Left Termination of the query pattern perm_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

append2(parts([], Y), is(sum(Y))).
append2(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append2(parts(X, Y), is(sum(Z))).
append1(parts([], Y), is(sum(Y))).
append1(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append1(parts(X, Y), is(sum(Z))).
perm([], []).
perm(L, .(H, T)) :- ','(append2(parts(V, .(H, U)), is(sum(L))), ','(append1(parts(V, U), is(sum(W))), perm(W, T))).

Queries:

perm(g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT 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: 4 | perm(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | perm([], T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: perm(T1, T2), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nperm(T1, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: perm([], T2), SCOPE: 1, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(append2(parts(X7, .(T7, X8)), is(sum([]))), ','(append1(parts(X7, X8), is(sum(X9))), perm(X9, T8)))\n\nX7 is free\nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(append2(parts(X7, .(T7, X8)), is(sum([]))), ','(append1(parts(X7, X8), is(sum(X9))), perm(X9, T8))), SCOPE: 2, CLAUSE: 0 | ','(append2(parts(X7, .(T7, X8)), is(sum([]))), ','(append1(parts(X7, X8), is(sum(X9))), perm(X9, T8))), SCOPE: 2, CLAUSE: 1\n\nX7 is free\nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(append2(parts(X7, .(T7, X8)), is(sum([]))), ','(append1(parts(X7, X8), is(sum(X9))), perm(X9, T8))), SCOPE: 2, CLAUSE: 1\n\nX7 is free\nX8 is free; \nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(append2(parts(X28, .(T17, X29)), is(sum(T14))), ','(append1(parts(X28, X29), is(sum(X30))), perm(X30, T18)))\n\nT14 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nperm(T14, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(append2(parts(X28, .(T17, X29)), is(sum(T14))), ','(append1(parts(X28, X29), is(sum(X30))), perm(X30, T18))), SCOPE: 3, CLAUSE: 0 | ','(append2(parts(X28, .(T17, X29)), is(sum(T14))), ','(append1(parts(X28, X29), is(sum(X30))), perm(X30, T18))), SCOPE: 3, CLAUSE: 1\n\nT14 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nperm(T14, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(append2(parts(X28, .(T17, X29)), is(sum(T14))), ','(append1(parts(X28, X29), is(sum(X30))), perm(X30, T18))), SCOPE: 3, CLAUSE: 0\n\nT14 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nperm(T14, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(append2(parts(X28, .(T17, X29)), is(sum(T14))), ','(append1(parts(X28, X29), is(sum(X30))), perm(X30, T18))), SCOPE: 3, CLAUSE: 1\n\nT14 is ground\nX28 is free; \nX29 is free; \nX30 is free; \nperm(T14, T2) !=? perm([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(append1(parts([], X43), is(sum(X30))), perm(X30, T24))\n\nX30 is free\nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: append1(parts([], X43), is(sum(X30)))\n\nX30 is free\nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: perm(T26, T24)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: append1(parts([], X43), is(sum(X30))), SCOPE: 4, CLAUSE: 2 | append1(parts([], X43), is(sum(X30))), SCOPE: 4, CLAUSE: 3\n\nX30 is free\nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: append1(parts([], X43), is(sum(X30))), SCOPE: 4, CLAUSE: 2\n\nX30 is free\nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: append1(parts([], X43), is(sum(X30))), SCOPE: 4, CLAUSE: 3\n\nX30 is free\nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\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(T26, T24), SCOPE: 5, CLAUSE: 4 | perm(T26, T24), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: perm(T26, T24), SCOPE: 5, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: perm(T26, T24), SCOPE: 5, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(append2(parts(X75, .(T36, X76)), is(sum(T37))), ','(append1(parts(X75, X76), is(sum(X77))), perm(X77, T38)))\n\nX75 is free\nX76 is free; \nX77 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: append2(parts(X75, .(T36, X76)), is(sum(T37)))\n\nX75 is free\nX76 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(append1(parts(T41, T42), is(sum(X77))), perm(X77, T43))\n\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: append2(parts(X75, .(T36, X76)), is(sum(T37))), SCOPE: 6, CLAUSE: 0 | append2(parts(X75, .(T36, X76)), is(sum(T37))), SCOPE: 6, CLAUSE: 1\n\nX75 is free\nX76 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: append2(parts(X75, .(T36, X76)), is(sum(T37))), SCOPE: 6, CLAUSE: 0\n\nX75 is free\nX76 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: append2(parts(X75, .(T36, X76)), is(sum(T37))), SCOPE: 6, CLAUSE: 1\n\nX75 is free\nX76 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: append2(parts(X119, .(T57, X120)), is(sum(T58)))\n\nX119 is free\nX120 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: append1(parts(T41, T42), is(sum(X77)))\n\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: perm(T63, T64)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: append1(parts(T41, T42), is(sum(X77))), SCOPE: 7, CLAUSE: 2 | append1(parts(T41, T42), is(sum(X77))), SCOPE: 7, CLAUSE: 3\n\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: append1(parts(T41, T42), is(sum(X77))), SCOPE: 7, CLAUSE: 2\n\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: append1(parts(T41, T42), is(sum(X77))), SCOPE: 7, CLAUSE: 3\n\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: append1(parts(T81, T82), is(sum(X153)))\n\nX153 is free", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(append2(parts(X179, .(T91, X180)), is(sum(T90))), ','(append1(parts(.(X178, X179), X180), is(sum(X30))), perm(X30, T92)))\n\nT90 is ground\nX30 is free; \nX178 is free; \nX179 is free; \nX180 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: append2(parts(X179, .(T91, X180)), is(sum(T90)))\n\nT90 is ground\nX179 is free; \nX180 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(append1(parts(.(X178, T95), T96), is(sum(X30))), perm(X30, T97))\n\nT95 is ground\nT96 is ground\nX30 is free; \nX178 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 58 [arrowhead = none ];
58 -> 2;

