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

(0) Obligation:

Clauses:

init_vars(E1, E2, E1init, E2init) :- ','(find_all_vars(E1, Vars1), ','(find_all_vars(E2, Vars2), ','(intersect(Vars1, Vars2, _X, Notin1, Notin2), ','(init_vars2(Notin1, E1, E1init), init_vars2(Notin2, E2, E2init))))).
find_all_vars(E, Vars) :- ','(find_all_vars2(E, Vars0), sort(Vars0, Vars)).
find_all_vars2([], []).
find_all_vars2(.(=(Vars, _Values), Es), AllVars) :- ','(append(Vars, AllVars1, AllVars), find_all_vars2(Es, AllVars1)).
init_vars2(Notin, E, Einit) :- ','(init_vars3(Notin, E, Einit0), sort(Einit0, Einit)).
init_vars3([], E, E).
init_vars3(.(Var, Vars), E, .(=(.(Var, []), .(unbound, [])), Es)) :- init_vars3(Vars, E, Es).
append([], A, A).
append(.(A, B), C, .(A, D)) :- append(B, C, D).
intersect(As, [], [], [], As) :- !.
intersect([], Bs, [], Bs, []) :- !.
intersect(.(A, As), .(B, Bs), Cs, Ds, Es) :- ;(;(->(=(A, B), ','(=(Cs, .(A, Cs2)), intersect(As, Bs, Cs2, Ds, Es))), ->(@<(A, B), ','(=(Es, .(A, Es2)), intersect(As, .(B, Bs), Cs, Ds, Es2)))), ','(=(Ds, .(B, Ds2)), intersect(.(A, As), Bs, Cs, Ds2, Es))).
get_query(E1, E2) :- ','(=(E1, .(=(X, .(a, [])), .(=(X, .(a, [])), .(=(X, .(a, [])), .(=(X, .(a, [])), []))))), ','(=(E2, .(=(Y, .(a, [])), .(=(Y, .(a, [])), .(=(Y, .(a, [])), .(=(Y, .(a, [])), []))))), ','(=(X, .(5, .(7, .(8, .(3, .(2, .(4, .(1, .(6, .(9, .(15, .(17, .(18, .(13, .(12, .(14, .(11, .(16, .(19, .(25, .(27, .(28, .(23, .(22, .(24, .(21, .(26, .(29, [])))))))))))))))))))))))))))), =(Y, .(15, .(17, .(18, .(13, .(12, .(14, .(11, .(16, .(19, .(35, .(37, .(38, .(33, .(32, .(34, .(5, .(7, .(8, .(3, .(2, .(4, .(1, .(6, .(9, .(31, .(36, .(39, []))))))))))))))))))))))))))))))).

Query: init_vars(g,g,a,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: init_vars(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: init_vars(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(find_all_vars(T9, X9), ','(find_all_vars(T10, X10), ','(intersect(X9, X10, X11, X12, X13), ','(init_vars2(X12, T9, T13), init_vars2(X13, T10, T14)))))\n\nT9 is ground\nT10 is ground\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(find_all_vars(T9, X9), ','(find_all_vars(T10, X10), ','(intersect(X9, X10, X11, X12, X13), ','(init_vars2(X12, T9, T13), init_vars2(X13, T10, T14))))), SCOPE: 2, CLAUSE: 1\n\nT9 is ground\nT10 is ground\nX9 is free\nX10 is free\nX11 is free\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(','(find_all_vars2(T17, X23), sort(X23, X24)), ','(find_all_vars(T10, X10), ','(intersect(X24, X10, X11, X12, X13), ','(init_vars2(X12, T17, T13), init_vars2(X13, T10, T14)))))\n\nT10 is ground\nT17 is ground\nX10 is free\nX11 is free\nX12 is free\nX13 is free\nX24 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: find_all_vars2(T17, X23)\n\nT17 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(sort(T18, X24), ','(find_all_vars(T10, X10), ','(intersect(X24, X10, X11, X12, X13), ','(init_vars2(X12, T17, T13), init_vars2(X13, T10, T14)))))\n\nT10 is ground\nT17 is ground\nX10 is free\nX11 is free\nX12 is free\nX13 is free\nX24 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: find_all_vars2(T17, X23), SCOPE: 3, CLAUSE: 2 | find_all_vars2(T17, X23), SCOPE: 3, CLAUSE: 3\n\nT17 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: find_all_vars2(T17, X23), SCOPE: 3, CLAUSE: 2\n\nT17 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: find_all_vars2(T17, X23), SCOPE: 3, CLAUSE: 3\n\nT17 is ground\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(append(T25, X40, X41), find_all_vars2(T27, X40))\n\nT25 is ground\nT27 is ground\nX41 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: append(T25, X40, X41)\n\nT25 is ground\nX41 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: find_all_vars2(T27, T28)\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: append(T25, X40, X41), SCOPE: 4, CLAUSE: 7 | append(T25, X40, X41), SCOPE: 4, CLAUSE: 8\n\nT25 is ground\nX41 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: append(T25, X40, X41), SCOPE: 4, CLAUSE: 7\n\nT25 is ground\nX41 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: append(T25, X40, X41), SCOPE: 4, CLAUSE: 8\n\nT25 is ground\nX41 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append(T35, X76, X77)\n\nT35 is ground\nX76 is free\nX77 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: find_all_vars2(T27, T28), SCOPE: 5, CLAUSE: 2 | find_all_vars2(T27, T28), SCOPE: 5, CLAUSE: 3\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: find_all_vars2(T27, T28), SCOPE: 5, CLAUSE: 2\n\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: find_all_vars2(T27, T28), SCOPE: 5, CLAUSE: 3\n\nT27 is ground", 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: ','(append(T44, X95, T48), find_all_vars2(T46, X95))\n\nT44 is ground\nT46 is ground\nX95 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: append(T44, X95, T48)\n\nT44 is ground\nX95 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: find_all_vars2(T46, T51)\n\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: append(T44, X95, T48), SCOPE: 6, CLAUSE: 7 | append(T44, X95, T48), SCOPE: 6, CLAUSE: 8\n\nT44 is ground\nX95 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: append(T44, X95, T48), SCOPE: 6, CLAUSE: 7\n\nT44 is ground\nX95 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: append(T44, X95, T48), SCOPE: 6, CLAUSE: 8\n\nT44 is ground\nX95 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: append(T66, X128, T68)\n\nT66 is ground\nX128 is free", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 45 [arrowhead = none ];
45 -> 2;

