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

(0) Obligation:

Clauses:

div(X1, 0, X2, X3) :- ','(!, failure(a)).
div(0, X4, Z, R) :- ','(!, ','(eq(Z, 0), eq(R, 0))).
div(X, Y, s(Z), R) :- ','(minus(X, Y, U), ','(!, div(U, Y, Z, R))).
div(X, X5, X6, X).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
failure(b).
eq(X, X).

Query: div(g,g,a,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: div(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(!_1, failure(a)) | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX10 is free\nX11 is free\nX12 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: failure(a), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(!_1, ','(eq(T17, 0), eq(T18, 0))) | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 2 | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT14 is ground\nX10 is free\nX11 is free\nX12 is free\ndiv(0, T14, T3, T4) !=? div(X10, 0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(eq(T17, 0), eq(T18, 0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(eq(T17, 0), eq(T18, 0)), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: eq(T22, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: eq(T22, 0), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))) | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)\ndiv(T1, T2, T3, T4) !=? div(X29, X30, s(X31), X32)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 4 | ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5 | ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5 | ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="B: ','(minus(T45, T46, X54), ','(!_1, div(X54, s(T46), T34, T35)))\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="C: minus(T45, T46, X54)\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="D: ','(!_1, div(T49, s(T46), T34, T35))\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: minus(T45, T46, X54), SCOPE: 6, CLAUSE: 4 | minus(T45, T46, X54), SCOPE: 6, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: minus(T45, T46, X54), SCOPE: 6, CLAUSE: 4\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: minus(T45, T46, X54), SCOPE: 6, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: minus(T61, T62, X75)\n\nT61 is ground\nT62 is ground\nX75 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: div(T49, s(T46), T34, T35)\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 111 [arrowhead = none ];
111 -> 6;

112 [label="EVAL with clause\ndiv(X10, 0, X11, X12) :- ','(!_1, failure(a)).\nand substitutionT1 -> T8,\nX10 -> T8,\nT2 -> 0,\nT3 -> T9,\nX11 -> T9,\nT4 -> T10,\nX12 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 112 [arrowhead = none ];
112 -> 8;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 113 [arrowhead = none ];
113 -> 10;

114 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 114 [arrowhead = none ];
114 -> 13;

115 [label="EVAL with clause\ndiv(0, X16, X17, X18) :- ','(!_1, ','(eq(X17, 0), eq(X18, 0))).\nand substitutionT1 -> 0,\nT2 -> T14,\nX16 -> T14,\nT3 -> T17,\nX17 -> T17,\nT4 -> T18,\nX18 -> T18,\nT15 -> T17,\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 115 [arrowhead = none ];
115 -> 22;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 116 [arrowhead = none ];
116 -> 24;

117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 117 [arrowhead = none ];
117 -> 17;

118 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 118 [arrowhead = none ];
118 -> 20;

119 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 119 [arrowhead = none ];
119 -> 26;

120 [label="EVAL with clause\ndiv(X29, X30, s(X31), X32) :- ','(minus(X29, X30, X33), ','(!_1, div(X33, X30, X31, X32))).\nand substitutionT1 -> T30,\nX29 -> T30,\nT2 -> T31,\nX30 -> T31,\nX31 -> T34,\nT3 -> s(T34),\nT4 -> T35,\nX32 -> T35,\nT32 -> T34,\nT33 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 120 [arrowhead = none ];
120 -> 52;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 121 [arrowhead = none ];
121 -> 54;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 122 [arrowhead = none ];
122 -> 28;

123 [label="EVAL with clause\neq(X21, X21).\nand substitutionT17 -> 0,\nX21 -> 0,\nT21 -> 0,\nT18 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 123 [arrowhead = none ];
123 -> 31;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 124 [arrowhead = none ];
124 -> 32;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 125 [arrowhead = none ];
125 -> 34;

126 [label="EVAL with clause\neq(X24, X24).\nand substitutionT22 -> 0,\nX24 -> 0,\nT25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 126 [arrowhead = none ];
126 -> 39;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 127 [arrowhead = none ];
127 -> 41;

128 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 128 [arrowhead = none ];
128 -> 42;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 129 [arrowhead = none ];
129 -> 56;