59 [label="EVAL with clause\nperm([], []).\nand substitution\nT1 -> []\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 59 [arrowhead = none ];
59 -> 3;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 60 [arrowhead = none ];
60 -> 4;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 61 [arrowhead = none ];
61 -> 5;

62 [label="EVAL with clause\nperm(X25, .(X26, X27)) :- ','(append2(parts(X28, .(X26, X29)), is(sum(X25))), ','(append1(parts(X28, X29), is(sum(X30))), perm(X30, X27))).\nand substitution\nT1 -> T14\nX25 -> T14\nX26 -> T17\nX27 -> T18\nT2 -> .(T17, T18)\nT15 -> T17\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="EVAL with clause\nperm(X4, .(X5, X6)) :- ','(append2(parts(X7, .(X5, X8)), is(sum(X4))), ','(append1(parts(X7, X8), is(sum(X9))), perm(X9, X6))).\nand substitution\nX4 -> []\nX5 -> T7\nX6 -> T8\nT2 -> .(T7, T8)\nT5 -> T7\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 64 [arrowhead = none ];
64 -> 6;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 65 [arrowhead = none ];
65 -> 7;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 66 [arrowhead = none ];
66 -> 8;

67 [label="BACKTRACK\nfor clause: append2(parts([], Y), is(sum(Y)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
8 -> 67 [arrowhead = none ];
67 -> 9;

68 [label="BACKTRACK\nfor clause: append2(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append2(parts(X, Y), is(sum(Z)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
9 -> 68 [arrowhead = none ];
68 -> 10;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
11 -> 69 [arrowhead = none ];
69 -> 13;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 70 [arrowhead = none ];
70 -> 14;

71 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
13 -> 71 [arrowhead = none ];
71 -> 15;

72 [label="EVAL with clause\nappend2(parts([], X42), is(sum(X42))).\nand substitution\nX28 -> []\nT17 -> T23\nX29 -> X43\nX42 -> .(T23, X43)\nT14 -> .(T23, X43)\nT18 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 72 [arrowhead = none ];
72 -> 16;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 73 [arrowhead = none ];
73 -> 17;

74 [label="EVAL with clause\nappend2(parts(.(X174, X175), X176), is(sum(.(X174, X177)))) :- append2(parts(X175, X176), is(sum(X177))).\nand substitution\nX174 -> X178\nX175 -> X179\nX28 -> .(X178, X179)\nT17 -> T91\nX29 -> X180\nX176 -> .(T91, X180)\nX177 -> T90\nT14 -> .(X178, T90)\nT89 -> T91\nT18 -> T92", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 74 [arrowhead = none ];
74 -> 54;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 75 [arrowhead = none ];
75 -> 55;

76 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 76 [arrowhead = none ];
76 -> 18;

77 [label="SPLIT 2\nreplacements:\nX43 -> T25\nX30 -> T26", fontsize=14, style = filled, fillcolor = violet];
16 -> 77 [arrowhead = none ];
77 -> 19;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 78 [arrowhead = none ];
78 -> 20;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 80 [arrowhead = none ];
80 -> 21;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 81 [arrowhead = none ];
81 -> 22;

82 [label="ONLY EVAL with clause\nappend1(parts([], X57), is(sum(X57))).\nand substitution\nX43 -> X58\nX57 -> X58\nX30 -> X58", fontsize=14, style = filled, fillcolor = white];
21 -> 82 [arrowhead = none ];
82 -> 23;

83 [label="BACKTRACK\nfor clause: append1(parts(.(H, X), Y), is(sum(.(H, Z)))) :- append1(parts(X, Y), is(sum(Z)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
22 -> 83 [arrowhead = none ];
83 -> 25;

84 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 84 [arrowhead = none ];
84 -> 24;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 85 [arrowhead = none ];
85 -> 27;

86 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 86 [arrowhead = none ];
86 -> 28;

