Left Termination of the query pattern
minus(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)).

Query: minus(g,a,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="F: 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(T36), X52), minus(X52, 0, T38))\n\nT36 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(s(T36), X52), minus(X52, 0, T38)), SCOPE: 5, CLAUSE: 0 | ','(p(s(T36), X52), minus(X52, 0, T38)), SCOPE: 5, CLAUSE: 1\n\nT36 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(s(T36), X52), minus(X52, 0, T38)), SCOPE: 5, CLAUSE: 1\n\nT36 is ground\nX52 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="C: minus(T44, 0, T38)\n\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: minus(T44, 0, T38), SCOPE: 6, CLAUSE: 5\n\nT44 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(le(T52, 0, X81), if(X81, T52, 0, T54))\n\nT52 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="D: le(T52, 0, X81)\n\nT52 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: if(T55, T52, 0, T54)\n\nT52 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: le(T52, 0, X81), SCOPE: 7, CLAUSE: 2 | le(T52, 0, X81), SCOPE: 7, CLAUSE: 3 | le(T52, 0, X81), SCOPE: 7, CLAUSE: 4\n\nT52 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: le(T52, 0, X81), SCOPE: 7, CLAUSE: 2\n\nT52 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: le(T52, 0, X81), SCOPE: 7, CLAUSE: 3 | le(T52, 0, X81), SCOPE: 7, CLAUSE: 4\n\nT52 is ground\nX81 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(T52, 0, X81), SCOPE: 7, CLAUSE: 3\n\nT52 is ground\nX81 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: le(T52, 0, X81), SCOPE: 7, CLAUSE: 4\n\nT52 is ground\nX81 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(T55, T52, 0, T54), SCOPE: 8, CLAUSE: 6 | if(T55, T52, 0, T54), SCOPE: 8, CLAUSE: 7\n\nT52 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: if(T55, T52, 0, T54), SCOPE: 8, CLAUSE: 6\n\nT52 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: if(T55, T52, 0, T54), SCOPE: 8, CLAUSE: 7\n\nT52 is ground\nT55 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(T73, X123), minus(X123, 0, T75))\n\nT73 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(p(T73, X123), minus(X123, 0, T75)), SCOPE: 9, CLAUSE: 0 | ','(p(T73, X123), minus(X123, 0, T75)), SCOPE: 9, CLAUSE: 1\n\nT73 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(p(T73, X123), minus(X123, 0, T75)), SCOPE: 9, CLAUSE: 0\n\nT73 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T73, X123), minus(X123, 0, T75)), SCOPE: 9, CLAUSE: 1\n\nT73 is ground\nX123 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="A: minus(0, 0, T75)\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, T75), SCOPE: 10, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(le(0, 0, X148), if(X148, 0, 0, T82))\n\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="B: le(0, 0, X148)\n\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: if(T83, 0, 0, T82)\n\nT83 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: le(0, 0, X148), SCOPE: 11, CLAUSE: 2 | le(0, 0, X148), SCOPE: 11, CLAUSE: 3 | le(0, 0, X148), SCOPE: 11, CLAUSE: 4\n\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: le(0, 0, X148), SCOPE: 11, CLAUSE: 2\n\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: le(0, 0, X148), SCOPE: 11, CLAUSE: 3 | le(0, 0, X148), SCOPE: 11, CLAUSE: 4\n\nX148 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, X148), SCOPE: 11, CLAUSE: 4\n\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: if(T83, 0, 0, T82), SCOPE: 12, CLAUSE: 6 | if(T83, 0, 0, T82), SCOPE: 12, CLAUSE: 7\n\nT83 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: if(T83, 0, 0, T82), SCOPE: 12, CLAUSE: 6\n\nT83 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: if(T83, 0, 0, T82), SCOPE: 12, CLAUSE: 