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

(0) Obligation:

Clauses:

preorder(X, Y) :- pdl(X, -(Y, [])).
pdl(nil, Y) :- ','(!, eq(Y, -(X, X))).
pdl(T, -(.(X, Xs), Zs)) :- ','(value(T, X), ','(left(T, L), ','(right(T, R), ','(pdl(L, -(Xs, Ys)), pdl(R, -(Ys, Zs)))))).
left(nil, nil).
left(tree(L, X1, X2), L).
right(nil, nil).
right(tree(X3, X4, R), R).
value(nil, X5).
value(tree(X6, X, X7), X).
eq(X, X).

Query: preorder(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: preorder(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: preorder(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="B: pdl(T5, -(T7, []))\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 1 | pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_2, eq(-(T11, []), -(X16, X16))) | pdl(nil, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: pdl(T5, -(T7, [])), SCOPE: 2, CLAUSE: 2\n\nT5 is ground\nX15 is free\npdl(T5, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: eq(-(T11, []), -(X16, X16))\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: eq(-(T11, []), -(X16, X16)), SCOPE: 3, CLAUSE: 9\n\nX16 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, []))))))\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 7 | ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(value(T21, T24), ','(left(T21, X35), ','(right(T21, X36), ','(pdl(X35, -(T25, X37)), pdl(X36, -(X37, [])))))), SCOPE: 4, CLAUSE: 8\n\nT21 is ground\nX15 is free\nX35 is free\nX36 is free\nX37 is free\npdl(T21, -(T7, [])) !=? pdl(nil, X15)", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, [])))))\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 3 | ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(left(tree(T34, T35, T36), X35), ','(right(tree(T34, T35, T36), X36), ','(pdl(X35, -(T37, X37)), pdl(X36, -(X37, []))))), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT35 is ground\nT36 is ground\nX35 is free\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, []))))\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 5 | ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(right(tree(T46, T47, T48), X36), ','(pdl(T46, -(T37, X37)), pdl(X36, -(X37, [])))), SCOPE: 6, CLAUSE: 6\n\nT46 is ground\nT47 is ground\nT48 is ground\nX36 is free\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="C: ','(pdl(T57, -(T37, X37)), pdl(T59, -(X37, [])))\n\nT57 is ground\nT59 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="D: pdl(T57, -(T37, X37))\n\nT57 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: pdl(T59, -(T61, []))\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 1 | pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nT57 is ground\nX37 is free", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: ','(!_7, eq(-(T66, X94), -(X93, X93))) | pdl(nil, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nX37 is free\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: pdl(T57, -(T37, X37)), SCOPE: 7, CLAUSE: 2\n\nT57 is ground\nX37 is free\nX92 is free\npdl(T57, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: eq(-(T66, X94), -(X93, X93))\n\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: eq(-(T66, X94), -(X93, X93)), SCOPE: 8, CLAUSE: 9\n\nX94 is free\nX93 is free", fontsize=16, style=filled, fillcolor=lightyellow];
118 [label="118: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121))))))\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
126 [label="126: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
127 [label="127: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 7 | ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 8\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
128 [label="128: ','(value(T76, T79), ','(left(T76, X118), ','(right(T76, X119), ','(pdl(X118, -(T80, X120)), pdl(X119, -(X120, X121)))))), SCOPE: 9, CLAUSE: 8\n\nT76 is ground\nX37 is free\nX92 is free\nX121 is free\nX118 is free\nX119 is free\nX120 is free\npdl(T76, -(T37, X37)) !=? pdl(nil, X92)", fontsize=16, style=filled, fillcolor=lightyellow];
129 [label="129: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121)))))\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 3 | ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 4\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="132: ','(left(tree(T89, T90, T91), X118), ','(right(tree(T89, T90, T91), X119), ','(pdl(X118, -(T92, X120)), pdl(X119, -(X120, X121))))), SCOPE: 10, CLAUSE: 4\n\nT89 is ground\nT90 is ground\nT91 is ground\nX121 is free\nX118 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
133 [label="133: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121))))\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
134 [label="134: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 5 | ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 6\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
135 [label="135: ','(right(tree(T101, T102, T103), X119), ','(pdl(T101, -(T92, X120)), pdl(X119, -(X120, X121)))), SCOPE: 11, CLAUSE: 6\n\nT101 is ground\nT102 is ground\nT103 is ground\nX121 is free\nX119 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
136 [label="E: ','(pdl(T112, -(T92, X120)), pdl(T114, -(X120, X121)))\n\nT112 is ground\nT114 is ground\nX121 is free\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
137 [label="137: pdl(T112, -(T92, X120))\n\nT112 is ground\nX120 is free", fontsize=16, style=filled, fillcolor=lightyellow];
138 [label="138: pdl(T114, -(T116, X121))\n\nT114 is ground\nX121 is free", fontsize=16, style=filled, fillcolor=lightyellow];
139 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 139 [arrowhead = none ];
139 -> 4;

