Left Termination proof of /tmp/tmpXS33O4/inorder-oi.pl

Left Termination of the query pattern in_order_in_2(a, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 UndefinedPredicateInTriplesTransformerProof (⇐)

↳4 TRIPLES

↳5 TriplesToPiDPProof (⇐)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇐)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇐)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 YES

    ↳23 PiDP

        ↳24 PiDPToQDPProof (⇐)

        ↳25 QDP

        ↳26 QDPOrderProof (⇔)

        ↳27 QDP

        ↳28 DependencyGraphProof (⇔)

        ↳29 TRUE

(0) Obligation:

Clauses:

in_order(void, L) :- ','(!, eq(L, [])).
in_order(T, Xs) :- ','(value(T, X), ','(app(Ls, .(X, Rs), Xs), ','(left(T, L), ','(in_order(L, Ls), ','(right(T, R), in_order(R, Rs)))))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
left(void, void).
left(node(L, X1, X2), L).
right(void, void).
right(node(X3, X4, R), R).
value(void, X5).
value(node(X6, X, X7), X).
eq(X, X).

Queries:

in_order(a,g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: in_order(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: in_order(T1, T2), SCOPE: 1, CLAUSE: 0 | in_order(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T4, [])) | in_order(T1, T4), SCOPE: 1, CLAUSE: 1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: in_order(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nin_order(T1, T2) !=? in_order(void, X9)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T4, [])\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T4, []), SCOPE: 2, CLAUSE: 10\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(value(T12, X19), ','(app(X20, .(X19, X21), T11), ','(left(T12, X22), ','(in_order(X22, X20), ','(right(T12, X23), in_order(X23, X21))))))\n\nT11 is ground\nX19 is free; \nX20 is free; \nX21 is free; \nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(value(T12, X19), ','(app(X20, .(X19, X21), T11), ','(left(T12, X22), ','(in_order(X22, X20), ','(right(T12, X23), in_order(X23, X21)))))), SCOPE: 3, CLAUSE: 8 | ','(value(T12, X19), ','(app(X20, .(X19, X21), T11), ','(left(T12, X22), ','(in_order(X22, X20), ','(right(T12, X23), in_order(X23, X21)))))), SCOPE: 3, CLAUSE: 9\n\nT11 is ground\nX19 is free; \nX20 is free; \nX21 is free; \nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(value(T12, X19), ','(app(X20, .(X19, X21), T11), ','(left(T12, X22), ','(in_order(X22, X20), ','(right(T12, X23), in_order(X23, X21)))))), SCOPE: 3, CLAUSE: 8\n\nT11 is ground\nX19 is free; \nX20 is free; \nX21 is free; \nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(value(T12, X19), ','(app(X20, .(X19, X21), T11), ','(left(T12, X22), ','(in_order(X22, X20), ','(right(T12, X23), in_order(X23, X21)))))), SCOPE: 3, CLAUSE: 9\n\nT11 is ground\nX19 is free; \nX20 is free; \nX21 is free; \nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(app(X20, .(X33, X21), T11), ','(left(void, X22), ','(in_order(X22, X20), ','(right(void, X23), in_order(X23, X21)))))\n\nT11 is ground\nX20 is free; \nX21 is free; \nX22 is free; \nX23 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app(X20, .(X33, X21), T11)\n\nT11 is ground\nX20 is free; \nX21 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(left(void, X22), ','(in_order(X22, T13), ','(right(void, X23), in_order(X23, T15))))\n\nT13 is ground\nT15 is ground\nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(X20, .(X33, X21), T11), SCOPE: 4, CLAUSE: 2 | app(X20, .(X33, X21), T11), SCOPE: 4, CLAUSE: 3\n\nT11 is ground\nX20 is free; \nX21 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(X20, .(X33, X21), T11), SCOPE: 4, CLAUSE: 2\n\nT11 is ground\nX20 is free; \nX21 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(X20, .(X33, X21), T11), SCOPE: 4, CLAUSE: 3\n\nT11 is ground\nX20 is free; \nX21 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: app(X84, .(X85, X86), T18)\n\nT18 is ground\nX84 is free; \nX85 is free; \nX86 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(left(void, X22), ','(in_order(X22, T13), ','(right(void, X23), in_order(X23, T15)))), SCOPE: 5, CLAUSE: 4 | ','(left(void, X22), ','(in_order(X22, T13), ','(right(void, X23), in_order(X23, T15)))), SCOPE: 5, CLAUSE: 5\n\nT13 is ground\nT15 is ground\nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(left(void, X22), ','(in_order(X22, T13), ','(right(void, X23), in_order(X23, T15)))), SCOPE: 5, CLAUSE: 4\n\nT13 is ground\nT15 is ground\nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(left(void, X22), ','(in_order(X22, T13), ','(right(void, X23), in_order(X23, T15)))), SCOPE: 5, CLAUSE: 5\n\nT13 is ground\nT15 is ground\nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(in_order(void, T13), ','(right(void, X23), in_order(X23, T15)))\n\nT13 is ground\nT15 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: in_order(void, T13)\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(right(void, X23), in_order(X23, T15))\n\nT15 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: in_order(void, T13), SCOPE: 6, CLAUSE: 0 | in_order(void, T13), SCOPE: 6, CLAUSE: 1\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(!_6, eq(T26, [])) | in_order(void, T26), SCOPE: 6, CLAUSE: 1\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: eq(T26, [])\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: eq(T26, []), SCOPE: 7, CLAUSE: 10\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(right(void, X23), in_order(X23, T15)), SCOPE: 8, CLAUSE: 6 | ','(right(void, X23), in_order(X23, T15)), SCOPE: 8, CLAUSE: 7\n\nT15 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(right(void, X23), in_order(X23, T15)), SCOPE: 8, CLAUSE: 6\n\nT15 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(right(void, X23), in_order(X23, T15)), SCOPE: 8, CLAUSE: 7\n\nT15 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: in_order(void, T15)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(app(X20, .(T43, X21), T11), ','(left(node(T44, T43, T45), X22), ','(in_order(X22, X20), ','(right(node(T44, T43, T45), X23), in_order(X23, X21)))))\n\nT11 is ground\nX20 is free; \nX21 is free; \nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: app(X20, .(T43, X21), T11)\n\nT11 is ground\nX20 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(left(node(T50, T51, T52), X22), ','(in_order(X22, T48), ','(right(node(T50, T51, T52), X23), in_order(X23, T49))))\n\nT48 is ground\nT51 is ground\nT49 is ground\nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: app(X20, .(T43, X21), T11), SCOPE: 9, CLAUSE: 2 | app(X20, .(T43, X21), T11), SCOPE: 9, CLAUSE: 3\n\nT11 is ground\nX20 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: app(X20, .(T43, X21), T11), SCOPE: 9, CLAUSE: 2\n\nT11 is ground\nX20 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: app(X20, .(T43, X21), T11), SCOPE: 9, CLAUSE: 3\n\nT11 is ground\nX20 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: app(X169, .(T66, X170), T65)\n\nT65 is ground\nX169 is free; \nX170 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(left(node(T50, T51, T52), X22), ','(in_order(X22, T48), ','(right(node(T50, T51, T52), X23), in_order(X23, T49)))), SCOPE: 10, CLAUSE: 4 | ','(left(node(T50, T51, T52), X22), ','(in_order(X22, T48), ','(right(node(T50, T51, T52), X23), in_order(X23, T49)))), SCOPE: 10, CLAUSE: 5\n\nT48 is ground\nT51 is ground\nT49 is ground\nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(left(node(T50, T51, T52), X22), ','(in_order(X22, T48), ','(right(node(T50, T51, T52), X23), in_order(X23, T49)))), SCOPE: 10, CLAUSE: 5\n\nT48 is ground\nT51 is ground\nT49 is ground\nX22 is free; \nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(in_order(T82, T48), ','(right(node(T82, T80, T83), X23), in_order(X23, T49)))\n\nT48 is ground\nT49 is ground\nT80 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: in_order(T82, T48)\n\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(right(node(T85, T80, T86), X23), in_order(X23, T49))\n\nT49 is ground\nT80 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(right(node(T85, T80, T86), X23), in_order(X23, T49)), SCOPE: 11, CLAUSE: 6 | ','(right(node(T85, T80, T86), X23), in_order(X23, T49)), SCOPE: 11, CLAUSE: 7\n\nT49 is ground\nT80 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(right(node(T85, T80, T86), X23), in_order(X23, T49)), SCOPE: 11, CLAUSE: 7\n\nT49 is ground\nT80 is ground\nX23 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: in_order(T100, T49)\n\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 65 [arrowhead = none ];
65 -> 2;

