minus1(s(T46), 0, s(T40)) :- minus26(T46, T40).
minus1(s(T103), s(T105), T106) :- le84(T103, T105, X249).
minus1(s(T165), s(T157), s(T156)) :- ','(lec84(T165, T157, false), minus1(T165, s(T157), T156)).

Clauses:

minusc54(0) :- lec58(true).
minusc54(s(T89)) :- ','(lec58(false), minusc54(T89)).
minusc26(T70, 0) :- lec29(T70, true).
minusc26(0, s(T77)) :- ','(lec29(0, false), minusc54(T77)).
minusc26(s(T96), s(T77)) :- ','(lec29(s(T96), false), minusc26(T96, T77)).
lec84(0, T118, true).
lec84(s(T123), 0, false).
lec84(s(T128), s(T130), X280) :- lec84(T128, T130, X280).
minusc1(0, T25, 0).
minusc1(s(T46), 0, s(T40)) :- minusc26(T46, T40).
minusc1(s(T145), s(T146), 0) :- lec84(T145, T146, true).
minusc1(s(T165), s(T157), s(T156)) :- ','(lec84(T165, T157, false), minusc1(T165, s(T157), T156)).
lec29(0, true).
lec29(s(T62), false).
lec58(true).

Afs:

minus1(x1, x2, x3)  =  minus1(x1)

(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:
minus1_in: (b,f,f)
minus26_in: (b,f)
minus54_in: (f)
le84_in: (b,f,f)
lec84_in: (b,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MINUS1_IN_GAA(s(T46), 0, s(T40)) → U8_GAA(T46, T40, minus26_in_ga(T46, T40))
MINUS1_IN_GAA(s(T46), 0, s(T40)) → MINUS26_IN_GA(T46, T40)
MINUS26_IN_GA(0, s(T77)) → U3_GA(T77, lec29_in_gg(0, false))
U3_GA(T77, lec29_out_gg(0, false)) → U4_GA(T77, minus54_in_a(T77))
U3_GA(T77, lec29_out_gg(0, false)) → MINUS54_IN_A(T77)
MINUS54_IN_A(s(T89)) → U1_A(T89, lec58_in_g(false))
U1_A(T89, lec58_out_g(false)) → U2_A(T89, minus54_in_a(T89))
U1_A(T89, lec58_out_g(false)) → MINUS54_IN_A(T89)
MINUS26_IN_GA(s(T96), s(T77)) → U5_GA(T96, T77, lec29_in_gg(s(T96), false))
U5_GA(T96, T77, lec29_out_gg(s(T96), false)) → U6_GA(T96, T77, minus26_in_ga(T96, T77))
U5_GA(T96, T77, lec29_out_gg(s(T96), false)) → MINUS26_IN_GA(T96, T77)
MINUS1_IN_GAA(s(T103), s(T105), T106) → U9_GAA(T103, T105, T106, le84_in_gaa(T103, T105, X249))
MINUS1_IN_GAA(s(T103), s(T105), T106) → LE84_IN_GAA(T103, T105, X249)
LE84_IN_GAA(s(T128), s(T130), X280) → U7_GAA(T128, T130, X280, le84_in_gaa(T128, T130, X280))
LE84_IN_GAA(s(T128), s(T130), X280) → LE84_IN_GAA(T128, T130, X280)
MINUS1_IN_GAA(s(T165), s(T157), s(T156)) → U10_GAA(T165, T157, T156, lec84_in_gag(T165, T157, false))
U10_GAA(T165, T157, T156, lec84_out_gag(T165, T157, false)) → U11_GAA(T165, T157, T156, minus1_in_gaa(T165, s(T157), T156))
U10_GAA(T165, T157, T156, lec84_out_gag(T165, T157, false)) → MINUS1_IN_GAA(T165, s(T157), T156)

