Left Termination of the query pattern
fold(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).

Query: fold(g,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: fold(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(!_1, eq(T6, T8)) | fold(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: 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];
14 [label="14: eq(T6, T8)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: eq(T6, T8), SCOPE: 2, CLAUSE: 7\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: 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];
28 [label="28: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18))))\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 3 | ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18)))\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 5 | ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(myop(T15, T31, X19), fold(X19, T32, T18))\n\nT15 is ground\nT31 is ground\nT32 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(myop(T15, T31, X19), fold(X19, T32, T18)), SCOPE: 5, CLAUSE: 2\n\nT15 is ground\nT31 is ground\nT32 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="B: fold(a, T32, T18)\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: fold(a, T32, T18), SCOPE: 6, CLAUSE: 0 | fold(a, T32, T18), SCOPE: 6, CLAUSE: 1\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(!_6, eq(a, T38)) | fold(a, [], T18), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: fold(a, T32, T18), SCOPE: 6, CLAUSE: 1\n\nT32 is ground\nX49 is free\nX50 is free\nfold(a, T32, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: eq(a, T38)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: eq(a, T38), SCOPE: 7, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47))))\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 3 | ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47)))\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 5 | ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 6\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 6\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: ','(myop(a, T61, X71), fold(X71, T62, T47))\n\nT61 is ground\nT62 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: ','(myop(a, T61, X71), fold(X71, T62, T47)), SCOPE: 10, CLAUSE: 2\n\nT61 is ground\nT62 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: fold(a, T62, T47)\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 103 [arrowhead = none ];
103 -> 6;

104 [label="EVAL with clause\nfold(X6, [], X7) :- ','(!_1, eq(X6, X7)).\nand substitutionT1 -> T6,\nX6 -> T6,\nT2 -> [],\nT3 -> T8,\nX7 -> T8,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 104 [arrowhead = none ];
104 -> 8;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 105 [arrowhead = none ];
105 -> 10;

106 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 106 [arrowhead = none ];
106 -> 14;

107 [label="ONLY EVAL with clause\nfold(X14, X15, X16) :- ','(head(X15, X17), ','(tail(X15, X18), ','(myop(X14, X17, X19), fold(X19, X18, X16)))).\nand substitutionT1 -> T15,\nX14 -> T15,\nT2 -> T16,\nX15 -> T16,\nT3 -> T18,\nX16 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 107 [arrowhead = none ];
107 -> 28;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 108 [arrowhead = none ];
108 -> 16;

109 [label="EVAL with clause\neq(X10, X10).\nand substitutionT6 -> T11,\nX10 -> T11,\nT8 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 109 [arrowhead = none ];
109 -> 19;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 110 [arrowhead = none ];
110 -> 22;

111 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 111 [arrowhead = none ];
111 -> 23;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 112 [arrowhead = none ];
112 -> 30;

113 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(T15, T16, T3), fold(X6, [], X7))", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 113 [arrowhead = none ];
113 -> 32;

114 [label="EVAL with clause\nhead(.(X26, X27), X26).\nand substitutionX26 -> T23,\nX27 -> T24,\nT16 -> .(T23, T24),\nX17 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 114 [arrowhead = none ];
114 -> 37;

115 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 115 [arrowhead = none ];
115 -> 38;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 116 [arrowhead = none ];
116 -> 39;

117 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 117 [arrowhead = none ];
117 -> 40;

118 [label="ONLY EVAL with clause\ntail(.(X38, X39), X39).\nand substitutionT23 -> T31,\nX38 -> T31,\nT24 -> T32,\nX39 -> T32,\nX18 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 118 [arrowhead = none ];
118 -> 45;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 119 [arrowhead = none ];
119 -> 46;

120 [label="EVAL with clause\nmyop(a, b, a).\nand substitutionT15 -> a,\nT31 -> b,\nX19 -> a", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 120 [arrowhead = none ];
120 -> 48;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 121 [arrowhead = none ];
121 -> 50;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
48 -> 122 [arrowhead = none ];
122 -> 60;

123 [label="EVAL with clause\nfold(X49, [], X50) :- ','(!_6, eq(X49, X50)).\nand substitutionX49 -> a,\nT32 -> [],\nT18 -> T38,\nX50 -> T38,\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 123 [arrowhead = none ];
123 -> 63;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 124 [arrowhead = none ];
124 -> 65;

125 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 125 [arrowhead = none ];
125 -> 66;

