Left Termination proof of /tmp/tmpXMuEVv/pl4.4.6a.pl

Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇐)

                ↳13 QDP

                ↳14 NonTerminationProof (⇔)

                ↳15 NO

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 NonTerminationProof (⇔)

                ↳22 NO

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 QDPSizeChangeProof (⇔)

                ↳29 YES

    ↳30 PrologToPiTRSProof (⇐)

        ↳31 PiTRS

        ↳32 DependencyPairsProof (⇔)

        ↳33 PiDP

        ↳34 DependencyGraphProof (⇔)

        ↳35 AND

            ↳36 PiDP

                ↳37 UsableRulesProof (⇔)

                ↳38 PiDP

                ↳39 PiDPToQDPProof (⇐)

                ↳40 QDP

                ↳41 NonTerminationProof (⇔)

                ↳42 NO

            ↳43 PiDP

                ↳44 UsableRulesProof (⇔)

                ↳45 PiDP

                ↳46 PiDPToQDPProof (⇐)

                ↳47 QDP

                ↳48 NonTerminationProof (⇔)

                ↳49 NO

            ↳50 PiDP

                ↳51 UsableRulesProof (⇔)

                ↳52 PiDP

                ↳53 PiDPToQDPProof (⇐)

                ↳54 QDP

                ↳55 QDPSizeChangeProof (⇔)

                ↳56 YES

(0) Obligation:

Clauses:

perm([], []).
perm(.(X, L), Z) :- ','(perm(L, Y), insert(X, Y, Z)).
insert(X, [], .(X, [])).
insert(X, L, .(X, L)).
insert(X, .(H, L1), .(H, L2)) :- insert(X, L1, L2).

Queries:

perm(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: perm(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: perm(T1, T2), SCOPE: 1, CLAUSE: 0 | perm(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | perm([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: perm(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nperm(T1, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: perm([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(perm(T7, X10), insert(T6, X10, T9))\n\nT6 is ground\nT7 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(perm(T7, X10), insert(T6, X10, T9)), SCOPE: 2, CLAUSE: 0 | ','(perm(T7, X10), insert(T6, X10, T9)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(perm(T7, X10), insert(T6, X10, T9)), SCOPE: 2, CLAUSE: 0\n\nT6 is ground\nT7 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(perm(T7, X10), insert(T6, X10, T9)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground\nT7 is ground\nX10 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: insert(T6, [], T9)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: insert(T6, [], T9), SCOPE: 3, CLAUSE: 2 | insert(T6, [], T9), SCOPE: 3, CLAUSE: 3 | insert(T6, [], T9), SCOPE: 3, CLAUSE: 4\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: insert(T6, [], T9), SCOPE: 3, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: insert(T6, [], T9), SCOPE: 3, CLAUSE: 3 | insert(T6, [], T9), SCOPE: 3, CLAUSE: 4\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: insert(T6, [], T9), SCOPE: 3, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: insert(T6, [], T9), SCOPE: 3, CLAUSE: 4\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(perm(T28, X50), ','(insert(T27, X50, X51), insert(T6, X51, T9)))\n\nT6 is ground\nT27 is ground\nT28 is ground\nX51 is free; \nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: perm(T28, X50)\n\nT28 is ground\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(insert(T27, T29, X51), insert(T6, X51, T9))\n\nT6 is ground\nT27 is ground\nT29 is ground\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: perm(T28, X50), SCOPE: 4, CLAUSE: 0 | perm(T28, X50), SCOPE: 4, CLAUSE: 1\n\nT28 is ground\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: perm(T28, X50), SCOPE: 4, CLAUSE: 0\n\nT28 is ground\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: perm(T28, X50), SCOPE: 4, CLAUSE: 1\n\nT28 is ground\nX50 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(perm(T35, X67), insert(T34, X67, X68))\n\nT34 is ground\nT35 is ground\nX68 is free; \nX67 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: perm(T35, X67)\n\nT35 is ground\nX67 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: insert(T34, T36, X68)\n\nT34 is ground\nT36 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: insert(T34, T36, X68), SCOPE: 5, CLAUSE: 2 | insert(T34, T36, X68), SCOPE: 5, CLAUSE: 3 | insert(T34, T36, X68), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT36 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: insert(T34, T36, X68), SCOPE: 5, CLAUSE: 2\n\nT34 is ground\nT36 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: insert(T34, T36, X68), SCOPE: 5, CLAUSE: 3 | insert(T34, T36, X68), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT36 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: insert(T34, T36, X68), SCOPE: 5, CLAUSE: 3\n\nT34 is ground\nT36 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: insert(T34, T36, X68), SCOPE: 5, CLAUSE: 4\n\nT34 is ground\nT36 is ground\nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: insert(T60, T62, X104)\n\nT60 is ground\nT62 is ground\nX104 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: insert(T27, T29, X51)\n\nT27 is ground\nT29 is ground\nX51 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: insert(T6, T67, T9)\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: insert(T6, T67, T9), SCOPE: 6, CLAUSE: 2 | insert(T6, T67, T9), SCOPE: 6, CLAUSE: 3 | insert(T6, T67, T9), SCOPE: 6, CLAUSE: 4\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: insert(T6, T67, T9), SCOPE: 6, CLAUSE: 2\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: insert(T6, T67, T9), SCOPE: 6, CLAUSE: 3 | insert(T6, T67, T9), SCOPE: 6, CLAUSE: 4\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: insert(T6, T67, T9), SCOPE: 6, CLAUSE: 3\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: insert(T6, T67, T9), SCOPE: 6, CLAUSE: 4\n\nT6 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: insert(T95, T97, T99)\n\nT95 is ground\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 67 [arrowhead = none ];
67 -> 2;