The TRS R consists of the following rules:

lec29_in_gg(0, true) → lec29_out_gg(0, true)
lec29_in_gg(s(T62), false) → lec29_out_gg(s(T62), false)
lec58_in_g(true) → lec58_out_g(true)
lec84_in_gag(0, T118, true) → lec84_out_gag(0, T118, true)
lec84_in_gag(s(T123), 0, false) → lec84_out_gag(s(T123), 0, false)
lec84_in_gag(s(T128), s(T130), X280) → U21_gag(T128, T130, X280, lec84_in_gag(T128, T130, X280))
U21_gag(T128, T130, X280, lec84_out_gag(T128, T130, X280)) → lec84_out_gag(s(T128), s(T130), X280)

The argument filtering Pi contains the following mapping:
minus1_in_gaa(x1, x2, x3)  =  minus1_in_gaa(x1)
s(x1)  =  s(x1)
minus26_in_ga(x1, x2)  =  minus26_in_ga(x1)
0  =  0
lec29_in_gg(x1, x2)  =  lec29_in_gg(x1, x2)
true  =  true
lec29_out_gg(x1, x2)  =  lec29_out_gg(x1, x2)
false  =  false
minus54_in_a(x1)  =  minus54_in_a
lec58_in_g(x1)  =  lec58_in_g(x1)
lec58_out_g(x1)  =  lec58_out_g(x1)
le84_in_gaa(x1, x2, x3)  =  le84_in_gaa(x1)
lec84_in_gag(x1, x2, x3)  =  lec84_in_gag(x1, x3)
lec84_out_gag(x1, x2, x3)  =  lec84_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUS1_IN_GAA(x1, x2, x3)  =  MINUS1_IN_GAA(x1)
U8_GAA(x1, x2, x3)  =  U8_GAA(x1, x3)
MINUS26_IN_GA(x1, x2)  =  MINUS26_IN_GA(x1)
U3_GA(x1, x2)  =  U3_GA(x2)
U4_GA(x1, x2)  =  U4_GA(x2)
MINUS54_IN_A(x1)  =  MINUS54_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
LE84_IN_GAA(x1, x2, x3)  =  LE84_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, 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:

MINUS1_IN_GAA(s(T46), 0, s(T40)) → U8_GAA(T46, T40, minus26_in_ga(T46, T40))
MINUS1_IN_GAA(s(T46), 0, s(T40)) → MINUS26_IN_GA(T46, T40)
MINUS26_IN_GA(0, s(T77)) → U3_GA(T77, lec29_in_gg(0, false))
U3_GA(T77, lec29_out_gg(0, false)) → U4_GA(T77, minus54_in_a(T77))
U3_GA(T77, lec29_out_gg(0, false)) → MINUS54_IN_A(T77)
MINUS54_IN_A(s(T89)) → U1_A(T89, lec58_in_g(false))
U1_A(T89, lec58_out_g(false)) → U2_A(T89, minus54_in_a(T89))
U1_A(T89, lec58_out_g(false)) → MINUS54_IN_A(T89)
MINUS26_IN_GA(s(T96), s(T77)) → U5_GA(T96, T77, lec29_in_gg(s(T96), false))
U5_GA(T96, T77, lec29_out_gg(s(T96), false)) → U6_GA(T96, T77, minus26_in_ga(T96, T77))
U5_GA(T96, T77, lec29_out_gg(s(T96), false)) → MINUS26_IN_GA(T96, T77)
MINUS1_IN_GAA(s(T103), s(T105), T106) → U9_GAA(T103, T105, T106, le84_in_gaa(T103, T105, X249))
MINUS1_IN_GAA(s(T103), s(T105), T106) → LE84_IN_GAA(T103, T105, X249)
LE84_IN_GAA(s(T128), s(T130), X280) → U7_GAA(T128, T130, X280, le84_in_gaa(T128, T130, X280))
LE84_IN_GAA(s(T128), s(T130), X280) → LE84_IN_GAA(T128, T130, X280)
MINUS1_IN_GAA(s(T165), s(T157), s(T156)) → U10_GAA(T165, T157, T156, lec84_in_gag(T165, T157, false))
U10_GAA(T165, T157, T156, lec84_out_gag(T165, T157, false)) → U11_GAA(T165, T157, T156, minus1_in_gaa(T165, s(T157), T156))
U10_GAA(T165, T157, T156, lec84_out_gag(T165, T157, false)) → MINUS1_IN_GAA(T165, s(T157), T156)

