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

(0) Obligation:

Clauses:

div(X, 0, Z) :- ','(!, fail).
div(0, Y, Z) :- ','(!, =(Z, 0)).
div(X, Y, s(Z)) :- ','(minus(X, Y, U), div(U, Y, Z)).
minus(0, Y, 0).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
=(X, X).

Query: div(g,g,a)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

div(X, 0, Z) :- ','(!, fail).
div(0, Y, Z) :- ','(!, user_defined_=(Z, 0)).
div(X, Y, s(Z)) :- ','(minus(X, Y, U), div(U, Y, Z)).
minus(0, Y, 0).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
user_defined_=(X, X).

Query: div(g,g,a)

(3) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
3 [label="A: div(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: div(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(!_1, fail) | div(T6, 0, T3), SCOPE: 1, CLAUSE: 1 | div(T6, 0, T3), SCOPE: 1, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: div(T1, T2, T3), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\ndiv(T1, T2, T3) !=? div(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(!_1, user_defined_=(T12, 0)) | div(0, T10, T3), SCOPE: 1, CLAUSE: 2\n\nT10 is ground\nX3 is free\nX4 is free\ndiv(0, T10, T3) !=? div(X3, 0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: div(T1, T2, T3), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\ndiv(T1, T2, T3) !=? div(X3, 0, X4)\ndiv(T1, T2, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: user_defined_=(T12, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: user_defined_=(T12, 0), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(minus(T19, T20, X18), div(X18, T20, T22))\n\nT19 is ground\nT20 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\nX18 is free\ndiv(T19, T20, T3) !=? div(X3, 0, X4)\ndiv(T19, T20, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 3 | ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 4 | ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 5\n\nT19 is ground\nT20 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\nX18 is free\ndiv(T19, T20, T3) !=? div(X3, 0, X4)\ndiv(T19, T20, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 4 | ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 5\n\nT19 is ground\nT20 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\nX18 is free\ndiv(T19, T20, T3) !=? div(X3, 0, X4)\ndiv(T19, T20, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: ','(minus(T19, T20, X18), div(X18, T20, T22)), SCOPE: 3, CLAUSE: 5\n\nT19 is ground\nT20 is ground\nX3 is free\nX4 is free\nX7 is free\nX8 is free\nX18 is free\ndiv(T19, T20, T3) !=? div(X3, 0, X4)\ndiv(T19, T20, T3) !=? div(0, X7, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="B: ','(minus(T29, T30, X32), div(X32, s(T30), T22))\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="C: minus(T29, T30, X32)\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: div(T33, s(T30), T22)\n\nT30 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: minus(T29, T30, X32), SCOPE: 4, CLAUSE: 3 | minus(T29, T30, X32), SCOPE: 4, CLAUSE: 4 | minus(T29, T30, X32), SCOPE: 4, CLAUSE: 5\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: minus(T29, T30, X32), SCOPE: 4, CLAUSE: 3\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: minus(T29, T30, X32), SCOPE: 4, CLAUSE: 4 | minus(T29, T30, X32), SCOPE: 4, CLAUSE: 5\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: minus(T29, T30, X32), SCOPE: 4, CLAUSE: 4\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: minus(T29, T30, X32), SCOPE: 4, CLAUSE: 5\n\nT29 is ground\nT30 is ground\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: minus(T50, T51, X58)\n\nT50 is ground\nT51 is ground\nX58 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 93 [arrowhead = none ];
93 -> 5;

94 [label="EVAL with clause\ndiv(X3, 0, X4) :- ','(!_1, fail).\nand substitutionT1 -> T6,\nX3 -> T6,\nT2 -> 0,\nT3 -> T7,\nX4 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 94 [arrowhead = none ];
94 -> 9;

95 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 95 [arrowhead = none ];
95 -> 12;

96 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 96 [arrowhead = none ];
96 -> 15;