68 [label="EVAL with clause\nperm([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 68 [arrowhead = none ];
68 -> 3;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 69 [arrowhead = none ];
69 -> 4;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 70 [arrowhead = none ];
70 -> 5;

71 [label="EVAL with clause\nperm(.(X7, X8), X9) :- ','(perm(X8, X10), insert(X7, X10, X9)).\nand substitution\nX7 -> T6\nX8 -> T7\nT1 -> .(T6, T7)\nT2 -> T9\nX9 -> T9\nT8 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 71 [arrowhead = none ];
71 -> 7;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 72 [arrowhead = none ];
72 -> 8;

73 [label="BACKTRACK\nfor clause: perm(.(X, L), Z) :- ','(perm(L, Y), insert(X, Y, Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 73 [arrowhead = none ];
73 -> 6;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 74 [arrowhead = none ];
74 -> 9;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 75 [arrowhead = none ];
75 -> 10;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 76 [arrowhead = none ];
76 -> 11;

77 [label="EVAL with clause\nperm([], []).\nand substitution\nT7 -> []\nX10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 77 [arrowhead = none ];
77 -> 12;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 78 [arrowhead = none ];
78 -> 13;

79 [label="EVAL with clause\nperm(.(X47, X48), X49) :- ','(perm(X48, X50), insert(X47, X50, X49)).\nand substitution\nX47 -> T27\nX48 -> T28\nT7 -> .(T27, T28)\nX10 -> X51\nX49 -> X51", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 80 [arrowhead = none ];
80 -> 27;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 81 [arrowhead = none ];
81 -> 14;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 82 [arrowhead = none ];
82 -> 15;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 83 [arrowhead = none ];
83 -> 16;

84 [label="EVAL with clause\ninsert(X20, [], .(X20, [])).\nand substitution\nT6 -> T16\nX20 -> T16\nT9 -> .(T16, [])", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 84 [arrowhead = none ];
84 -> 17;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 85 [arrowhead = none ];
85 -> 18;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 86 [arrowhead = none ];
86 -> 20;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 87 [arrowhead = none ];
87 -> 21;

88 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 88 [arrowhead = none ];
88 -> 19;

89 [label="EVAL with clause\ninsert(X30, X31, .(X30, X31)).\nand substitution\nT6 -> T21\nX30 -> T21\nX31 -> []\nT9 -> .(T21, [])", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 89 [arrowhead = none ];
89 -> 22;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 90 [arrowhead = none ];
90 -> 23;

91 [label="BACKTRACK\nfor clause: insert(X, .(H, L1), .(H, L2)) :- insert(X, L1, L2)because of non-unification", fontsize=14, style = filled, fillcolor = white];
21 -> 91 [arrowhead = none ];
91 -> 25;