46 [label="ONLY EVAL with clause\ninit_vars(X5, X6, X7, X8) :- ','(find_all_vars(X5, X9), ','(find_all_vars(X6, X10), ','(intersect(X9, X10, X11, X12, X13), ','(init_vars2(X12, X5, X7), init_vars2(X13, X6, X8))))).\nand substitutionT1 -> T9,\nX5 -> T9,\nT2 -> T10,\nX6 -> T10,\nT3 -> T13,\nX7 -> T13,\nT4 -> T14,\nX8 -> T14,\nT11 -> T13,\nT12 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 46 [arrowhead = none ];
46 -> 3;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 47 [arrowhead = none ];
47 -> 4;

48 [label="ONLY EVAL with clause\nfind_all_vars(X21, X22) :- ','(find_all_vars2(X21, X23), sort(X23, X22)).\nand substitutionT9 -> T17,\nX21 -> T17,\nX9 -> X24,\nX22 -> X24", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 48 [arrowhead = none ];
48 -> 5;

49 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
5 -> 49 [arrowhead = none ];
49 -> 6;

50 [label="SPLIT 2\nnew knowledge:\nT17 is ground\nreplacements:X23 -> T18", fontsize=14, style = filled, fillcolor = violet];
5 -> 50 [arrowhead = none ];
50 -> 7;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 51 [arrowhead = none ];
51 -> 8;

52 [label="UNDEFINED ERROR", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 52 [arrowhead = none ];
52 -> 44;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 53 [arrowhead = none ];
53 -> 9;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 54 [arrowhead = none ];
54 -> 10;

55 [label="EVAL with clause\nfind_all_vars2([], []).\nand substitutionT17 -> [],\nX23 -> []", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 55 [arrowhead = none ];
55 -> 11;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 56 [arrowhead = none ];
56 -> 12;

57 [label="EVAL with clause\nfind_all_vars2(.(=(X36, X37), X38), X39) :- ','(append(X36, X40, X39), find_all_vars2(X38, X40)).\nand substitutionX36 -> T25,\nX37 -> T26,\nX38 -> T27,\nT17 -> .(=(T25, T26), T27),\nX23 -> X41,\nX39 -> X41", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 57 [arrowhead = none ];
57 -> 14;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 58 [arrowhead = none ];
58 -> 15;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 59 [arrowhead = none ];
59 -> 13;

60 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
14 -> 60 [arrowhead = none ];
60 -> 16;

61 [label="SPLIT 2\nnew knowledge:\nT25 is ground\nreplacements:X40 -> T28,\nX41 -> T29", fontsize=14, style = filled, fillcolor = violet];
14 -> 61 [arrowhead = none ];
61 -> 17;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 62 [arrowhead = none ];
62 -> 18;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 63 [arrowhead = none ];
63 -> 26;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 64 [arrowhead = none ];
64 -> 19;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 65 [arrowhead = none ];
65 -> 20;

66 [label="EVAL with clause\nappend([], X58, X58).\nand substitutionT25 -> [],\nX40 -> X59,\nX58 -> X59,\nX41 -> X59", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 66 [arrowhead = none ];
66 -> 21;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 67 [arrowhead = none ];
67 -> 22;

68 [label="EVAL with clause\nappend(.(X72, X73), X74, .(X72, X75)) :- append(X73, X74, X75).\nand substitutionX72 -> T34,\nX73 -> T35,\nT25 -> .(T34, T35),\nX40 -> X76,\nX74 -> X76,\nX75 -> X77,\nX41 -> .(T34, X77)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 68 [arrowhead = none ];
68 -> 24;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 69 [arrowhead = none ];
69 -> 25;

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 70 [arrowhead = none ];
70 -> 23;

71 [label="INSTANCE with matching:\nT25 -> T35\nX40 -> X76\nX41 -> X77", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 71 [arrowhead = none , style=dashed];
71 -> 16[style=dashed];

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 72 [arrowhead = none ];
72 -> 27;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 73 [arrowhead = none ];
73 -> 28;

74 [label="EVAL with clause\nfind_all_vars2([], []).\nand substitutionT27 -> [],\nT28 -> []", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 74 [arrowhead = none ];
74 -> 29;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 75 [arrowhead = none ];
75 -> 30;