97 [label="EVAL with clause\ndiv(0, X7, X8) :- ','(!_1, user_defined_=(X8, 0)).\nand substitutionT1 -> 0,\nT2 -> T10,\nX7 -> T10,\nT3 -> T12,\nX8 -> T12,\nT11 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 97 [arrowhead = none ];
97 -> 20;

98 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 98 [arrowhead = none ];
98 -> 23;

99 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 99 [arrowhead = none ];
99 -> 17;

100 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 100 [arrowhead = none ];
100 -> 26;

101 [label="EVAL with clause\ndiv(X15, X16, s(X17)) :- ','(minus(X15, X16, X18), div(X18, X16, X17)).\nand substitutionT1 -> T19,\nX15 -> T19,\nT2 -> T20,\nX16 -> T20,\nX17 -> T22,\nT3 -> s(T22),\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 101 [arrowhead = none ];
101 -> 41;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 102 [arrowhead = none ];
102 -> 46;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 103 [arrowhead = none ];
103 -> 27;

104 [label="EVAL with clause\nuser_defined_=(X11, X11).\nand substitutionT12 -> 0,\nX11 -> 0,\nT15 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 104 [arrowhead = none ];
104 -> 32;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 105 [arrowhead = none ];
105 -> 35;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 106 [arrowhead = none ];
106 -> 36;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 107 [arrowhead = none ];
107 -> 47;

108 [label="BACKTRACK\nfor clause: minus(0, Y, 0)\nwith clash: (div(T19, T20, T3), div(0, X7, X8))", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 108 [arrowhead = none ];
108 -> 49;

109 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T19, T20, T3), div(X3, 0, X4))", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 109 [arrowhead = none ];
109 -> 50;

110 [label="EVAL with clause\nminus(s(X29), s(X30), X31) :- minus(X29, X30, X31).\nand substitutionX29 -> T29,\nT19 -> s(T29),\nX30 -> T30,\nT20 -> s(T30),\nX18 -> X32,\nX31 -> X32", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 110 [arrowhead = none ];
110 -> 53;

111 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 111 [arrowhead = none ];
111 -> 54;

112 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
53 -> 112 [arrowhead = none ];
112 -> 57;

113 [label="SPLIT 2\nnew knowledge:\nT29 is ground\nT30 is ground\nT33 is ground\nreplacements:X32 -> T33", fontsize=14, style = filled, fillcolor = violet];
53 -> 113 [arrowhead = none ];
113 -> 59;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
57 -> 114 [arrowhead = none ];
114 -> 62;

115 [label="INSTANCE with matching:\nT1 -> T33\nT2 -> s(T30)\nT3 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 115 [arrowhead = none , style=dashed];
115 -> 3[style=dashed];

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 116 [arrowhead = none ];
116 -> 64;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
62 -> 117 [arrowhead = none ];
117 -> 66;

118 [label="EVAL with clause\nminus(0, X41, 0).\nand substitutionT29 -> 0,\nT30 -> T40,\nX41 -> T40,\nX32 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 118 [arrowhead = none ];
118 -> 70;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 119 [arrowhead = none ];
119 -> 71;

120 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 120 [arrowhead = none ];
120 -> 77;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
66 -> 121 [arrowhead = none ];
121 -> 79;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 122 [arrowhead = none ];
122 -> 73;

123 [label="EVAL with clause\nminus(X46, 0, X46).\nand substitutionT29 -> T45,\nX46 -> T45,\nT30 -> 0,\nX32 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
77 -> 123 [arrowhead = none ];
123 -> 84;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 124 [arrowhead = none ];
124 -> 85;

125 [label="EVAL with clause\nminus(s(X55), s(X56), X57) :- minus(X55, X56, X57).\nand substitutionX55 -> T50,\nT29 -> s(T50),\nX56 -> T51,\nT30 -> s(T51),\nX32 -> X58,\nX57 -> X58", fontsize=14, style = filled, fillcolor = lightblue];
79 -> 125 [arrowhead = none ];
125 -> 91;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 126 [arrowhead = none ];
126 -> 92;