7\n\nT83 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, X188), minus(X188, 0, T87))\n\nX188 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(p(0, X188), minus(X188, 0, T87)), SCOPE: 13, CLAUSE: 0 | ','(p(0, X188), minus(X188, 0, T87)), SCOPE: 13, CLAUSE: 1\n\nX188 is free", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(p(0, X188), minus(X188, 0, T87)), SCOPE: 13, CLAUSE: 0\n\nX188 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(p(0, X188), minus(X188, 0, T87)), SCOPE: 13, CLAUSE: 1\n\nX188 is free", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: minus(0, 0, T87)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: minus(T94, 0, T75)\n\nT94 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(le(T101, T103, X225), if(X225, s(T101), s(T103), T104))\n\nT101 is ground\nX225 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="E: le(T101, T103, X225)\n\nT101 is ground\nX225 is free", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: if(T107, s(T101), s(T108), T109)\n\nT101 is ground\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: le(T101, T103, X225), SCOPE: 14, CLAUSE: 2 | le(T101, T103, X225), SCOPE: 14, CLAUSE: 3 | le(T101, T103, X225), SCOPE: 14, CLAUSE: 4\n\nT101 is ground\nX225 is free", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: le(T101, T103, X225), SCOPE: 14, CLAUSE: 2\n\nT101 is ground\nX225 is free", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: le(T101, T103, X225), SCOPE: 14, CLAUSE: 3 | le(T101, T103, X225), SCOPE: 14, CLAUSE: 4\n\nT101 is ground\nX225 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(T101, T103, X225), SCOPE: 14, CLAUSE: 3\n\nT101 is ground\nX225 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: le(T101, T103, X225), SCOPE: 14, CLAUSE: 4\n\nT101 is ground\nX225 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(T126, T128, X251)\n\nT126 is ground\nX251 is free", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: if(T107, s(T101), s(T108), T109), SCOPE: 15, CLAUSE: 6 | if(T107, s(T101), s(T108), T109), SCOPE: 15, CLAUSE: 7\n\nT101 is ground\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: if(T107, s(T101), s(T108), T109), SCOPE: 15, CLAUSE: 6\n\nT101 is ground\nT107 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: if(T107, s(T101), s(T108), T109), SCOPE: 15, CLAUSE: 7\n\nT101 is ground\nT107 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(T151), X278), minus(X278, s(T155), T154))\n\nT151 is ground\nX278 is free", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: ','(p(s(T151), X278), minus(X278, s(T155), T154)), SCOPE: 16, CLAUSE: 0 | ','(p(s(T151), X278), minus(X278, s(T155), T154)), SCOPE: 16, CLAUSE: 1\n\nT151 is ground\nX278 is free", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: ','(p(s(T151), X278), minus(X278, s(T155), T154)), SCOPE: 16, CLAUSE: 1\n\nT151 is ground\nX278 is free", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: minus(T163, s(T155), T154)\n\nT163 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 substitutionT1 -> T7,\nX5 -> T7,\nT2 -> T10,\nX6 -> T10,\nT3 -> T11,\nX7 -> T11,\nT8 -> T10,\nT9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
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 substitutionT7 -> 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 = lightgreen];
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, X26, X27, 0).\nand substitutionX26 -> 0,\nT17 -> T25,\nX27 -> T25,\nT18 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 122 [arrowhead = none ];
122 -> 12;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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(X35), 0, false).\nand substitutionX35 -> 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 = lightgreen];
16 -> 127 [arrowhead = none ];
127 -> 19;

