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

(0) Obligation:

Clauses:

app([], Y, Y).
app(X, Y, .(H, Z)) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), app(T, Y, Z)))).
head([], X1).
head(.(X, X2), X).
tail([], []).
tail(.(X3, Xs), Xs).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Query: app(g,a,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: app(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: app(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | app([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX6 is free\napp(T1, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14))))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 7 | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(','(call(empty([])), ','(!_2, failure(a))), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))) | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14))))) | ?_3 | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14))))), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(','(!_2, failure(a)), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))) | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 2, CLAUSE: 8\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(failure(a), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14))))\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(failure(a), ','(head([], T12), ','(tail([], X15), app(X15, T13, T14)))), SCOPE: 5, CLAUSE: 9\n\nX15 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(no(empty(T21)), ','(head(T21, T25), ','(tail(T21, X29), app(X29, T26, T27))))\n\nT21 is ground\nX6 is free\nX29 is free\napp(T21, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(no(empty(T21)), ','(head(T21, T25), ','(tail(T21, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 7 | ','(no(empty(T21)), ','(head(T21, T25), ','(tail(T21, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT21 is ground\nX6 is free\nX29 is free\napp(T21, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(','(call(empty(T30)), ','(!_6, failure(a))), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))) | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(empty(T30), ','(','(!_6, failure(a)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27))))) | ?_7 | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(empty(T30), ','(','(!_6, failure(a)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27))))), SCOPE: 8, CLAUSE: 6 | ?_8 | ?_7 | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: ','(','(!_6, failure(a)), ','(head([], T25), ','(tail([], X29), app(X29, T26, T27)))) | ?_8 | ?_7 | ','(no(empty([])), ','(head([], T25), ','(tail([], X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: ?_8 | ?_7 | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)\nempty(T30) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(failure(a), ','(head([], T25), ','(tail([], X29), app(X29, T26, T27))))\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(failure(a), ','(head([], T25), ','(tail([], X29), app(X29, T26, T27)))), SCOPE: 9, CLAUSE: 9\n\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: ?_7 | ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)\nempty(T30) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(no(empty(T30)), ','(head(T30, T25), ','(tail(T30, X29), app(X29, T26, T27)))), SCOPE: 6, CLAUSE: 8\n\nT30 is ground\nX6 is free\nX29 is free\napp(T30, T2, T3) !=? app([], X6, X6)\nempty(T30) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(head(T37, T25), ','(tail(T37, X29), app(X29, T26, T27)))\n\nT37 is ground\nX6 is free\nX29 is free\napp(T37, T2, T3) !=? app([], X6, X6)\nempty(T37) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(head(T37, T25), ','(tail(T37, X29), app(X29, T26, T27))), SCOPE: 10, CLAUSE: 2 | ','(head(T37, T25), ','(tail(T37, X29), app(X29, T26, T27))), SCOPE: 10, CLAUSE: 3\n\nT37 is ground\nX6 is free\nX29 is free\napp(T37, T2, T3) !=? app([], X6, X6)\nempty(T37) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: ','(head(T37, T25), ','(tail(T37, X29), app(X29, T26, T27))), SCOPE: 10, CLAUSE: 3\n\nT37 is ground\nX6 is free\nX29 is free\napp(T37, T2, T3) !=? app([], X6, X6)\nempty(T37) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: ','(tail(.(T45, T46), X29), app(X29, T47, T48))\n\nT45 is ground\nT46 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(tail(.(T45, T46), X29), app(X29, T47, T48)), SCOPE: 11, CLAUSE: 4 | ','(tail(.(T45, T46), X29), app(X29, T47, T48)), SCOPE: 11, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: ','(tail(.(T45, T46), X29), app(X29, T47, T48)), SCOPE: 11, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX29 is free", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: app(T58, T47, T48)\n\nT58 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 101 [arrowhead = none ];
101 -> 4;

102 [label="EVAL with clause\napp([], X6, X6).\nand substitutionT1 -> [],\nT2 -> T5,\nX6 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 102 [arrowhead = none ];
102 -> 10;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 103 [arrowhead = none ];
103 -> 16;

104 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 104 [arrowhead = none ];
104 -> 17;

105 [label="EVAL with clause\napp(X25, X26, .(X27, X28)) :- ','(no(empty(X25)), ','(head(X25, X27), ','(tail(X25, X29), app(X29, X26, X28)))).\nand substitutionT1 -> T21,\nX25 -> T21,\nT2 -> T26,\nX26 -> T26,\nX27 -> T25,\nX28 -> T27,\nT3 -> .(T25, T27),\nT23 -> T25,\nT22 -> T26,\nT24 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 105 [arrowhead = none ];
105 -> 51;

106 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 106 [arrowhead = none ];
106 -> 53;