87 [label="EVAL with clause\nperm([], []).\nand substitution\nT26 -> []\nT24 -> []", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 87 [arrowhead = none ];
87 -> 29;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 88 [arrowhead = none ];
88 -> 30;

89 [label="EVAL with clause\nperm(X72, .(X73, X74)) :- ','(append2(parts(X75, .(X73, X76)), is(sum(X72))), ','(append1(parts(X75, X76), is(sum(X77))), perm(X77, X74))).\nand substitution\nT26 -> T37\nX72 -> T37\nX73 -> T36\nX74 -> T38\nT24 -> .(T36, T38)\nT34 -> T36\nT33 -> T37\nT35 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 89 [arrowhead = none ];
89 -> 32;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 90 [arrowhead = none ];
90 -> 33;

91 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 91 [arrowhead = none ];
91 -> 31;

92 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
32 -> 92 [arrowhead = none ];
92 -> 34;

93 [label="SPLIT 2\nreplacements:\nX75 -> T41\nX76 -> T42\nT38 -> T43", fontsize=14, style = filled, fillcolor = violet];
32 -> 93 [arrowhead = none ];
93 -> 35;

94 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 94 [arrowhead = none ];
94 -> 36;

95 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 95 [arrowhead = none ];
95 -> 44;

96 [label="SPLIT 2\nreplacements:\nX77 -> T63\nT43 -> T64", fontsize=14, style = filled, fillcolor = violet];
35 -> 96 [arrowhead = none ];
96 -> 45;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 97 [arrowhead = none ];
97 -> 37;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 98 [arrowhead = none ];
98 -> 38;

99 [label="EVAL with clause\nappend2(parts([], X97), is(sum(X97))).\nand substitution\nX75 -> []\nT36 -> T50\nX76 -> X98\nX97 -> .(T50, X98)\nT37 -> .(T50, X98)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 99 [arrowhead = none ];
99 -> 39;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 100 [arrowhead = none ];
100 -> 40;

101 [label="EVAL with clause\nappend2(parts(.(X114, X115), X116), is(sum(.(X114, X117)))) :- append2(parts(X115, X116), is(sum(X117))).\nand substitution\nX114 -> X118\nX115 -> X119\nX75 -> .(X118, X119)\nT36 -> T57\nX76 -> X120\nX116 -> .(T57, X120)\nX117 -> T58\nT37 -> .(X118, T58)\nT55 -> T57\nT56 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 101 [arrowhead = none ];
101 -> 42;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 102 [arrowhead = none ];
102 -> 43;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 103 [arrowhead = none ];
103 -> 41;

104 [label="INSTANCE with matching:\nX75 -> X119\nT36 -> T57\nX76 -> X120\nT37 -> T58", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 104 [arrowhead = none , style=dashed];
104 -> 34[style=dashed];

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 105 [arrowhead = none ];
105 -> 46;

106 [label="INSTANCE with matching:\nT26 -> T63\nT24 -> T64", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 106 [arrowhead = none , style=dashed];
106 -> 19[style=dashed];

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 107 [arrowhead = none ];
107 -> 47;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
46 -> 108 [arrowhead = none ];
108 -> 48;

109 [label="EVAL with clause\nappend1(parts([], X137), is(sum(X137))).\nand substitution\nT41 -> []\nT42 -> T71\nX137 -> T71\nX77 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 109 [arrowhead = none ];
109 -> 49;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 110 [arrowhead = none ];
110 -> 50;

111 [label="EVAL with clause\nappend1(parts(.(X149, X150), X151), is(sum(.(X149, X152)))) :- append1(parts(X150, X151), is(sum(X152))).\nand substitution\nX149 -> T78\nX150 -> T81\nT41 -> .(T78, T81)\nT42 -> T82\nX151 -> T82\nX152 -> X153\nX77 -> .(T78, X153)\nT79 -> T81\nT80 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 111 [arrowhead = none ];
111 -> 52;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 112 [arrowhead = none ];
112 -> 53;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 113 [arrowhead = none ];
113 -> 51;

114 [label="INSTANCE with matching:\nT41 -> T81\nT42 -> T82\nX77 -> X153", fontsize=14, style = filled, fillcolor = lightgrey];
52 -> 114 [arrowhead = none , style=dashed];
114 -> 44[style=dashed];

115 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
54 -> 115 [arrowhead = none ];
115 -> 56;

116 [label="SPLIT 2\nnew knowledge:\nT95 is ground\nT91 is ground\nT96 is ground\nreplacements:\nX179 -> T95\nX180 -> T96\nT92 -> T97", fontsize=14, style = filled, fillcolor = violet];
54 -> 116 [arrowhead = none ];
116 -> 57;

117 [label="INSTANCE with matching:\nX75 -> X179\nT36 -> T91\nX76 -> X180\nT37 -> T90", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 117 [arrowhead = none , style=dashed];
117 -> 34[style=dashed];