128 [label="EVAL with clause\nle(s(X222), s(X223), X224) :- le(X222, X223, X224).\nand substitutionX222 -> T101,\nT7 -> s(T101),\nX223 -> T103,\nT10 -> s(T103),\nX8 -> X225,\nX224 -> X225,\nT102 -> T103,\nT11 -> T104", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 128 [arrowhead = none ];
128 -> 82;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
20 -> 131 [arrowhead = none ];
131 -> 21;

132 [label="EVAL with clause\nif(false, X49, X50, s(X51)) :- ','(p(X49, X52), minus(X52, X50, X51)).\nand substitutionT30 -> T36,\nX49 -> s(T36),\nX50 -> 0,\nX51 -> T38,\nT31 -> s(T38),\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 132 [arrowhead = none ];
132 -> 22;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
24 -> 135 [arrowhead = none ];
135 -> 25;

136 [label="ONLY EVAL with clause\np(s(X64), X64).\nand substitutionT36 -> T44,\nX64 -> T44,\nX52 -> T44", fontsize=14, style = filled, fillcolor = lightblue];
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(X78, X79, X80) :- ','(le(X78, X79, X81), if(X81, X78, X79, X80)).\nand substitutionT44 -> T52,\nX78 -> T52,\nX79 -> 0,\nT38 -> T54,\nX80 -> T54,\nT53 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT52 is ground\nT55 is ground\nreplacements:X81 -> T55", 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, X90, true).\nand substitutionT52 -> 0,\nX90 -> 0,\nX81 -> true", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 145 [arrowhead = none ];
145 -> 34;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X95), 0, false).\nand substitutionX95 -> T60,\nT52 -> s(T60),\nX81 -> false", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 150 [arrowhead = none ];
150 -> 39;

151 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
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, X111, X112, 0).\nand substitutionT55 -> true,\nT52 -> T68,\nX111 -> T68,\nX112 -> 0,\nT54 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 156 [arrowhead = none ];
156 -> 46;

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

158 [label="EVAL with clause\nif(false, X120, X121, s(X122)) :- ','(p(X120, X123), minus(X123, X121, X122)).\nand substitutionT55 -> false,\nT52 -> T73,\nX120 -> T73,\nX121 -> 0,\nX122 -> T75,\nT54 -> s(T75),\nT74 -> T75", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 158 [arrowhead = none ];
158 -> 49;

159 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionT73 -> 0,\nX123 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 164 [arrowhead = none ];
164 -> 54;

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

166 [label="EVAL with clause\np(s(X209), X209).\nand substitutionX209 -> T94,\nT73 -> s(T94),\nX123 -> T94", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 166 [arrowhead = none ];
166 -> 80;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X145, X146, X147) :- ','(le(X145, X146, X148), if(X148, X145, X146, X147)).\nand substitutionX145 -> 0,\nX146 -> 0,\nT75 -> T82,\nX147 -> T82,\nT81 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
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:\nT83 is ground\nreplacements:X148 -> T83", 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, X159, true).\nand substitutionX159 -> 0,\nX148 -> true", fontsize=14, style = filled, fillcolor = lightblue];
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 = lightgreen];
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 = lightgreen];
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, X176, X177, 0).\nand substitutionT83 -> true,\nX176 -> 0,\nX177 -> 0,\nT82 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 182 [arrowhead = none ];
182 -> 70;

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

184 [label="EVAL with clause\nif(false, X185, X186, s(X187)) :- ','(p(X185, X188), minus(X188, X186, X187)).\nand substitutionT83 -> false,\nX185 -> 0,\nX186 -> 0,\nX187 -> T87,\nT82 -> s(T87),\nT86 -> T87", fontsize=14, style = filled, fillcolor = lightblue];
69 -> 184 [arrowhead = none ];
184 -> 73;

185 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 substitutionX188 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
76 -> 190 [arrowhead = none ];
190 -> 78;

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

192 [label="INSTANCE with matching:\nT75 -> T87", fontsize=14, style = filled, fillcolor = lightgrey];
78 -> 192 [arrowhead = none , style=dashed];
192 -> 54[style=dashed];

193 [label="INSTANCE with matching:\nT44 -> T94\nT38 -> T75", 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:\nT101 is ground\nT107 is ground\nreplacements:X225 -> T107,\nT103 -> T108,\nT104 -> T109", 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, X234, true).\nand substitutionT101 -> 0,\nT103 -> T116,\nX234 -> T116,\nX225 -> true", fontsize=14, style = filled, fillcolor = lightblue];
87 -> 200 [arrowhead = none ];
200 -> 89;

201 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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(X239), 0, false).\nand substitutionX239 -> T121,\nT101 -> s(T121),\nT103 -> 0,\nX225 -> false", fontsize=14, style = filled, fillcolor = lightblue];
92 -> 205 [arrowhead = none ];
205 -> 94;

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

207 [label="EVAL with clause\nle(s(X248), s(X249), X250) :- le(X248, X249, X250).\nand substitutionX248 -> T126,\nT101 -> s(T126),\nX249 -> T128,\nT103 -> s(T128),\nX225 -> X251,\nX250 -> X251,\nT127 -> T128", fontsize=14, style = filled, fillcolor = lightblue];
93 -> 207 [arrowhead = none ];
207 -> 97;

