Left Termination of the query pattern fold_in_3(g, g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

fold(X, [], Z) :- ','(!, eq(X, Z)).
fold(X, Y, Z) :- ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z)))).
myop(a, b, a).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
eq(X, X).

Queries:

fold(g,g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: fold(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T4, T6)) | fold(T4, [], T3), SCOPE: 1, CLAUSE: 1 | ?_1\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX6 is free; \nX7 is free; \nfold(T1, T2, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T4, T6)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T4, T6), SCOPE: 2, CLAUSE: 7\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(head(T9, X16), ','(tail(T9, X17), ','(myop(T8, X16, X18), fold(X18, X17, T11))))\n\nT8 is ground\nT9 is ground\nX6 is free; \nX7 is free; \nX16 is free; \nX17 is free; \nX18 is free; \nfold(T8, T9, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T9, X16), ','(tail(T9, X17), ','(myop(T8, X16, X18), fold(X18, X17, T11)))), SCOPE: 3, CLAUSE: 3 | ','(head(T9, X16), ','(tail(T9, X17), ','(myop(T8, X16, X18), fold(X18, X17, T11)))), SCOPE: 3, CLAUSE: 4\n\nT8 is ground\nT9 is ground\nX6 is free; \nX7 is free; \nX16 is free; \nX17 is free; \nX18 is free; \nfold(T8, T9, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(head(T9, X16), ','(tail(T9, X17), ','(myop(T8, X16, X18), fold(X18, X17, T11)))), SCOPE: 3, CLAUSE: 4\n\nT8 is ground\nT9 is ground\nX6 is free; \nX7 is free; \nX16 is free; \nX17 is free; \nX18 is free; \nX20 is free; \nfold(T8, T9, T3) !=? fold(X6, [], X7)\nhead(T9, X16) !=? head([], X20)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(tail(.(T12, T13), X17), ','(myop(T8, T12, X18), fold(X18, X17, T11)))\n\nT8 is ground\nT12 is ground\nT13 is ground\nX6 is free; \nX7 is free; \nX16 is free; \nX17 is free; \nX18 is free; \nX20 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\nX6 is free\nX7 is free; \nX16 is free; \nX20 is free; \nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(tail(.(T12, T13), X17), ','(myop(T8, T12, X18), fold(X18, X17, T11))), SCOPE: 4, CLAUSE: 5 | ','(tail(.(T12, T13), X17), ','(myop(T8, T12, X18), fold(X18, X17, T11))), SCOPE: 4, CLAUSE: 6\n\nT8 is ground\nT12 is ground\nT13 is ground\nX17 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(.(T12, T13), X17), ','(myop(T8, T12, X18), fold(X18, X17, T11))), SCOPE: 4, CLAUSE: 6\n\nT8 is ground\nT12 is ground\nT13 is ground\nX17 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(myop(T8, T14, X18), fold(X18, T15, T11))\n\nT8 is ground\nT14 is ground\nT15 is ground\nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(myop(T8, T14, X18), fold(X18, T15, T11)), SCOPE: 5, CLAUSE: 2\n\nT8 is ground\nT14 is ground\nT15 is ground\nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: fold(a, T15, T11)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: fold(a, T15, T11), SCOPE: 6, CLAUSE: 0 | fold(a, T15, T11), SCOPE: 6, CLAUSE: 1 | ?_6\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(!_6, eq(a, T17)) | fold(a, [], T11), SCOPE: 6, CLAUSE: 1 | ?_6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: fold(a, T15, T11), SCOPE: 6, CLAUSE: 1\n\nT15 is ground\nX31 is free; \nX32 is free; \nfold(a, T15, T11) !=? fold(X31, [], X32)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: eq(a, T17)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: eq(a, T17), SCOPE: 7, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\nX34 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T18, X41), ','(tail(T18, X42), ','(myop(a, X41, X43), fold(X43, X42, T20))))\n\nT18 is ground\nX31 is free; \nX32 is free; \nX41 is free; \nX42 is free; \nX43 is free; \nfold(a, T18, T11) !=? fold(X31, [], X32)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(head(T18, X41), ','(tail(T18, X42), ','(myop(a, X41, X43), fold(X43, X42, T20)))), SCOPE: 8, CLAUSE: 3 | ','(head(T18, X41), ','(tail(T18, X42), ','(myop(a, X41, X43), fold(X43, X42, T20)))), SCOPE: 8, CLAUSE: 4\n\nT18 is ground\nX31 is free; \nX32 is free; \nX41 is free; \nX42 is free; \nX43 is free; \nfold(a, T18, T11) !=? fold(X31, [], X32)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(head(T18, X41), ','(tail(T18, X42), ','(myop(a, X41, X43), fold(X43, X42, T20)))), SCOPE: 8, CLAUSE: 4\n\nT18 is ground\nX31 is free; \nX32 is free; \nX41 is free; \nX42 is free; \nX43 is free; \nX45 is free; \nfold(a, T18, T11) !=? fold(X31, [], X32)\nhead(T18, X41) !=? head([], X45)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(.(T21, T22), X42), ','(myop(a, T21, X43), fold(X43, X42, T20)))\n\nT21 is ground\nT22 is ground\nX31 is free; \nX32 is free; \nX41 is free; \nX42 is free; \nX43 is free; \nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\nX31 is free\nX32 is free; \nX41 is free; \nX45 is free; \nX48 is free; \nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tail(.(T21, T22), X42), ','(myop(a, T21, X43), fold(X43, X42, T20))), SCOPE: 9, CLAUSE: 5 | ','(tail(.(T21, T22), X42), ','(myop(a, T21, X43), fold(X43, X42, T20))), SCOPE: 9, CLAUSE: 6\n\nT21 is ground\nT22 is ground\nX42 is free; \nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(tail(.(T21, T22), X42), ','(myop(a, T21, X43), fold(X43, X42, T20))), SCOPE: 9, CLAUSE: 6\n\nT21 is ground\nT22 is ground\nX42 is free; \nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(myop(a, T23, X43), fold(X43, T24, T20))\n\nT23 is ground\nT24 is ground\nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(myop(a, T23, X43), fold(X43, T24, T20)), SCOPE: 10, CLAUSE: 2\n\nT23 is ground\nT24 is ground\nX43 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: fold(a, T24, T20)\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\nX43 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 40 [arrowhead = none ];
40 -> 2;