118 [label="INSTANCE with matching:\nT41 -> .(X178, T95)\nT42 -> T96\nX77 -> X30\nT43 -> T97", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 118 [arrowhead = none , style=dashed];
118 -> 35[style=dashed];

}

(2) Obligation:

Triples:

append234(.(X118, X119), T57, X120, .(X118, T58)) :- append234(X119, T57, X120, T58).
perm19(T37, .(T36, T38)) :- append234(X75, T36, X76, T37).
perm19(T37, .(T36, T43)) :- ','(append2c34(T41, T36, T42, T37), p35(T41, T42, X77, T43)).
append144(.(T78, T81), T82, .(T78, X153)) :- append144(T81, T82, X153).
p35(T41, T42, X77, T43) :- append144(T41, T42, X77).
p35(T41, T42, T63, T64) :- ','(append1c44(T41, T42, T63), perm19(T63, T64)).
perm1(.(T23, T25), .(T23, T24)) :- ','(append1c18(T25, T26), perm19(T26, T24)).
perm1(.(X178, T90), .(T91, T92)) :- append234(X179, T91, X180, T90).
perm1(.(X178, T90), .(T91, T97)) :- ','(append2c34(T95, T91, T96, T90), p35(.(X178, T95), T96, X30, T97)).

Clauses:

append2c34([], T50, X98, .(T50, X98)).
append2c34(.(X118, X119), T57, X120, .(X118, T58)) :- append2c34(X119, T57, X120, T58).
permc19([], []).
permc19(T37, .(T36, T43)) :- ','(append2c34(T41, T36, T42, T37), qc35(T41, T42, X77, T43)).
append1c44([], T71, T71).
append1c44(.(T78, T81), T82, .(T78, X153)) :- append1c44(T81, T82, X153).
qc35(T41, T42, T63, T64) :- ','(append1c44(T41, T42, T63), permc19(T63, T64)).
append1c18(X58, X58).

Afs:

perm1(x1, x2)  =  perm1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
perm1_in: (b,f)
perm19_in: (b,f)
append234_in: (f,f,f,b)
append2c34_in: (f,f,f,b)
p35_in: (b,b,f,f)
append144_in: (b,b,f)
append1c44_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

PERM1_IN_GA(.(T23, T25), .(T23, T24)) → U9_GA(T23, T25, T24, append1c18_in_ga(T25, T26))
U9_GA(T23, T25, T24, append1c18_out_ga(T25, T26)) → U10_GA(T23, T25, T24, perm19_in_ga(T26, T24))
U9_GA(T23, T25, T24, append1c18_out_ga(T25, T26)) → PERM19_IN_GA(T26, T24)
PERM19_IN_GA(T37, .(T36, T38)) → U2_GA(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
PERM19_IN_GA(T37, .(T36, T38)) → APPEND234_IN_AAAG(X75, T36, X76, T37)
APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → U1_AAAG(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → APPEND234_IN_AAAG(X119, T57, X120, T58)
PERM19_IN_GA(T37, .(T36, T43)) → U3_GA(T37, T36, T43, append2c34_in_aaag(T41, T36, T42, T37))
U3_GA(T37, T36, T43, append2c34_out_aaag(T41, T36, T42, T37)) → U4_GA(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
U3_GA(T37, T36, T43, append2c34_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42, X77, T43)
P35_IN_GGAA(T41, T42, X77, T43) → U6_GGAA(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
P35_IN_GGAA(T41, T42, X77, T43) → APPEND144_IN_GGA(T41, T42, X77)
APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → U5_GGA(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → APPEND144_IN_GGA(T81, T82, X153)
P35_IN_GGAA(T41, T42, T63, T64) → U7_GGAA(T41, T42, T63, T64, append1c44_in_gga(T41, T42, T63))
U7_GGAA(T41, T42, T63, T64, append1c44_out_gga(T41, T42, T63)) → U8_GGAA(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U7_GGAA(T41, T42, T63, T64, append1c44_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63, T64)
PERM1_IN_GA(.(X178, T90), .(T91, T92)) → U11_GA(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
PERM1_IN_GA(.(X178, T90), .(T91, T92)) → APPEND234_IN_AAAG(X179, T91, X180, T90)
PERM1_IN_GA(.(X178, T90), .(T91, T97)) → U12_GA(X178, T90, T91, T97, append2c34_in_aaag(T95, T91, T96, T90))
U12_GA(X178, T90, T91, T97, append2c34_out_aaag(T95, T91, T96, T90)) → U13_GA(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U12_GA(X178, T90, T91, T97, append2c34_out_aaag(T95, T91, T96, T90)) → P35_IN_GGAA(.(X178, T95), T96, X30, T97)