92 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 92 [arrowhead = none ];
92 -> 24;

93 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
26 -> 93 [arrowhead = none ];
93 -> 28;

94 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nreplacements:\nX50 -> T29", fontsize=14, style = filled, fillcolor = violet];
26 -> 94 [arrowhead = none ];
94 -> 29;

95 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 95 [arrowhead = none ];
95 -> 30;

96 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 96 [arrowhead = none ];
96 -> 52;

97 [label="SPLIT 2\nnew knowledge:\nT67 is ground\nreplacements:\nX51 -> T67", fontsize=14, style = filled, fillcolor = violet];
29 -> 97 [arrowhead = none ];
97 -> 53;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 98 [arrowhead = none ];
98 -> 31;

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

100 [label="EVAL with clause\nperm([], []).\nand substitution\nT28 -> []\nX50 -> []", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 100 [arrowhead = none ];
100 -> 33;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 101 [arrowhead = none ];
101 -> 34;

102 [label="EVAL with clause\nperm(.(X64, X65), X66) :- ','(perm(X65, X67), insert(X64, X67, X66)).\nand substitution\nX64 -> T34\nX65 -> T35\nT28 -> .(T34, T35)\nX50 -> X68\nX66 -> X68", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 102 [arrowhead = none ];
102 -> 36;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 103 [arrowhead = none ];
103 -> 37;

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 104 [arrowhead = none ];
104 -> 35;

105 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
36 -> 105 [arrowhead = none ];
105 -> 38;

106 [label="SPLIT 2\nnew knowledge:\nT36 is ground\nreplacements:\nX67 -> T36", fontsize=14, style = filled, fillcolor = violet];
36 -> 106 [arrowhead = none ];
106 -> 39;

107 [label="INSTANCE with matching:\nT28 -> T35\nX50 -> X67", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 107 [arrowhead = none , style=dashed];
107 -> 28[style=dashed];

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 108 [arrowhead = none ];
108 -> 40;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 109 [arrowhead = none ];
109 -> 41;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 110 [arrowhead = none ];
110 -> 42;

111 [label="EVAL with clause\ninsert(X78, [], .(X78, [])).\nand substitution\nT34 -> T43\nX78 -> T43\nT36 -> []\nX68 -> .(T43, [])", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 111 [arrowhead = none ];
111 -> 43;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 112 [arrowhead = none ];
112 -> 44;

113 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 113 [arrowhead = none ];
113 -> 46;

114 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 114 [arrowhead = none ];
114 -> 47;

115 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 115 [arrowhead = none ];
115 -> 45;

116 [label="ONLY EVAL with clause\ninsert(X88, X89, .(X88, X89)).\nand substitution\nT34 -> T52\nX88 -> T52\nT36 -> T53\nX89 -> T53\nX68 -> .(T52, T53)", fontsize=14, style = filled, fillcolor = white];
46 -> 116 [arrowhead = none ];
116 -> 48;

117 [label="EVAL with clause\ninsert(X100, .(X101, X102), .(X101, X103)) :- insert(X100, X102, X103).\nand substitution\nT34 -> T60\nX100 -> T60\nX101 -> T61\nX102 -> T62\nT36 -> .(T61, T62)\nX103 -> X104\nX68 -> .(T61, X104)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 117 [arrowhead = none ];
117 -> 50;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 118 [arrowhead = none ];
118 -> 51;

119 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 119 [arrowhead = none ];
119 -> 49;

120 [label="INSTANCE with matching:\nT34 -> T60\nT36 -> T62\nX68 -> X104", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 120 [arrowhead = none , style=dashed];
120 -> 39[style=dashed];

121 [label="INSTANCE with matching:\nT34 -> T27\nT36 -> T29\nX68 -> X51", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 121 [arrowhead = none , style=dashed];
121 -> 39[style=dashed];

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 122 [arrowhead = none ];
122 -> 54;

123 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 123 [arrowhead = none ];
123 -> 55;

124 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 124 [arrowhead = none ];
124 -> 56;

125 [label="EVAL with clause\ninsert(X121, [], .(X121, [])).\nand substitution\nT6 -> T76\nX121 -> T76\nT67 -> []\nT9 -> .(T76, [])", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 125 [arrowhead = none ];
125 -> 57;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 126 [arrowhead = none ];
126 -> 58;