41 [label="EVAL with clause\nfold(X6, [], X7) :- ','(!, eq(X6, X7)).\nand substitution\nT1 -> T4\nX6 -> T4\nT2 -> []\nT3 -> T6\nX7 -> T6\nT5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 41 [arrowhead = none ];
41 -> 3;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 42 [arrowhead = none ];
42 -> 4;

43 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 43 [arrowhead = none ];
43 -> 5;

44 [label="EVAL with clause\nfold(X13, X14, X15) :- ','(head(X14, X16), ','(tail(X14, X17), ','(myop(X13, X16, X18), fold(X18, X17, X15)))).\nand substitution\nT1 -> T8\nX13 -> T8\nT2 -> T9\nX14 -> T9\nT3 -> T11\nX15 -> T11\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 44 [arrowhead = none ];
44 -> 10;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 45 [arrowhead = none ];
45 -> 6;

46 [label="EVAL with clause\neq(X9, X9).\nand substitution\nT4 -> T7\nX9 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 46 [arrowhead = none ];
46 -> 7;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 47 [arrowhead = none ];
47 -> 8;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 48 [arrowhead = none ];
48 -> 9;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 49 [arrowhead = none ];
49 -> 11;

50 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(T8, T9, T3), fold(X6, [], X7))", fontsize=14, style = filled, fillcolor = white];
11 -> 50 [arrowhead = none ];
50 -> 12;

51 [label="EVAL with clause\nhead(.(X23, X24), X23).\nand substitution\nX23 -> T12\nX24 -> T13\nT9 -> .(T12, T13)\nX16 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 51 [arrowhead = none ];
51 -> 13;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 52 [arrowhead = none ];
52 -> 14;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 53 [arrowhead = none ];
53 -> 15;

54 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
15 -> 54 [arrowhead = none ];
54 -> 16;

55 [label="EVAL with clause\ntail(.(X27, X28), X28).\nand substitution\nT12 -> T14\nX27 -> T14\nT13 -> T15\nX28 -> T15\nX17 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 55 [arrowhead = none ];
55 -> 17;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="EVAL with clause\nmyop(a, b, a).\nand substitution\nT8 -> a\nT14 -> b\nX18 -> a", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 57 [arrowhead = none ];
57 -> 19;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 58 [arrowhead = none ];
58 -> 20;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 59 [arrowhead = none ];
59 -> 21;

60 [label="EVAL with clause\nfold(X31, [], X32) :- ','(!, eq(X31, X32)).\nand substitution\nX31 -> a\nT15 -> []\nT11 -> T17\nX32 -> T17\nT16 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 60 [arrowhead = none ];
60 -> 22;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 61 [arrowhead = none ];
61 -> 23;

62 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 62 [arrowhead = none ];
62 -> 24;