The TRS R consists of the following rules:

lec29_in_gg(0, true) → lec29_out_gg(0, true)
lec29_in_gg(s(T62), false) → lec29_out_gg(s(T62), false)
lec58_in_g(true) → lec58_out_g(true)
lec84_in_gag(0, T118, true) → lec84_out_gag(0, T118, true)
lec84_in_gag(s(T123), 0, false) → lec84_out_gag(s(T123), 0, false)
lec84_in_gag(s(T128), s(T130), X280) → U21_gag(T128, T130, X280, lec84_in_gag(T128, T130, X280))
U21_gag(T128, T130, X280, lec84_out_gag(T128, T130, X280)) → lec84_out_gag(s(T128), s(T130), X280)

The argument filtering Pi contains the following mapping:
minus1_in_gaa(x1, x2, x3)  =  minus1_in_gaa(x1)
s(x1)  =  s(x1)
minus26_in_ga(x1, x2)  =  minus26_in_ga(x1)
0  =  0
lec29_in_gg(x1, x2)  =  lec29_in_gg(x1, x2)
true  =  true
lec29_out_gg(x1, x2)  =  lec29_out_gg(x1, x2)
false  =  false
minus54_in_a(x1)  =  minus54_in_a
lec58_in_g(x1)  =  lec58_in_g(x1)
lec58_out_g(x1)  =  lec58_out_g(x1)
le84_in_gaa(x1, x2, x3)  =  le84_in_gaa(x1)
lec84_in_gag(x1, x2, x3)  =  lec84_in_gag(x1, x3)
lec84_out_gag(x1, x2, x3)  =  lec84_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUS1_IN_GAA(x1, x2, x3)  =  MINUS1_IN_GAA(x1)
U8_GAA(x1, x2, x3)  =  U8_GAA(x1, x3)
MINUS26_IN_GA(x1, x2)  =  MINUS26_IN_GA(x1)
U3_GA(x1, x2)  =  U3_GA(x2)
U4_GA(x1, x2)  =  U4_GA(x2)
MINUS54_IN_A(x1)  =  MINUS54_IN_A
U1_A(x1, x2)  =  U1_A(x2)
U2_A(x1, x2)  =  U2_A(x2)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)
U6_GA(x1, x2, x3)  =  U6_GA(x1, x3)
U9_GAA(x1, x2, x3, x4)  =  U9_GAA(x1, x4)
LE84_IN_GAA(x1, x2, x3)  =  LE84_IN_GAA(x1)
U7_GAA(x1, x2, x3, x4)  =  U7_GAA(x1, x4)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)
U11_GAA(x1, x2, x3, x4)  =  U11_GAA(x1, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 13 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

LE84_IN_GAA(s(T128), s(T130), X280) → LE84_IN_GAA(T128, T130, X280)

The TRS R consists of the following rules:

lec29_in_gg(0, true) → lec29_out_gg(0, true)
lec29_in_gg(s(T62), false) → lec29_out_gg(s(T62), false)
lec58_in_g(true) → lec58_out_g(true)
lec84_in_gag(0, T118, true) → lec84_out_gag(0, T118, true)
lec84_in_gag(s(T123), 0, false) → lec84_out_gag(s(T123), 0, false)
lec84_in_gag(s(T128), s(T130), X280) → U21_gag(T128, T130, X280, lec84_in_gag(T128, T130, X280))
U21_gag(T128, T130, X280, lec84_out_gag(T128, T130, X280)) → lec84_out_gag(s(T128), s(T130), X280)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lec29_in_gg(x1, x2)  =  lec29_in_gg(x1, x2)
true  =  true
lec29_out_gg(x1, x2)  =  lec29_out_gg(x1, x2)
false  =  false
lec58_in_g(x1)  =  lec58_in_g(x1)
lec58_out_g(x1)  =  lec58_out_g(x1)
lec84_in_gag(x1, x2, x3)  =  lec84_in_gag(x1, x3)
lec84_out_gag(x1, x2, x3)  =  lec84_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
LE84_IN_GAA(x1, x2, x3)  =  LE84_IN_GAA(x1)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