76 [label="EVAL with clause\nfind_all_vars2(.(=(X91, X92), X93), X94) :- ','(append(X91, X95, X94), find_all_vars2(X93, X95)).\nand substitutionX91 -> T44,\nX92 -> T45,\nX93 -> T46,\nT27 -> .(=(T44, T45), T46),\nT28 -> T48,\nX94 -> T48,\nT47 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 76 [arrowhead = none ];
76 -> 32;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 77 [arrowhead = none ];
77 -> 33;

78 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 78 [arrowhead = none ];
78 -> 31;

79 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
32 -> 79 [arrowhead = none ];
79 -> 34;

80 [label="SPLIT 2\nnew knowledge:\nT44 is ground\nreplacements:X95 -> T51", fontsize=14, style = filled, fillcolor = violet];
32 -> 80 [arrowhead = none ];
80 -> 35;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 81 [arrowhead = none ];
81 -> 36;

82 [label="INSTANCE with matching:\nT27 -> T46\nT28 -> T51", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 82 [arrowhead = none , style=dashed];
82 -> 17[style=dashed];

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 83 [arrowhead = none ];
83 -> 37;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 84 [arrowhead = none ];
84 -> 38;

85 [label="EVAL with clause\nappend([], X112, X112).\nand substitutionT44 -> [],\nX95 -> T58,\nX112 -> T58,\nT48 -> T58,\nX113 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 85 [arrowhead = none ];
85 -> 39;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 86 [arrowhead = none ];
86 -> 40;

87 [label="EVAL with clause\nappend(.(X124, X125), X126, .(X124, X127)) :- append(X125, X126, X127).\nand substitutionX124 -> T65,\nX125 -> T66,\nT44 -> .(T65, T66),\nX95 -> X128,\nX126 -> X128,\nX127 -> T68,\nT48 -> .(T65, T68),\nT67 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 87 [arrowhead = none ];
87 -> 42;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 88 [arrowhead = none ];
88 -> 43;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 89 [arrowhead = none ];
89 -> 41;

90 [label="INSTANCE with matching:\nT44 -> T66\nX95 -> X128\nT48 -> T68", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 90 [arrowhead = none , style=dashed];
90 -> 34[style=dashed];

}

(2) Obligation:

Clauses:

appendA([], X59, X59).
appendA(.(T34, T35), X76, .(T34, X77)) :- appendA(T35, X76, X77).
find_all_vars2B([], []).
find_all_vars2B(.(=(T44, T45), T46), T48) :- appendC(T44, X95, T48).
find_all_vars2B(.(=(T44, T45), T46), T48) :- ','(appendC(T44, T51, T48), find_all_vars2B(T46, T51)).
appendC([], T58, T58).
appendC(.(T65, T66), X128, .(T65, T68)) :- appendC(T66, X128, T68).
find_all_vars2D([], []).
find_all_vars2D(.(=(T25, T26), T27), X41) :- appendA(T25, X40, X41).
find_all_vars2D(.(=(T25, T26), T27), T29) :- ','(appendA(T25, T28, T29), find_all_vars2B(T27, T28)).
init_varsE(T17, T10, T13, T14) :- find_all_vars2D(T17, X23).

Query: init_varsE(g,g,a,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
init_varsE_in: (b,b,f,f)
find_all_vars2D_in: (b,f)
appendA_in: (b,f,f)
find_all_vars2B_in: (b,f)
appendC_in: (b,f,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

init_varsE_in_ggaa(T17, T10, T13, T14) → U9_ggaa(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
find_all_vars2D_in_ga([], []) → find_all_vars2D_out_ga([], [])
find_all_vars2D_in_ga(.(=(T25, T26), T27), X41) → U6_ga(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
appendA_in_gaa([], X59, X59) → appendA_out_gaa([], X59, X59)
appendA_in_gaa(.(T34, T35), X76, .(T34, X77)) → U1_gaa(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
U1_gaa(T34, T35, X76, X77, appendA_out_gaa(T35, X76, X77)) → appendA_out_gaa(.(T34, T35), X76, .(T34, X77))
U6_ga(T25, T26, T27, X41, appendA_out_gaa(T25, X40, X41)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), X41)
find_all_vars2D_in_ga(.(=(T25, T26), T27), T29) → U7_ga(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_ga(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_ga(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
find_all_vars2B_in_ga([], []) → find_all_vars2B_out_ga([], [])
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U2_ga(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
appendC_in_gaa([], T58, T58) → appendC_out_gaa([], T58, T58)
appendC_in_gaa(.(T65, T66), X128, .(T65, T68)) → U5_gaa(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
U5_gaa(T65, T66, X128, T68, appendC_out_gaa(T66, X128, T68)) → appendC_out_gaa(.(T65, T66), X128, .(T65, T68))
U2_ga(T44, T45, T46, T48, appendC_out_gaa(T44, X95, T48)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U3_ga(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_ga(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_ga(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U4_ga(T44, T45, T46, T48, find_all_vars2B_out_ga(T46, T51)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
U8_ga(T25, T26, T27, T29, find_all_vars2B_out_ga(T27, T28)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), T29)
U9_ggaa(T17, T10, T13, T14, find_all_vars2D_out_ga(T17, X23)) → init_varsE_out_ggaa(T17, T10, T13, T14)