127 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 127 [arrowhead = none ];
127 -> 60;

128 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
56 -> 128 [arrowhead = none ];
128 -> 61;

129 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 129 [arrowhead = none ];
129 -> 59;

130 [label="EVAL with clause\ninsert(X131, X132, .(X131, X132)).\nand substitution\nT6 -> T85\nX131 -> T85\nT67 -> T86\nX132 -> T86\nT9 -> .(T85, T86)", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 130 [arrowhead = none ];
130 -> 62;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 131 [arrowhead = none ];
131 -> 63;

132 [label="EVAL with clause\ninsert(X143, .(X144, X145), .(X144, X146)) :- insert(X143, X145, X146).\nand substitution\nT6 -> T95\nX143 -> T95\nX144 -> T96\nX145 -> T97\nT67 -> .(T96, T97)\nX146 -> T99\nT9 -> .(T96, T99)\nT98 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 132 [arrowhead = none ];
132 -> 65;

133 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
61 -> 133 [arrowhead = none ];
133 -> 66;

134 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 134 [arrowhead = none ];
134 -> 64;

135 [label="INSTANCE with matching:\nT6 -> T95\nT67 -> T97\nT9 -> T99", fontsize=14, style = filled, fillcolor = lightgrey];
65 -> 135 [arrowhead = none , style=dashed];
135 -> 53[style=dashed];

}

(2) Obligation:

Clauses:

perm28([], []).
perm28(.(T34, T35), X68) :- perm28(T35, X67).
perm28(.(T34, T35), X68) :- ','(perm28(T35, T36), insert39(T34, T36, X68)).
insert39(T43, [], .(T43, [])).
insert39(T52, T53, .(T52, T53)).
insert39(T60, .(T61, T62), .(T61, X104)) :- insert39(T60, T62, X104).
insert53(T76, [], .(T76, [])).
insert53(T85, T86, .(T85, T86)).
insert53(T95, .(T96, T97), .(T96, T99)) :- insert53(T95, T97, T99).
perm1([], []).
perm1(.(T16, []), .(T16, [])).
perm1(.(T21, []), .(T21, [])).
perm1(.(T6, .(T27, T28)), T9) :- perm28(T28, X50).
perm1(.(T6, .(T27, T28)), T9) :- ','(perm28(T28, T29), insert39(T27, T29, X51)).
perm1(.(T6, .(T27, T28)), T9) :- ','(perm28(T28, T29), ','(insert39(T27, T29, T67), insert53(T6, T67, T9))).

Queries:

perm1(g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm1_in: (b,f)
perm28_in: (b,f)
insert39_in: (b,f,f)
insert53_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(5) 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:

PERM1_IN_GA(.(T6, .(T27, T28)), T9) → U6_GA(T6, T27, T28, T9, perm28_in_ga(T28, X50))
PERM1_IN_GA(.(T6, .(T27, T28)), T9) → PERM28_IN_GA(T28, X50)
PERM28_IN_GA(.(T34, T35), X68) → U1_GA(T34, T35, X68, perm28_in_ga(T35, X67))
PERM28_IN_GA(.(T34, T35), X68) → PERM28_IN_GA(T35, X67)
PERM28_IN_GA(.(T34, T35), X68) → U2_GA(T34, T35, X68, perm28_in_ga(T35, T36))
U2_GA(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_GA(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
U2_GA(T34, T35, X68, perm28_out_ga(T35, T36)) → INSERT39_IN_GAA(T34, T36, X68)
INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → U4_GAA(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → INSERT39_IN_GAA(T60, T62, X104)
PERM1_IN_GA(.(T6, .(T27, T28)), T9) → U7_GA(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_GA(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → INSERT39_IN_GAA(T27, T29, X51)
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_GA(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_GA(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_GA(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
U9_GA(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → INSERT53_IN_GAA(T6, T67, T9)
INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → U5_GAA(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → INSERT53_IN_GAA(T95, T97, T99)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2) = perm1_in_ga(x1)
[] = []
perm1_out_ga(x1, x2) = perm1_out_ga(x1)
.(x1, x2) = .(x1, x2)
U6_ga(x1, x2, x3, x4, x5) = U6_ga(x1, x2, x3, x5)
perm28_in_ga(x1, x2) = perm28_in_ga(x1)
perm28_out_ga(x1, x2) = perm28_out_ga(x1)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x2, x4)
U2_ga(x1, x2, x3, x4) = U2_ga(x1, x2, x4)
U3_ga(x1, x2, x3, x4) = U3_ga(x1, x2, x4)
insert39_in_gaa(x1, x2, x3) = insert39_in_gaa(x1)
insert39_out_gaa(x1, x2, x3) = insert39_out_gaa(x1)
U4_gaa(x1, x2, x3, x4, x5) = U4_gaa(x1, x5)
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x1, x2, x3, x5)
U8_ga(x1, x2, x3, x4, x5) = U8_ga(x1, x2, x3, x5)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x1, x2, x3, x5)
U10_ga(x1, x2, x3, x4, x5) = U10_ga(x1, x2, x3, x5)
insert53_in_gaa(x1, x2, x3) = insert53_in_gaa(x1)
insert53_out_gaa(x1, x2, x3) = insert53_out_gaa(x1)
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x1, x5)
PERM1_IN_GA(x1, x2) = PERM1_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5) = U6_GA(x1, x2, x3, x5)
PERM28_IN_GA(x1, x2) = PERM28_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x2, x4)
U2_GA(x1, x2, x3, x4) = U2_GA(x1, x2, x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x1, x2, x4)
INSERT39_IN_GAA(x1, x2, x3) = INSERT39_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x1, x5)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x1, x2, x3, x5)
U8_GA(x1, x2, x3, x4, x5) = U8_GA(x1, x2, x3, x5)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x2, x3, x5)
U10_GA(x1, x2, x3, x4, x5) = U10_GA(x1, x2, x3, x5)
INSERT53_IN_GAA(x1, x2, x3) = INSERT53_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x1, x5)

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

(6) Obligation:

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

PERM1_IN_GA(.(T6, .(T27, T28)), T9) → U6_GA(T6, T27, T28, T9, perm28_in_ga(T28, X50))
PERM1_IN_GA(.(T6, .(T27, T28)), T9) → PERM28_IN_GA(T28, X50)
PERM28_IN_GA(.(T34, T35), X68) → U1_GA(T34, T35, X68, perm28_in_ga(T35, X67))
PERM28_IN_GA(.(T34, T35), X68) → PERM28_IN_GA(T35, X67)
PERM28_IN_GA(.(T34, T35), X68) → U2_GA(T34, T35, X68, perm28_in_ga(T35, T36))
U2_GA(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_GA(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
U2_GA(T34, T35, X68, perm28_out_ga(T35, T36)) → INSERT39_IN_GAA(T34, T36, X68)
INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → U4_GAA(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → INSERT39_IN_GAA(T60, T62, X104)
PERM1_IN_GA(.(T6, .(T27, T28)), T9) → U7_GA(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_GA(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → INSERT39_IN_GAA(T27, T29, X51)
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_GA(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_GA(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_GA(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
U9_GA(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → INSERT53_IN_GAA(T6, T67, T9)
INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → U5_GAA(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → INSERT53_IN_GAA(T95, T97, T99)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → INSERT53_IN_GAA(T95, T97, T99)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(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:

INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → INSERT53_IN_GAA(T95, T97, T99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
INSERT53_IN_GAA(x1, x2, x3) = INSERT53_IN_GAA(x1)

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:

INSERT53_IN_GAA(T95) → INSERT53_IN_GAA(T95)

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

(14) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = INSERT53_IN_GAA(T95) evaluates to t =INSERT53_IN_GAA(T95)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from INSERT53_IN_GAA(T95) to INSERT53_IN_GAA(T95).

(15) NO

(16) Obligation:

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

INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → INSERT39_IN_GAA(T60, T62, X104)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(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:

INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → INSERT39_IN_GAA(T60, T62, X104)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
INSERT39_IN_GAA(x1, x2, x3) = INSERT39_IN_GAA(x1)

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:

INSERT39_IN_GAA(T60) → INSERT39_IN_GAA(T60)

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

(21) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = INSERT39_IN_GAA(T60) evaluates to t =INSERT39_IN_GAA(T60)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from INSERT39_IN_GAA(T60) to INSERT39_IN_GAA(T60).