LE84_IN_GAA(s(T128), s(T130), X280) → LE84_IN_GAA(T128, T130, X280)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
LE84_IN_GAA(x1, x2, x3)  =  LE84_IN_GAA(x1)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

LE84_IN_GAA(s(T128)) → LE84_IN_GAA(T128)

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

(12) 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:

• LE84_IN_GAA(s(T128)) → LE84_IN_GAA(T128)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

MINUS1_IN_GAA(s(T165), s(T157), s(T156)) → U10_GAA(T165, T157, T156, lec84_in_gag(T165, T157, false))
U10_GAA(T165, T157, T156, lec84_out_gag(T165, T157, false)) → MINUS1_IN_GAA(T165, s(T157), T156)

The TRS R consists of the following rules:

lec29_in_gg(0, true) → lec29_out_gg(0, true)
lec29_in_gg(s(T62), false) → lec29_out_gg(s(T62), false)
lec58_in_g(true) → lec58_out_g(true)
lec84_in_gag(0, T118, true) → lec84_out_gag(0, T118, true)
lec84_in_gag(s(T123), 0, false) → lec84_out_gag(s(T123), 0, false)
lec84_in_gag(s(T128), s(T130), X280) → U21_gag(T128, T130, X280, lec84_in_gag(T128, T130, X280))
U21_gag(T128, T130, X280, lec84_out_gag(T128, T130, X280)) → lec84_out_gag(s(T128), s(T130), X280)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lec29_in_gg(x1, x2)  =  lec29_in_gg(x1, x2)
true  =  true
lec29_out_gg(x1, x2)  =  lec29_out_gg(x1, x2)
false  =  false
lec58_in_g(x1)  =  lec58_in_g(x1)
lec58_out_g(x1)  =  lec58_out_g(x1)
lec84_in_gag(x1, x2, x3)  =  lec84_in_gag(x1, x3)
lec84_out_gag(x1, x2, x3)  =  lec84_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUS1_IN_GAA(x1, x2, x3)  =  MINUS1_IN_GAA(x1)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

MINUS1_IN_GAA(s(T165), s(T157), s(T156)) → U10_GAA(T165, T157, T156, lec84_in_gag(T165, T157, false))
U10_GAA(T165, T157, T156, lec84_out_gag(T165, T157, false)) → MINUS1_IN_GAA(T165, s(T157), T156)

The TRS R consists of the following rules:

lec84_in_gag(s(T123), 0, false) → lec84_out_gag(s(T123), 0, false)
lec84_in_gag(s(T128), s(T130), X280) → U21_gag(T128, T130, X280, lec84_in_gag(T128, T130, X280))
U21_gag(T128, T130, X280, lec84_out_gag(T128, T130, X280)) → lec84_out_gag(s(T128), s(T130), X280)
lec84_in_gag(0, T118, true) → lec84_out_gag(0, T118, true)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
true  =  true
false  =  false
lec84_in_gag(x1, x2, x3)  =  lec84_in_gag(x1, x3)
lec84_out_gag(x1, x2, x3)  =  lec84_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUS1_IN_GAA(x1, x2, x3)  =  MINUS1_IN_GAA(x1)
U10_GAA(x1, x2, x3, x4)  =  U10_GAA(x1, x4)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

MINUS1_IN_GAA(s(T165)) → U10_GAA(T165, lec84_in_gag(T165, false))
U10_GAA(T165, lec84_out_gag(T165, false)) → MINUS1_IN_GAA(T165)