66 [label="EVAL with clause\nin_order(void, X9) :- ','(!_1, eq(X9, [])).\nand substitution\nT1 -> void\nT2 -> T4\nX9 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 66 [arrowhead = none ];
66 -> 3;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 67 [arrowhead = none ];
67 -> 4;

68 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 68 [arrowhead = none ];
68 -> 5;

69 [label="ONLY EVAL with clause\nin_order(X17, X18) :- ','(value(X17, X19), ','(app(X20, .(X19, X21), X18), ','(left(X17, X22), ','(in_order(X22, X20), ','(right(X17, X23), in_order(X23, X21)))))).\nand substitution\nT1 -> T12\nX17 -> T12\nT2 -> T11\nX18 -> T11\nT10 -> T12", fontsize=14, style = filled, fillcolor = white];
4 -> 69 [arrowhead = none ];
69 -> 10;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 70 [arrowhead = none ];
70 -> 6;

71 [label="EVAL with clause\neq(X13, X13).\nand substitution\nT4 -> []\nX13 -> []\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 71 [arrowhead = none ];
71 -> 7;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 72 [arrowhead = none ];
72 -> 8;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 73 [arrowhead = none ];
73 -> 9;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 74 [arrowhead = none ];
74 -> 11;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 75 [arrowhead = none ];
75 -> 12;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 76 [arrowhead = none ];
76 -> 13;

77 [label="EVAL with clause\nvalue(void, X32).\nand substitution\nT12 -> void\nX19 -> X33\nX32 -> X33", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 77 [arrowhead = none ];
77 -> 14;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 78 [arrowhead = none ];
78 -> 15;

79 [label="EVAL with clause\nvalue(node(X125, X126, X127), X126).\nand substitution\nX125 -> T44\nX126 -> T43\nX127 -> T45\nT12 -> node(T44, T43, T45)\nX19 -> T43\nT41 -> T43\nT40 -> T44\nT42 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 79 [arrowhead = none ];
79 -> 45;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 80 [arrowhead = none ];
80 -> 46;

81 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
14 -> 81 [arrowhead = none ];
81 -> 16;

82 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nT14 is ground\nT15 is ground\nreplacements:\nX20 -> T13\nX33 -> T14\nX21 -> T15", fontsize=14, style = filled, fillcolor = violet];
14 -> 82 [arrowhead = none ];
82 -> 17;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 83 [arrowhead = none ];
83 -> 18;

84 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 84 [arrowhead = none ];
84 -> 26;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 85 [arrowhead = none ];
85 -> 19;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 86 [arrowhead = none ];
86 -> 20;

87 [label="EVAL with clause\napp([], X59, X59).\nand substitution\nX20 -> []\nX33 -> X60\nX21 -> X61\nX59 -> .(X60, X61)\nT11 -> .(X60, X61)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 87 [arrowhead = none ];
87 -> 21;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 88 [arrowhead = none ];
88 -> 22;

89 [label="EVAL with clause\napp(.(X79, X80), X81, .(X79, X82)) :- app(X80, X81, X82).\nand substitution\nX79 -> X83\nX80 -> X84\nX20 -> .(X83, X84)\nX33 -> X85\nX21 -> X86\nX81 -> .(X85, X86)\nX82 -> T18\nT11 -> .(X83, T18)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 89 [arrowhead = none ];
89 -> 24;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 90 [arrowhead = none ];
90 -> 25;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 91 [arrowhead = none ];
91 -> 23;

92 [label="INSTANCE with matching:\nX20 -> X84\nX33 -> X85\nX21 -> X86\nT11 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 92 [arrowhead = none , style=dashed];
92 -> 16[style=dashed];

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 93 [arrowhead = none ];
93 -> 27;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 94 [arrowhead = none ];
94 -> 28;

95 [label="ONLY EVAL with clause\nleft(void, void).\nand substitution\nX22 -> void", fontsize=14, style = filled, fillcolor = white];
27 -> 95 [arrowhead = none ];
95 -> 29;

96 [label="BACKTRACK\nfor clause: left(node(L, X1, X2), L)because of non-unification", fontsize=14, style = filled, fillcolor = white];
28 -> 96 [arrowhead = none ];
96 -> 44;

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 97 [arrowhead = none ];
97 -> 30;

98 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
29 -> 98 [arrowhead = none ];
98 -> 31;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 99 [arrowhead = none ];
99 -> 32;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 100 [arrowhead = none ];
100 -> 39;

101 [label="ONLY EVAL with clause\nin_order(void, X104) :- ','(!_6, eq(X104, [])).\nand substitution\nT13 -> T26\nX104 -> T26", fontsize=14, style = filled, fillcolor = white];
32 -> 101 [arrowhead = none ];
101 -> 33;

102 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 102 [arrowhead = none ];
102 -> 34;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 103 [arrowhead = none ];
103 -> 35;

104 [label="EVAL with clause\neq(X107, X107).\nand substitution\nT26 -> []\nX107 -> []\nT29 -> []", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 104 [arrowhead = none ];
104 -> 36;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 105 [arrowhead = none ];
105 -> 37;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 106 [arrowhead = none ];
106 -> 38;

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 107 [arrowhead = none ];
107 -> 40;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
39 -> 108 [arrowhead = none ];
108 -> 41;

109 [label="ONLY EVAL with clause\nright(void, void).\nand substitution\nX23 -> void", fontsize=14, style = filled, fillcolor = white];
40 -> 109 [arrowhead = none ];
109 -> 42;

110 [label="BACKTRACK\nfor clause: right(node(X3, X4, R), R)because of non-unification", fontsize=14, style = filled, fillcolor = white];
41 -> 110 [arrowhead = none ];
110 -> 43;

111 [label="INSTANCE with matching:\nT13 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 111 [arrowhead = none , style=dashed];
111 -> 30[style=dashed];