The argument filtering Pi contains the following mapping:
init_varsE_in_ggaa(x1, x2, x3, x4)  =  init_varsE_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x5)
find_all_vars2D_in_ga(x1, x2)  =  find_all_vars2D_in_ga(x1)
[]  =  []
find_all_vars2D_out_ga(x1, x2)  =  find_all_vars2D_out_ga
.(x1, x2)  =  .(x1, x2)
=(x1, x2)  =  =(x1, x2)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
appendA_out_gaa(x1, x2, x3)  =  appendA_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
find_all_vars2B_in_ga(x1, x2)  =  find_all_vars2B_in_ga(x1)
find_all_vars2B_out_ga(x1, x2)  =  find_all_vars2B_out_ga
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U5_gaa(x1, x2, x3, x4, x5)  =  U5_gaa(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
init_varsE_out_ggaa(x1, x2, x3, x4)  =  init_varsE_out_ggaa

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

init_varsE_in_ggaa(T17, T10, T13, T14) → U9_ggaa(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
find_all_vars2D_in_ga([], []) → find_all_vars2D_out_ga([], [])
find_all_vars2D_in_ga(.(=(T25, T26), T27), X41) → U6_ga(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
appendA_in_gaa([], X59, X59) → appendA_out_gaa([], X59, X59)
appendA_in_gaa(.(T34, T35), X76, .(T34, X77)) → U1_gaa(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
U1_gaa(T34, T35, X76, X77, appendA_out_gaa(T35, X76, X77)) → appendA_out_gaa(.(T34, T35), X76, .(T34, X77))
U6_ga(T25, T26, T27, X41, appendA_out_gaa(T25, X40, X41)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), X41)
find_all_vars2D_in_ga(.(=(T25, T26), T27), T29) → U7_ga(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_ga(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_ga(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
find_all_vars2B_in_ga([], []) → find_all_vars2B_out_ga([], [])
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U2_ga(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
appendC_in_gaa([], T58, T58) → appendC_out_gaa([], T58, T58)
appendC_in_gaa(.(T65, T66), X128, .(T65, T68)) → U5_gaa(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
U5_gaa(T65, T66, X128, T68, appendC_out_gaa(T66, X128, T68)) → appendC_out_gaa(.(T65, T66), X128, .(T65, T68))
U2_ga(T44, T45, T46, T48, appendC_out_gaa(T44, X95, T48)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U3_ga(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_ga(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_ga(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U4_ga(T44, T45, T46, T48, find_all_vars2B_out_ga(T46, T51)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
U8_ga(T25, T26, T27, T29, find_all_vars2B_out_ga(T27, T28)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), T29)
U9_ggaa(T17, T10, T13, T14, find_all_vars2D_out_ga(T17, X23)) → init_varsE_out_ggaa(T17, T10, T13, T14)

The argument filtering Pi contains the following mapping:
init_varsE_in_ggaa(x1, x2, x3, x4)  =  init_varsE_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x5)
find_all_vars2D_in_ga(x1, x2)  =  find_all_vars2D_in_ga(x1)
[]  =  []
find_all_vars2D_out_ga(x1, x2)  =  find_all_vars2D_out_ga
.(x1, x2)  =  .(x1, x2)
=(x1, x2)  =  =(x1, x2)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
appendA_out_gaa(x1, x2, x3)  =  appendA_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
find_all_vars2B_in_ga(x1, x2)  =  find_all_vars2B_in_ga(x1)
find_all_vars2B_out_ga(x1, x2)  =  find_all_vars2B_out_ga
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U5_gaa(x1, x2, x3, x4, x5)  =  U5_gaa(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
init_varsE_out_ggaa(x1, x2, x3, x4)  =  init_varsE_out_ggaa

(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:

INIT_VARSE_IN_GGAA(T17, T10, T13, T14) → U9_GGAA(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
INIT_VARSE_IN_GGAA(T17, T10, T13, T14) → FIND_ALL_VARS2D_IN_GA(T17, X23)
FIND_ALL_VARS2D_IN_GA(.(=(T25, T26), T27), X41) → U6_GA(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
FIND_ALL_VARS2D_IN_GA(.(=(T25, T26), T27), X41) → APPENDA_IN_GAA(T25, X40, X41)
APPENDA_IN_GAA(.(T34, T35), X76, .(T34, X77)) → U1_GAA(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
APPENDA_IN_GAA(.(T34, T35), X76, .(T34, X77)) → APPENDA_IN_GAA(T35, X76, X77)
FIND_ALL_VARS2D_IN_GA(.(=(T25, T26), T27), T29) → U7_GA(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_GA(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_GA(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
U7_GA(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → FIND_ALL_VARS2B_IN_GA(T27, T28)
FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46), T48) → U2_GA(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46), T48) → APPENDC_IN_GAA(T44, X95, T48)
APPENDC_IN_GAA(.(T65, T66), X128, .(T65, T68)) → U5_GAA(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
APPENDC_IN_GAA(.(T65, T66), X128, .(T65, T68)) → APPENDC_IN_GAA(T66, X128, T68)
FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46), T48) → U3_GA(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_GA(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_GA(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U3_GA(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → FIND_ALL_VARS2B_IN_GA(T46, T51)