126 [label="ONLY EVAL with clause\nfold(X66, X67, X68) :- ','(head(X67, X69), ','(tail(X67, X70), ','(myop(X66, X69, X71), fold(X71, X70, X68)))).\nand substitutionX66 -> a,\nT32 -> T45,\nX67 -> T45,\nT18 -> T47,\nX68 -> T47,\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 126 [arrowhead = none ];
126 -> 85;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 127 [arrowhead = none ];
127 -> 68;

128 [label="EVAL with clause\neq(X53, X53).\nand substitutionX53 -> a,\nT38 -> a", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 128 [arrowhead = none ];
128 -> 72;

129 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 129 [arrowhead = none ];
129 -> 73;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
72 -> 130 [arrowhead = none ];
130 -> 74;

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 131 [arrowhead = none ];
131 -> 87;

132 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(a, T45, T18), fold(X49, [], X50))", fontsize=14, style = filled, fillcolor = lightgreen];
87 -> 132 [arrowhead = none ];
132 -> 88;

133 [label="EVAL with clause\nhead(.(X81, X82), X81).\nand substitutionX81 -> T53,\nX82 -> T54,\nT45 -> .(T53, T54),\nX69 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 133 [arrowhead = none ];
133 -> 91;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 134 [arrowhead = none ];
134 -> 92;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
91 -> 135 [arrowhead = none ];
135 -> 93;

136 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
93 -> 136 [arrowhead = none ];
136 -> 95;

137 [label="ONLY EVAL with clause\ntail(.(X93, X94), X94).\nand substitutionT53 -> T61,\nX93 -> T61,\nT54 -> T62,\nX94 -> T62,\nX70 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
95 -> 137 [arrowhead = none ];
137 -> 99;

138 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
99 -> 138 [arrowhead = none ];
138 -> 100;

139 [label="EVAL with clause\nmyop(a, b, a).\nand substitutionT61 -> b,\nX71 -> a", fontsize=14, style = filled, fillcolor = lightblue];
100 -> 139 [arrowhead = none ];
139 -> 101;

140 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 140 [arrowhead = none ];
140 -> 102;

141 [label="INSTANCE with matching:\nT32 -> T62\nT18 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
101 -> 141 [arrowhead = none , style=dashed];
141 -> 48[style=dashed];

}

(2) Obligation:

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

foldA_in_gga(T11, [], T11) → foldA_out_gga(T11, [], T11)
foldA_in_gga(a, .(b, T32), T18) → U1_gga(T32, T18, foldB_in_ga(T32, T18))
foldB_in_ga([], a) → foldB_out_ga([], a)
foldB_in_ga(.(b, T62), T47) → U2_ga(T62, T47, foldB_in_ga(T62, T47))
U2_ga(T62, T47, foldB_out_ga(T62, T47)) → foldB_out_ga(.(b, T62), T47)
U1_gga(T32, T18, foldB_out_ga(T32, T18)) → foldA_out_gga(a, .(b, T32), T18)

The argument filtering Pi contains the following mapping:
foldA_in_gga(x1, x2, x3)  =  foldA_in_gga(x1, x2)
[]  =  []
foldA_out_gga(x1, x2, x3)  =  foldA_out_gga(x1, x2, x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
foldB_in_ga(x1, x2)  =  foldB_in_ga(x1)
foldB_out_ga(x1, x2)  =  foldB_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)

(3) 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:

FOLDA_IN_GGA(a, .(b, T32), T18) → U1_GGA(T32, T18, foldB_in_ga(T32, T18))
FOLDA_IN_GGA(a, .(b, T32), T18) → FOLDB_IN_GA(T32, T18)
FOLDB_IN_GA(.(b, T62), T47) → U2_GA(T62, T47, foldB_in_ga(T62, T47))
FOLDB_IN_GA(.(b, T62), T47) → FOLDB_IN_GA(T62, T47)

The TRS R consists of the following rules:

foldA_in_gga(T11, [], T11) → foldA_out_gga(T11, [], T11)
foldA_in_gga(a, .(b, T32), T18) → U1_gga(T32, T18, foldB_in_ga(T32, T18))
foldB_in_ga([], a) → foldB_out_ga([], a)
foldB_in_ga(.(b, T62), T47) → U2_ga(T62, T47, foldB_in_ga(T62, T47))
U2_ga(T62, T47, foldB_out_ga(T62, T47)) → foldB_out_ga(.(b, T62), T47)
U1_gga(T32, T18, foldB_out_ga(T32, T18)) → foldA_out_gga(a, .(b, T32), T18)