127 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 127 [arrowhead = none ];
127 -> 86;

128 [label="INSTANCE with matching:\nT29 -> T50\nT30 -> T51\nX32 -> X58", fontsize=14, style = filled, fillcolor = lightgrey];
91 -> 128 [arrowhead = none , style=dashed];
128 -> 57[style=dashed];

}

(4) Obligation:

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

divA_in_gga(0, T10, 0) → divA_out_gga(0, T10, 0)
divA_in_gga(s(T29), s(T30), s(T22)) → U1_gga(T29, T30, T22, pB_in_ggaa(T29, T30, X32, T22))
pB_in_ggaa(T29, T30, T33, T22) → U3_ggaa(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
minusC_in_gga(0, T40, 0) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0, T45) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51), X58) → U2_gga(T50, T51, X58, minusC_in_gga(T50, T51, X58))
U2_gga(T50, T51, X58, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)
U3_ggaa(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → U4_ggaa(T29, T30, T33, T22, divA_in_gga(T33, s(T30), T22))
U4_ggaa(T29, T30, T33, T22, divA_out_gga(T33, s(T30), T22)) → pB_out_ggaa(T29, T30, T33, T22)
U1_gga(T29, T30, T22, pB_out_ggaa(T29, T30, X32, T22)) → divA_out_gga(s(T29), s(T30), s(T22))

The argument filtering Pi contains the following mapping:
divA_in_gga(x1, x2, x3)  =  divA_in_gga(x1, x2)
0  =  0
divA_out_gga(x1, x2, x3)  =  divA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaa(x1, x2, x3, x4)  =  pB_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
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)
U4_ggaa(x1, x2, x3, x4, x5)  =  U4_ggaa(x1, x2, x3, x5)
pB_out_ggaa(x1, x2, x3, x4)  =  pB_out_ggaa(x1, x2, x3, x4)

(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:

DIVA_IN_GGA(s(T29), s(T30), s(T22)) → U1_GGA(T29, T30, T22, pB_in_ggaa(T29, T30, X32, T22))
DIVA_IN_GGA(s(T29), s(T30), s(T22)) → PB_IN_GGAA(T29, T30, X32, T22)
PB_IN_GGAA(T29, T30, T33, T22) → U3_GGAA(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
PB_IN_GGAA(T29, T30, T33, T22) → MINUSC_IN_GGA(T29, T30, T33)
MINUSC_IN_GGA(s(T50), s(T51), X58) → U2_GGA(T50, T51, X58, minusC_in_gga(T50, T51, X58))
MINUSC_IN_GGA(s(T50), s(T51), X58) → MINUSC_IN_GGA(T50, T51, X58)
U3_GGAA(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → U4_GGAA(T29, T30, T33, T22, divA_in_gga(T33, s(T30), T22))
U3_GGAA(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → DIVA_IN_GGA(T33, s(T30), T22)

The TRS R consists of the following rules:

divA_in_gga(0, T10, 0) → divA_out_gga(0, T10, 0)
divA_in_gga(s(T29), s(T30), s(T22)) → U1_gga(T29, T30, T22, pB_in_ggaa(T29, T30, X32, T22))
pB_in_ggaa(T29, T30, T33, T22) → U3_ggaa(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
minusC_in_gga(0, T40, 0) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0, T45) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51), X58) → U2_gga(T50, T51, X58, minusC_in_gga(T50, T51, X58))
U2_gga(T50, T51, X58, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)
U3_ggaa(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → U4_ggaa(T29, T30, T33, T22, divA_in_gga(T33, s(T30), T22))
U4_ggaa(T29, T30, T33, T22, divA_out_gga(T33, s(T30), T22)) → pB_out_ggaa(T29, T30, T33, T22)
U1_gga(T29, T30, T22, pB_out_ggaa(T29, T30, X32, T22)) → divA_out_gga(s(T29), s(T30), s(T22))