The TRS R consists of the following rules:

append1c18_in_ga(X58, X58) → append1c18_out_ga(X58, X58)
append2c34_in_aaag([], T50, X98, .(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U15_aaag(X118, X119, T57, X120, T58, append2c34_in_aaag(X119, T57, X120, T58))
U15_aaag(X118, X119, T57, X120, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
append1c44_in_gga([], T71, T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82, .(T78, X153)) → U18_gga(T78, T81, T82, X153, append1c44_in_gga(T81, T82, X153))
U18_gga(T78, T81, T82, X153, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
append1c18_in_ga(x1, x2)  =  append1c18_in_ga(x1)
append1c18_out_ga(x1, x2)  =  append1c18_out_ga(x1, x2)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append2c34_in_aaag(x1, x2, x3, x4)  =  append2c34_in_aaag(x4)
append2c34_out_aaag(x1, x2, x3, x4)  =  append2c34_out_aaag(x1, x2, x3, x4)
U15_aaag(x1, x2, x3, x4, x5, x6)  =  U15_aaag(x1, x5, x6)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append1c44_in_gga(x1, x2, x3)  =  append1c44_in_gga(x1, x2)
[]  =  []
append1c44_out_gga(x1, x2, x3)  =  append1c44_out_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x2, x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
PERM19_IN_GA(x1, x2)  =  PERM19_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
APPEND234_IN_AAAG(x1, x2, x3, x4)  =  APPEND234_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, x6)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
P35_IN_GGAA(x1, x2, x3, x4)  =  P35_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5)  =  U6_GGAA(x1, x2, x5)
APPEND144_IN_GGA(x1, x2, x3)  =  APPEND144_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(x1, x2, x5)
U8_GGAA(x1, x2, x3, x4, x5)  =  U8_GGAA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x5)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

PERM1_IN_GA(.(T23, T25), .(T23, T24)) → U9_GA(T23, T25, T24, append1c18_in_ga(T25, T26))
U9_GA(T23, T25, T24, append1c18_out_ga(T25, T26)) → U10_GA(T23, T25, T24, perm19_in_ga(T26, T24))
U9_GA(T23, T25, T24, append1c18_out_ga(T25, T26)) → PERM19_IN_GA(T26, T24)
PERM19_IN_GA(T37, .(T36, T38)) → U2_GA(T37, T36, T38, append234_in_aaag(X75, T36, X76, T37))
PERM19_IN_GA(T37, .(T36, T38)) → APPEND234_IN_AAAG(X75, T36, X76, T37)
APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → U1_AAAG(X118, X119, T57, X120, T58, append234_in_aaag(X119, T57, X120, T58))
APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → APPEND234_IN_AAAG(X119, T57, X120, T58)
PERM19_IN_GA(T37, .(T36, T43)) → U3_GA(T37, T36, T43, append2c34_in_aaag(T41, T36, T42, T37))
U3_GA(T37, T36, T43, append2c34_out_aaag(T41, T36, T42, T37)) → U4_GA(T37, T36, T43, p35_in_ggaa(T41, T42, X77, T43))
U3_GA(T37, T36, T43, append2c34_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42, X77, T43)
P35_IN_GGAA(T41, T42, X77, T43) → U6_GGAA(T41, T42, X77, T43, append144_in_gga(T41, T42, X77))
P35_IN_GGAA(T41, T42, X77, T43) → APPEND144_IN_GGA(T41, T42, X77)
APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → U5_GGA(T78, T81, T82, X153, append144_in_gga(T81, T82, X153))
APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → APPEND144_IN_GGA(T81, T82, X153)
P35_IN_GGAA(T41, T42, T63, T64) → U7_GGAA(T41, T42, T63, T64, append1c44_in_gga(T41, T42, T63))
U7_GGAA(T41, T42, T63, T64, append1c44_out_gga(T41, T42, T63)) → U8_GGAA(T41, T42, T63, T64, perm19_in_ga(T63, T64))
U7_GGAA(T41, T42, T63, T64, append1c44_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63, T64)
PERM1_IN_GA(.(X178, T90), .(T91, T92)) → U11_GA(X178, T90, T91, T92, append234_in_aaag(X179, T91, X180, T90))
PERM1_IN_GA(.(X178, T90), .(T91, T92)) → APPEND234_IN_AAAG(X179, T91, X180, T90)
PERM1_IN_GA(.(X178, T90), .(T91, T97)) → U12_GA(X178, T90, T91, T97, append2c34_in_aaag(T95, T91, T96, T90))
U12_GA(X178, T90, T91, T97, append2c34_out_aaag(T95, T91, T96, T90)) → U13_GA(X178, T90, T91, T97, p35_in_ggaa(.(X178, T95), T96, X30, T97))
U12_GA(X178, T90, T91, T97, append2c34_out_aaag(T95, T91, T96, T90)) → P35_IN_GGAA(.(X178, T95), T96, X30, T97)