112 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
45 -> 112 [arrowhead = none ];
112 -> 47;

113 [label="SPLIT 2\nnew knowledge:\nT48 is ground\nT51 is ground\nT49 is ground\nreplacements:\nX20 -> T48\nX21 -> T49\nT44 -> T50\nT43 -> T51\nT45 -> T52", fontsize=14, style = filled, fillcolor = violet];
45 -> 113 [arrowhead = none ];
113 -> 48;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
47 -> 114 [arrowhead = none ];
114 -> 49;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 115 [arrowhead = none ];
115 -> 57;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 116 [arrowhead = none ];
116 -> 50;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
49 -> 117 [arrowhead = none ];
117 -> 51;

118 [label="EVAL with clause\napp([], X147, X147).\nand substitution\nX20 -> []\nT43 -> T59\nX21 -> X148\nX147 -> .(T59, X148)\nT11 -> .(T59, X148)", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 118 [arrowhead = none ];
118 -> 52;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 119 [arrowhead = none ];
119 -> 53;

120 [label="EVAL with clause\napp(.(X164, X165), X166, .(X164, X167)) :- app(X165, X166, X167).\nand substitution\nX164 -> X168\nX165 -> X169\nX20 -> .(X168, X169)\nT43 -> T66\nX21 -> X170\nX166 -> .(T66, X170)\nX167 -> T65\nT11 -> .(X168, T65)\nT64 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 120 [arrowhead = none ];
120 -> 55;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 121 [arrowhead = none ];
121 -> 56;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 122 [arrowhead = none ];
122 -> 54;

123 [label="INSTANCE with matching:\nX20 -> X169\nT43 -> T66\nX21 -> X170\nT11 -> T65", fontsize=14, style = filled, fillcolor = lightgrey];
55 -> 123 [arrowhead = none , style=dashed];
123 -> 47[style=dashed];

124 [label="BACKTRACK\nfor clause: left(void, void)because of non-unification", fontsize=14, style = filled, fillcolor = white];
57 -> 124 [arrowhead = none ];
124 -> 58;

125 [label="ONLY EVAL with clause\nleft(node(X187, X188, X189), X187).\nand substitution\nT50 -> T82\nX187 -> T82\nT51 -> T80\nX188 -> T80\nT52 -> T83\nX189 -> T83\nX22 -> T82\nT79 -> T82\nT81 -> T83", fontsize=14, style = filled, fillcolor = white];
58 -> 125 [arrowhead = none ];
125 -> 59;

126 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
59 -> 126 [arrowhead = none ];
126 -> 60;

127 [label="SPLIT 2\nreplacements:\nT82 -> T85\nT83 -> T86", fontsize=14, style = filled, fillcolor = violet];
59 -> 127 [arrowhead = none ];
127 -> 61;

128 [label="INSTANCE with matching:\nT1 -> T82\nT2 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 128 [arrowhead = none , style=dashed];
128 -> 1[style=dashed];

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 129 [arrowhead = none ];
129 -> 62;

130 [label="BACKTRACK\nfor clause: right(void, void)because of non-unification", fontsize=14, style = filled, fillcolor = white];
62 -> 130 [arrowhead = none ];
130 -> 63;

131 [label="ONLY EVAL with clause\nright(node(X200, X201, X202), X202).\nand substitution\nT85 -> T97\nX200 -> T97\nT80 -> T98\nX201 -> T98\nT86 -> T100\nX202 -> T100\nX23 -> T100\nT99 -> T100", fontsize=14, style = filled, fillcolor = white];
63 -> 131 [arrowhead = none ];
131 -> 64;

132 [label="INSTANCE with matching:\nT1 -> T100\nT2 -> T49", fontsize=14, style = filled, fillcolor = lightgrey];
64 -> 132 [arrowhead = none , style=dashed];
132 -> 1[style=dashed];

}

(2) Obligation:

Triples:

app16(.(X83, X84), X85, X86, .(X83, T18)) :- app16(X84, X85, X86, T18).
app47(.(X168, X169), T66, X170, .(X168, T65)) :- app47(X169, T66, X170, T65).
in_order1(void, T11) :- app16(X20, X33, X21, T11).
in_order1(void, T11) :- ','(appc16(T13, T14, T15, T11), in_order30(T13)).
in_order1(void, T11) :- ','(appc16(T13, T14, T15, T11), ','(in_orderc30(T13), in_order30(T15))).
in_order1(node(T44, T43, T45), T11) :- app47(X20, T43, X21, T11).
in_order1(node(T82, T80, T83), T11) :- ','(appc47(T48, T80, T49, T11), in_order1(T82, T48)).
in_order1(node(T97, T98, T100), T11) :- ','(appc47(T48, T98, T49, T11), ','(in_orderc1(T97, T48), in_order1(T100, T49))).

Clauses:

appc16([], X60, X61, .(X60, X61)).
appc16(.(X83, X84), X85, X86, .(X83, T18)) :- appc16(X84, X85, X86, T18).
in_orderc30([]).
appc47([], T59, X148, .(T59, X148)).
appc47(.(X168, X169), T66, X170, .(X168, T65)) :- appc47(X169, T66, X170, T65).
in_orderc1(void, []).
in_orderc1(void, T11) :- ','(appc16(T13, T14, T15, T11), ','(in_orderc30(T13), in_orderc30(T15))).
in_orderc1(node(T97, T98, T100), T11) :- ','(appc47(T48, T98, T49, T11), ','(in_orderc1(T97, T48), in_orderc1(T100, T49))).

Afs:

in_order1(x1, x2) = in_order1(x2)

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

Deleted triples and predicates having undefined goals [UNKNOWN].

(4) Obligation:

Triples:

app16(.(X83, X84), X85, X86, .(X83, T18)) :- app16(X84, X85, X86, T18).
app47(.(X168, X169), T66, X170, .(X168, T65)) :- app47(X169, T66, X170, T65).
in_order1(void, T11) :- app16(X20, X33, X21, T11).
in_order1(node(T44, T43, T45), T11) :- app47(X20, T43, X21, T11).
in_order1(node(T82, T80, T83), T11) :- ','(appc47(T48, T80, T49, T11), in_order1(T82, T48)).
in_order1(node(T97, T98, T100), T11) :- ','(appc47(T48, T98, T49, T11), ','(in_orderc1(T97, T48), in_order1(T100, T49))).

Clauses:

appc16([], X60, X61, .(X60, X61)).
appc16(.(X83, X84), X85, X86, .(X83, T18)) :- appc16(X84, X85, X86, T18).
in_orderc30([]).
appc47([], T59, X148, .(T59, X148)).
appc47(.(X168, X169), T66, X170, .(X168, T65)) :- appc47(X169, T66, X170, T65).
in_orderc1(void, []).
in_orderc1(void, T11) :- ','(appc16(T13, T14, T15, T11), ','(in_orderc30(T13), in_orderc30(T15))).
in_orderc1(node(T97, T98, T100), T11) :- ','(appc47(T48, T98, T49, T11), ','(in_orderc1(T97, T48), in_orderc1(T100, T49))).