The argument filtering Pi contains the following mapping:
divA_in_gga(x1, x2, x3)  =  divA_in_gga(x1, x2)
0  =  0
divA_out_gga(x1, x2, x3)  =  divA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaa(x1, x2, x3, x4)  =  pB_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
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)
U4_ggaa(x1, x2, x3, x4, x5)  =  U4_ggaa(x1, x2, x3, x5)
pB_out_ggaa(x1, x2, x3, x4)  =  pB_out_ggaa(x1, x2, x3, x4)
DIVA_IN_GGA(x1, x2, x3)  =  DIVA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
PB_IN_GGAA(x1, x2, x3, x4)  =  PB_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5)  =  U3_GGAA(x1, x2, x5)
MINUSC_IN_GGA(x1, x2, x3)  =  MINUSC_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U4_GGAA(x1, x2, x3, x4, x5)  =  U4_GGAA(x1, x2, x3, x5)

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

(6) Obligation:

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

DIVA_IN_GGA(s(T29), s(T30), s(T22)) → U1_GGA(T29, T30, T22, pB_in_ggaa(T29, T30, X32, T22))
DIVA_IN_GGA(s(T29), s(T30), s(T22)) → PB_IN_GGAA(T29, T30, X32, T22)
PB_IN_GGAA(T29, T30, T33, T22) → U3_GGAA(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
PB_IN_GGAA(T29, T30, T33, T22) → MINUSC_IN_GGA(T29, T30, T33)
MINUSC_IN_GGA(s(T50), s(T51), X58) → U2_GGA(T50, T51, X58, minusC_in_gga(T50, T51, X58))
MINUSC_IN_GGA(s(T50), s(T51), X58) → MINUSC_IN_GGA(T50, T51, X58)
U3_GGAA(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → U4_GGAA(T29, T30, T33, T22, divA_in_gga(T33, s(T30), T22))
U3_GGAA(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → DIVA_IN_GGA(T33, s(T30), T22)

The TRS R consists of the following rules:

divA_in_gga(0, T10, 0) → divA_out_gga(0, T10, 0)
divA_in_gga(s(T29), s(T30), s(T22)) → U1_gga(T29, T30, T22, pB_in_ggaa(T29, T30, X32, T22))
pB_in_ggaa(T29, T30, T33, T22) → U3_ggaa(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
minusC_in_gga(0, T40, 0) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0, T45) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51), X58) → U2_gga(T50, T51, X58, minusC_in_gga(T50, T51, X58))
U2_gga(T50, T51, X58, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)
U3_ggaa(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → U4_ggaa(T29, T30, T33, T22, divA_in_gga(T33, s(T30), T22))
U4_ggaa(T29, T30, T33, T22, divA_out_gga(T33, s(T30), T22)) → pB_out_ggaa(T29, T30, T33, T22)
U1_gga(T29, T30, T22, pB_out_ggaa(T29, T30, X32, T22)) → divA_out_gga(s(T29), s(T30), s(T22))

The argument filtering Pi contains the following mapping:
divA_in_gga(x1, x2, x3)  =  divA_in_gga(x1, x2)
0  =  0
divA_out_gga(x1, x2, x3)  =  divA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaa(x1, x2, x3, x4)  =  pB_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
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)
U4_ggaa(x1, x2, x3, x4, x5)  =  U4_ggaa(x1, x2, x3, x5)
pB_out_ggaa(x1, x2, x3, x4)  =  pB_out_ggaa(x1, x2, x3, x4)
DIVA_IN_GGA(x1, x2, x3)  =  DIVA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4)  =  U1_GGA(x1, x2, x4)
PB_IN_GGAA(x1, x2, x3, x4)  =  PB_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5)  =  U3_GGAA(x1, x2, x5)
MINUSC_IN_GGA(x1, x2, x3)  =  MINUSC_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4)  =  U2_GGA(x1, x2, x4)
U4_GGAA(x1, x2, x3, x4, x5)  =  U4_GGAA(x1, x2, x3, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MINUSC_IN_GGA(s(T50), s(T51), X58) → MINUSC_IN_GGA(T50, T51, X58)