208 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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:\nT101 -> T126\nT103 -> T128\nX225 -> X251", 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, X266, X267, 0).\nand substitutionT107 -> true,\nT101 -> T143,\nX266 -> s(T143),\nT108 -> T144,\nX267 -> s(T144),\nT109 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 213 [arrowhead = none ];
213 -> 102;

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

215 [label="EVAL with clause\nif(false, X275, X276, s(X277)) :- ','(p(X275, X278), minus(X278, X276, X277)).\nand substitutionT107 -> false,\nT101 -> T151,\nX275 -> s(T151),\nT108 -> T155,\nX276 -> s(T155),\nX277 -> T154,\nT109 -> s(T154),\nT153 -> T154,\nT152 -> T155", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 215 [arrowhead = none ];
215 -> 105;

216 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
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 = lightgreen];
107 -> 219 [arrowhead = none ];
219 -> 108;

220 [label="ONLY EVAL with clause\np(s(X290), X290).\nand substitutionT151 -> T163,\nX290 -> T163,\nX278 -> T163", fontsize=14, style = filled, fillcolor = lightblue];
108 -> 220 [arrowhead = none ];
220 -> 109;

221 [label="INSTANCE with matching:\nT1 -> T163\nT2 -> s(T155)\nT3 -> T154", fontsize=14, style = filled, fillcolor = lightgrey];
109 -> 221 [arrowhead = none , style=dashed];
221 -> 1[style=dashed];

}

(2) Obligation:

Triples:

minusA(s(X1)) :- ','(lecB(false), minusA(X1)).
minusC(0, s(X1)) :- ','(lecD(0, false), minusA(X1)).
minusC(s(X1), s(X2)) :- ','(lecD(s(X1), false), minusC(X1, X2)).
leE(s(X1), s(X2), X3) :- leE(X1, X2, X3).
minusF(s(X1), 0, s(X2)) :- minusC(X1, X2).
minusF(s(X1), s(X2), X3) :- leE(X1, X2, X4).
minusF(s(X1), s(X2), s(X3)) :- ','(lecE(X1, X2, false), minusF(X1, s(X2), X3)).

Clauses:

minuscA(0) :- lecB(true).
minuscA(s(X1)) :- ','(lecB(false), minuscA(X1)).
minuscC(X1, 0) :- lecD(X1, true).
minuscC(0, s(X1)) :- ','(lecD(0, false), minuscA(X1)).
minuscC(s(X1), s(X2)) :- ','(lecD(s(X1), false), minuscC(X1, X2)).
lecE(0, X1, true).
lecE(s(X1), 0, false).
lecE(s(X1), s(X2), X3) :- lecE(X1, X2, X3).
minuscF(0, X1, 0).
minuscF(s(X1), 0, s(X2)) :- minuscC(X1, X2).
minuscF(s(X1), s(X2), 0) :- lecE(X1, X2, true).
minuscF(s(X1), s(X2), s(X3)) :- ','(lecE(X1, X2, false), minuscF(X1, s(X2), X3)).
lecD(0, true).
lecD(s(X1), false).
lecB(true).

Afs:

minusF(x1, x2, x3)  =  minusF(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
minusF_in: (b,f,f)
minusC_in: (b,f)
minusA_in: (f)
leE_in: (b,f,f)
lecE_in: (b,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MINUSF_IN_GAA(s(X1), 0, s(X2)) → U8_GAA(X1, X2, minusC_in_ga(X1, X2))
MINUSF_IN_GAA(s(X1), 0, s(X2)) → MINUSC_IN_GA(X1, X2)
MINUSC_IN_GA(0, s(X1)) → U3_GA(X1, lecD_in_gg(0, false))
U3_GA(X1, lecD_out_gg(0, false)) → U4_GA(X1, minusA_in_a(X1))
U3_GA(X1, lecD_out_gg(0, false)) → MINUSA_IN_A(X1)
MINUSA_IN_A(s(X1)) → U1_A(X1, lecB_in_g(false))
U1_A(X1, lecB_out_g(false)) → U2_A(X1, minusA_in_a(X1))
U1_A(X1, lecB_out_g(false)) → MINUSA_IN_A(X1)
MINUSC_IN_GA(s(X1), s(X2)) → U5_GA(X1, X2, lecD_in_gg(s(X1), false))
U5_GA(X1, X2, lecD_out_gg(s(X1), false)) → U6_GA(X1, X2, minusC_in_ga(X1, X2))
U5_GA(X1, X2, lecD_out_gg(s(X1), false)) → MINUSC_IN_GA(X1, X2)
MINUSF_IN_GAA(s(X1), s(X2), X3) → U9_GAA(X1, X2, X3, leE_in_gaa(X1, X2, X4))
MINUSF_IN_GAA(s(X1), s(X2), X3) → LEE_IN_GAA(X1, X2, X4)
LEE_IN_GAA(s(X1), s(X2), X3) → U7_GAA(X1, X2, X3, leE_in_gaa(X1, X2, X3))
LEE_IN_GAA(s(X1), s(X2), X3) → LEE_IN_GAA(X1, X2, X3)
MINUSF_IN_GAA(s(X1), s(X2), s(X3)) → U10_GAA(X1, X2, X3, lecE_in_gag(X1, X2, false))
U10_GAA(X1, X2, X3, lecE_out_gag(X1, X2, false)) → U11_GAA(X1, X2, X3, minusF_in_gaa(X1, s(X2), X3))
U10_GAA(X1, X2, X3, lecE_out_gag(X1, X2, false)) → MINUSF_IN_GAA(X1, s(X2), X3)

The TRS R consists of the following rules:

lecD_in_gg(0, true) → lecD_out_gg(0, true)
lecD_in_gg(s(X1), false) → lecD_out_gg(s(X1), false)
lecB_in_g(true) → lecB_out_g(true)
lecE_in_gag(0, X1, true) → lecE_out_gag(0, X1, true)
lecE_in_gag(s(X1), 0, false) → lecE_out_gag(s(X1), 0, false)
lecE_in_gag(s(X1), s(X2), X3) → U21_gag(X1, X2, X3, lecE_in_gag(X1, X2, X3))
U21_gag(X1, X2, X3, lecE_out_gag(X1, X2, X3)) → lecE_out_gag(s(X1), s(X2), X3)

The argument filtering Pi contains the following mapping:
minusF_in_gaa(x1, x2, x3)  =  minusF_in_gaa(x1)
s(x1)  =  s(x1)
minusC_in_ga(x1, x2)  =  minusC_in_ga(x1)
0  =  0
lecD_in_gg(x1, x2)  =  lecD_in_gg(x1, x2)
true  =  true
lecD_out_gg(x1, x2)  =  lecD_out_gg(x1, x2)
false  =  false
minusA_in_a(x1)  =  minusA_in_a
lecB_in_g(x1)  =  lecB_in_g(x1)
lecB_out_g(x1)  =  lecB_out_g(x1)
leE_in_gaa(x1, x2, x3)  =  leE_in_gaa(x1)
lecE_in_gag(x1, x2, x3)  =  lecE_in_gag(x1, x3)
lecE_out_gag(x1, x2, x3)  =  lecE_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUSF_IN_GAA(x1, x2, x3)  =  MINUSF_IN_GAA(x1)
U8_GAA(x1, x2, x3)  =  U8_GAA(x1, x3)
MINUSC_IN_GA(x1, x2)  =  MINUSC_IN_GA(x1)
U3_GA(x1, x2)  =  U3_GA(x2)
U4_GA(x1, x2)  =  U4_GA(x2)
MINUSA_IN_A(x1)  =  MINUSA_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)
LEE_IN_GAA(x1, x2, x3)  =  LEE_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:

MINUSF_IN_GAA(s(X1), 0, s(X2)) → U8_GAA(X1, X2, minusC_in_ga(X1, X2))
MINUSF_IN_GAA(s(X1), 0, s(X2)) → MINUSC_IN_GA(X1, X2)
MINUSC_IN_GA(0, s(X1)) → U3_GA(X1, lecD_in_gg(0, false))
U3_GA(X1, lecD_out_gg(0, false)) → U4_GA(X1, minusA_in_a(X1))
U3_GA(X1, lecD_out_gg(0, false)) → MINUSA_IN_A(X1)
MINUSA_IN_A(s(X1)) → U1_A(X1, lecB_in_g(false))
U1_A(X1, lecB_out_g(false)) → U2_A(X1, minusA_in_a(X1))
U1_A(X1, lecB_out_g(false)) → MINUSA_IN_A(X1)
MINUSC_IN_GA(s(X1), s(X2)) → U5_GA(X1, X2, lecD_in_gg(s(X1), false))
U5_GA(X1, X2, lecD_out_gg(s(X1), false)) → U6_GA(X1, X2, minusC_in_ga(X1, X2))
U5_GA(X1, X2, lecD_out_gg(s(X1), false)) → MINUSC_IN_GA(X1, X2)
MINUSF_IN_GAA(s(X1), s(X2), X3) → U9_GAA(X1, X2, X3, leE_in_gaa(X1, X2, X4))
MINUSF_IN_GAA(s(X1), s(X2), X3) → LEE_IN_GAA(X1, X2, X4)
LEE_IN_GAA(s(X1), s(X2), X3) → U7_GAA(X1, X2, X3, leE_in_gaa(X1, X2, X3))
LEE_IN_GAA(s(X1), s(X2), X3) → LEE_IN_GAA(X1, X2, X3)
MINUSF_IN_GAA(s(X1), s(X2), s(X3)) → U10_GAA(X1, X2, X3, lecE_in_gag(X1, X2, false))
U10_GAA(X1, X2, X3, lecE_out_gag(X1, X2, false)) → U11_GAA(X1, X2, X3, minusF_in_gaa(X1, s(X2), X3))
U10_GAA(X1, X2, X3, lecE_out_gag(X1, X2, false)) → MINUSF_IN_GAA(X1, s(X2), X3)

The TRS R consists of the following rules:

lecD_in_gg(0, true) → lecD_out_gg(0, true)
lecD_in_gg(s(X1), false) → lecD_out_gg(s(X1), false)
lecB_in_g(true) → lecB_out_g(true)
lecE_in_gag(0, X1, true) → lecE_out_gag(0, X1, true)
lecE_in_gag(s(X1), 0, false) → lecE_out_gag(s(X1), 0, false)
lecE_in_gag(s(X1), s(X2), X3) → U21_gag(X1, X2, X3, lecE_in_gag(X1, X2, X3))
U21_gag(X1, X2, X3, lecE_out_gag(X1, X2, X3)) → lecE_out_gag(s(X1), s(X2), X3)

The argument filtering Pi contains the following mapping:
minusF_in_gaa(x1, x2, x3)  =  minusF_in_gaa(x1)
s(x1)  =  s(x1)
minusC_in_ga(x1, x2)  =  minusC_in_ga(x1)
0  =  0
lecD_in_gg(x1, x2)  =  lecD_in_gg(x1, x2)
true  =  true
lecD_out_gg(x1, x2)  =  lecD_out_gg(x1, x2)
false  =  false
minusA_in_a(x1)  =  minusA_in_a
lecB_in_g(x1)  =  lecB_in_g(x1)
lecB_out_g(x1)  =  lecB_out_g(x1)
leE_in_gaa(x1, x2, x3)  =  leE_in_gaa(x1)
lecE_in_gag(x1, x2, x3)  =  lecE_in_gag(x1, x3)
lecE_out_gag(x1, x2, x3)  =  lecE_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUSF_IN_GAA(x1, x2, x3)  =  MINUSF_IN_GAA(x1)
U8_GAA(x1, x2, x3)  =  U8_GAA(x1, x3)
MINUSC_IN_GA(x1, x2)  =  MINUSC_IN_GA(x1)
U3_GA(x1, x2)  =  U3_GA(x2)
U4_GA(x1, x2)  =  U4_GA(x2)
MINUSA_IN_A(x1)  =  MINUSA_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)
LEE_IN_GAA(x1, x2, x3)  =  LEE_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:

LEE_IN_GAA(s(X1), s(X2), X3) → LEE_IN_GAA(X1, X2, X3)