Afs:

in_order1(x1, x2) = in_order1(x2)

(5) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
in_order1_in: (f,b)
app16_in: (f,f,f,b)
app47_in: (f,f,f,b)
appc47_in: (f,f,f,b)
in_orderc1_in: (f,b)
appc16_in: (f,f,f,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

IN_ORDER1_IN_AG(void, T11) → U3_AG(T11, app16_in_aaag(X20, X33, X21, T11))
IN_ORDER1_IN_AG(void, T11) → APP16_IN_AAAG(X20, X33, X21, T11)
APP16_IN_AAAG(.(X83, X84), X85, X86, .(X83, T18)) → U1_AAAG(X83, X84, X85, X86, T18, app16_in_aaag(X84, X85, X86, T18))
APP16_IN_AAAG(.(X83, X84), X85, X86, .(X83, T18)) → APP16_IN_AAAG(X84, X85, X86, T18)
IN_ORDER1_IN_AG(node(T44, T43, T45), T11) → U4_AG(T44, T43, T45, T11, app47_in_aaag(X20, T43, X21, T11))
IN_ORDER1_IN_AG(node(T44, T43, T45), T11) → APP47_IN_AAAG(X20, T43, X21, T11)
APP47_IN_AAAG(.(X168, X169), T66, X170, .(X168, T65)) → U2_AAAG(X168, X169, T66, X170, T65, app47_in_aaag(X169, T66, X170, T65))
APP47_IN_AAAG(.(X168, X169), T66, X170, .(X168, T65)) → APP47_IN_AAAG(X169, T66, X170, T65)
IN_ORDER1_IN_AG(node(T82, T80, T83), T11) → U5_AG(T82, T80, T83, T11, appc47_in_aaag(T48, T80, T49, T11))
U5_AG(T82, T80, T83, T11, appc47_out_aaag(T48, T80, T49, T11)) → U6_AG(T82, T80, T83, T11, in_order1_in_ag(T82, T48))
U5_AG(T82, T80, T83, T11, appc47_out_aaag(T48, T80, T49, T11)) → IN_ORDER1_IN_AG(T82, T48)
IN_ORDER1_IN_AG(node(T97, T98, T100), T11) → U7_AG(T97, T98, T100, T11, appc47_in_aaag(T48, T98, T49, T11))
U7_AG(T97, T98, T100, T11, appc47_out_aaag(T48, T98, T49, T11)) → U8_AG(T97, T98, T100, T11, T49, in_orderc1_in_ag(T97, T48))
U8_AG(T97, T98, T100, T11, T49, in_orderc1_out_ag(T97, T48)) → U9_AG(T97, T98, T100, T11, in_order1_in_ag(T100, T49))
U8_AG(T97, T98, T100, T11, T49, in_orderc1_out_ag(T97, T48)) → IN_ORDER1_IN_AG(T100, T49)

The TRS R consists of the following rules:

appc47_in_aaag([], T59, X148, .(T59, X148)) → appc47_out_aaag([], T59, X148, .(T59, X148))
appc47_in_aaag(.(X168, X169), T66, X170, .(X168, T65)) → U12_aaag(X168, X169, T66, X170, T65, appc47_in_aaag(X169, T66, X170, T65))
U12_aaag(X168, X169, T66, X170, T65, appc47_out_aaag(X169, T66, X170, T65)) → appc47_out_aaag(.(X168, X169), T66, X170, .(X168, T65))
in_orderc1_in_ag(void, []) → in_orderc1_out_ag(void, [])
in_orderc1_in_ag(void, T11) → U13_ag(T11, appc16_in_aaag(T13, T14, T15, T11))
appc16_in_aaag([], X60, X61, .(X60, X61)) → appc16_out_aaag([], X60, X61, .(X60, X61))
appc16_in_aaag(.(X83, X84), X85, X86, .(X83, T18)) → U11_aaag(X83, X84, X85, X86, T18, appc16_in_aaag(X84, X85, X86, T18))
U11_aaag(X83, X84, X85, X86, T18, appc16_out_aaag(X84, X85, X86, T18)) → appc16_out_aaag(.(X83, X84), X85, X86, .(X83, T18))
U13_ag(T11, appc16_out_aaag(T13, T14, T15, T11)) → U14_ag(T11, T13, T14, T15, in_orderc30_in_g(T13))
in_orderc30_in_g([]) → in_orderc30_out_g([])
U14_ag(T11, T13, T14, T15, in_orderc30_out_g(T13)) → U15_ag(T11, in_orderc30_in_g(T15))
U15_ag(T11, in_orderc30_out_g(T15)) → in_orderc1_out_ag(void, T11)
in_orderc1_in_ag(node(T97, T98, T100), T11) → U16_ag(T97, T98, T100, T11, appc47_in_aaag(T48, T98, T49, T11))
U16_ag(T97, T98, T100, T11, appc47_out_aaag(T48, T98, T49, T11)) → U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_in_ag(T97, T48))
U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_out_ag(T97, T48)) → U18_ag(T97, T98, T100, T11, in_orderc1_in_ag(T100, T49))
U18_ag(T97, T98, T100, T11, in_orderc1_out_ag(T100, T49)) → in_orderc1_out_ag(node(T97, T98, T100), T11)

The argument filtering Pi contains the following mapping:
in_order1_in_ag(x1, x2) = in_order1_in_ag(x2)
app16_in_aaag(x1, x2, x3, x4) = app16_in_aaag(x4)
.(x1, x2) = .(x1, x2)
app47_in_aaag(x1, x2, x3, x4) = app47_in_aaag(x4)
appc47_in_aaag(x1, x2, x3, x4) = appc47_in_aaag(x4)
appc47_out_aaag(x1, x2, x3, x4) = appc47_out_aaag(x1, x2, x3, x4)
U12_aaag(x1, x2, x3, x4, x5, x6) = U12_aaag(x1, x5, x6)
in_orderc1_in_ag(x1, x2) = in_orderc1_in_ag(x2)
[] = []
in_orderc1_out_ag(x1, x2) = in_orderc1_out_ag(x1, x2)
U13_ag(x1, x2) = U13_ag(x1, x2)
appc16_in_aaag(x1, x2, x3, x4) = appc16_in_aaag(x4)
appc16_out_aaag(x1, x2, x3, x4) = appc16_out_aaag(x1, x2, x3, x4)
U11_aaag(x1, x2, x3, x4, x5, x6) = U11_aaag(x1, x5, x6)
U14_ag(x1, x2, x3, x4, x5) = U14_ag(x1, x4, x5)
in_orderc30_in_g(x1) = in_orderc30_in_g(x1)
in_orderc30_out_g(x1) = in_orderc30_out_g(x1)
U15_ag(x1, x2) = U15_ag(x1, x2)
U16_ag(x1, x2, x3, x4, x5) = U16_ag(x4, x5)
U17_ag(x1, x2, x3, x4, x5, x6, x7) = U17_ag(x2, x4, x6, x7)
U18_ag(x1, x2, x3, x4, x5) = U18_ag(x1, x2, x4, x5)
void = void
node(x1, x2, x3) = node(x1, x2, x3)
IN_ORDER1_IN_AG(x1, x2) = IN_ORDER1_IN_AG(x2)
U3_AG(x1, x2) = U3_AG(x1, x2)
APP16_IN_AAAG(x1, x2, x3, x4) = APP16_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6) = U1_AAAG(x1, x5, x6)
U4_AG(x1, x2, x3, x4, x5) = U4_AG(x4, x5)
APP47_IN_AAAG(x1, x2, x3, x4) = APP47_IN_AAAG(x4)
U2_AAAG(x1, x2, x3, x4, x5, x6) = U2_AAAG(x1, x5, x6)
U5_AG(x1, x2, x3, x4, x5) = U5_AG(x4, x5)
U6_AG(x1, x2, x3, x4, x5) = U6_AG(x4, x5)
U7_AG(x1, x2, x3, x4, x5) = U7_AG(x4, x5)
U8_AG(x1, x2, x3, x4, x5, x6) = U8_AG(x4, x5, x6)
U9_AG(x1, x2, x3, x4, x5) = U9_AG(x4, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(6) Obligation:

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

IN_ORDER1_IN_AG(void, T11) → U3_AG(T11, app16_in_aaag(X20, X33, X21, T11))
IN_ORDER1_IN_AG(void, T11) → APP16_IN_AAAG(X20, X33, X21, T11)
APP16_IN_AAAG(.(X83, X84), X85, X86, .(X83, T18)) → U1_AAAG(X83, X84, X85, X86, T18, app16_in_aaag(X84, X85, X86, T18))
APP16_IN_AAAG(.(X83, X84), X85, X86, .(X83, T18)) → APP16_IN_AAAG(X84, X85, X86, T18)
IN_ORDER1_IN_AG(node(T44, T43, T45), T11) → U4_AG(T44, T43, T45, T11, app47_in_aaag(X20, T43, X21, T11))
IN_ORDER1_IN_AG(node(T44, T43, T45), T11) → APP47_IN_AAAG(X20, T43, X21, T11)
APP47_IN_AAAG(.(X168, X169), T66, X170, .(X168, T65)) → U2_AAAG(X168, X169, T66, X170, T65, app47_in_aaag(X169, T66, X170, T65))
APP47_IN_AAAG(.(X168, X169), T66, X170, .(X168, T65)) → APP47_IN_AAAG(X169, T66, X170, T65)
IN_ORDER1_IN_AG(node(T82, T80, T83), T11) → U5_AG(T82, T80, T83, T11, appc47_in_aaag(T48, T80, T49, T11))
U5_AG(T82, T80, T83, T11, appc47_out_aaag(T48, T80, T49, T11)) → U6_AG(T82, T80, T83, T11, in_order1_in_ag(T82, T48))
U5_AG(T82, T80, T83, T11, appc47_out_aaag(T48, T80, T49, T11)) → IN_ORDER1_IN_AG(T82, T48)
IN_ORDER1_IN_AG(node(T97, T98, T100), T11) → U7_AG(T97, T98, T100, T11, appc47_in_aaag(T48, T98, T49, T11))
U7_AG(T97, T98, T100, T11, appc47_out_aaag(T48, T98, T49, T11)) → U8_AG(T97, T98, T100, T11, T49, in_orderc1_in_ag(T97, T48))
U8_AG(T97, T98, T100, T11, T49, in_orderc1_out_ag(T97, T48)) → U9_AG(T97, T98, T100, T11, in_order1_in_ag(T100, T49))
U8_AG(T97, T98, T100, T11, T49, in_orderc1_out_ag(T97, T48)) → IN_ORDER1_IN_AG(T100, T49)

The TRS R consists of the following rules:

appc47_in_aaag([], T59, X148, .(T59, X148)) → appc47_out_aaag([], T59, X148, .(T59, X148))
appc47_in_aaag(.(X168, X169), T66, X170, .(X168, T65)) → U12_aaag(X168, X169, T66, X170, T65, appc47_in_aaag(X169, T66, X170, T65))
U12_aaag(X168, X169, T66, X170, T65, appc47_out_aaag(X169, T66, X170, T65)) → appc47_out_aaag(.(X168, X169), T66, X170, .(X168, T65))
in_orderc1_in_ag(void, []) → in_orderc1_out_ag(void, [])
in_orderc1_in_ag(void, T11) → U13_ag(T11, appc16_in_aaag(T13, T14, T15, T11))
appc16_in_aaag([], X60, X61, .(X60, X61)) → appc16_out_aaag([], X60, X61, .(X60, X61))
appc16_in_aaag(.(X83, X84), X85, X86, .(X83, T18)) → U11_aaag(X83, X84, X85, X86, T18, appc16_in_aaag(X84, X85, X86, T18))
U11_aaag(X83, X84, X85, X86, T18, appc16_out_aaag(X84, X85, X86, T18)) → appc16_out_aaag(.(X83, X84), X85, X86, .(X83, T18))
U13_ag(T11, appc16_out_aaag(T13, T14, T15, T11)) → U14_ag(T11, T13, T14, T15, in_orderc30_in_g(T13))
in_orderc30_in_g([]) → in_orderc30_out_g([])
U14_ag(T11, T13, T14, T15, in_orderc30_out_g(T13)) → U15_ag(T11, in_orderc30_in_g(T15))
U15_ag(T11, in_orderc30_out_g(T15)) → in_orderc1_out_ag(void, T11)
in_orderc1_in_ag(node(T97, T98, T100), T11) → U16_ag(T97, T98, T100, T11, appc47_in_aaag(T48, T98, T49, T11))
U16_ag(T97, T98, T100, T11, appc47_out_aaag(T48, T98, T49, T11)) → U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_in_ag(T97, T48))
U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_out_ag(T97, T48)) → U18_ag(T97, T98, T100, T11, in_orderc1_in_ag(T100, T49))
U18_ag(T97, T98, T100, T11, in_orderc1_out_ag(T100, T49)) → in_orderc1_out_ag(node(T97, T98, T100), T11)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP47_IN_AAAG(.(X168, X169), T66, X170, .(X168, T65)) → APP47_IN_AAAG(X169, T66, X170, T65)

The TRS R consists of the following rules:

appc47_in_aaag([], T59, X148, .(T59, X148)) → appc47_out_aaag([], T59, X148, .(T59, X148))
appc47_in_aaag(.(X168, X169), T66, X170, .(X168, T65)) → U12_aaag(X168, X169, T66, X170, T65, appc47_in_aaag(X169, T66, X170, T65))
U12_aaag(X168, X169, T66, X170, T65, appc47_out_aaag(X169, T66, X170, T65)) → appc47_out_aaag(.(X168, X169), T66, X170, .(X168, T65))
in_orderc1_in_ag(void, []) → in_orderc1_out_ag(void, [])
in_orderc1_in_ag(void, T11) → U13_ag(T11, appc16_in_aaag(T13, T14, T15, T11))
appc16_in_aaag([], X60, X61, .(X60, X61)) → appc16_out_aaag([], X60, X61, .(X60, X61))
appc16_in_aaag(.(X83, X84), X85, X86, .(X83, T18)) → U11_aaag(X83, X84, X85, X86, T18, appc16_in_aaag(X84, X85, X86, T18))
U11_aaag(X83, X84, X85, X86, T18, appc16_out_aaag(X84, X85, X86, T18)) → appc16_out_aaag(.(X83, X84), X85, X86, .(X83, T18))
U13_ag(T11, appc16_out_aaag(T13, T14, T15, T11)) → U14_ag(T11, T13, T14, T15, in_orderc30_in_g(T13))
in_orderc30_in_g([]) → in_orderc30_out_g([])
U14_ag(T11, T13, T14, T15, in_orderc30_out_g(T13)) → U15_ag(T11, in_orderc30_in_g(T15))
U15_ag(T11, in_orderc30_out_g(T15)) → in_orderc1_out_ag(void, T11)
in_orderc1_in_ag(node(T97, T98, T100), T11) → U16_ag(T97, T98, T100, T11, appc47_in_aaag(T48, T98, T49, T11))
U16_ag(T97, T98, T100, T11, appc47_out_aaag(T48, T98, T49, T11)) → U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_in_ag(T97, T48))
U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_out_ag(T97, T48)) → U18_ag(T97, T98, T100, T11, in_orderc1_in_ag(T100, T49))
U18_ag(T97, T98, T100, T11, in_orderc1_out_ag(T100, T49)) → in_orderc1_out_ag(node(T97, T98, T100), T11)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
appc47_in_aaag(x1, x2, x3, x4) = appc47_in_aaag(x4)
appc47_out_aaag(x1, x2, x3, x4) = appc47_out_aaag(x1, x2, x3, x4)
U12_aaag(x1, x2, x3, x4, x5, x6) = U12_aaag(x1, x5, x6)
in_orderc1_in_ag(x1, x2) = in_orderc1_in_ag(x2)
[] = []
in_orderc1_out_ag(x1, x2) = in_orderc1_out_ag(x1, x2)
U13_ag(x1, x2) = U13_ag(x1, x2)
appc16_in_aaag(x1, x2, x3, x4) = appc16_in_aaag(x4)
appc16_out_aaag(x1, x2, x3, x4) = appc16_out_aaag(x1, x2, x3, x4)
U11_aaag(x1, x2, x3, x4, x5, x6) = U11_aaag(x1, x5, x6)
U14_ag(x1, x2, x3, x4, x5) = U14_ag(x1, x4, x5)
in_orderc30_in_g(x1) = in_orderc30_in_g(x1)
in_orderc30_out_g(x1) = in_orderc30_out_g(x1)
U15_ag(x1, x2) = U15_ag(x1, x2)
U16_ag(x1, x2, x3, x4, x5) = U16_ag(x4, x5)
U17_ag(x1, x2, x3, x4, x5, x6, x7) = U17_ag(x2, x4, x6, x7)
U18_ag(x1, x2, x3, x4, x5) = U18_ag(x1, x2, x4, x5)
void = void
node(x1, x2, x3) = node(x1, x2, x3)
APP47_IN_AAAG(x1, x2, x3, x4) = APP47_IN_AAAG(x4)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APP47_IN_AAAG(.(X168, X169), T66, X170, .(X168, T65)) → APP47_IN_AAAG(X169, T66, X170, T65)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP47_IN_AAAG(x1, x2, x3, x4) = APP47_IN_AAAG(x4)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

APP47_IN_AAAG(.(X168, T65)) → APP47_IN_AAAG(T65)

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

(14) 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:

APP47_IN_AAAG(.(X168, T65)) → APP47_IN_AAAG(T65)
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

APP16_IN_AAAG(.(X83, X84), X85, X86, .(X83, T18)) → APP16_IN_AAAG(X84, X85, X86, T18)

The TRS R consists of the following rules:

appc47_in_aaag([], T59, X148, .(T59, X148)) → appc47_out_aaag([], T59, X148, .(T59, X148))
appc47_in_aaag(.(X168, X169), T66, X170, .(X168, T65)) → U12_aaag(X168, X169, T66, X170, T65, appc47_in_aaag(X169, T66, X170, T65))
U12_aaag(X168, X169, T66, X170, T65, appc47_out_aaag(X169, T66, X170, T65)) → appc47_out_aaag(.(X168, X169), T66, X170, .(X168, T65))
in_orderc1_in_ag(void, []) → in_orderc1_out_ag(void, [])
in_orderc1_in_ag(void, T11) → U13_ag(T11, appc16_in_aaag(T13, T14, T15, T11))
appc16_in_aaag([], X60, X61, .(X60, X61)) → appc16_out_aaag([], X60, X61, .(X60, X61))
appc16_in_aaag(.(X83, X84), X85, X86, .(X83, T18)) → U11_aaag(X83, X84, X85, X86, T18, appc16_in_aaag(X84, X85, X86, T18))
U11_aaag(X83, X84, X85, X86, T18, appc16_out_aaag(X84, X85, X86, T18)) → appc16_out_aaag(.(X83, X84), X85, X86, .(X83, T18))
U13_ag(T11, appc16_out_aaag(T13, T14, T15, T11)) → U14_ag(T11, T13, T14, T15, in_orderc30_in_g(T13))
in_orderc30_in_g([]) → in_orderc30_out_g([])
U14_ag(T11, T13, T14, T15, in_orderc30_out_g(T13)) → U15_ag(T11, in_orderc30_in_g(T15))
U15_ag(T11, in_orderc30_out_g(T15)) → in_orderc1_out_ag(void, T11)
in_orderc1_in_ag(node(T97, T98, T100), T11) → U16_ag(T97, T98, T100, T11, appc47_in_aaag(T48, T98, T49, T11))
U16_ag(T97, T98, T100, T11, appc47_out_aaag(T48, T98, T49, T11)) → U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_in_ag(T97, T48))
U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_out_ag(T97, T48)) → U18_ag(T97, T98, T100, T11, in_orderc1_in_ag(T100, T49))
U18_ag(T97, T98, T100, T11, in_orderc1_out_ag(T100, T49)) → in_orderc1_out_ag(node(T97, T98, T100), T11)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
appc47_in_aaag(x1, x2, x3, x4) = appc47_in_aaag(x4)
appc47_out_aaag(x1, x2, x3, x4) = appc47_out_aaag(x1, x2, x3, x4)
U12_aaag(x1, x2, x3, x4, x5, x6) = U12_aaag(x1, x5, x6)
in_orderc1_in_ag(x1, x2) = in_orderc1_in_ag(x2)
[] = []
in_orderc1_out_ag(x1, x2) = in_orderc1_out_ag(x1, x2)
U13_ag(x1, x2) = U13_ag(x1, x2)
appc16_in_aaag(x1, x2, x3, x4) = appc16_in_aaag(x4)
appc16_out_aaag(x1, x2, x3, x4) = appc16_out_aaag(x1, x2, x3, x4)
U11_aaag(x1, x2, x3, x4, x5, x6) = U11_aaag(x1, x5, x6)
U14_ag(x1, x2, x3, x4, x5) = U14_ag(x1, x4, x5)
in_orderc30_in_g(x1) = in_orderc30_in_g(x1)
in_orderc30_out_g(x1) = in_orderc30_out_g(x1)
U15_ag(x1, x2) = U15_ag(x1, x2)
U16_ag(x1, x2, x3, x4, x5) = U16_ag(x4, x5)
U17_ag(x1, x2, x3, x4, x5, x6, x7) = U17_ag(x2, x4, x6, x7)
U18_ag(x1, x2, x3, x4, x5) = U18_ag(x1, x2, x4, x5)
void = void
node(x1, x2, x3) = node(x1, x2, x3)
APP16_IN_AAAG(x1, x2, x3, x4) = APP16_IN_AAAG(x4)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