107 [label="EVAL with clause\napp(X11, X12, .(X13, X14)) :- ','(no(empty(X11)), ','(head(X11, X13), ','(tail(X11, X15), app(X15, X12, X14)))).\nand substitutionX11 -> [],\nT2 -> T13,\nX12 -> T13,\nX13 -> T12,\nX14 -> T14,\nT3 -> .(T12, T14),\nT10 -> T12,\nT9 -> T13,\nT11 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 107 [arrowhead = none ];
107 -> 20;

108 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 108 [arrowhead = none ];
108 -> 22;

109 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 109 [arrowhead = none ];
109 -> 23;

110 [label="ONLY EVAL with clause\nno(X18) :- ','(call(X18), ','(!_2, failure(a))).\nand substitutionX18 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 110 [arrowhead = none ];
110 -> 27;

111 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 111 [arrowhead = none ];
111 -> 29;

112 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 112 [arrowhead = none ];
112 -> 33;

113 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 113 [arrowhead = none ];
113 -> 36;

114 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 114 [arrowhead = none ];
114 -> 38;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 115 [arrowhead = none ];
115 -> 46;

116 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 116 [arrowhead = none ];
116 -> 48;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 117 [arrowhead = none ];
117 -> 55;

118 [label="ONLY EVAL with clause\nno(X32) :- ','(call(X32), ','(!_6, failure(a))).\nand substitutionT21 -> T30,\nX32 -> empty(T30)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 118 [arrowhead = none ];
118 -> 58;

119 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
58 -> 119 [arrowhead = none ];
119 -> 61;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 120 [arrowhead = none ];
120 -> 64;

121 [label="EVAL with clause\nempty([]).\nand substitutionT30 -> []", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 121 [arrowhead = none ];
121 -> 66;

122 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 122 [arrowhead = none ];
122 -> 69;

123 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 123 [arrowhead = none ];
123 -> 71;

124 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 124 [arrowhead = none ];
124 -> 81;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
71 -> 125 [arrowhead = none ];
125 -> 75;

126 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 126 [arrowhead = none ];
126 -> 77;

127 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 127 [arrowhead = none ];
127 -> 83;

128 [label="ONLY EVAL with clause\nno(X39).\nand substitutionT30 -> T37,\nX39 -> empty(T37)", fontsize=14, style = filled, fillcolor = lightblue];
83 -> 128 [arrowhead = none ];
128 -> 85;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 129 [arrowhead = none ];
129 -> 88;

130 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T37), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 130 [arrowhead = none ];
130 -> 90;

131 [label="EVAL with clause\nhead(.(X47, X48), X47).\nand substitutionX47 -> T45,\nX48 -> T46,\nT37 -> .(T45, T46),\nT25 -> T45,\nT26 -> T47,\nT27 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
90 -> 131 [arrowhead = none ];
131 -> 93;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 132 [arrowhead = none ];
132 -> 94;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
93 -> 133 [arrowhead = none ];
133 -> 97;

134 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
97 -> 134 [arrowhead = none ];
134 -> 98;

135 [label="ONLY EVAL with clause\ntail(.(X57, X58), X58).\nand substitutionT45 -> T57,\nX57 -> T57,\nT46 -> T58,\nX58 -> T58,\nX29 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
98 -> 135 [arrowhead = none ];
135 -> 100;

136 [label="INSTANCE with matching:\nT1 -> T58\nT2 -> T47\nT3 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
100 -> 136 [arrowhead = none , style=dashed];
136 -> 2[style=dashed];

}

(2) Obligation:

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

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa(x1)
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x2, x5)

(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:

APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → U1_GAA(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa(x1)
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x2, x5)
APPA_IN_GAA(x1, x2, x3)  =  APPA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, x5)

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

(4) Obligation:

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

APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → U1_GAA(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa(x1)
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x2, x5)
APPA_IN_GAA(x1, x2, x3)  =  APPA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5)  =  U1_GAA(x1, x2, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T57, T58), T47, .(T57, T48)) → U1_gaa(T57, T58, T47, T48, appA_in_gaa(T58, T47, T48))
U1_gaa(T57, T58, T47, T48, appA_out_gaa(T58, T47, T48)) → appA_out_gaa(.(T57, T58), T47, .(T57, T48))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3)  =  appA_in_gaa(x1)
[]  =  []
appA_out_gaa(x1, x2, x3)  =  appA_out_gaa(x1)
.(x1, x2)  =  .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5)  =  U1_gaa(x1, x2, x5)
APPA_IN_GAA(x1, x2, x3)  =  APPA_IN_GAA(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:

APPA_IN_GAA(.(T57, T58), T47, .(T57, T48)) → APPA_IN_GAA(T58, T47, T48)

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

APPA_IN_GAA(.(T57, T58)) → APPA_IN_GAA(T58)

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:

• APPA_IN_GAA(.(T57, T58)) → APPA_IN_GAA(T58)
  The graph contains the following edges 1 > 1

(12) YES