63 [label="EVAL with clause\nfold(X38, X39, X40) :- ','(head(X39, X41), ','(tail(X39, X42), ','(myop(X38, X41, X43), fold(X43, X42, X40)))).\nand substitution\nX38 -> a\nT15 -> T18\nX39 -> T18\nT11 -> T20\nX40 -> T20\nT19 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 63 [arrowhead = none ];
63 -> 29;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 64 [arrowhead = none ];
64 -> 25;

65 [label="EVAL with clause\neq(X34, X34).\nand substitution\nX34 -> a\nT17 -> a", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 65 [arrowhead = none ];
65 -> 26;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 66 [arrowhead = none ];
66 -> 27;

67 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 67 [arrowhead = none ];
67 -> 28;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 68 [arrowhead = none ];
68 -> 30;

69 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(a, T18, T11), fold(X31, [], X32))", fontsize=14, style = filled, fillcolor = white];
30 -> 69 [arrowhead = none ];
69 -> 31;

70 [label="EVAL with clause\nhead(.(X48, X49), X48).\nand substitution\nX48 -> T21\nX49 -> T22\nT18 -> .(T21, T22)\nX41 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 70 [arrowhead = none ];
70 -> 32;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 71 [arrowhead = none ];
71 -> 33;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 72 [arrowhead = none ];
72 -> 34;

73 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = white];
34 -> 73 [arrowhead = none ];
73 -> 35;

74 [label="EVAL with clause\ntail(.(X52, X53), X53).\nand substitution\nT21 -> T23\nX52 -> T23\nT22 -> T24\nX53 -> T24\nX42 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 74 [arrowhead = none ];
74 -> 36;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 75 [arrowhead = none ];
75 -> 37;

76 [label="EVAL with clause\nmyop(a, b, a).\nand substitution\nT23 -> b\nX43 -> a", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 76 [arrowhead = none ];
76 -> 38;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 77 [arrowhead = none ];
77 -> 39;

78 [label="INSTANCE with matching:\nT15 -> T24\nT11 -> T20", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 78 [arrowhead = none , style=dashed];
78 -> 19[style=dashed];

}

(2) Obligation:

Clauses:

fold19([], a).
fold19(.(b, T24), T20) :- fold19(T24, T20).
fold1(T7, [], T7).
fold1(a, .(b, []), a).
fold1(a, .(b, .(b, T24)), T20) :- fold19(T24, T20).

Queries:

fold1(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
fold1_in: (b,b,f)
fold19_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), a) → fold1_out_gga(a, .(b, []), a)
fold1_in_gga(a, .(b, .(b, T24)), T20) → U2_gga(T24, T20, fold19_in_ga(T24, T20))
fold19_in_ga([], a) → fold19_out_ga([], a)
fold19_in_ga(.(b, T24), T20) → U1_ga(T24, T20, fold19_in_ga(T24, T20))
U1_ga(T24, T20, fold19_out_ga(T24, T20)) → fold19_out_ga(.(b, T24), T20)
U2_gga(T24, T20, fold19_out_ga(T24, T20)) → fold1_out_gga(a, .(b, .(b, T24)), T20)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_out_gga(x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U2_gga(x1, x2, x3)  =  U2_gga(x3)
fold19_in_ga(x1, x2)  =  fold19_in_ga(x1)
fold19_out_ga(x1, x2)  =  fold19_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), a) → fold1_out_gga(a, .(b, []), a)
fold1_in_gga(a, .(b, .(b, T24)), T20) → U2_gga(T24, T20, fold19_in_ga(T24, T20))
fold19_in_ga([], a) → fold19_out_ga([], a)
fold19_in_ga(.(b, T24), T20) → U1_ga(T24, T20, fold19_in_ga(T24, T20))
U1_ga(T24, T20, fold19_out_ga(T24, T20)) → fold19_out_ga(.(b, T24), T20)
U2_gga(T24, T20, fold19_out_ga(T24, T20)) → fold1_out_gga(a, .(b, .(b, T24)), T20)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_out_gga(x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U2_gga(x1, x2, x3)  =  U2_gga(x3)
fold19_in_ga(x1, x2)  =  fold19_in_ga(x1)
fold19_out_ga(x1, x2)  =  fold19_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)

(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:

FOLD1_IN_GGA(a, .(b, .(b, T24)), T20) → U2_GGA(T24, T20, fold19_in_ga(T24, T20))
FOLD1_IN_GGA(a, .(b, .(b, T24)), T20) → FOLD19_IN_GA(T24, T20)
FOLD19_IN_GA(.(b, T24), T20) → U1_GA(T24, T20, fold19_in_ga(T24, T20))
FOLD19_IN_GA(.(b, T24), T20) → FOLD19_IN_GA(T24, T20)

The TRS R consists of the following rules:

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), a) → fold1_out_gga(a, .(b, []), a)
fold1_in_gga(a, .(b, .(b, T24)), T20) → U2_gga(T24, T20, fold19_in_ga(T24, T20))
fold19_in_ga([], a) → fold19_out_ga([], a)
fold19_in_ga(.(b, T24), T20) → U1_ga(T24, T20, fold19_in_ga(T24, T20))
U1_ga(T24, T20, fold19_out_ga(T24, T20)) → fold19_out_ga(.(b, T24), T20)
U2_gga(T24, T20, fold19_out_ga(T24, T20)) → fold1_out_gga(a, .(b, .(b, T24)), T20)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_out_gga(x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U2_gga(x1, x2, x3)  =  U2_gga(x3)
fold19_in_ga(x1, x2)  =  fold19_in_ga(x1)
fold19_out_ga(x1, x2)  =  fold19_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
FOLD1_IN_GGA(x1, x2, x3)  =  FOLD1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
FOLD19_IN_GA(x1, x2)  =  FOLD19_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

(6) Obligation:

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

FOLD1_IN_GGA(a, .(b, .(b, T24)), T20) → U2_GGA(T24, T20, fold19_in_ga(T24, T20))
FOLD1_IN_GGA(a, .(b, .(b, T24)), T20) → FOLD19_IN_GA(T24, T20)
FOLD19_IN_GA(.(b, T24), T20) → U1_GA(T24, T20, fold19_in_ga(T24, T20))
FOLD19_IN_GA(.(b, T24), T20) → FOLD19_IN_GA(T24, T20)

The TRS R consists of the following rules:

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), a) → fold1_out_gga(a, .(b, []), a)
fold1_in_gga(a, .(b, .(b, T24)), T20) → U2_gga(T24, T20, fold19_in_ga(T24, T20))
fold19_in_ga([], a) → fold19_out_ga([], a)
fold19_in_ga(.(b, T24), T20) → U1_ga(T24, T20, fold19_in_ga(T24, T20))
U1_ga(T24, T20, fold19_out_ga(T24, T20)) → fold19_out_ga(.(b, T24), T20)
U2_gga(T24, T20, fold19_out_ga(T24, T20)) → fold1_out_gga(a, .(b, .(b, T24)), T20)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_out_gga(x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U2_gga(x1, x2, x3)  =  U2_gga(x3)
fold19_in_ga(x1, x2)  =  fold19_in_ga(x1)
fold19_out_ga(x1, x2)  =  fold19_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
FOLD1_IN_GGA(x1, x2, x3)  =  FOLD1_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3)  =  U2_GGA(x3)
FOLD19_IN_GA(x1, x2)  =  FOLD19_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

FOLD19_IN_GA(.(b, T24), T20) → FOLD19_IN_GA(T24, T20)

The TRS R consists of the following rules:

fold1_in_gga(T7, [], T7) → fold1_out_gga(T7, [], T7)
fold1_in_gga(a, .(b, []), a) → fold1_out_gga(a, .(b, []), a)
fold1_in_gga(a, .(b, .(b, T24)), T20) → U2_gga(T24, T20, fold19_in_ga(T24, T20))
fold19_in_ga([], a) → fold19_out_ga([], a)
fold19_in_ga(.(b, T24), T20) → U1_ga(T24, T20, fold19_in_ga(T24, T20))
U1_ga(T24, T20, fold19_out_ga(T24, T20)) → fold19_out_ga(.(b, T24), T20)
U2_gga(T24, T20, fold19_out_ga(T24, T20)) → fold1_out_gga(a, .(b, .(b, T24)), T20)

The argument filtering Pi contains the following mapping:
fold1_in_gga(x1, x2, x3)  =  fold1_in_gga(x1, x2)
[]  =  []
fold1_out_gga(x1, x2, x3)  =  fold1_out_gga(x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U2_gga(x1, x2, x3)  =  U2_gga(x3)
fold19_in_ga(x1, x2)  =  fold19_in_ga(x1)
fold19_out_ga(x1, x2)  =  fold19_out_ga(x2)
U1_ga(x1, x2, x3)  =  U1_ga(x3)
FOLD19_IN_GA(x1, x2)  =  FOLD19_IN_GA(x1)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

FOLD19_IN_GA(.(b, T24), T20) → FOLD19_IN_GA(T24, T20)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
b  =  b
FOLD19_IN_GA(x1, x2)  =  FOLD19_IN_GA(x1)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

FOLD19_IN_GA(.(b, T24)) → FOLD19_IN_GA(T24)

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

(13) 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:

• FOLD19_IN_GA(.(b, T24)) → FOLD19_IN_GA(T24)
  The graph contains the following edges 1 > 1

(14) TRUE