The TRS R consists of the following rules:

append1c18_in_ga(X58, X58) → append1c18_out_ga(X58, X58)
append2c34_in_aaag([], T50, X98, .(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U15_aaag(X118, X119, T57, X120, T58, append2c34_in_aaag(X119, T57, X120, T58))
U15_aaag(X118, X119, T57, X120, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
append1c44_in_gga([], T71, T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82, .(T78, X153)) → U18_gga(T78, T81, T82, X153, append1c44_in_gga(T81, T82, X153))
U18_gga(T78, T81, T82, X153, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
append1c18_in_ga(x1, x2)  =  append1c18_in_ga(x1)
append1c18_out_ga(x1, x2)  =  append1c18_out_ga(x1, x2)
perm19_in_ga(x1, x2)  =  perm19_in_ga(x1)
append234_in_aaag(x1, x2, x3, x4)  =  append234_in_aaag(x4)
append2c34_in_aaag(x1, x2, x3, x4)  =  append2c34_in_aaag(x4)
append2c34_out_aaag(x1, x2, x3, x4)  =  append2c34_out_aaag(x1, x2, x3, x4)
U15_aaag(x1, x2, x3, x4, x5, x6)  =  U15_aaag(x1, x5, x6)
p35_in_ggaa(x1, x2, x3, x4)  =  p35_in_ggaa(x1, x2)
append144_in_gga(x1, x2, x3)  =  append144_in_gga(x1, x2)
append1c44_in_gga(x1, x2, x3)  =  append1c44_in_gga(x1, x2)
[]  =  []
append1c44_out_gga(x1, x2, x3)  =  append1c44_out_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
PERM1_IN_GA(x1, x2)  =  PERM1_IN_GA(x1)
U9_GA(x1, x2, x3, x4)  =  U9_GA(x1, x2, x4)
U10_GA(x1, x2, x3, x4)  =  U10_GA(x1, x2, x4)
PERM19_IN_GA(x1, x2)  =  PERM19_IN_GA(x1)
U2_GA(x1, x2, x3, x4)  =  U2_GA(x1, x4)
APPEND234_IN_AAAG(x1, x2, x3, x4)  =  APPEND234_IN_AAAG(x4)
U1_AAAG(x1, x2, x3, x4, x5, x6)  =  U1_AAAG(x1, x5, x6)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
U4_GA(x1, x2, x3, x4)  =  U4_GA(x1, x4)
P35_IN_GGAA(x1, x2, x3, x4)  =  P35_IN_GGAA(x1, x2)
U6_GGAA(x1, x2, x3, x4, x5)  =  U6_GGAA(x1, x2, x5)
APPEND144_IN_GGA(x1, x2, x3)  =  APPEND144_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3, x4, x5)  =  U5_GGA(x1, x2, x3, x5)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(x1, x2, x5)
U8_GGAA(x1, x2, x3, x4, x5)  =  U8_GGAA(x1, x2, x3, x5)
U11_GA(x1, x2, x3, x4, x5)  =  U11_GA(x1, x2, x5)
U12_GA(x1, x2, x3, x4, x5)  =  U12_GA(x1, x2, x5)
U13_GA(x1, x2, x3, x4, x5)  =  U13_GA(x1, x2, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 16 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → APPEND144_IN_GGA(T81, T82, X153)

The TRS R consists of the following rules:

append1c18_in_ga(X58, X58) → append1c18_out_ga(X58, X58)
append2c34_in_aaag([], T50, X98, .(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U15_aaag(X118, X119, T57, X120, T58, append2c34_in_aaag(X119, T57, X120, T58))
U15_aaag(X118, X119, T57, X120, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
append1c44_in_gga([], T71, T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82, .(T78, X153)) → U18_gga(T78, T81, T82, X153, append1c44_in_gga(T81, T82, X153))
U18_gga(T78, T81, T82, X153, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
append1c18_in_ga(x1, x2)  =  append1c18_in_ga(x1)
append1c18_out_ga(x1, x2)  =  append1c18_out_ga(x1, x2)
append2c34_in_aaag(x1, x2, x3, x4)  =  append2c34_in_aaag(x4)
append2c34_out_aaag(x1, x2, x3, x4)  =  append2c34_out_aaag(x1, x2, x3, x4)
U15_aaag(x1, x2, x3, x4, x5, x6)  =  U15_aaag(x1, x5, x6)
append1c44_in_gga(x1, x2, x3)  =  append1c44_in_gga(x1, x2)
[]  =  []
append1c44_out_gga(x1, x2, x3)  =  append1c44_out_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
APPEND144_IN_GGA(x1, x2, x3)  =  APPEND144_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

APPEND144_IN_GGA(.(T78, T81), T82, .(T78, X153)) → APPEND144_IN_GGA(T81, T82, X153)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND144_IN_GGA(x1, x2, x3)  =  APPEND144_IN_GGA(x1, x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

APPEND144_IN_GGA(.(T78, T81), T82) → APPEND144_IN_GGA(T81, T82)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• APPEND144_IN_GGA(.(T78, T81), T82) → APPEND144_IN_GGA(T81, T82)
  The graph contains the following edges 1 > 1, 2 >= 2