(22) NO

(23) Obligation:

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

PERM28_IN_GA(.(T34, T35), X68) → PERM28_IN_GA(T35, X67)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

PERM28_IN_GA(.(T34, T35), X68) → PERM28_IN_GA(T35, X67)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
PERM28_IN_GA(x1, x2) = PERM28_IN_GA(x1)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

PERM28_IN_GA(.(T34, T35)) → PERM28_IN_GA(T35)

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

(28) 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:

PERM28_IN_GA(.(T34, T35)) → PERM28_IN_GA(T35)
The graph contains the following edges 1 > 1

(29) YES

(30) PrologToPiTRSProof (SOUND transformation)

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(31) Obligation:

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

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(32) 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:

PERM1_IN_GA(.(T6, .(T27, T28)), T9) → U6_GA(T6, T27, T28, T9, perm28_in_ga(T28, X50))
PERM1_IN_GA(.(T6, .(T27, T28)), T9) → PERM28_IN_GA(T28, X50)
PERM28_IN_GA(.(T34, T35), X68) → U1_GA(T34, T35, X68, perm28_in_ga(T35, X67))
PERM28_IN_GA(.(T34, T35), X68) → PERM28_IN_GA(T35, X67)
PERM28_IN_GA(.(T34, T35), X68) → U2_GA(T34, T35, X68, perm28_in_ga(T35, T36))
U2_GA(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_GA(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
U2_GA(T34, T35, X68, perm28_out_ga(T35, T36)) → INSERT39_IN_GAA(T34, T36, X68)
INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → U4_GAA(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → INSERT39_IN_GAA(T60, T62, X104)
PERM1_IN_GA(.(T6, .(T27, T28)), T9) → U7_GA(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_GA(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → INSERT39_IN_GAA(T27, T29, X51)
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_GA(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_GA(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_GA(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
U9_GA(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → INSERT53_IN_GAA(T6, T67, T9)
INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → U5_GAA(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → INSERT53_IN_GAA(T95, T97, T99)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

The argument filtering Pi contains the following mapping:
perm1_in_ga(x1, x2) = perm1_in_ga(x1)
[] = []
perm1_out_ga(x1, x2) = perm1_out_ga
.(x1, x2) = .(x1, x2)
U6_ga(x1, x2, x3, x4, x5) = U6_ga(x5)
perm28_in_ga(x1, x2) = perm28_in_ga(x1)
perm28_out_ga(x1, x2) = perm28_out_ga
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
U2_ga(x1, x2, x3, x4) = U2_ga(x1, x4)
U3_ga(x1, x2, x3, x4) = U3_ga(x4)
insert39_in_gaa(x1, x2, x3) = insert39_in_gaa(x1)
insert39_out_gaa(x1, x2, x3) = insert39_out_gaa
U4_gaa(x1, x2, x3, x4, x5) = U4_gaa(x5)
U7_ga(x1, x2, x3, x4, x5) = U7_ga(x1, x2, x5)
U8_ga(x1, x2, x3, x4, x5) = U8_ga(x5)
U9_ga(x1, x2, x3, x4, x5) = U9_ga(x1, x5)
U10_ga(x1, x2, x3, x4, x5) = U10_ga(x5)
insert53_in_gaa(x1, x2, x3) = insert53_in_gaa(x1)
insert53_out_gaa(x1, x2, x3) = insert53_out_gaa
U5_gaa(x1, x2, x3, x4, x5) = U5_gaa(x5)
PERM1_IN_GA(x1, x2) = PERM1_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5) = U6_GA(x5)
PERM28_IN_GA(x1, x2) = PERM28_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x4)
U2_GA(x1, x2, x3, x4) = U2_GA(x1, x4)
U3_GA(x1, x2, x3, x4) = U3_GA(x4)
INSERT39_IN_GAA(x1, x2, x3) = INSERT39_IN_GAA(x1)
U4_GAA(x1, x2, x3, x4, x5) = U4_GAA(x5)
U7_GA(x1, x2, x3, x4, x5) = U7_GA(x1, x2, x5)
U8_GA(x1, x2, x3, x4, x5) = U8_GA(x5)
U9_GA(x1, x2, x3, x4, x5) = U9_GA(x1, x5)
U10_GA(x1, x2, x3, x4, x5) = U10_GA(x5)
INSERT53_IN_GAA(x1, x2, x3) = INSERT53_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5) = U5_GAA(x5)