APP16_IN_AAAG(.(X83, X84), X85, X86, .(X83, T18)) → APP16_IN_AAAG(X84, X85, X86, T18)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP16_IN_AAAG(x1, x2, x3, x4) = APP16_IN_AAAG(x4)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

APP16_IN_AAAG(.(X83, T18)) → APP16_IN_AAAG(T18)

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

(21) 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:

APP16_IN_AAAG(.(X83, T18)) → APP16_IN_AAAG(T18)
The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

IN_ORDER1_IN_AG(node(T82, T80, T83), T11) → U5_AG(T82, T80, T83, T11, appc47_in_aaag(T48, T80, T49, T11))
U5_AG(T82, T80, T83, T11, appc47_out_aaag(T48, T80, T49, T11)) → IN_ORDER1_IN_AG(T82, T48)
IN_ORDER1_IN_AG(node(T97, T98, T100), T11) → U7_AG(T97, T98, T100, T11, appc47_in_aaag(T48, T98, T49, T11))
U7_AG(T97, T98, T100, T11, appc47_out_aaag(T48, T98, T49, T11)) → U8_AG(T97, T98, T100, T11, T49, in_orderc1_in_ag(T97, T48))
U8_AG(T97, T98, T100, T11, T49, in_orderc1_out_ag(T97, T48)) → IN_ORDER1_IN_AG(T100, T49)

The TRS R consists of the following rules:

appc47_in_aaag([], T59, X148, .(T59, X148)) → appc47_out_aaag([], T59, X148, .(T59, X148))
appc47_in_aaag(.(X168, X169), T66, X170, .(X168, T65)) → U12_aaag(X168, X169, T66, X170, T65, appc47_in_aaag(X169, T66, X170, T65))
U12_aaag(X168, X169, T66, X170, T65, appc47_out_aaag(X169, T66, X170, T65)) → appc47_out_aaag(.(X168, X169), T66, X170, .(X168, T65))
in_orderc1_in_ag(void, []) → in_orderc1_out_ag(void, [])
in_orderc1_in_ag(void, T11) → U13_ag(T11, appc16_in_aaag(T13, T14, T15, T11))
appc16_in_aaag([], X60, X61, .(X60, X61)) → appc16_out_aaag([], X60, X61, .(X60, X61))
appc16_in_aaag(.(X83, X84), X85, X86, .(X83, T18)) → U11_aaag(X83, X84, X85, X86, T18, appc16_in_aaag(X84, X85, X86, T18))
U11_aaag(X83, X84, X85, X86, T18, appc16_out_aaag(X84, X85, X86, T18)) → appc16_out_aaag(.(X83, X84), X85, X86, .(X83, T18))
U13_ag(T11, appc16_out_aaag(T13, T14, T15, T11)) → U14_ag(T11, T13, T14, T15, in_orderc30_in_g(T13))
in_orderc30_in_g([]) → in_orderc30_out_g([])
U14_ag(T11, T13, T14, T15, in_orderc30_out_g(T13)) → U15_ag(T11, in_orderc30_in_g(T15))
U15_ag(T11, in_orderc30_out_g(T15)) → in_orderc1_out_ag(void, T11)
in_orderc1_in_ag(node(T97, T98, T100), T11) → U16_ag(T97, T98, T100, T11, appc47_in_aaag(T48, T98, T49, T11))
U16_ag(T97, T98, T100, T11, appc47_out_aaag(T48, T98, T49, T11)) → U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_in_ag(T97, T48))
U17_ag(T97, T98, T100, T11, T48, T49, in_orderc1_out_ag(T97, T48)) → U18_ag(T97, T98, T100, T11, in_orderc1_in_ag(T100, T49))
U18_ag(T97, T98, T100, T11, in_orderc1_out_ag(T100, T49)) → in_orderc1_out_ag(node(T97, T98, T100), T11)

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
appc47_in_aaag(x1, x2, x3, x4) = appc47_in_aaag(x4)
appc47_out_aaag(x1, x2, x3, x4) = appc47_out_aaag(x1, x2, x3, x4)
U12_aaag(x1, x2, x3, x4, x5, x6) = U12_aaag(x1, x5, x6)
in_orderc1_in_ag(x1, x2) = in_orderc1_in_ag(x2)
[] = []
in_orderc1_out_ag(x1, x2) = in_orderc1_out_ag(x1, x2)
U13_ag(x1, x2) = U13_ag(x1, x2)
appc16_in_aaag(x1, x2, x3, x4) = appc16_in_aaag(x4)
appc16_out_aaag(x1, x2, x3, x4) = appc16_out_aaag(x1, x2, x3, x4)
U11_aaag(x1, x2, x3, x4, x5, x6) = U11_aaag(x1, x5, x6)
U14_ag(x1, x2, x3, x4, x5) = U14_ag(x1, x4, x5)
in_orderc30_in_g(x1) = in_orderc30_in_g(x1)
in_orderc30_out_g(x1) = in_orderc30_out_g(x1)
U15_ag(x1, x2) = U15_ag(x1, x2)
U16_ag(x1, x2, x3, x4, x5) = U16_ag(x4, x5)
U17_ag(x1, x2, x3, x4, x5, x6, x7) = U17_ag(x2, x4, x6, x7)
U18_ag(x1, x2, x3, x4, x5) = U18_ag(x1, x2, x4, x5)
void = void
node(x1, x2, x3) = node(x1, x2, x3)
IN_ORDER1_IN_AG(x1, x2) = IN_ORDER1_IN_AG(x2)
U5_AG(x1, x2, x3, x4, x5) = U5_AG(x4, x5)
U7_AG(x1, x2, x3, x4, x5) = U7_AG(x4, x5)
U8_AG(x1, x2, x3, x4, x5, x6) = U8_AG(x4, x5, x6)

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:

IN_ORDER1_IN_AG(T11) → U5_AG(T11, appc47_in_aaag(T11))
U5_AG(T11, appc47_out_aaag(T48, T80, T49, T11)) → IN_ORDER1_IN_AG(T48)
IN_ORDER1_IN_AG(T11) → U7_AG(T11, appc47_in_aaag(T11))
U7_AG(T11, appc47_out_aaag(T48, T98, T49, T11)) → U8_AG(T11, T49, in_orderc1_in_ag(T48))
U8_AG(T11, T49, in_orderc1_out_ag(T97, T48)) → IN_ORDER1_IN_AG(T49)