The TRS R consists of the following rules:

divA_in_gga(0, T10, 0) → divA_out_gga(0, T10, 0)
divA_in_gga(s(T29), s(T30), s(T22)) → U1_gga(T29, T30, T22, pB_in_ggaa(T29, T30, X32, T22))
pB_in_ggaa(T29, T30, T33, T22) → U3_ggaa(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
minusC_in_gga(0, T40, 0) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0, T45) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51), X58) → U2_gga(T50, T51, X58, minusC_in_gga(T50, T51, X58))
U2_gga(T50, T51, X58, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)
U3_ggaa(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → U4_ggaa(T29, T30, T33, T22, divA_in_gga(T33, s(T30), T22))
U4_ggaa(T29, T30, T33, T22, divA_out_gga(T33, s(T30), T22)) → pB_out_ggaa(T29, T30, T33, T22)
U1_gga(T29, T30, T22, pB_out_ggaa(T29, T30, X32, T22)) → divA_out_gga(s(T29), s(T30), s(T22))

The argument filtering Pi contains the following mapping:
divA_in_gga(x1, x2, x3)  =  divA_in_gga(x1, x2)
0  =  0
divA_out_gga(x1, x2, x3)  =  divA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaa(x1, x2, x3, x4)  =  pB_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
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)
U4_ggaa(x1, x2, x3, x4, x5)  =  U4_ggaa(x1, x2, x3, x5)
pB_out_ggaa(x1, x2, x3, x4)  =  pB_out_ggaa(x1, x2, x3, x4)
MINUSC_IN_GGA(x1, x2, x3)  =  MINUSC_IN_GGA(x1, x2)

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

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

MINUSC_IN_GGA(s(T50), s(T51), X58) → MINUSC_IN_GGA(T50, T51, X58)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

MINUSC_IN_GGA(s(T50), s(T51)) → MINUSC_IN_GGA(T50, T51)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• MINUSC_IN_GGA(s(T50), s(T51)) → MINUSC_IN_GGA(T50, T51)
  The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

DIVA_IN_GGA(s(T29), s(T30), s(T22)) → PB_IN_GGAA(T29, T30, X32, T22)
PB_IN_GGAA(T29, T30, T33, T22) → U3_GGAA(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
U3_GGAA(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → DIVA_IN_GGA(T33, s(T30), T22)

The TRS R consists of the following rules:

divA_in_gga(0, T10, 0) → divA_out_gga(0, T10, 0)
divA_in_gga(s(T29), s(T30), s(T22)) → U1_gga(T29, T30, T22, pB_in_ggaa(T29, T30, X32, T22))
pB_in_ggaa(T29, T30, T33, T22) → U3_ggaa(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
minusC_in_gga(0, T40, 0) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0, T45) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51), X58) → U2_gga(T50, T51, X58, minusC_in_gga(T50, T51, X58))
U2_gga(T50, T51, X58, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)
U3_ggaa(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → U4_ggaa(T29, T30, T33, T22, divA_in_gga(T33, s(T30), T22))
U4_ggaa(T29, T30, T33, T22, divA_out_gga(T33, s(T30), T22)) → pB_out_ggaa(T29, T30, T33, T22)
U1_gga(T29, T30, T22, pB_out_ggaa(T29, T30, X32, T22)) → divA_out_gga(s(T29), s(T30), s(T22))