140 [label="ONLY EVAL with clause\npreorder(X10, X11) :- pdl(X10, -(X11, [])).\nand substitutionT1 -> T5,\nX10 -> T5,\nT2 -> T7,\nX11 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 140 [arrowhead = none ];
140 -> 8;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 141 [arrowhead = none ];
141 -> 10;

142 [label="EVAL with clause\npdl(nil, X15) :- ','(!_2, eq(X15, -(X16, X16))).\nand substitutionT5 -> nil,\nT7 -> T11,\nX15 -> -(T11, []),\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 142 [arrowhead = none ];
142 -> 13;

143 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 143 [arrowhead = none ];
143 -> 14;

144 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 144 [arrowhead = none ];
144 -> 17;

145 [label="EVAL with clause\npdl(X31, -(.(X32, X33), X34)) :- ','(value(X31, X32), ','(left(X31, X35), ','(right(X31, X36), ','(pdl(X35, -(X33, X37)), pdl(X36, -(X37, X34)))))).\nand substitutionT5 -> T21,\nX31 -> T21,\nX32 -> T24,\nX33 -> T25,\nT7 -> .(T24, T25),\nX34 -> [],\nT22 -> T24,\nT23 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 145 [arrowhead = none ];
145 -> 36;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 146 [arrowhead = none ];
146 -> 37;

147 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 147 [arrowhead = none ];
147 -> 19;

148 [label="EVAL with clause\neq(X19, X19).\nand substitutionT11 -> [],\nX19 -> -([], []),\nX16 -> [],\nT14 -> []", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 148 [arrowhead = none ];
148 -> 28;

149 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 149 [arrowhead = none ];
149 -> 29;

150 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 150 [arrowhead = none ];
150 -> 30;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 151 [arrowhead = none ];
151 -> 40;

152 [label="BACKTRACK\nfor clause: value(nil, X5)\nwith clash: (pdl(T21, -(T7, [])), pdl(nil, X15))", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 152 [arrowhead = none ];
152 -> 43;

153 [label="EVAL with clause\nvalue(tree(X48, X49, X50), X49).\nand substitutionX48 -> T34,\nX49 -> T35,\nX50 -> T36,\nT21 -> tree(T34, T35, T36),\nT24 -> T35,\nT25 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 153 [arrowhead = none ];
153 -> 57;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 154 [arrowhead = none ];
154 -> 58;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 155 [arrowhead = none ];
155 -> 61;

156 [label="BACKTRACK\nfor clause: left(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 156 [arrowhead = none ];
156 -> 62;

157 [label="ONLY EVAL with clause\nleft(tree(X63, X64, X65), X63).\nand substitutionT34 -> T46,\nX63 -> T46,\nT35 -> T47,\nX64 -> T47,\nT36 -> T48,\nX65 -> T48,\nX35 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 157 [arrowhead = none ];
157 -> 67;

158 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
67 -> 158 [arrowhead = none ];
158 -> 83;

159 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
83 -> 159 [arrowhead = none ];
159 -> 84;

160 [label="ONLY EVAL with clause\nright(tree(X78, X79, X80), X80).\nand substitutionT46 -> T57,\nX78 -> T57,\nT47 -> T58,\nX79 -> T58,\nT48 -> T59,\nX80 -> T59,\nX36 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
84 -> 160 [arrowhead = none ];
160 -> 89;

161 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
89 -> 161 [arrowhead = none ];
161 -> 105;

162 [label="SPLIT 2\nnew knowledge:\nT57 is ground\nreplacements:X37 -> T61", fontsize=14, style = filled, fillcolor = violet];
89 -> 162 [arrowhead = none ];
162 -> 106;

163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
105 -> 163 [arrowhead = none ];
163 -> 109;

164 [label="INSTANCE with matching:\nT5 -> T59\nT7 -> T61", fontsize=14, style = filled, fillcolor = lightgrey];
106 -> 164 [arrowhead = none , style=dashed];
164 -> 8[style=dashed];

165 [label="EVAL with clause\npdl(nil, X92) :- ','(!_7, eq(X92, -(X93, X93))).\nand substitutionT57 -> nil,\nT37 -> T66,\nX37 -> X94,\nX92 -> -(T66, X94),\nT65 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 165 [arrowhead = none ];
165 -> 111;