The TRS R consists of the following rules:

lec84_in_gag(s(T123), false) → lec84_out_gag(s(T123), false)
lec84_in_gag(s(T128), X280) → U21_gag(T128, X280, lec84_in_gag(T128, X280))
U21_gag(T128, X280, lec84_out_gag(T128, X280)) → lec84_out_gag(s(T128), X280)
lec84_in_gag(0, true) → lec84_out_gag(0, true)

The set Q consists of the following terms:

lec84_in_gag(x0, x1)
U21_gag(x0, x1, x2)

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

(19) 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:

• U10_GAA(T165, lec84_out_gag(T165, false)) → MINUS1_IN_GAA(T165)
  The graph contains the following edges 1 >= 1, 2 > 1

• MINUS1_IN_GAA(s(T165)) → U10_GAA(T165, lec84_in_gag(T165, false))
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

MINUS26_IN_GA(s(T96), s(T77)) → U5_GA(T96, T77, lec29_in_gg(s(T96), false))
U5_GA(T96, T77, lec29_out_gg(s(T96), false)) → MINUS26_IN_GA(T96, T77)

The TRS R consists of the following rules:

lec29_in_gg(0, true) → lec29_out_gg(0, true)
lec29_in_gg(s(T62), false) → lec29_out_gg(s(T62), false)
lec58_in_g(true) → lec58_out_g(true)
lec84_in_gag(0, T118, true) → lec84_out_gag(0, T118, true)
lec84_in_gag(s(T123), 0, false) → lec84_out_gag(s(T123), 0, false)
lec84_in_gag(s(T128), s(T130), X280) → U21_gag(T128, T130, X280, lec84_in_gag(T128, T130, X280))
U21_gag(T128, T130, X280, lec84_out_gag(T128, T130, X280)) → lec84_out_gag(s(T128), s(T130), X280)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lec29_in_gg(x1, x2)  =  lec29_in_gg(x1, x2)
true  =  true
lec29_out_gg(x1, x2)  =  lec29_out_gg(x1, x2)
false  =  false
lec58_in_g(x1)  =  lec58_in_g(x1)
lec58_out_g(x1)  =  lec58_out_g(x1)
lec84_in_gag(x1, x2, x3)  =  lec84_in_gag(x1, x3)
lec84_out_gag(x1, x2, x3)  =  lec84_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUS26_IN_GA(x1, x2)  =  MINUS26_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

MINUS26_IN_GA(s(T96), s(T77)) → U5_GA(T96, T77, lec29_in_gg(s(T96), false))
U5_GA(T96, T77, lec29_out_gg(s(T96), false)) → MINUS26_IN_GA(T96, T77)

The TRS R consists of the following rules:

lec29_in_gg(s(T62), false) → lec29_out_gg(s(T62), false)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lec29_in_gg(x1, x2)  =  lec29_in_gg(x1, x2)
lec29_out_gg(x1, x2)  =  lec29_out_gg(x1, x2)
false  =  false
MINUS26_IN_GA(x1, x2)  =  MINUS26_IN_GA(x1)
U5_GA(x1, x2, x3)  =  U5_GA(x1, x3)

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

(24) PiDPToQDPProof (SOUND transformation)

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

(25) Obligation:

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

MINUS26_IN_GA(s(T96)) → U5_GA(T96, lec29_in_gg(s(T96), false))
U5_GA(T96, lec29_out_gg(s(T96), false)) → MINUS26_IN_GA(T96)

The TRS R consists of the following rules:

lec29_in_gg(s(T62), false) → lec29_out_gg(s(T62), false)

The set Q consists of the following terms:

lec29_in_gg(x0, x1)

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

(26) 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:

• U5_GA(T96, lec29_out_gg(s(T96), false)) → MINUS26_IN_GA(T96)
  The graph contains the following edges 1 >= 1, 2 > 1

• MINUS26_IN_GA(s(T96)) → U5_GA(T96, lec29_in_gg(s(T96), false))
  The graph contains the following edges 1 > 1

(27) YES