130 [label="EVAL with clause\ndiv(X93, X94, X95, X93).\nand substitutionT1 -> T78,\nX93 -> T78,\nT2 -> T79,\nX94 -> T79,\nT3 -> T80,\nX95 -> T80,\nT4 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 130 [arrowhead = none ];
130 -> 105;

131 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 131 [arrowhead = none ];
131 -> 109;

132 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T30, T31, T3, T4), div(X10, 0, X11, X12))", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 132 [arrowhead = none ];
132 -> 60;

133 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 133 [arrowhead = none ];
133 -> 62;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
60 -> 134 [arrowhead = none ];
134 -> 64;

135 [label="EVAL with clause\nminus(s(X51), s(X52), X53) :- minus(X51, X52, X53).\nand substitutionX51 -> T45,\nT30 -> s(T45),\nX52 -> T46,\nT31 -> s(T46),\nX33 -> X54,\nX53 -> X54", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 135 [arrowhead = none ];
135 -> 68;

136 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
62 -> 136 [arrowhead = none ];
136 -> 70;

137 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 137 [arrowhead = none ];
137 -> 97;

138 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
68 -> 138 [arrowhead = none ];
138 -> 72;

139 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nT46 is ground\nT49 is ground\nreplacements:X54 -> T49", fontsize=14, style = filled, fillcolor = violet];
68 -> 139 [arrowhead = none ];
139 -> 73;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
72 -> 140 [arrowhead = none ];
140 -> 76;

141 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
73 -> 141 [arrowhead = none ];
141 -> 94;

142 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 142 [arrowhead = none ];
142 -> 79;

143 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
76 -> 143 [arrowhead = none ];
143 -> 80;

144 [label="EVAL with clause\nminus(X63, 0, X63).\nand substitutionT45 -> T56,\nX63 -> T56,\nT46 -> 0,\nX54 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 144 [arrowhead = none ];
144 -> 82;

145 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 145 [arrowhead = none ];
145 -> 83;

146 [label="EVAL with clause\nminus(s(X72), s(X73), X74) :- minus(X72, X73, X74).\nand substitutionX72 -> T61,\nT45 -> s(T61),\nX73 -> T62,\nT46 -> s(T62),\nX54 -> X75,\nX74 -> X75", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 146 [arrowhead = none ];
146 -> 90;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 147 [arrowhead = none ];
147 -> 91;

148 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 148 [arrowhead = none ];
148 -> 85;

149 [label="INSTANCE with matching:\nT45 -> T61\nT46 -> T62\nX54 -> X75", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 149 [arrowhead = none , style=dashed];
149 -> 72[style=dashed];

150 [label="INSTANCE with matching:\nT1 -> T49\nT2 -> s(T46)\nT3 -> T34\nT4 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
94 -> 150 [arrowhead = none , style=dashed];
150 -> 1[style=dashed];

151 [label="EVAL with clause\ndiv(X87, X88, X89, X87).\nand substitutionT30 -> T72,\nX87 -> T72,\nT31 -> T73,\nX88 -> T73,\nT3 -> T74,\nX89 -> T74,\nT4 -> T72", fontsize=14, style = filled, fillcolor = lightblue];
97 -> 151 [arrowhead = none ];
151 -> 100;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
97 -> 152 [arrowhead = none ];
152 -> 101;

153 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
100 -> 153 [arrowhead = none ];
153 -> 102;

154 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
105 -> 154 [arrowhead = none ];
154 -> 110;

}

(2) Obligation:

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

divA_in_ggaa(0, T14, 0, 0) → divA_out_ggaa(0, T14, 0, 0)
divA_in_ggaa(s(T45), s(T46), s(T34), T35) → U1_ggaa(T45, T46, T34, T35, pB_in_ggaaa(T45, T46, X54, T34, T35))
pB_in_ggaaa(T45, T46, T49, T34, T35) → U4_ggaaa(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
minusC_in_gga(T56, 0, T56) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62), X75) → U2_gga(T61, T62, X75, minusC_in_gga(T61, T62, X75))
U2_gga(T61, T62, X75, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)
U4_ggaaa(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → U5_ggaaa(T45, T46, T49, T34, T35, pD_in_ggaa(T49, T46, T34, T35))
pD_in_ggaa(T49, T46, T34, T35) → U3_ggaa(T49, T46, T34, T35, divA_in_ggaa(T49, s(T46), T34, T35))
divA_in_ggaa(T72, T73, T74, T72) → divA_out_ggaa(T72, T73, T74, T72)
U3_ggaa(T49, T46, T34, T35, divA_out_ggaa(T49, s(T46), T34, T35)) → pD_out_ggaa(T49, T46, T34, T35)
U5_ggaaa(T45, T46, T49, T34, T35, pD_out_ggaa(T49, T46, T34, T35)) → pB_out_ggaaa(T45, T46, T49, T34, T35)
U1_ggaa(T45, T46, T34, T35, pB_out_ggaaa(T45, T46, X54, T34, T35)) → divA_out_ggaa(s(T45), s(T46), s(T34), T35)

The argument filtering Pi contains the following mapping:
divA_in_ggaa(x1, x2, x3, x4)  =  divA_in_ggaa(x1, x2)
0  =  0
divA_out_ggaa(x1, x2, x3, x4)  =  divA_out_ggaa(x1, x2, x4)
s(x1)  =  s(x1)
U1_ggaa(x1, x2, x3, x4, x5)  =  U1_ggaa(x1, x2, x5)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U4_ggaaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaaa(x1, x2, x6)
minusC_in_gga(x1, x2, x3)  =  minusC_in_gga(x1, x2)
minusC_out_gga(x1, x2, x3)  =  minusC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x3, x6)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, 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:

DIVA_IN_GGAA(s(T45), s(T46), s(T34), T35) → U1_GGAA(T45, T46, T34, T35, pB_in_ggaaa(T45, T46, X54, T34, T35))
DIVA_IN_GGAA(s(T45), s(T46), s(T34), T35) → PB_IN_GGAAA(T45, T46, X54, T34, T35)
PB_IN_GGAAA(T45, T46, T49, T34, T35) → U4_GGAAA(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
PB_IN_GGAAA(T45, T46, T49, T34, T35) → MINUSC_IN_GGA(T45, T46, T49)
MINUSC_IN_GGA(s(T61), s(T62), X75) → U2_GGA(T61, T62, X75, minusC_in_gga(T61, T62, X75))
MINUSC_IN_GGA(s(T61), s(T62), X75) → MINUSC_IN_GGA(T61, T62, X75)
U4_GGAAA(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → U5_GGAAA(T45, T46, T49, T34, T35, pD_in_ggaa(T49, T46, T34, T35))
U4_GGAAA(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → PD_IN_GGAA(T49, T46, T34, T35)
PD_IN_GGAA(T49, T46, T34, T35) → U3_GGAA(T49, T46, T34, T35, divA_in_ggaa(T49, s(T46), T34, T35))
PD_IN_GGAA(T49, T46, T34, T35) → DIVA_IN_GGAA(T49, s(T46), T34, T35)

The TRS R consists of the following rules:

divA_in_ggaa(0, T14, 0, 0) → divA_out_ggaa(0, T14, 0, 0)
divA_in_ggaa(s(T45), s(T46), s(T34), T35) → U1_ggaa(T45, T46, T34, T35, pB_in_ggaaa(T45, T46, X54, T34, T35))
pB_in_ggaaa(T45, T46, T49, T34, T35) → U4_ggaaa(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
minusC_in_gga(T56, 0, T56) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62), X75) → U2_gga(T61, T62, X75, minusC_in_gga(T61, T62, X75))
U2_gga(T61, T62, X75, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)
U4_ggaaa(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → U5_ggaaa(T45, T46, T49, T34, T35, pD_in_ggaa(T49, T46, T34, T35))
pD_in_ggaa(T49, T46, T34, T35) → U3_ggaa(T49, T46, T34, T35, divA_in_ggaa(T49, s(T46), T34, T35))
divA_in_ggaa(T72, T73, T74, T72) → divA_out_ggaa(T72, T73, T74, T72)
U3_ggaa(T49, T46, T34, T35, divA_out_ggaa(T49, s(T46), T34, T35)) → pD_out_ggaa(T49, T46, T34, T35)
U5_ggaaa(T45, T46, T49, T34, T35, pD_out_ggaa(T49, T46, T34, T35)) → pB_out_ggaaa(T45, T46, T49, T34, T35)
U1_ggaa(T45, T46, T34, T35, pB_out_ggaaa(T45, T46, X54, T34, T35)) → divA_out_ggaa(s(T45), s(T46), s(T34), T35)