The argument filtering Pi contains the following mapping:
foldA_in_gga(x1, x2, x3)  =  foldA_in_gga(x1, x2)
[]  =  []
foldA_out_gga(x1, x2, x3)  =  foldA_out_gga(x1, x2, x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
foldB_in_ga(x1, x2)  =  foldB_in_ga(x1)
foldB_out_ga(x1, x2)  =  foldB_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
FOLDA_IN_GGA(x1, x2, x3)  =  FOLDA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
FOLDB_IN_GA(x1, x2)  =  FOLDB_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(4) Obligation:

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

FOLDA_IN_GGA(a, .(b, T32), T18) → U1_GGA(T32, T18, foldB_in_ga(T32, T18))
FOLDA_IN_GGA(a, .(b, T32), T18) → FOLDB_IN_GA(T32, T18)
FOLDB_IN_GA(.(b, T62), T47) → U2_GA(T62, T47, foldB_in_ga(T62, T47))
FOLDB_IN_GA(.(b, T62), T47) → FOLDB_IN_GA(T62, T47)

The TRS R consists of the following rules:

foldA_in_gga(T11, [], T11) → foldA_out_gga(T11, [], T11)
foldA_in_gga(a, .(b, T32), T18) → U1_gga(T32, T18, foldB_in_ga(T32, T18))
foldB_in_ga([], a) → foldB_out_ga([], a)
foldB_in_ga(.(b, T62), T47) → U2_ga(T62, T47, foldB_in_ga(T62, T47))
U2_ga(T62, T47, foldB_out_ga(T62, T47)) → foldB_out_ga(.(b, T62), T47)
U1_gga(T32, T18, foldB_out_ga(T32, T18)) → foldA_out_gga(a, .(b, T32), T18)

The argument filtering Pi contains the following mapping:
foldA_in_gga(x1, x2, x3)  =  foldA_in_gga(x1, x2)
[]  =  []
foldA_out_gga(x1, x2, x3)  =  foldA_out_gga(x1, x2, x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
foldB_in_ga(x1, x2)  =  foldB_in_ga(x1)
foldB_out_ga(x1, x2)  =  foldB_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
FOLDA_IN_GGA(x1, x2, x3)  =  FOLDA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3)  =  U1_GGA(x1, x3)
FOLDB_IN_GA(x1, x2)  =  FOLDB_IN_GA(x1)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

FOLDB_IN_GA(.(b, T62), T47) → FOLDB_IN_GA(T62, T47)

The TRS R consists of the following rules:

foldA_in_gga(T11, [], T11) → foldA_out_gga(T11, [], T11)
foldA_in_gga(a, .(b, T32), T18) → U1_gga(T32, T18, foldB_in_ga(T32, T18))
foldB_in_ga([], a) → foldB_out_ga([], a)
foldB_in_ga(.(b, T62), T47) → U2_ga(T62, T47, foldB_in_ga(T62, T47))
U2_ga(T62, T47, foldB_out_ga(T62, T47)) → foldB_out_ga(.(b, T62), T47)
U1_gga(T32, T18, foldB_out_ga(T32, T18)) → foldA_out_gga(a, .(b, T32), T18)

The argument filtering Pi contains the following mapping:
foldA_in_gga(x1, x2, x3)  =  foldA_in_gga(x1, x2)
[]  =  []
foldA_out_gga(x1, x2, x3)  =  foldA_out_gga(x1, x2, x3)
a  =  a
.(x1, x2)  =  .(x1, x2)
b  =  b
U1_gga(x1, x2, x3)  =  U1_gga(x1, x3)
foldB_in_ga(x1, x2)  =  foldB_in_ga(x1)
foldB_out_ga(x1, x2)  =  foldB_out_ga(x1, x2)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
FOLDB_IN_GA(x1, x2)  =  FOLDB_IN_GA(x1)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

FOLDB_IN_GA(.(b, T62), T47) → FOLDB_IN_GA(T62, T47)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

FOLDB_IN_GA(.(b, T62)) → FOLDB_IN_GA(T62)

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

(11) 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:

• FOLDB_IN_GA(.(b, T62)) → FOLDB_IN_GA(T62)
  The graph contains the following edges 1 > 1

(12) YES