The TRS R consists of the following rules:

appc47_in_aaag(.(T59, X148)) → appc47_out_aaag([], T59, X148, .(T59, X148))
appc47_in_aaag(.(X168, T65)) → U12_aaag(X168, T65, appc47_in_aaag(T65))
U12_aaag(X168, T65, appc47_out_aaag(X169, T66, X170, T65)) → appc47_out_aaag(.(X168, X169), T66, X170, .(X168, T65))
in_orderc1_in_ag([]) → in_orderc1_out_ag(void, [])
in_orderc1_in_ag(T11) → U13_ag(T11, appc16_in_aaag(T11))
appc16_in_aaag(.(X60, X61)) → appc16_out_aaag([], X60, X61, .(X60, X61))
appc16_in_aaag(.(X83, T18)) → U11_aaag(X83, T18, appc16_in_aaag(T18))
U11_aaag(X83, T18, appc16_out_aaag(X84, X85, X86, T18)) → appc16_out_aaag(.(X83, X84), X85, X86, .(X83, T18))
U13_ag(T11, appc16_out_aaag(T13, T14, T15, T11)) → U14_ag(T11, T15, in_orderc30_in_g(T13))
in_orderc30_in_g([]) → in_orderc30_out_g([])
U14_ag(T11, T15, in_orderc30_out_g(T13)) → U15_ag(T11, in_orderc30_in_g(T15))
U15_ag(T11, in_orderc30_out_g(T15)) → in_orderc1_out_ag(void, T11)
in_orderc1_in_ag(T11) → U16_ag(T11, appc47_in_aaag(T11))
U16_ag(T11, appc47_out_aaag(T48, T98, T49, T11)) → U17_ag(T98, T11, T49, in_orderc1_in_ag(T48))
U17_ag(T98, T11, T49, in_orderc1_out_ag(T97, T48)) → U18_ag(T97, T98, T11, in_orderc1_in_ag(T49))
U18_ag(T97, T98, T11, in_orderc1_out_ag(T100, T49)) → in_orderc1_out_ag(node(T97, T98, T100), T11)

The set Q consists of the following terms:

appc47_in_aaag(x0)
U12_aaag(x0, x1, x2)
in_orderc1_in_ag(x0)
appc16_in_aaag(x0)
U11_aaag(x0, x1, x2)
U13_ag(x0, x1)
in_orderc30_in_g(x0)
U14_ag(x0, x1, x2)
U15_ag(x0, x1)
U16_ag(x0, x1)
U17_ag(x0, x1, x2, x3)
U18_ag(x0, x1, x2, x3)

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

(26) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U5_AG(T11, appc47_out_aaag(T48, T80, T49, T11)) → IN_ORDER1_IN_AG(T48)
U7_AG(T11, appc47_out_aaag(T48, T98, T49, T11)) → U8_AG(T11, T49, in_orderc1_in_ag(T48))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₁ + x₂
POL(IN_ORDER1_IN_AG(x₁)) = x₁
POL(U11_aaag(x₁, x₂, x₃)) = 0
POL(U12_aaag(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(U13_ag(x₁, x₂)) = 0
POL(U14_ag(x₁, x₂, x₃)) = 0
POL(U15_ag(x₁, x₂)) = 0
POL(U16_ag(x₁, x₂)) = 0
POL(U17_ag(x₁, x₂, x₃, x₄)) = 0
POL(U18_ag(x₁, x₂, x₃, x₄)) = 0
POL(U5_AG(x₁, x₂)) = x₂
POL(U7_AG(x₁, x₂)) = x₂
POL(U8_AG(x₁, x₂, x₃)) = x₂
POL([]) = 0
POL(appc16_in_aaag(x₁)) = 0
POL(appc16_out_aaag(x₁, x₂, x₃, x₄)) = 0
POL(appc47_in_aaag(x₁)) = x₁
POL(appc47_out_aaag(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₃
POL(in_orderc1_in_ag(x₁)) = 0
POL(in_orderc1_out_ag(x₁, x₂)) = 0
POL(in_orderc30_in_g(x₁)) = 0
POL(in_orderc30_out_g(x₁)) = 0
POL(node(x₁, x₂, x₃)) = 0
POL(void) = 0

The following usable rules [FROCOS05] were oriented:

appc47_in_aaag(.(T59, X148)) → appc47_out_aaag([], T59, X148, .(T59, X148))
appc47_in_aaag(.(X168, T65)) → U12_aaag(X168, T65, appc47_in_aaag(T65))
U12_aaag(X168, T65, appc47_out_aaag(X169, T66, X170, T65)) → appc47_out_aaag(.(X168, X169), T66, X170, .(X168, T65))

(27) Obligation:

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

IN_ORDER1_IN_AG(T11) → U5_AG(T11, appc47_in_aaag(T11))
IN_ORDER1_IN_AG(T11) → U7_AG(T11, appc47_in_aaag(T11))
U8_AG(T11, T49, in_orderc1_out_ag(T97, T48)) → IN_ORDER1_IN_AG(T49)

The TRS R consists of the following rules:

appc47_in_aaag(.(T59, X148)) → appc47_out_aaag([], T59, X148, .(T59, X148))
appc47_in_aaag(.(X168, T65)) → U12_aaag(X168, T65, appc47_in_aaag(T65))
U12_aaag(X168, T65, appc47_out_aaag(X169, T66, X170, T65)) → appc47_out_aaag(.(X168, X169), T66, X170, .(X168, T65))
in_orderc1_in_ag([]) → in_orderc1_out_ag(void, [])
in_orderc1_in_ag(T11) → U13_ag(T11, appc16_in_aaag(T11))
appc16_in_aaag(.(X60, X61)) → appc16_out_aaag([], X60, X61, .(X60, X61))
appc16_in_aaag(.(X83, T18)) → U11_aaag(X83, T18, appc16_in_aaag(T18))
U11_aaag(X83, T18, appc16_out_aaag(X84, X85, X86, T18)) → appc16_out_aaag(.(X83, X84), X85, X86, .(X83, T18))
U13_ag(T11, appc16_out_aaag(T13, T14, T15, T11)) → U14_ag(T11, T15, in_orderc30_in_g(T13))
in_orderc30_in_g([]) → in_orderc30_out_g([])
U14_ag(T11, T15, in_orderc30_out_g(T13)) → U15_ag(T11, in_orderc30_in_g(T15))
U15_ag(T11, in_orderc30_out_g(T15)) → in_orderc1_out_ag(void, T11)
in_orderc1_in_ag(T11) → U16_ag(T11, appc47_in_aaag(T11))
U16_ag(T11, appc47_out_aaag(T48, T98, T49, T11)) → U17_ag(T98, T11, T49, in_orderc1_in_ag(T48))
U17_ag(T98, T11, T49, in_orderc1_out_ag(T97, T48)) → U18_ag(T97, T98, T11, in_orderc1_in_ag(T49))
U18_ag(T97, T98, T11, in_orderc1_out_ag(T100, T49)) → in_orderc1_out_ag(node(T97, T98, T100), T11)

The set Q consists of the following terms:

appc47_in_aaag(x0)
U12_aaag(x0, x1, x2)
in_orderc1_in_ag(x0)
appc16_in_aaag(x0)
U11_aaag(x0, x1, x2)
U13_ag(x0, x1)
in_orderc30_in_g(x0)
U14_ag(x0, x1, x2)
U15_ag(x0, x1)
U16_ag(x0, x1)
U17_ag(x0, x1, x2, x3)
U18_ag(x0, x1, x2, x3)

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

(28) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) UndefinedPredicateInTriplesTransformerProof (SOUND transformation)

(4) Obligation:

(5) TriplesToPiDPProof (SOUND transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) PiDPToQDPProof (SOUND transformation)

(25) Obligation:

(26) QDPOrderProof (EQUIVALENT transformation)

(27) Obligation:

(28) DependencyGraphProof (EQUIVALENT transformation)

(29) TRUE