(13) YES

(14) Obligation:

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

APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → APPEND234_IN_AAAG(X119, T57, X120, T58)

The TRS R consists of the following rules:

append1c18_in_ga(X58, X58) → append1c18_out_ga(X58, X58)
append2c34_in_aaag([], T50, X98, .(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U15_aaag(X118, X119, T57, X120, T58, append2c34_in_aaag(X119, T57, X120, T58))
U15_aaag(X118, X119, T57, X120, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
append1c44_in_gga([], T71, T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82, .(T78, X153)) → U18_gga(T78, T81, T82, X153, append1c44_in_gga(T81, T82, X153))
U18_gga(T78, T81, T82, X153, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
append1c18_in_ga(x1, x2)  =  append1c18_in_ga(x1)
append1c18_out_ga(x1, x2)  =  append1c18_out_ga(x1, x2)
append2c34_in_aaag(x1, x2, x3, x4)  =  append2c34_in_aaag(x4)
append2c34_out_aaag(x1, x2, x3, x4)  =  append2c34_out_aaag(x1, x2, x3, x4)
U15_aaag(x1, x2, x3, x4, x5, x6)  =  U15_aaag(x1, x5, x6)
append1c44_in_gga(x1, x2, x3)  =  append1c44_in_gga(x1, x2)
[]  =  []
append1c44_out_gga(x1, x2, x3)  =  append1c44_out_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
APPEND234_IN_AAAG(x1, x2, x3, x4)  =  APPEND234_IN_AAAG(x4)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

APPEND234_IN_AAAG(.(X118, X119), T57, X120, .(X118, T58)) → APPEND234_IN_AAAG(X119, T57, X120, T58)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APPEND234_IN_AAAG(x1, x2, x3, x4)  =  APPEND234_IN_AAAG(x4)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

APPEND234_IN_AAAG(.(X118, T58)) → APPEND234_IN_AAAG(T58)

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

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

• APPEND234_IN_AAAG(.(X118, T58)) → APPEND234_IN_AAAG(T58)
  The graph contains the following edges 1 > 1

(20) YES

(21) Obligation:

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

PERM19_IN_GA(T37, .(T36, T43)) → U3_GA(T37, T36, T43, append2c34_in_aaag(T41, T36, T42, T37))
U3_GA(T37, T36, T43, append2c34_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42, X77, T43)
P35_IN_GGAA(T41, T42, T63, T64) → U7_GGAA(T41, T42, T63, T64, append1c44_in_gga(T41, T42, T63))
U7_GGAA(T41, T42, T63, T64, append1c44_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63, T64)

The TRS R consists of the following rules:

append1c18_in_ga(X58, X58) → append1c18_out_ga(X58, X58)
append2c34_in_aaag([], T50, X98, .(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U15_aaag(X118, X119, T57, X120, T58, append2c34_in_aaag(X119, T57, X120, T58))
U15_aaag(X118, X119, T57, X120, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
append1c44_in_gga([], T71, T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82, .(T78, X153)) → U18_gga(T78, T81, T82, X153, append1c44_in_gga(T81, T82, X153))
U18_gga(T78, T81, T82, X153, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
append1c18_in_ga(x1, x2)  =  append1c18_in_ga(x1)
append1c18_out_ga(x1, x2)  =  append1c18_out_ga(x1, x2)
append2c34_in_aaag(x1, x2, x3, x4)  =  append2c34_in_aaag(x4)
append2c34_out_aaag(x1, x2, x3, x4)  =  append2c34_out_aaag(x1, x2, x3, x4)
U15_aaag(x1, x2, x3, x4, x5, x6)  =  U15_aaag(x1, x5, x6)
append1c44_in_gga(x1, x2, x3)  =  append1c44_in_gga(x1, x2)
[]  =  []
append1c44_out_gga(x1, x2, x3)  =  append1c44_out_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
PERM19_IN_GA(x1, x2)  =  PERM19_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
P35_IN_GGAA(x1, x2, x3, x4)  =  P35_IN_GGAA(x1, x2)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(x1, x2, x5)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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

PERM19_IN_GA(T37, .(T36, T43)) → U3_GA(T37, T36, T43, append2c34_in_aaag(T41, T36, T42, T37))
U3_GA(T37, T36, T43, append2c34_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42, X77, T43)
P35_IN_GGAA(T41, T42, T63, T64) → U7_GGAA(T41, T42, T63, T64, append1c44_in_gga(T41, T42, T63))
U7_GGAA(T41, T42, T63, T64, append1c44_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63, T64)