The TRS R consists of the following rules:

init_varsE_in_ggaa(T17, T10, T13, T14) → U9_ggaa(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
find_all_vars2D_in_ga([], []) → find_all_vars2D_out_ga([], [])
find_all_vars2D_in_ga(.(=(T25, T26), T27), X41) → U6_ga(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
appendA_in_gaa([], X59, X59) → appendA_out_gaa([], X59, X59)
appendA_in_gaa(.(T34, T35), X76, .(T34, X77)) → U1_gaa(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
U1_gaa(T34, T35, X76, X77, appendA_out_gaa(T35, X76, X77)) → appendA_out_gaa(.(T34, T35), X76, .(T34, X77))
U6_ga(T25, T26, T27, X41, appendA_out_gaa(T25, X40, X41)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), X41)
find_all_vars2D_in_ga(.(=(T25, T26), T27), T29) → U7_ga(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_ga(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_ga(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
find_all_vars2B_in_ga([], []) → find_all_vars2B_out_ga([], [])
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U2_ga(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
appendC_in_gaa([], T58, T58) → appendC_out_gaa([], T58, T58)
appendC_in_gaa(.(T65, T66), X128, .(T65, T68)) → U5_gaa(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
U5_gaa(T65, T66, X128, T68, appendC_out_gaa(T66, X128, T68)) → appendC_out_gaa(.(T65, T66), X128, .(T65, T68))
U2_ga(T44, T45, T46, T48, appendC_out_gaa(T44, X95, T48)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U3_ga(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_ga(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_ga(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U4_ga(T44, T45, T46, T48, find_all_vars2B_out_ga(T46, T51)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
U8_ga(T25, T26, T27, T29, find_all_vars2B_out_ga(T27, T28)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), T29)
U9_ggaa(T17, T10, T13, T14, find_all_vars2D_out_ga(T17, X23)) → init_varsE_out_ggaa(T17, T10, T13, T14)

The argument filtering Pi contains the following mapping:
init_varsE_in_ggaa(x1, x2, x3, x4)  =  init_varsE_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x5)
find_all_vars2D_in_ga(x1, x2)  =  find_all_vars2D_in_ga(x1)
[]  =  []
find_all_vars2D_out_ga(x1, x2)  =  find_all_vars2D_out_ga
.(x1, x2)  =  .(x1, x2)
=(x1, x2)  =  =(x1, x2)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
appendA_out_gaa(x1, x2, x3)  =  appendA_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
find_all_vars2B_in_ga(x1, x2)  =  find_all_vars2B_in_ga(x1)
find_all_vars2B_out_ga(x1, x2)  =  find_all_vars2B_out_ga
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U5_gaa(x1, x2, x3, x4, x5)  =  U5_gaa(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
init_varsE_out_ggaa(x1, x2, x3, x4)  =  init_varsE_out_ggaa
INIT_VARSE_IN_GGAA(x1, x2, x3, x4)  =  INIT_VARSE_IN_GGAA(x1, x2)
U9_GGAA(x1, x2, x3, x4, x5)  =  U9_GGAA(x5)
FIND_ALL_VARS2D_IN_GA(x1, x2)  =  FIND_ALL_VARS2D_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x5)
APPENDA_IN_GAA(x1, x2, x3)  =  APPENDA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
FIND_ALL_VARS2B_IN_GA(x1, x2)  =  FIND_ALL_VARS2B_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5)  =  U5_GAA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

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

(6) Obligation:

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

INIT_VARSE_IN_GGAA(T17, T10, T13, T14) → U9_GGAA(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
INIT_VARSE_IN_GGAA(T17, T10, T13, T14) → FIND_ALL_VARS2D_IN_GA(T17, X23)
FIND_ALL_VARS2D_IN_GA(.(=(T25, T26), T27), X41) → U6_GA(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
FIND_ALL_VARS2D_IN_GA(.(=(T25, T26), T27), X41) → APPENDA_IN_GAA(T25, X40, X41)
APPENDA_IN_GAA(.(T34, T35), X76, .(T34, X77)) → U1_GAA(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
APPENDA_IN_GAA(.(T34, T35), X76, .(T34, X77)) → APPENDA_IN_GAA(T35, X76, X77)
FIND_ALL_VARS2D_IN_GA(.(=(T25, T26), T27), T29) → U7_GA(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_GA(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_GA(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
U7_GA(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → FIND_ALL_VARS2B_IN_GA(T27, T28)
FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46), T48) → U2_GA(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46), T48) → APPENDC_IN_GAA(T44, X95, T48)
APPENDC_IN_GAA(.(T65, T66), X128, .(T65, T68)) → U5_GAA(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
APPENDC_IN_GAA(.(T65, T66), X128, .(T65, T68)) → APPENDC_IN_GAA(T66, X128, T68)
FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46), T48) → U3_GA(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_GA(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_GA(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U3_GA(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → FIND_ALL_VARS2B_IN_GA(T46, T51)

The TRS R consists of the following rules:

init_varsE_in_ggaa(T17, T10, T13, T14) → U9_ggaa(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
find_all_vars2D_in_ga([], []) → find_all_vars2D_out_ga([], [])
find_all_vars2D_in_ga(.(=(T25, T26), T27), X41) → U6_ga(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
appendA_in_gaa([], X59, X59) → appendA_out_gaa([], X59, X59)
appendA_in_gaa(.(T34, T35), X76, .(T34, X77)) → U1_gaa(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
U1_gaa(T34, T35, X76, X77, appendA_out_gaa(T35, X76, X77)) → appendA_out_gaa(.(T34, T35), X76, .(T34, X77))
U6_ga(T25, T26, T27, X41, appendA_out_gaa(T25, X40, X41)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), X41)
find_all_vars2D_in_ga(.(=(T25, T26), T27), T29) → U7_ga(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_ga(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_ga(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
find_all_vars2B_in_ga([], []) → find_all_vars2B_out_ga([], [])
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U2_ga(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
appendC_in_gaa([], T58, T58) → appendC_out_gaa([], T58, T58)
appendC_in_gaa(.(T65, T66), X128, .(T65, T68)) → U5_gaa(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
U5_gaa(T65, T66, X128, T68, appendC_out_gaa(T66, X128, T68)) → appendC_out_gaa(.(T65, T66), X128, .(T65, T68))
U2_ga(T44, T45, T46, T48, appendC_out_gaa(T44, X95, T48)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U3_ga(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_ga(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_ga(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U4_ga(T44, T45, T46, T48, find_all_vars2B_out_ga(T46, T51)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
U8_ga(T25, T26, T27, T29, find_all_vars2B_out_ga(T27, T28)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), T29)
U9_ggaa(T17, T10, T13, T14, find_all_vars2D_out_ga(T17, X23)) → init_varsE_out_ggaa(T17, T10, T13, T14)