The argument filtering Pi contains the following mapping:
divA_in_ggaa(x1, x2, x3, x4)  =  divA_in_ggaa(x1, x2)
0  =  0
divA_out_ggaa(x1, x2, x3, x4)  =  divA_out_ggaa(x1, x2, x4)
s(x1)  =  s(x1)
U1_ggaa(x1, x2, x3, x4, x5)  =  U1_ggaa(x1, x2, x5)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U4_ggaaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaaa(x1, x2, x6)
minusC_in_gga(x1, x2, x3)  =  minusC_in_gga(x1, x2)
minusC_out_gga(x1, x2, x3)  =  minusC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x3, x6)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x5)
DIVA_IN_GGAA(x1, x2, x3, x4)  =  DIVA_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5)  =  U1_GGAA(x1, x2, x5)
PB_IN_GGAAA(x1, x2, x3, x4, x5)  =  PB_IN_GGAAA(x1, x2)
U4_GGAAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAA(x1, x2, x6)
MINUSC_IN_GGA(x1, x2, x3)  =  MINUSC_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U5_GGAAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAAA(x1, x2, x3, x6)
PD_IN_GGAA(x1, x2, x3, x4)  =  PD_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5)  =  U3_GGAA(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:

DIVA_IN_GGAA(s(T45), s(T46), s(T34), T35) → U1_GGAA(T45, T46, T34, T35, pB_in_ggaaa(T45, T46, X54, T34, T35))
DIVA_IN_GGAA(s(T45), s(T46), s(T34), T35) → PB_IN_GGAAA(T45, T46, X54, T34, T35)
PB_IN_GGAAA(T45, T46, T49, T34, T35) → U4_GGAAA(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
PB_IN_GGAAA(T45, T46, T49, T34, T35) → MINUSC_IN_GGA(T45, T46, T49)
MINUSC_IN_GGA(s(T61), s(T62), X75) → U2_GGA(T61, T62, X75, minusC_in_gga(T61, T62, X75))
MINUSC_IN_GGA(s(T61), s(T62), X75) → MINUSC_IN_GGA(T61, T62, X75)
U4_GGAAA(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → U5_GGAAA(T45, T46, T49, T34, T35, pD_in_ggaa(T49, T46, T34, T35))
U4_GGAAA(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → PD_IN_GGAA(T49, T46, T34, T35)
PD_IN_GGAA(T49, T46, T34, T35) → U3_GGAA(T49, T46, T34, T35, divA_in_ggaa(T49, s(T46), T34, T35))
PD_IN_GGAA(T49, T46, T34, T35) → DIVA_IN_GGAA(T49, s(T46), T34, T35)

The TRS R consists of the following rules:

divA_in_ggaa(0, T14, 0, 0) → divA_out_ggaa(0, T14, 0, 0)
divA_in_ggaa(s(T45), s(T46), s(T34), T35) → U1_ggaa(T45, T46, T34, T35, pB_in_ggaaa(T45, T46, X54, T34, T35))
pB_in_ggaaa(T45, T46, T49, T34, T35) → U4_ggaaa(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
minusC_in_gga(T56, 0, T56) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62), X75) → U2_gga(T61, T62, X75, minusC_in_gga(T61, T62, X75))
U2_gga(T61, T62, X75, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)
U4_ggaaa(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → U5_ggaaa(T45, T46, T49, T34, T35, pD_in_ggaa(T49, T46, T34, T35))
pD_in_ggaa(T49, T46, T34, T35) → U3_ggaa(T49, T46, T34, T35, divA_in_ggaa(T49, s(T46), T34, T35))
divA_in_ggaa(T72, T73, T74, T72) → divA_out_ggaa(T72, T73, T74, T72)
U3_ggaa(T49, T46, T34, T35, divA_out_ggaa(T49, s(T46), T34, T35)) → pD_out_ggaa(T49, T46, T34, T35)
U5_ggaaa(T45, T46, T49, T34, T35, pD_out_ggaa(T49, T46, T34, T35)) → pB_out_ggaaa(T45, T46, T49, T34, T35)
U1_ggaa(T45, T46, T34, T35, pB_out_ggaaa(T45, T46, X54, T34, T35)) → divA_out_ggaa(s(T45), s(T46), s(T34), T35)