166 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
109 -> 166 [arrowhead = none ];
166 -> 112;

167 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
111 -> 167 [arrowhead = none ];
167 -> 113;

168 [label="EVAL with clause\npdl(X114, -(.(X115, X116), X117)) :- ','(value(X114, X115), ','(left(X114, X118), ','(right(X114, X119), ','(pdl(X118, -(X116, X120)), pdl(X119, -(X120, X117)))))).\nand substitutionT57 -> T76,\nX114 -> T76,\nX115 -> T79,\nX116 -> T80,\nT37 -> .(T79, T80),\nX37 -> X121,\nX117 -> X121,\nT77 -> T79,\nT78 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
112 -> 168 [arrowhead = none ];
168 -> 125;

169 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
112 -> 169 [arrowhead = none ];
169 -> 126;

170 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
113 -> 170 [arrowhead = none ];
170 -> 116;

171 [label="ONLY EVAL with clause\neq(X99, X99).\nand substitutionT66 -> T69,\nX94 -> T69,\nX99 -> -(T69, T69),\nX93 -> T69,\nX100 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
116 -> 171 [arrowhead = none ];
171 -> 118;

172 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
118 -> 172 [arrowhead = none ];
172 -> 119;

173 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
125 -> 173 [arrowhead = none ];
173 -> 127;

174 [label="BACKTRACK\nfor clause: value(nil, X5)\nwith clash: (pdl(T76, -(T37, X37)), pdl(nil, X92))", fontsize=14, style = filled, fillcolor = lightgreen];
127 -> 174 [arrowhead = none ];
174 -> 128;

175 [label="EVAL with clause\nvalue(tree(X132, X133, X134), X133).\nand substitutionX132 -> T89,\nX133 -> T90,\nX134 -> T91,\nT76 -> tree(T89, T90, T91),\nT79 -> T90,\nT80 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
128 -> 175 [arrowhead = none ];
175 -> 129;

176 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
128 -> 176 [arrowhead = none ];
176 -> 130;

177 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
129 -> 177 [arrowhead = none ];
177 -> 131;

178 [label="BACKTRACK\nfor clause: left(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
131 -> 178 [arrowhead = none ];
178 -> 132;

179 [label="ONLY EVAL with clause\nleft(tree(X147, X148, X149), X147).\nand substitutionT89 -> T101,\nX147 -> T101,\nT90 -> T102,\nX148 -> T102,\nT91 -> T103,\nX149 -> T103,\nX118 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
132 -> 179 [arrowhead = none ];
179 -> 133;

180 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
133 -> 180 [arrowhead = none ];
180 -> 134;

181 [label="BACKTRACK\nfor clause: right(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
134 -> 181 [arrowhead = none ];
181 -> 135;

182 [label="ONLY EVAL with clause\nright(tree(X162, X163, X164), X164).\nand substitutionT101 -> T112,\nX162 -> T112,\nT102 -> T113,\nX163 -> T113,\nT103 -> T114,\nX164 -> T114,\nX119 -> T114", fontsize=14, style = filled, fillcolor = lightblue];
135 -> 182 [arrowhead = none ];
182 -> 136;

183 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
136 -> 183 [arrowhead = none ];
183 -> 137;

184 [label="SPLIT 2\nnew knowledge:\nT112 is ground\nreplacements:X120 -> T116", fontsize=14, style = filled, fillcolor = violet];
136 -> 184 [arrowhead = none ];
184 -> 138;

185 [label="INSTANCE with matching:\nT57 -> T112\nT37 -> T92\nX37 -> X120", fontsize=14, style = filled, fillcolor = lightgrey];
137 -> 185 [arrowhead = none , style=dashed];
185 -> 105[style=dashed];

186 [label="INSTANCE with matching:\nT57 -> T114\nT37 -> T116\nX37 -> X121", fontsize=14, style = filled, fillcolor = lightgrey];
138 -> 186 [arrowhead = none , style=dashed];
186 -> 105[style=dashed];

}

(2) Obligation:

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

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, pdlB_in_ga(T5, T7))
pdlB_in_ga(nil, []) → pdlB_out_ga(nil, [])
pdlB_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pC_in_gaag(T57, T37, X37, T59))
pC_in_gaag(T57, T37, T61, T59) → U4_gaag(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
pdlD_in_gaa(nil, T69, T69) → pdlD_out_gaa(nil, T69, T69)
pdlD_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U3_gaa(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
pE_in_gaaga(T112, T92, T116, T114, X121) → U6_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U7_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T114, T116, X121)) → pE_out_gaaga(T112, T92, T116, T114, X121)
U3_gaa(T112, T113, T114, T92, X121, pE_out_gaaga(T112, T92, X120, T114, X121)) → pdlD_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaag(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → U5_gaag(T57, T37, T61, T59, pdlB_in_ga(T59, T61))
U5_gaag(T57, T37, T61, T59, pdlB_out_ga(T59, T61)) → pC_out_gaag(T57, T37, T61, T59)
U2_ga(T57, T58, T59, T37, pC_out_gaag(T57, T37, X37, T59)) → pdlB_out_ga(tree(T57, T58, T59), .(T58, T37))
U1_ga(T5, T7, pdlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pdlB_in_ga(x1, x2)  =  pdlB_in_ga(x1)
nil  =  nil
pdlB_out_ga(x1, x2)  =  pdlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
pdlD_in_gaa(x1, x2, x3)  =  pdlD_in_gaa(x1)
pdlD_out_gaa(x1, x2, x3)  =  pdlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

PREORDERA_IN_GA(T5, T7) → U1_GA(T5, T7, pdlB_in_ga(T5, T7))
PREORDERA_IN_GA(T5, T7) → PDLB_IN_GA(T5, T7)
PDLB_IN_GA(tree(T57, T58, T59), .(T58, T37)) → U2_GA(T57, T58, T59, T37, pC_in_gaag(T57, T37, X37, T59))
PDLB_IN_GA(tree(T57, T58, T59), .(T58, T37)) → PC_IN_GAAG(T57, T37, X37, T59)
PC_IN_GAAG(T57, T37, T61, T59) → U4_GAAG(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
PC_IN_GAAG(T57, T37, T61, T59) → PDLD_IN_GAA(T57, T37, T61)
PDLD_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → U3_GAA(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
PDLD_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → PE_IN_GAAGA(T112, T92, X120, T114, X121)
PE_IN_GAAGA(T112, T92, T116, T114, X121) → U6_GAAGA(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
PE_IN_GAAGA(T112, T92, T116, T114, X121) → PDLD_IN_GAA(T112, T92, T116)
U6_GAAGA(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_GAAGA(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U6_GAAGA(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → PDLD_IN_GAA(T114, T116, X121)
U4_GAAG(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → U5_GAAG(T57, T37, T61, T59, pdlB_in_ga(T59, T61))
U4_GAAG(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → PDLB_IN_GA(T59, T61)

The TRS R consists of the following rules:

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, pdlB_in_ga(T5, T7))
pdlB_in_ga(nil, []) → pdlB_out_ga(nil, [])
pdlB_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pC_in_gaag(T57, T37, X37, T59))
pC_in_gaag(T57, T37, T61, T59) → U4_gaag(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
pdlD_in_gaa(nil, T69, T69) → pdlD_out_gaa(nil, T69, T69)
pdlD_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U3_gaa(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
pE_in_gaaga(T112, T92, T116, T114, X121) → U6_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U7_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T114, T116, X121)) → pE_out_gaaga(T112, T92, T116, T114, X121)
U3_gaa(T112, T113, T114, T92, X121, pE_out_gaaga(T112, T92, X120, T114, X121)) → pdlD_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaag(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → U5_gaag(T57, T37, T61, T59, pdlB_in_ga(T59, T61))
U5_gaag(T57, T37, T61, T59, pdlB_out_ga(T59, T61)) → pC_out_gaag(T57, T37, T61, T59)
U2_ga(T57, T58, T59, T37, pC_out_gaag(T57, T37, X37, T59)) → pdlB_out_ga(tree(T57, T58, T59), .(T58, T37))
U1_ga(T5, T7, pdlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pdlB_in_ga(x1, x2)  =  pdlB_in_ga(x1)
nil  =  nil
pdlB_out_ga(x1, x2)  =  pdlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
pdlD_in_gaa(x1, x2, x3)  =  pdlD_in_gaa(x1)
pdlD_out_gaa(x1, x2, x3)  =  pdlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)
PREORDERA_IN_GA(x1, x2)  =  PREORDERA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PDLB_IN_GA(x1, x2)  =  PDLB_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
PC_IN_GAAG(x1, x2, x3, x4)  =  PC_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5)  =  U4_GAAG(x1, x4, x5)
PDLD_IN_GAA(x1, x2, x3)  =  PDLD_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5, x6)  =  U3_GAA(x1, x2, x3, x6)
PE_IN_GAAGA(x1, x2, x3, x4, x5)  =  PE_IN_GAAGA(x1, x4)
U6_GAAGA(x1, x2, x3, x4, x5, x6)  =  U6_GAAGA(x1, x4, x6)
U7_GAAGA(x1, x2, x3, x4, x5, x6)  =  U7_GAAGA(x1, x4, x6)
U5_GAAG(x1, x2, x3, x4, x5)  =  U5_GAAG(x1, x4, x5)