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

(33) Obligation:

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

PERM1_IN_GA(.(T6, .(T27, T28)), T9) → U6_GA(T6, T27, T28, T9, perm28_in_ga(T28, X50))
PERM1_IN_GA(.(T6, .(T27, T28)), T9) → PERM28_IN_GA(T28, X50)
PERM28_IN_GA(.(T34, T35), X68) → U1_GA(T34, T35, X68, perm28_in_ga(T35, X67))
PERM28_IN_GA(.(T34, T35), X68) → PERM28_IN_GA(T35, X67)
PERM28_IN_GA(.(T34, T35), X68) → U2_GA(T34, T35, X68, perm28_in_ga(T35, T36))
U2_GA(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_GA(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
U2_GA(T34, T35, X68, perm28_out_ga(T35, T36)) → INSERT39_IN_GAA(T34, T36, X68)
INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → U4_GAA(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → INSERT39_IN_GAA(T60, T62, X104)
PERM1_IN_GA(.(T6, .(T27, T28)), T9) → U7_GA(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_GA(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → INSERT39_IN_GAA(T27, T29, X51)
U7_GA(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_GA(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_GA(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_GA(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
U9_GA(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → INSERT53_IN_GAA(T6, T67, T9)
INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → U5_GAA(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → INSERT53_IN_GAA(T95, T97, T99)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(34) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 14 less nodes.

(35) Complex Obligation (AND)

(36) Obligation:

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

INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → INSERT53_IN_GAA(T95, T97, T99)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(37) UsableRulesProof (EQUIVALENT transformation)

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

(38) Obligation:

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

INSERT53_IN_GAA(T95, .(T96, T97), .(T96, T99)) → INSERT53_IN_GAA(T95, T97, T99)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
INSERT53_IN_GAA(x1, x2, x3) = INSERT53_IN_GAA(x1)

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

(39) PiDPToQDPProof (SOUND transformation)

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

(40) Obligation:

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

INSERT53_IN_GAA(T95) → INSERT53_IN_GAA(T95)

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

(41) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from INSERT53_IN_GAA(T95) to INSERT53_IN_GAA(T95).

(42) NO

(43) Obligation:

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

INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → INSERT39_IN_GAA(T60, T62, X104)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(44) UsableRulesProof (EQUIVALENT transformation)

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

(45) Obligation:

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

INSERT39_IN_GAA(T60, .(T61, T62), .(T61, X104)) → INSERT39_IN_GAA(T60, T62, X104)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
INSERT39_IN_GAA(x1, x2, x3) = INSERT39_IN_GAA(x1)

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

(46) PiDPToQDPProof (SOUND transformation)

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

(47) Obligation:

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

INSERT39_IN_GAA(T60) → INSERT39_IN_GAA(T60)

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

(48) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from INSERT39_IN_GAA(T60) to INSERT39_IN_GAA(T60).

(49) NO

(50) Obligation:

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

PERM28_IN_GA(.(T34, T35), X68) → PERM28_IN_GA(T35, X67)

The TRS R consists of the following rules:

perm1_in_ga([], []) → perm1_out_ga([], [])
perm1_in_ga(.(T16, []), .(T16, [])) → perm1_out_ga(.(T16, []), .(T16, []))
perm1_in_ga(.(T6, .(T27, T28)), T9) → U6_ga(T6, T27, T28, T9, perm28_in_ga(T28, X50))
perm28_in_ga([], []) → perm28_out_ga([], [])
perm28_in_ga(.(T34, T35), X68) → U1_ga(T34, T35, X68, perm28_in_ga(T35, X67))
perm28_in_ga(.(T34, T35), X68) → U2_ga(T34, T35, X68, perm28_in_ga(T35, T36))
U2_ga(T34, T35, X68, perm28_out_ga(T35, T36)) → U3_ga(T34, T35, X68, insert39_in_gaa(T34, T36, X68))
insert39_in_gaa(T43, [], .(T43, [])) → insert39_out_gaa(T43, [], .(T43, []))
insert39_in_gaa(T52, T53, .(T52, T53)) → insert39_out_gaa(T52, T53, .(T52, T53))
insert39_in_gaa(T60, .(T61, T62), .(T61, X104)) → U4_gaa(T60, T61, T62, X104, insert39_in_gaa(T60, T62, X104))
U4_gaa(T60, T61, T62, X104, insert39_out_gaa(T60, T62, X104)) → insert39_out_gaa(T60, .(T61, T62), .(T61, X104))
U3_ga(T34, T35, X68, insert39_out_gaa(T34, T36, X68)) → perm28_out_ga(.(T34, T35), X68)
U1_ga(T34, T35, X68, perm28_out_ga(T35, X67)) → perm28_out_ga(.(T34, T35), X68)
U6_ga(T6, T27, T28, T9, perm28_out_ga(T28, X50)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
perm1_in_ga(.(T6, .(T27, T28)), T9) → U7_ga(T6, T27, T28, T9, perm28_in_ga(T28, T29))
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U8_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, X51))
U8_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, X51)) → perm1_out_ga(.(T6, .(T27, T28)), T9)
U7_ga(T6, T27, T28, T9, perm28_out_ga(T28, T29)) → U9_ga(T6, T27, T28, T9, insert39_in_gaa(T27, T29, T67))
U9_ga(T6, T27, T28, T9, insert39_out_gaa(T27, T29, T67)) → U10_ga(T6, T27, T28, T9, insert53_in_gaa(T6, T67, T9))
insert53_in_gaa(T76, [], .(T76, [])) → insert53_out_gaa(T76, [], .(T76, []))
insert53_in_gaa(T85, T86, .(T85, T86)) → insert53_out_gaa(T85, T86, .(T85, T86))
insert53_in_gaa(T95, .(T96, T97), .(T96, T99)) → U5_gaa(T95, T96, T97, T99, insert53_in_gaa(T95, T97, T99))
U5_gaa(T95, T96, T97, T99, insert53_out_gaa(T95, T97, T99)) → insert53_out_gaa(T95, .(T96, T97), .(T96, T99))
U10_ga(T6, T27, T28, T9, insert53_out_gaa(T6, T67, T9)) → perm1_out_ga(.(T6, .(T27, T28)), T9)

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

PERM28_IN_GA(.(T34, T35), X68) → PERM28_IN_GA(T35, X67)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
PERM28_IN_GA(x1, x2) = PERM28_IN_GA(x1)

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

(53) PiDPToQDPProof (SOUND transformation)

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

(54) Obligation:

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

PERM28_IN_GA(.(T34, T35)) → PERM28_IN_GA(T35)

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

(55) 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:

PERM28_IN_GA(.(T34, T35)) → PERM28_IN_GA(T35)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT 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) NonTerminationProof (EQUIVALENT transformation)

(15) NO

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (SOUND transformation)

(20) Obligation:

(21) NonTerminationProof (EQUIVALENT transformation)

(22) NO

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) PrologToPiTRSProof (SOUND transformation)

(31) Obligation:

(32) DependencyPairsProof (EQUIVALENT transformation)

(33) Obligation:

(34) DependencyGraphProof (EQUIVALENT transformation)

(35) Complex Obligation (AND)

(36) Obligation:

(37) UsableRulesProof (EQUIVALENT transformation)

(38) Obligation:

(39) PiDPToQDPProof (SOUND transformation)

(40) Obligation:

(41) NonTerminationProof (EQUIVALENT transformation)

(42) NO

(43) Obligation:

(44) UsableRulesProof (EQUIVALENT transformation)

(45) Obligation:

(46) PiDPToQDPProof (SOUND transformation)

(47) Obligation:

(48) NonTerminationProof (EQUIVALENT transformation)

(49) NO

(50) Obligation:

(51) UsableRulesProof (EQUIVALENT transformation)

(52) Obligation:

(53) PiDPToQDPProof (SOUND transformation)

(54) Obligation:

(55) QDPSizeChangeProof (EQUIVALENT transformation)

(56) YES