The argument filtering Pi contains the following mapping:
divA_in_ggaa(x1, x2, x3, x4)  =  divA_in_ggaa(x1, x2)
0  =  0
divA_out_ggaa(x1, x2, x3, x4)  =  divA_out_ggaa(x1, x2, x4)
s(x1)  =  s(x1)
U1_ggaa(x1, x2, x3, x4, x5)  =  U1_ggaa(x1, x2, x5)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U4_ggaaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaaa(x1, x2, x6)
minusC_in_gga(x1, x2, x3)  =  minusC_in_gga(x1, x2)
minusC_out_gga(x1, x2, x3)  =  minusC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x3, x6)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x5)
DIVA_IN_GGAA(x1, x2, x3, x4)  =  DIVA_IN_GGAA(x1, x2)
U1_GGAA(x1, x2, x3, x4, x5)  =  U1_GGAA(x1, x2, x5)
PB_IN_GGAAA(x1, x2, x3, x4, x5)  =  PB_IN_GGAAA(x1, x2)
U4_GGAAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAA(x1, x2, x6)
MINUSC_IN_GGA(x1, x2, x3)  =  MINUSC_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U5_GGAAA(x1, x2, x3, x4, x5, x6)  =  U5_GGAAA(x1, x2, x3, x6)
PD_IN_GGAA(x1, x2, x3, x4)  =  PD_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5)  =  U3_GGAA(x1, x2, x5)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 5 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MINUSC_IN_GGA(s(T61), s(T62), X75) → MINUSC_IN_GGA(T61, T62, X75)

The TRS R consists of the following rules:

divA_in_ggaa(0, T14, 0, 0) → divA_out_ggaa(0, T14, 0, 0)
divA_in_ggaa(s(T45), s(T46), s(T34), T35) → U1_ggaa(T45, T46, T34, T35, pB_in_ggaaa(T45, T46, X54, T34, T35))
pB_in_ggaaa(T45, T46, T49, T34, T35) → U4_ggaaa(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
minusC_in_gga(T56, 0, T56) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62), X75) → U2_gga(T61, T62, X75, minusC_in_gga(T61, T62, X75))
U2_gga(T61, T62, X75, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)
U4_ggaaa(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → U5_ggaaa(T45, T46, T49, T34, T35, pD_in_ggaa(T49, T46, T34, T35))
pD_in_ggaa(T49, T46, T34, T35) → U3_ggaa(T49, T46, T34, T35, divA_in_ggaa(T49, s(T46), T34, T35))
divA_in_ggaa(T72, T73, T74, T72) → divA_out_ggaa(T72, T73, T74, T72)
U3_ggaa(T49, T46, T34, T35, divA_out_ggaa(T49, s(T46), T34, T35)) → pD_out_ggaa(T49, T46, T34, T35)
U5_ggaaa(T45, T46, T49, T34, T35, pD_out_ggaa(T49, T46, T34, T35)) → pB_out_ggaaa(T45, T46, T49, T34, T35)
U1_ggaa(T45, T46, T34, T35, pB_out_ggaaa(T45, T46, X54, T34, T35)) → divA_out_ggaa(s(T45), s(T46), s(T34), T35)

The argument filtering Pi contains the following mapping:
divA_in_ggaa(x1, x2, x3, x4)  =  divA_in_ggaa(x1, x2)
0  =  0
divA_out_ggaa(x1, x2, x3, x4)  =  divA_out_ggaa(x1, x2, x4)
s(x1)  =  s(x1)
U1_ggaa(x1, x2, x3, x4, x5)  =  U1_ggaa(x1, x2, x5)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U4_ggaaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaaa(x1, x2, x6)
minusC_in_gga(x1, x2, x3)  =  minusC_in_gga(x1, x2)
minusC_out_gga(x1, x2, x3)  =  minusC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x3, x6)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x5)
MINUSC_IN_GGA(x1, x2, x3)  =  MINUSC_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