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

(4) Obligation:

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

PREORDERA_IN_GA(T5, T7) → U1_GA(T5, T7, pdlB_in_ga(T5, T7))
PREORDERA_IN_GA(T5, T7) → PDLB_IN_GA(T5, T7)
PDLB_IN_GA(tree(T57, T58, T59), .(T58, T37)) → U2_GA(T57, T58, T59, T37, pC_in_gaag(T57, T37, X37, T59))
PDLB_IN_GA(tree(T57, T58, T59), .(T58, T37)) → PC_IN_GAAG(T57, T37, X37, T59)
PC_IN_GAAG(T57, T37, T61, T59) → U4_GAAG(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
PC_IN_GAAG(T57, T37, T61, T59) → PDLD_IN_GAA(T57, T37, T61)
PDLD_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → U3_GAA(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
PDLD_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → PE_IN_GAAGA(T112, T92, X120, T114, X121)
PE_IN_GAAGA(T112, T92, T116, T114, X121) → U6_GAAGA(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
PE_IN_GAAGA(T112, T92, T116, T114, X121) → PDLD_IN_GAA(T112, T92, T116)
U6_GAAGA(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_GAAGA(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U6_GAAGA(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → PDLD_IN_GAA(T114, T116, X121)
U4_GAAG(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → U5_GAAG(T57, T37, T61, T59, pdlB_in_ga(T59, T61))
U4_GAAG(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → PDLB_IN_GA(T59, T61)

The TRS R consists of the following rules:

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, pdlB_in_ga(T5, T7))
pdlB_in_ga(nil, []) → pdlB_out_ga(nil, [])
pdlB_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pC_in_gaag(T57, T37, X37, T59))
pC_in_gaag(T57, T37, T61, T59) → U4_gaag(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
pdlD_in_gaa(nil, T69, T69) → pdlD_out_gaa(nil, T69, T69)
pdlD_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U3_gaa(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
pE_in_gaaga(T112, T92, T116, T114, X121) → U6_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U7_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T114, T116, X121)) → pE_out_gaaga(T112, T92, T116, T114, X121)
U3_gaa(T112, T113, T114, T92, X121, pE_out_gaaga(T112, T92, X120, T114, X121)) → pdlD_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaag(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → U5_gaag(T57, T37, T61, T59, pdlB_in_ga(T59, T61))
U5_gaag(T57, T37, T61, T59, pdlB_out_ga(T59, T61)) → pC_out_gaag(T57, T37, T61, T59)
U2_ga(T57, T58, T59, T37, pC_out_gaag(T57, T37, X37, T59)) → pdlB_out_ga(tree(T57, T58, T59), .(T58, T37))
U1_ga(T5, T7, pdlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pdlB_in_ga(x1, x2)  =  pdlB_in_ga(x1)
nil  =  nil
pdlB_out_ga(x1, x2)  =  pdlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
pdlD_in_gaa(x1, x2, x3)  =  pdlD_in_gaa(x1)
pdlD_out_gaa(x1, x2, x3)  =  pdlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)
PREORDERA_IN_GA(x1, x2)  =  PREORDERA_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
PDLB_IN_GA(x1, x2)  =  PDLB_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)
PC_IN_GAAG(x1, x2, x3, x4)  =  PC_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5)  =  U4_GAAG(x1, x4, x5)
PDLD_IN_GAA(x1, x2, x3)  =  PDLD_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4, x5, x6)  =  U3_GAA(x1, x2, x3, x6)
PE_IN_GAAGA(x1, x2, x3, x4, x5)  =  PE_IN_GAAGA(x1, x4)
U6_GAAGA(x1, x2, x3, x4, x5, x6)  =  U6_GAAGA(x1, x4, x6)
U7_GAAGA(x1, x2, x3, x4, x5, x6)  =  U7_GAAGA(x1, x4, x6)
U5_GAAG(x1, x2, x3, x4, x5)  =  U5_GAAG(x1, x4, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