The argument filtering Pi contains the following mapping:
init_varsE_in_ggaa(x1, x2, x3, x4)  =  init_varsE_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x5)
find_all_vars2D_in_ga(x1, x2)  =  find_all_vars2D_in_ga(x1)
[]  =  []
find_all_vars2D_out_ga(x1, x2)  =  find_all_vars2D_out_ga
.(x1, x2)  =  .(x1, x2)
=(x1, x2)  =  =(x1, x2)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
appendA_out_gaa(x1, x2, x3)  =  appendA_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
find_all_vars2B_in_ga(x1, x2)  =  find_all_vars2B_in_ga(x1)
find_all_vars2B_out_ga(x1, x2)  =  find_all_vars2B_out_ga
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U5_gaa(x1, x2, x3, x4, x5)  =  U5_gaa(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
init_varsE_out_ggaa(x1, x2, x3, x4)  =  init_varsE_out_ggaa
INIT_VARSE_IN_GGAA(x1, x2, x3, x4)  =  INIT_VARSE_IN_GGAA(x1, x2)
U9_GGAA(x1, x2, x3, x4, x5)  =  U9_GGAA(x5)
FIND_ALL_VARS2D_IN_GA(x1, x2)  =  FIND_ALL_VARS2D_IN_GA(x1)
U6_GA(x1, x2, x3, x4, x5)  =  U6_GA(x5)
APPENDA_IN_GAA(x1, x2, x3)  =  APPENDA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x5)
U7_GA(x1, x2, x3, x4, x5)  =  U7_GA(x3, x5)
U8_GA(x1, x2, x3, x4, x5)  =  U8_GA(x5)
FIND_ALL_VARS2B_IN_GA(x1, x2)  =  FIND_ALL_VARS2B_IN_GA(x1)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x5)
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)
U5_GAA(x1, x2, x3, x4, x5)  =  U5_GAA(x5)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x3, x5)
U4_GA(x1, x2, x3, x4, x5)  =  U4_GA(x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 12 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APPENDC_IN_GAA(.(T65, T66), X128, .(T65, T68)) → APPENDC_IN_GAA(T66, X128, T68)

The TRS R consists of the following rules:

init_varsE_in_ggaa(T17, T10, T13, T14) → U9_ggaa(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
find_all_vars2D_in_ga([], []) → find_all_vars2D_out_ga([], [])
find_all_vars2D_in_ga(.(=(T25, T26), T27), X41) → U6_ga(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
appendA_in_gaa([], X59, X59) → appendA_out_gaa([], X59, X59)
appendA_in_gaa(.(T34, T35), X76, .(T34, X77)) → U1_gaa(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
U1_gaa(T34, T35, X76, X77, appendA_out_gaa(T35, X76, X77)) → appendA_out_gaa(.(T34, T35), X76, .(T34, X77))
U6_ga(T25, T26, T27, X41, appendA_out_gaa(T25, X40, X41)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), X41)
find_all_vars2D_in_ga(.(=(T25, T26), T27), T29) → U7_ga(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_ga(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_ga(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
find_all_vars2B_in_ga([], []) → find_all_vars2B_out_ga([], [])
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U2_ga(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
appendC_in_gaa([], T58, T58) → appendC_out_gaa([], T58, T58)
appendC_in_gaa(.(T65, T66), X128, .(T65, T68)) → U5_gaa(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
U5_gaa(T65, T66, X128, T68, appendC_out_gaa(T66, X128, T68)) → appendC_out_gaa(.(T65, T66), X128, .(T65, T68))
U2_ga(T44, T45, T46, T48, appendC_out_gaa(T44, X95, T48)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U3_ga(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_ga(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_ga(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U4_ga(T44, T45, T46, T48, find_all_vars2B_out_ga(T46, T51)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
U8_ga(T25, T26, T27, T29, find_all_vars2B_out_ga(T27, T28)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), T29)
U9_ggaa(T17, T10, T13, T14, find_all_vars2D_out_ga(T17, X23)) → init_varsE_out_ggaa(T17, T10, T13, T14)

The argument filtering Pi contains the following mapping:
init_varsE_in_ggaa(x1, x2, x3, x4)  =  init_varsE_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x5)
find_all_vars2D_in_ga(x1, x2)  =  find_all_vars2D_in_ga(x1)
[]  =  []
find_all_vars2D_out_ga(x1, x2)  =  find_all_vars2D_out_ga
.(x1, x2)  =  .(x1, x2)
=(x1, x2)  =  =(x1, x2)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
appendA_out_gaa(x1, x2, x3)  =  appendA_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
find_all_vars2B_in_ga(x1, x2)  =  find_all_vars2B_in_ga(x1)
find_all_vars2B_out_ga(x1, x2)  =  find_all_vars2B_out_ga
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U5_gaa(x1, x2, x3, x4, x5)  =  U5_gaa(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
init_varsE_out_ggaa(x1, x2, x3, x4)  =  init_varsE_out_ggaa
APPENDC_IN_GAA(x1, x2, x3)  =  APPENDC_IN_GAA(x1)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