MINUSC_IN_GGA(s(T61), s(T62), X75) → MINUSC_IN_GGA(T61, T62, X75)

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

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

MINUSC_IN_GGA(s(T61), s(T62)) → MINUSC_IN_GGA(T61, T62)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• MINUSC_IN_GGA(s(T61), s(T62)) → MINUSC_IN_GGA(T61, T62)
  The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

DIVA_IN_GGAA(s(T45), s(T46), s(T34), T35) → PB_IN_GGAAA(T45, T46, X54, T34, T35)
PB_IN_GGAAA(T45, T46, T49, T34, T35) → U4_GGAAA(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
U4_GGAAA(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → PD_IN_GGAA(T49, T46, T34, T35)
PD_IN_GGAA(T49, T46, T34, T35) → DIVA_IN_GGAA(T49, s(T46), T34, T35)

The TRS R consists of the following rules:

divA_in_ggaa(0, T14, 0, 0) → divA_out_ggaa(0, T14, 0, 0)
divA_in_ggaa(s(T45), s(T46), s(T34), T35) → U1_ggaa(T45, T46, T34, T35, pB_in_ggaaa(T45, T46, X54, T34, T35))
pB_in_ggaaa(T45, T46, T49, T34, T35) → U4_ggaaa(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
minusC_in_gga(T56, 0, T56) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62), X75) → U2_gga(T61, T62, X75, minusC_in_gga(T61, T62, X75))
U2_gga(T61, T62, X75, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)
U4_ggaaa(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → U5_ggaaa(T45, T46, T49, T34, T35, pD_in_ggaa(T49, T46, T34, T35))
pD_in_ggaa(T49, T46, T34, T35) → U3_ggaa(T49, T46, T34, T35, divA_in_ggaa(T49, s(T46), T34, T35))
divA_in_ggaa(T72, T73, T74, T72) → divA_out_ggaa(T72, T73, T74, T72)
U3_ggaa(T49, T46, T34, T35, divA_out_ggaa(T49, s(T46), T34, T35)) → pD_out_ggaa(T49, T46, T34, T35)
U5_ggaaa(T45, T46, T49, T34, T35, pD_out_ggaa(T49, T46, T34, T35)) → pB_out_ggaaa(T45, T46, T49, T34, T35)
U1_ggaa(T45, T46, T34, T35, pB_out_ggaaa(T45, T46, X54, T34, T35)) → divA_out_ggaa(s(T45), s(T46), s(T34), T35)

The argument filtering Pi contains the following mapping:
divA_in_ggaa(x1, x2, x3, x4)  =  divA_in_ggaa(x1, x2)
0  =  0
divA_out_ggaa(x1, x2, x3, x4)  =  divA_out_ggaa(x1, x2, x4)
s(x1)  =  s(x1)
U1_ggaa(x1, x2, x3, x4, x5)  =  U1_ggaa(x1, x2, x5)
pB_in_ggaaa(x1, x2, x3, x4, x5)  =  pB_in_ggaaa(x1, x2)
U4_ggaaa(x1, x2, x3, x4, x5, x6)  =  U4_ggaaa(x1, x2, x6)
minusC_in_gga(x1, x2, x3)  =  minusC_in_gga(x1, x2)
minusC_out_gga(x1, x2, x3)  =  minusC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
U5_ggaaa(x1, x2, x3, x4, x5, x6)  =  U5_ggaaa(x1, x2, x3, x6)
pD_in_ggaa(x1, x2, x3, x4)  =  pD_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
pD_out_ggaa(x1, x2, x3, x4)  =  pD_out_ggaa(x1, x2, x4)
pB_out_ggaaa(x1, x2, x3, x4, x5)  =  pB_out_ggaaa(x1, x2, x3, x5)
DIVA_IN_GGAA(x1, x2, x3, x4)  =  DIVA_IN_GGAA(x1, x2)
PB_IN_GGAAA(x1, x2, x3, x4, x5)  =  PB_IN_GGAAA(x1, x2)
U4_GGAAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAA(x1, x2, x6)
PD_IN_GGAA(x1, x2, x3, x4)  =  PD_IN_GGAA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