PDLD_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → PE_IN_GAAGA(T112, T92, X120, T114, X121)
PE_IN_GAAGA(T112, T92, T116, T114, X121) → U6_GAAGA(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_GAAGA(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → PDLD_IN_GAA(T114, T116, X121)
PE_IN_GAAGA(T112, T92, T116, T114, X121) → PDLD_IN_GAA(T112, T92, T116)

The TRS R consists of the following rules:

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, pdlB_in_ga(T5, T7))
pdlB_in_ga(nil, []) → pdlB_out_ga(nil, [])
pdlB_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pC_in_gaag(T57, T37, X37, T59))
pC_in_gaag(T57, T37, T61, T59) → U4_gaag(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
pdlD_in_gaa(nil, T69, T69) → pdlD_out_gaa(nil, T69, T69)
pdlD_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U3_gaa(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
pE_in_gaaga(T112, T92, T116, T114, X121) → U6_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U7_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T114, T116, X121)) → pE_out_gaaga(T112, T92, T116, T114, X121)
U3_gaa(T112, T113, T114, T92, X121, pE_out_gaaga(T112, T92, X120, T114, X121)) → pdlD_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaag(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → U5_gaag(T57, T37, T61, T59, pdlB_in_ga(T59, T61))
U5_gaag(T57, T37, T61, T59, pdlB_out_ga(T59, T61)) → pC_out_gaag(T57, T37, T61, T59)
U2_ga(T57, T58, T59, T37, pC_out_gaag(T57, T37, X37, T59)) → pdlB_out_ga(tree(T57, T58, T59), .(T58, T37))
U1_ga(T5, T7, pdlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pdlB_in_ga(x1, x2)  =  pdlB_in_ga(x1)
nil  =  nil
pdlB_out_ga(x1, x2)  =  pdlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
pdlD_in_gaa(x1, x2, x3)  =  pdlD_in_gaa(x1)
pdlD_out_gaa(x1, x2, x3)  =  pdlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)
PDLD_IN_GAA(x1, x2, x3)  =  PDLD_IN_GAA(x1)
PE_IN_GAAGA(x1, x2, x3, x4, x5)  =  PE_IN_GAAGA(x1, x4)
U6_GAAGA(x1, x2, x3, x4, x5, x6)  =  U6_GAAGA(x1, x4, x6)

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:

PDLD_IN_GAA(tree(T112, T113, T114), .(T113, T92), X121) → PE_IN_GAAGA(T112, T92, X120, T114, X121)
PE_IN_GAAGA(T112, T92, T116, T114, X121) → U6_GAAGA(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_GAAGA(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → PDLD_IN_GAA(T114, T116, X121)
PE_IN_GAAGA(T112, T92, T116, T114, X121) → PDLD_IN_GAA(T112, T92, T116)

The TRS R consists of the following rules:

pdlD_in_gaa(nil, T69, T69) → pdlD_out_gaa(nil, T69, T69)
pdlD_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U3_gaa(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
U3_gaa(T112, T113, T114, T92, X121, pE_out_gaaga(T112, T92, X120, T114, X121)) → pdlD_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
pE_in_gaaga(T112, T92, T116, T114, X121) → U6_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U7_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T114, T116, X121)) → pE_out_gaaga(T112, T92, T116, T114, X121)

The argument filtering Pi contains the following mapping:
nil  =  nil
tree(x1, x2, x3)  =  tree(x1, x2, x3)
pdlD_in_gaa(x1, x2, x3)  =  pdlD_in_gaa(x1)
pdlD_out_gaa(x1, x2, x3)  =  pdlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
PDLD_IN_GAA(x1, x2, x3)  =  PDLD_IN_GAA(x1)
PE_IN_GAAGA(x1, x2, x3, x4, x5)  =  PE_IN_GAAGA(x1, x4)
U6_GAAGA(x1, x2, x3, x4, x5, x6)  =  U6_GAAGA(x1, x4, x6)

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:

PDLD_IN_GAA(tree(T112, T113, T114)) → PE_IN_GAAGA(T112, T114)
PE_IN_GAAGA(T112, T114) → U6_GAAGA(T112, T114, pdlD_in_gaa(T112))
U6_GAAGA(T112, T114, pdlD_out_gaa(T112)) → PDLD_IN_GAA(T114)
PE_IN_GAAGA(T112, T114) → PDLD_IN_GAA(T112)