The argument filtering Pi contains the following mapping:
divA_in_gga(x1, x2, x3)  =  divA_in_gga(x1, x2)
0  =  0
divA_out_gga(x1, x2, x3)  =  divA_out_gga(x1, x2, x3)
s(x1)  =  s(x1)
U1_gga(x1, x2, x3, x4)  =  U1_gga(x1, x2, x4)
pB_in_ggaa(x1, x2, x3, x4)  =  pB_in_ggaa(x1, x2)
U3_ggaa(x1, x2, x3, x4, x5)  =  U3_ggaa(x1, x2, x5)
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)
U4_ggaa(x1, x2, x3, x4, x5)  =  U4_ggaa(x1, x2, x3, x5)
pB_out_ggaa(x1, x2, x3, x4)  =  pB_out_ggaa(x1, x2, x3, x4)
DIVA_IN_GGA(x1, x2, x3)  =  DIVA_IN_GGA(x1, x2)
PB_IN_GGAA(x1, x2, x3, x4)  =  PB_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

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

DIVA_IN_GGA(s(T29), s(T30), s(T22)) → PB_IN_GGAA(T29, T30, X32, T22)
PB_IN_GGAA(T29, T30, T33, T22) → U3_GGAA(T29, T30, T33, T22, minusC_in_gga(T29, T30, T33))
U3_GGAA(T29, T30, T33, T22, minusC_out_gga(T29, T30, T33)) → DIVA_IN_GGA(T33, s(T30), T22)

The TRS R consists of the following rules:

minusC_in_gga(0, T40, 0) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0, T45) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51), X58) → U2_gga(T50, T51, X58, minusC_in_gga(T50, T51, X58))
U2_gga(T50, T51, X58, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)

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_GGA(x1, x2, x3)  =  DIVA_IN_GGA(x1, x2)
PB_IN_GGAA(x1, x2, x3, x4)  =  PB_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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

DIVA_IN_GGA(s(T29), s(T30)) → PB_IN_GGAA(T29, T30)
PB_IN_GGAA(T29, T30) → U3_GGAA(T29, T30, minusC_in_gga(T29, T30))
U3_GGAA(T29, T30, minusC_out_gga(T29, T30, T33)) → DIVA_IN_GGA(T33, s(T30))

The TRS R consists of the following rules:

minusC_in_gga(0, T40) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51)) → U2_gga(T50, T51, minusC_in_gga(T50, T51))
U2_gga(T50, T51, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)

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) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

PB_IN_GGAA(T29, T30) → U3_GGAA(T29, T30, minusC_in_gga(T29, T30))

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

POL(0) = 0   
POL(DIVA_IN_GGA(x₁, x₂)) = x₁   
POL(PB_IN_GGAA(x₁, x₂)) = 1 + x₁   
POL(U2_gga(x₁, x₂, x₃)) = x₃   
POL(U3_GGAA(x₁, x₂, x₃)) = x₃   
POL(minusC_in_gga(x₁, x₂)) = x₁   
POL(minusC_out_gga(x₁, x₂, x₃)) = 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(0, T40) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51)) → U2_gga(T50, T51, minusC_in_gga(T50, T51))
U2_gga(T50, T51, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)

(22) Obligation:

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

DIVA_IN_GGA(s(T29), s(T30)) → PB_IN_GGAA(T29, T30)
U3_GGAA(T29, T30, minusC_out_gga(T29, T30, T33)) → DIVA_IN_GGA(T33, s(T30))

The TRS R consists of the following rules:

minusC_in_gga(0, T40) → minusC_out_gga(0, T40, 0)
minusC_in_gga(T45, 0) → minusC_out_gga(T45, 0, T45)
minusC_in_gga(s(T50), s(T51)) → U2_gga(T50, T51, minusC_in_gga(T50, T51))
U2_gga(T50, T51, minusC_out_gga(T50, T51, X58)) → minusC_out_gga(s(T50), s(T51), X58)

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.

(23) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(24) TRUE