DIVA_IN_GGAA(s(T45), s(T46), s(T34), T35) → PB_IN_GGAAA(T45, T46, X54, T34, T35)
PB_IN_GGAAA(T45, T46, T49, T34, T35) → U4_GGAAA(T45, T46, T49, T34, T35, minusC_in_gga(T45, T46, T49))
U4_GGAAA(T45, T46, T49, T34, T35, minusC_out_gga(T45, T46, T49)) → PD_IN_GGAA(T49, T46, T34, T35)
PD_IN_GGAA(T49, T46, T34, T35) → DIVA_IN_GGAA(T49, s(T46), T34, T35)

The TRS R consists of the following rules:

minusC_in_gga(T56, 0, T56) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62), X75) → U2_gga(T61, T62, X75, minusC_in_gga(T61, T62, X75))
U2_gga(T61, T62, X75, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)

The argument filtering Pi contains the following mapping:
0  =  0
s(x1)  =  s(x1)
minusC_in_gga(x1, x2, x3)  =  minusC_in_gga(x1, x2)
minusC_out_gga(x1, x2, x3)  =  minusC_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4)  =  U2_gga(x1, x2, x4)
DIVA_IN_GGAA(x1, x2, x3, x4)  =  DIVA_IN_GGAA(x1, x2)
PB_IN_GGAAA(x1, x2, x3, x4, x5)  =  PB_IN_GGAAA(x1, x2)
U4_GGAAA(x1, x2, x3, x4, x5, x6)  =  U4_GGAAA(x1, x2, x6)
PD_IN_GGAA(x1, x2, x3, x4)  =  PD_IN_GGAA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

DIVA_IN_GGAA(s(T45), s(T46)) → PB_IN_GGAAA(T45, T46)
PB_IN_GGAAA(T45, T46) → U4_GGAAA(T45, T46, minusC_in_gga(T45, T46))
U4_GGAAA(T45, T46, minusC_out_gga(T45, T46, T49)) → PD_IN_GGAA(T49, T46)
PD_IN_GGAA(T49, T46) → DIVA_IN_GGAA(T49, s(T46))

The TRS R consists of the following rules:

minusC_in_gga(T56, 0) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62)) → U2_gga(T61, T62, minusC_in_gga(T61, T62))
U2_gga(T61, T62, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)

The set Q consists of the following terms:

minusC_in_gga(x0, x1)
U2_gga(x0, x1, x2)

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

(19) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

DIVA_IN_GGAA(s(T45), s(T46)) → PB_IN_GGAAA(T45, T46)

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

POL(0) = 0   
POL(DIVA_IN_GGAA(x₁, x₂)) = 1 + x₁   
POL(PB_IN_GGAAA(x₁, x₂)) = 1 + x₁   
POL(PD_IN_GGAA(x₁, x₂)) = 1 + x₁   
POL(U2_gga(x₁, x₂, x₃)) = x₃   
POL(U4_GGAAA(x₁, x₂, x₃)) = x₃   
POL(minusC_in_gga(x₁, x₂)) = 1 + x₁   
POL(minusC_out_gga(x₁, x₂, x₃)) = 1 + x₃   
POL(s(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

minusC_in_gga(T56, 0) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62)) → U2_gga(T61, T62, minusC_in_gga(T61, T62))
U2_gga(T61, T62, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)

(20) Obligation:

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

PB_IN_GGAAA(T45, T46) → U4_GGAAA(T45, T46, minusC_in_gga(T45, T46))
U4_GGAAA(T45, T46, minusC_out_gga(T45, T46, T49)) → PD_IN_GGAA(T49, T46)
PD_IN_GGAA(T49, T46) → DIVA_IN_GGAA(T49, s(T46))

The TRS R consists of the following rules:

minusC_in_gga(T56, 0) → minusC_out_gga(T56, 0, T56)
minusC_in_gga(s(T61), s(T62)) → U2_gga(T61, T62, minusC_in_gga(T61, T62))
U2_gga(T61, T62, minusC_out_gga(T61, T62, X75)) → minusC_out_gga(s(T61), s(T62), X75)

The set Q consists of the following terms:

minusC_in_gga(x0, x1)
U2_gga(x0, x1, x2)

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

(21) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(22) TRUE