The TRS R consists of the following rules:

pdlD_in_gaa(nil) → pdlD_out_gaa(nil)
pdlD_in_gaa(tree(T112, T113, T114)) → U3_gaa(T112, T113, T114, pE_in_gaaga(T112, T114))
U3_gaa(T112, T113, T114, pE_out_gaaga(T112, T114)) → pdlD_out_gaa(tree(T112, T113, T114))
pE_in_gaaga(T112, T114) → U6_gaaga(T112, T114, pdlD_in_gaa(T112))
U6_gaaga(T112, T114, pdlD_out_gaa(T112)) → U7_gaaga(T112, T114, pdlD_in_gaa(T114))
U7_gaaga(T112, T114, pdlD_out_gaa(T114)) → pE_out_gaaga(T112, T114)

The set Q consists of the following terms:

pdlD_in_gaa(x0)
U3_gaa(x0, x1, x2, x3)
pE_in_gaaga(x0, x1)
U6_gaaga(x0, x1, x2)
U7_gaaga(x0, x1, x2)

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:

• PE_IN_GAAGA(T112, T114) → PDLD_IN_GAA(T112)
  The graph contains the following edges 1 >= 1

• PE_IN_GAAGA(T112, T114) → U6_GAAGA(T112, T114, pdlD_in_gaa(T112))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U6_GAAGA(T112, T114, pdlD_out_gaa(T112)) → PDLD_IN_GAA(T114)
  The graph contains the following edges 2 >= 1

• PDLD_IN_GAA(tree(T112, T113, T114)) → PE_IN_GAAGA(T112, T114)
  The graph contains the following edges 1 > 1, 1 > 2

(13) YES

(14) Obligation:

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

PDLB_IN_GA(tree(T57, T58, T59), .(T58, T37)) → PC_IN_GAAG(T57, T37, X37, T59)
PC_IN_GAAG(T57, T37, T61, T59) → U4_GAAG(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
U4_GAAG(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → PDLB_IN_GA(T59, T61)

The TRS R consists of the following rules:

preorderA_in_ga(T5, T7) → U1_ga(T5, T7, pdlB_in_ga(T5, T7))
pdlB_in_ga(nil, []) → pdlB_out_ga(nil, [])
pdlB_in_ga(tree(T57, T58, T59), .(T58, T37)) → U2_ga(T57, T58, T59, T37, pC_in_gaag(T57, T37, X37, T59))
pC_in_gaag(T57, T37, T61, T59) → U4_gaag(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
pdlD_in_gaa(nil, T69, T69) → pdlD_out_gaa(nil, T69, T69)
pdlD_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U3_gaa(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
pE_in_gaaga(T112, T92, T116, T114, X121) → U6_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U7_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T114, T116, X121)) → pE_out_gaaga(T112, T92, T116, T114, X121)
U3_gaa(T112, T113, T114, T92, X121, pE_out_gaaga(T112, T92, X120, T114, X121)) → pdlD_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
U4_gaag(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → U5_gaag(T57, T37, T61, T59, pdlB_in_ga(T59, T61))
U5_gaag(T57, T37, T61, T59, pdlB_out_ga(T59, T61)) → pC_out_gaag(T57, T37, T61, T59)
U2_ga(T57, T58, T59, T37, pC_out_gaag(T57, T37, X37, T59)) → pdlB_out_ga(tree(T57, T58, T59), .(T58, T37))
U1_ga(T5, T7, pdlB_out_ga(T5, T7)) → preorderA_out_ga(T5, T7)

The argument filtering Pi contains the following mapping:
preorderA_in_ga(x1, x2)  =  preorderA_in_ga(x1)
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
pdlB_in_ga(x1, x2)  =  pdlB_in_ga(x1)
nil  =  nil
pdlB_out_ga(x1, x2)  =  pdlB_out_ga(x1)
tree(x1, x2, x3)  =  tree(x1, x2, x3)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
pC_in_gaag(x1, x2, x3, x4)  =  pC_in_gaag(x1, x4)
U4_gaag(x1, x2, x3, x4, x5)  =  U4_gaag(x1, x4, x5)
pdlD_in_gaa(x1, x2, x3)  =  pdlD_in_gaa(x1)
pdlD_out_gaa(x1, x2, x3)  =  pdlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
U5_gaag(x1, x2, x3, x4, x5)  =  U5_gaag(x1, x4, x5)
pC_out_gaag(x1, x2, x3, x4)  =  pC_out_gaag(x1, x4)
preorderA_out_ga(x1, x2)  =  preorderA_out_ga(x1)
PDLB_IN_GA(x1, x2)  =  PDLB_IN_GA(x1)
PC_IN_GAAG(x1, x2, x3, x4)  =  PC_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5)  =  U4_GAAG(x1, x4, x5)