APPENDC_IN_GAA(.(T65, T66), X128, .(T65, T68)) → APPENDC_IN_GAA(T66, X128, T68)

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

APPENDC_IN_GAA(.(T65, T66)) → APPENDC_IN_GAA(T66)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• APPENDC_IN_GAA(.(T65, T66)) → APPENDC_IN_GAA(T66)
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46), T48) → U3_GA(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_GA(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → FIND_ALL_VARS2B_IN_GA(T46, T51)

The TRS R consists of the following rules:

init_varsE_in_ggaa(T17, T10, T13, T14) → U9_ggaa(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
find_all_vars2D_in_ga([], []) → find_all_vars2D_out_ga([], [])
find_all_vars2D_in_ga(.(=(T25, T26), T27), X41) → U6_ga(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
appendA_in_gaa([], X59, X59) → appendA_out_gaa([], X59, X59)
appendA_in_gaa(.(T34, T35), X76, .(T34, X77)) → U1_gaa(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
U1_gaa(T34, T35, X76, X77, appendA_out_gaa(T35, X76, X77)) → appendA_out_gaa(.(T34, T35), X76, .(T34, X77))
U6_ga(T25, T26, T27, X41, appendA_out_gaa(T25, X40, X41)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), X41)
find_all_vars2D_in_ga(.(=(T25, T26), T27), T29) → U7_ga(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_ga(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_ga(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
find_all_vars2B_in_ga([], []) → find_all_vars2B_out_ga([], [])
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U2_ga(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
appendC_in_gaa([], T58, T58) → appendC_out_gaa([], T58, T58)
appendC_in_gaa(.(T65, T66), X128, .(T65, T68)) → U5_gaa(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
U5_gaa(T65, T66, X128, T68, appendC_out_gaa(T66, X128, T68)) → appendC_out_gaa(.(T65, T66), X128, .(T65, T68))
U2_ga(T44, T45, T46, T48, appendC_out_gaa(T44, X95, T48)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U3_ga(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_ga(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_ga(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U4_ga(T44, T45, T46, T48, find_all_vars2B_out_ga(T46, T51)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
U8_ga(T25, T26, T27, T29, find_all_vars2B_out_ga(T27, T28)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), T29)
U9_ggaa(T17, T10, T13, T14, find_all_vars2D_out_ga(T17, X23)) → init_varsE_out_ggaa(T17, T10, T13, T14)

The argument filtering Pi contains the following mapping:
init_varsE_in_ggaa(x1, x2, x3, x4)  =  init_varsE_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x5)
find_all_vars2D_in_ga(x1, x2)  =  find_all_vars2D_in_ga(x1)
[]  =  []
find_all_vars2D_out_ga(x1, x2)  =  find_all_vars2D_out_ga
.(x1, x2)  =  .(x1, x2)
=(x1, x2)  =  =(x1, x2)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
appendA_out_gaa(x1, x2, x3)  =  appendA_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
find_all_vars2B_in_ga(x1, x2)  =  find_all_vars2B_in_ga(x1)
find_all_vars2B_out_ga(x1, x2)  =  find_all_vars2B_out_ga
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U5_gaa(x1, x2, x3, x4, x5)  =  U5_gaa(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
init_varsE_out_ggaa(x1, x2, x3, x4)  =  init_varsE_out_ggaa
FIND_ALL_VARS2B_IN_GA(x1, x2)  =  FIND_ALL_VARS2B_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x3, x5)

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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46), T48) → U3_GA(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_GA(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → FIND_ALL_VARS2B_IN_GA(T46, T51)

The TRS R consists of the following rules:

appendC_in_gaa([], T58, T58) → appendC_out_gaa([], T58, T58)
appendC_in_gaa(.(T65, T66), X128, .(T65, T68)) → U5_gaa(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
U5_gaa(T65, T66, X128, T68, appendC_out_gaa(T66, X128, T68)) → appendC_out_gaa(.(T65, T66), X128, .(T65, T68))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
=(x1, x2)  =  =(x1, x2)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U5_gaa(x1, x2, x3, x4, x5)  =  U5_gaa(x5)
FIND_ALL_VARS2B_IN_GA(x1, x2)  =  FIND_ALL_VARS2B_IN_GA(x1)
U3_GA(x1, x2, x3, x4, x5)  =  U3_GA(x3, x5)

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:

FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46)) → U3_GA(T46, appendC_in_gaa(T44))
U3_GA(T46, appendC_out_gaa) → FIND_ALL_VARS2B_IN_GA(T46)

The TRS R consists of the following rules:

appendC_in_gaa([]) → appendC_out_gaa
appendC_in_gaa(.(T65, T66)) → U5_gaa(appendC_in_gaa(T66))
U5_gaa(appendC_out_gaa) → appendC_out_gaa

The set Q consists of the following terms:

appendC_in_gaa(x0)
U5_gaa(x0)

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

(21) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• U3_GA(T46, appendC_out_gaa) → FIND_ALL_VARS2B_IN_GA(T46)
  The graph contains the following edges 1 >= 1