The TRS R consists of the following rules:

append2c34_in_aaag([], T50, X98, .(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, X119), T57, X120, .(X118, T58)) → U15_aaag(X118, X119, T57, X120, T58, append2c34_in_aaag(X119, T57, X120, T58))
append1c44_in_gga([], T71, T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82, .(T78, X153)) → U18_gga(T78, T81, T82, X153, append1c44_in_gga(T81, T82, X153))
U15_aaag(X118, X119, T57, X120, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U18_gga(T78, T81, T82, X153, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
append2c34_in_aaag(x1, x2, x3, x4)  =  append2c34_in_aaag(x4)
append2c34_out_aaag(x1, x2, x3, x4)  =  append2c34_out_aaag(x1, x2, x3, x4)
U15_aaag(x1, x2, x3, x4, x5, x6)  =  U15_aaag(x1, x5, x6)
append1c44_in_gga(x1, x2, x3)  =  append1c44_in_gga(x1, x2)
[]  =  []
append1c44_out_gga(x1, x2, x3)  =  append1c44_out_gga(x1, x2, x3)
U18_gga(x1, x2, x3, x4, x5)  =  U18_gga(x1, x2, x3, x5)
PERM19_IN_GA(x1, x2)  =  PERM19_IN_GA(x1)
U3_GA(x1, x2, x3, x4)  =  U3_GA(x1, x4)
P35_IN_GGAA(x1, x2, x3, x4)  =  P35_IN_GGAA(x1, x2)
U7_GGAA(x1, x2, x3, x4, x5)  =  U7_GGAA(x1, x2, x5)

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:

PERM19_IN_GA(T37) → U3_GA(T37, append2c34_in_aaag(T37))
U3_GA(T37, append2c34_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42)
P35_IN_GGAA(T41, T42) → U7_GGAA(T41, T42, append1c44_in_gga(T41, T42))
U7_GGAA(T41, T42, append1c44_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63)

The TRS R consists of the following rules:

append2c34_in_aaag(.(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, T58)) → U15_aaag(X118, T58, append2c34_in_aaag(T58))
append1c44_in_gga([], T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82) → U18_gga(T78, T81, T82, append1c44_in_gga(T81, T82))
U15_aaag(X118, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U18_gga(T78, T81, T82, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

The set Q consists of the following terms:

append2c34_in_aaag(x0)
append1c44_in_gga(x0, x1)
U15_aaag(x0, x1, x2)
U18_gga(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.

P35_IN_GGAA(T41, T42) → U7_GGAA(T41, T42, append1c44_in_gga(T41, T42))

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(P35_IN_GGAA(x₁, x₂)) = 1 + x₁ + x₂   
POL(PERM19_IN_GA(x₁)) = x₁   
POL(U15_aaag(x₁, x₂, x₃)) = 1 + x₃   
POL(U18_gga(x₁, x₂, x₃, x₄)) = 1 + x₄   
POL(U3_GA(x₁, x₂)) = x₂   
POL(U7_GGAA(x₁, x₂, x₃)) = x₃   
POL([]) = 0   
POL(append1c44_in_gga(x₁, x₂)) = x₁ + x₂   
POL(append1c44_out_gga(x₁, x₂, x₃)) = x₃   
POL(append2c34_in_aaag(x₁)) = x₁   
POL(append2c34_out_aaag(x₁, x₂, x₃, x₄)) = 1 + x₁ + x₃   

The following usable rules [FROCOS05] were oriented:

append2c34_in_aaag(.(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, T58)) → U15_aaag(X118, T58, append2c34_in_aaag(T58))
append1c44_in_gga([], T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82) → U18_gga(T78, T81, T82, append1c44_in_gga(T81, T82))
U15_aaag(X118, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U18_gga(T78, T81, T82, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

(27) Obligation:

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

PERM19_IN_GA(T37) → U3_GA(T37, append2c34_in_aaag(T37))
U3_GA(T37, append2c34_out_aaag(T41, T36, T42, T37)) → P35_IN_GGAA(T41, T42)
U7_GGAA(T41, T42, append1c44_out_gga(T41, T42, T63)) → PERM19_IN_GA(T63)

The TRS R consists of the following rules:

append2c34_in_aaag(.(T50, X98)) → append2c34_out_aaag([], T50, X98, .(T50, X98))
append2c34_in_aaag(.(X118, T58)) → U15_aaag(X118, T58, append2c34_in_aaag(T58))
append1c44_in_gga([], T71) → append1c44_out_gga([], T71, T71)
append1c44_in_gga(.(T78, T81), T82) → U18_gga(T78, T81, T82, append1c44_in_gga(T81, T82))
U15_aaag(X118, T58, append2c34_out_aaag(X119, T57, X120, T58)) → append2c34_out_aaag(.(X118, X119), T57, X120, .(X118, T58))
U18_gga(T78, T81, T82, append1c44_out_gga(T81, T82, X153)) → append1c44_out_gga(.(T78, T81), T82, .(T78, X153))

The set Q consists of the following terms:

append2c34_in_aaag(x0)
append1c44_in_gga(x0, x1)
U15_aaag(x0, x1, x2)
U18_gga(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.

(29) TRUE