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:

PDLB_IN_GA(tree(T57, T58, T59), .(T58, T37)) → PC_IN_GAAG(T57, T37, X37, T59)
PC_IN_GAAG(T57, T37, T61, T59) → U4_GAAG(T57, T37, T61, T59, pdlD_in_gaa(T57, T37, T61))
U4_GAAG(T57, T37, T61, T59, pdlD_out_gaa(T57, T37, T61)) → PDLB_IN_GA(T59, T61)

The TRS R consists of the following rules:

pdlD_in_gaa(nil, T69, T69) → pdlD_out_gaa(nil, T69, T69)
pdlD_in_gaa(tree(T112, T113, T114), .(T113, T92), X121) → U3_gaa(T112, T113, T114, T92, X121, pE_in_gaaga(T112, T92, X120, T114, X121))
U3_gaa(T112, T113, T114, T92, X121, pE_out_gaaga(T112, T92, X120, T114, X121)) → pdlD_out_gaa(tree(T112, T113, T114), .(T113, T92), X121)
pE_in_gaaga(T112, T92, T116, T114, X121) → U6_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T112, T92, T116))
U6_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T112, T92, T116)) → U7_gaaga(T112, T92, T116, T114, X121, pdlD_in_gaa(T114, T116, X121))
U7_gaaga(T112, T92, T116, T114, X121, pdlD_out_gaa(T114, T116, X121)) → pE_out_gaaga(T112, T92, T116, T114, X121)

The argument filtering Pi contains the following mapping:
nil  =  nil
tree(x1, x2, x3)  =  tree(x1, x2, x3)
pdlD_in_gaa(x1, x2, x3)  =  pdlD_in_gaa(x1)
pdlD_out_gaa(x1, x2, x3)  =  pdlD_out_gaa(x1)
U3_gaa(x1, x2, x3, x4, x5, x6)  =  U3_gaa(x1, x2, x3, x6)
pE_in_gaaga(x1, x2, x3, x4, x5)  =  pE_in_gaaga(x1, x4)
U6_gaaga(x1, x2, x3, x4, x5, x6)  =  U6_gaaga(x1, x4, x6)
U7_gaaga(x1, x2, x3, x4, x5, x6)  =  U7_gaaga(x1, x4, x6)
pE_out_gaaga(x1, x2, x3, x4, x5)  =  pE_out_gaaga(x1, x4)
PDLB_IN_GA(x1, x2)  =  PDLB_IN_GA(x1)
PC_IN_GAAG(x1, x2, x3, x4)  =  PC_IN_GAAG(x1, x4)
U4_GAAG(x1, x2, x3, x4, x5)  =  U4_GAAG(x1, x4, x5)

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:

PDLB_IN_GA(tree(T57, T58, T59)) → PC_IN_GAAG(T57, T59)
PC_IN_GAAG(T57, T59) → U4_GAAG(T57, T59, pdlD_in_gaa(T57))
U4_GAAG(T57, T59, pdlD_out_gaa(T57)) → PDLB_IN_GA(T59)

The TRS R consists of the following rules:

pdlD_in_gaa(nil) → pdlD_out_gaa(nil)
pdlD_in_gaa(tree(T112, T113, T114)) → U3_gaa(T112, T113, T114, pE_in_gaaga(T112, T114))
U3_gaa(T112, T113, T114, pE_out_gaaga(T112, T114)) → pdlD_out_gaa(tree(T112, T113, T114))
pE_in_gaaga(T112, T114) → U6_gaaga(T112, T114, pdlD_in_gaa(T112))
U6_gaaga(T112, T114, pdlD_out_gaa(T112)) → U7_gaaga(T112, T114, pdlD_in_gaa(T114))
U7_gaaga(T112, T114, pdlD_out_gaa(T114)) → pE_out_gaaga(T112, T114)

The set Q consists of the following terms:

pdlD_in_gaa(x0)
U3_gaa(x0, x1, x2, x3)
pE_in_gaaga(x0, x1)
U6_gaaga(x0, x1, x2)
U7_gaaga(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:

• PC_IN_GAAG(T57, T59) → U4_GAAG(T57, T59, pdlD_in_gaa(T57))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U4_GAAG(T57, T59, pdlD_out_gaa(T57)) → PDLB_IN_GA(T59)
  The graph contains the following edges 2 >= 1

• PDLB_IN_GA(tree(T57, T58, T59)) → PC_IN_GAAG(T57, T59)
  The graph contains the following edges 1 > 1, 1 > 2

(20) YES