• FIND_ALL_VARS2B_IN_GA(.(=(T44, T45), T46)) → U3_GA(T46, appendC_in_gaa(T44))
  The graph contains the following edges 1 > 1

(22) YES

(23) Obligation:

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

APPENDA_IN_GAA(.(T34, T35), X76, .(T34, X77)) → APPENDA_IN_GAA(T35, X76, X77)

The TRS R consists of the following rules:

init_varsE_in_ggaa(T17, T10, T13, T14) → U9_ggaa(T17, T10, T13, T14, find_all_vars2D_in_ga(T17, X23))
find_all_vars2D_in_ga([], []) → find_all_vars2D_out_ga([], [])
find_all_vars2D_in_ga(.(=(T25, T26), T27), X41) → U6_ga(T25, T26, T27, X41, appendA_in_gaa(T25, X40, X41))
appendA_in_gaa([], X59, X59) → appendA_out_gaa([], X59, X59)
appendA_in_gaa(.(T34, T35), X76, .(T34, X77)) → U1_gaa(T34, T35, X76, X77, appendA_in_gaa(T35, X76, X77))
U1_gaa(T34, T35, X76, X77, appendA_out_gaa(T35, X76, X77)) → appendA_out_gaa(.(T34, T35), X76, .(T34, X77))
U6_ga(T25, T26, T27, X41, appendA_out_gaa(T25, X40, X41)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), X41)
find_all_vars2D_in_ga(.(=(T25, T26), T27), T29) → U7_ga(T25, T26, T27, T29, appendA_in_gaa(T25, T28, T29))
U7_ga(T25, T26, T27, T29, appendA_out_gaa(T25, T28, T29)) → U8_ga(T25, T26, T27, T29, find_all_vars2B_in_ga(T27, T28))
find_all_vars2B_in_ga([], []) → find_all_vars2B_out_ga([], [])
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U2_ga(T44, T45, T46, T48, appendC_in_gaa(T44, X95, T48))
appendC_in_gaa([], T58, T58) → appendC_out_gaa([], T58, T58)
appendC_in_gaa(.(T65, T66), X128, .(T65, T68)) → U5_gaa(T65, T66, X128, T68, appendC_in_gaa(T66, X128, T68))
U5_gaa(T65, T66, X128, T68, appendC_out_gaa(T66, X128, T68)) → appendC_out_gaa(.(T65, T66), X128, .(T65, T68))
U2_ga(T44, T45, T46, T48, appendC_out_gaa(T44, X95, T48)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
find_all_vars2B_in_ga(.(=(T44, T45), T46), T48) → U3_ga(T44, T45, T46, T48, appendC_in_gaa(T44, T51, T48))
U3_ga(T44, T45, T46, T48, appendC_out_gaa(T44, T51, T48)) → U4_ga(T44, T45, T46, T48, find_all_vars2B_in_ga(T46, T51))
U4_ga(T44, T45, T46, T48, find_all_vars2B_out_ga(T46, T51)) → find_all_vars2B_out_ga(.(=(T44, T45), T46), T48)
U8_ga(T25, T26, T27, T29, find_all_vars2B_out_ga(T27, T28)) → find_all_vars2D_out_ga(.(=(T25, T26), T27), T29)
U9_ggaa(T17, T10, T13, T14, find_all_vars2D_out_ga(T17, X23)) → init_varsE_out_ggaa(T17, T10, T13, T14)

The argument filtering Pi contains the following mapping:
init_varsE_in_ggaa(x1, x2, x3, x4)  =  init_varsE_in_ggaa(x1, x2)
U9_ggaa(x1, x2, x3, x4, x5)  =  U9_ggaa(x5)
find_all_vars2D_in_ga(x1, x2)  =  find_all_vars2D_in_ga(x1)
[]  =  []
find_all_vars2D_out_ga(x1, x2)  =  find_all_vars2D_out_ga
.(x1, x2)  =  .(x1, x2)
=(x1, x2)  =  =(x1, x2)
U6_ga(x1, x2, x3, x4, x5)  =  U6_ga(x5)
appendA_in_gaa(x1, x2, x3)  =  appendA_in_gaa(x1)
appendA_out_gaa(x1, x2, x3)  =  appendA_out_gaa
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x5)
U7_ga(x1, x2, x3, x4, x5)  =  U7_ga(x3, x5)
U8_ga(x1, x2, x3, x4, x5)  =  U8_ga(x5)
find_all_vars2B_in_ga(x1, x2)  =  find_all_vars2B_in_ga(x1)
find_all_vars2B_out_ga(x1, x2)  =  find_all_vars2B_out_ga
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x5)
appendC_in_gaa(x1, x2, x3)  =  appendC_in_gaa(x1)
appendC_out_gaa(x1, x2, x3)  =  appendC_out_gaa
U5_gaa(x1, x2, x3, x4, x5)  =  U5_gaa(x5)
U3_ga(x1, x2, x3, x4, x5)  =  U3_ga(x3, x5)
U4_ga(x1, x2, x3, x4, x5)  =  U4_ga(x5)
init_varsE_out_ggaa(x1, x2, x3, x4)  =  init_varsE_out_ggaa
APPENDA_IN_GAA(x1, x2, x3)  =  APPENDA_IN_GAA(x1)

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

(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:

APPENDA_IN_GAA(.(T34, T35), X76, .(T34, X77)) → APPENDA_IN_GAA(T35, X76, X77)

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

APPENDA_IN_GAA(.(T34, T35)) → APPENDA_IN_GAA(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:

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

(29) YES