The TRS R consists of the following rules:

lecD_in_gg(0, true) → lecD_out_gg(0, true)
lecD_in_gg(s(X1), false) → lecD_out_gg(s(X1), false)
lecB_in_g(true) → lecB_out_g(true)
lecE_in_gag(0, X1, true) → lecE_out_gag(0, X1, true)
lecE_in_gag(s(X1), 0, false) → lecE_out_gag(s(X1), 0, false)
lecE_in_gag(s(X1), s(X2), X3) → U21_gag(X1, X2, X3, lecE_in_gag(X1, X2, X3))
U21_gag(X1, X2, X3, lecE_out_gag(X1, X2, X3)) → lecE_out_gag(s(X1), s(X2), X3)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lecD_in_gg(x1, x2)  =  lecD_in_gg(x1, x2)
true  =  true
lecD_out_gg(x1, x2)  =  lecD_out_gg(x1, x2)
false  =  false
lecB_in_g(x1)  =  lecB_in_g(x1)
lecB_out_g(x1)  =  lecB_out_g(x1)
lecE_in_gag(x1, x2, x3)  =  lecE_in_gag(x1, x3)
lecE_out_gag(x1, x2, x3)  =  lecE_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
LEE_IN_GAA(x1, x2, x3)  =  LEE_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:

LEE_IN_GAA(s(X1), s(X2), X3) → LEE_IN_GAA(X1, X2, X3)

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

LEE_IN_GAA(s(X1)) → LEE_IN_GAA(X1)

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:

• LEE_IN_GAA(s(X1)) → LEE_IN_GAA(X1)
  The graph contains the following edges 1 > 1

(13) YES

(14) Obligation:

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

MINUSF_IN_GAA(s(X1), s(X2), s(X3)) → U10_GAA(X1, X2, X3, lecE_in_gag(X1, X2, false))
U10_GAA(X1, X2, X3, lecE_out_gag(X1, X2, false)) → MINUSF_IN_GAA(X1, s(X2), X3)

The TRS R consists of the following rules:

lecD_in_gg(0, true) → lecD_out_gg(0, true)
lecD_in_gg(s(X1), false) → lecD_out_gg(s(X1), false)
lecB_in_g(true) → lecB_out_g(true)
lecE_in_gag(0, X1, true) → lecE_out_gag(0, X1, true)
lecE_in_gag(s(X1), 0, false) → lecE_out_gag(s(X1), 0, false)
lecE_in_gag(s(X1), s(X2), X3) → U21_gag(X1, X2, X3, lecE_in_gag(X1, X2, X3))
U21_gag(X1, X2, X3, lecE_out_gag(X1, X2, X3)) → lecE_out_gag(s(X1), s(X2), X3)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lecD_in_gg(x1, x2)  =  lecD_in_gg(x1, x2)
true  =  true
lecD_out_gg(x1, x2)  =  lecD_out_gg(x1, x2)
false  =  false
lecB_in_g(x1)  =  lecB_in_g(x1)
lecB_out_g(x1)  =  lecB_out_g(x1)
lecE_in_gag(x1, x2, x3)  =  lecE_in_gag(x1, x3)
lecE_out_gag(x1, x2, x3)  =  lecE_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUSF_IN_GAA(x1, x2, x3)  =  MINUSF_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:

MINUSF_IN_GAA(s(X1), s(X2), s(X3)) → U10_GAA(X1, X2, X3, lecE_in_gag(X1, X2, false))
U10_GAA(X1, X2, X3, lecE_out_gag(X1, X2, false)) → MINUSF_IN_GAA(X1, s(X2), X3)

The TRS R consists of the following rules:

lecE_in_gag(s(X1), 0, false) → lecE_out_gag(s(X1), 0, false)
lecE_in_gag(s(X1), s(X2), X3) → U21_gag(X1, X2, X3, lecE_in_gag(X1, X2, X3))
U21_gag(X1, X2, X3, lecE_out_gag(X1, X2, X3)) → lecE_out_gag(s(X1), s(X2), X3)
lecE_in_gag(0, X1, true) → lecE_out_gag(0, X1, true)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
true  =  true
false  =  false
lecE_in_gag(x1, x2, x3)  =  lecE_in_gag(x1, x3)
lecE_out_gag(x1, x2, x3)  =  lecE_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUSF_IN_GAA(x1, x2, x3)  =  MINUSF_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:

MINUSF_IN_GAA(s(X1)) → U10_GAA(X1, lecE_in_gag(X1, false))
U10_GAA(X1, lecE_out_gag(X1, false)) → MINUSF_IN_GAA(X1)

The TRS R consists of the following rules:

lecE_in_gag(s(X1), false) → lecE_out_gag(s(X1), false)
lecE_in_gag(s(X1), X3) → U21_gag(X1, X3, lecE_in_gag(X1, X3))
U21_gag(X1, X3, lecE_out_gag(X1, X3)) → lecE_out_gag(s(X1), X3)
lecE_in_gag(0, true) → lecE_out_gag(0, true)

The set Q consists of the following terms:

lecE_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(X1, lecE_out_gag(X1, false)) → MINUSF_IN_GAA(X1)
  The graph contains the following edges 1 >= 1, 2 > 1

• MINUSF_IN_GAA(s(X1)) → U10_GAA(X1, lecE_in_gag(X1, false))
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

MINUSC_IN_GA(s(X1), s(X2)) → U5_GA(X1, X2, lecD_in_gg(s(X1), false))
U5_GA(X1, X2, lecD_out_gg(s(X1), false)) → MINUSC_IN_GA(X1, X2)

The TRS R consists of the following rules:

lecD_in_gg(0, true) → lecD_out_gg(0, true)
lecD_in_gg(s(X1), false) → lecD_out_gg(s(X1), false)
lecB_in_g(true) → lecB_out_g(true)
lecE_in_gag(0, X1, true) → lecE_out_gag(0, X1, true)
lecE_in_gag(s(X1), 0, false) → lecE_out_gag(s(X1), 0, false)
lecE_in_gag(s(X1), s(X2), X3) → U21_gag(X1, X2, X3, lecE_in_gag(X1, X2, X3))
U21_gag(X1, X2, X3, lecE_out_gag(X1, X2, X3)) → lecE_out_gag(s(X1), s(X2), X3)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
0  =  0
lecD_in_gg(x1, x2)  =  lecD_in_gg(x1, x2)
true  =  true
lecD_out_gg(x1, x2)  =  lecD_out_gg(x1, x2)
false  =  false
lecB_in_g(x1)  =  lecB_in_g(x1)
lecB_out_g(x1)  =  lecB_out_g(x1)
lecE_in_gag(x1, x2, x3)  =  lecE_in_gag(x1, x3)
lecE_out_gag(x1, x2, x3)  =  lecE_out_gag(x1, x3)
U21_gag(x1, x2, x3, x4)  =  U21_gag(x1, x3, x4)
MINUSC_IN_GA(x1, x2)  =  MINUSC_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:

MINUSC_IN_GA(s(X1), s(X2)) → U5_GA(X1, X2, lecD_in_gg(s(X1), false))
U5_GA(X1, X2, lecD_out_gg(s(X1), false)) → MINUSC_IN_GA(X1, X2)

The TRS R consists of the following rules:

lecD_in_gg(s(X1), false) → lecD_out_gg(s(X1), false)

The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
lecD_in_gg(x1, x2)  =  lecD_in_gg(x1, x2)
lecD_out_gg(x1, x2)  =  lecD_out_gg(x1, x2)
false  =  false
MINUSC_IN_GA(x1, x2)  =  MINUSC_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:

MINUSC_IN_GA(s(X1)) → U5_GA(X1, lecD_in_gg(s(X1), false))
U5_GA(X1, lecD_out_gg(s(X1), false)) → MINUSC_IN_GA(X1)

The TRS R consists of the following rules:

lecD_in_gg(s(X1), false) → lecD_out_gg(s(X1), false)

The set Q consists of the following terms:

lecD_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(X1, lecD_out_gg(s(X1), false)) → MINUSC_IN_GA(X1)
  The graph contains the following edges 1 >= 1, 2 > 1

• MINUSC_IN_GA(s(X1)) → U5_GA(X1, lecD_in_gg(s(X1), false))
  The graph contains the following edges 1 > 1

(27) YES