Left Termination proof of /tmp/tmpWWfhAL/dis

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 89 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 98 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 0 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 10 ms)

↳14 YES

(0) Obligation:

Clauses:

dis(or(B1, B2)) :- ','(con(B1), dis(B2)).
dis(B) :- con(B).
con(and(B1, B2)) :- ','(dis(B1), con(B2)).
con(B) :- bool(B).
bool(0).
bool(1).

Query: dis(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: dis(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: dis(T1), SCOPE: 1, CLAUSE: 0 | dis(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(con(T4), dis(T5)) | dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: dis(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX3 is free\nX4 is free\ndis(T1) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 2 | ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 3 | ?_2 | dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 2\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 3 | ?_2 | dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(','(dis(T14), con(T15)), dis(T5))\n\nT5 is ground\nT14 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: dis(T14)\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(con(T15), dis(T5))\n\nT5 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: con(T15)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: dis(T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: con(T15), SCOPE: 3, CLAUSE: 2 | con(T15), SCOPE: 3, CLAUSE: 3\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: con(T15), SCOPE: 3, CLAUSE: 2\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: con(T15), SCOPE: 3, CLAUSE: 3\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(dis(T32), con(T33))\n\nT32 is ground\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: dis(T32)\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: con(T33)\n\nT33 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: bool(T42)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: bool(T42), SCOPE: 4, CLAUSE: 4 | bool(T42), SCOPE: 4, CLAUSE: 5\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: bool(T42), SCOPE: 4, CLAUSE: 4\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: bool(T42), SCOPE: 4, CLAUSE: 5\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(con(T4), dis(T5)), SCOPE: 2, CLAUSE: 3\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ?_2 | dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(bool(T53), dis(T5))\n\nT5 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(bool(T53), dis(T5)), SCOPE: 5, CLAUSE: 4 | ','(bool(T53), dis(T5)), SCOPE: 5, CLAUSE: 5\n\nT5 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(bool(T53), dis(T5)), SCOPE: 5, CLAUSE: 4\n\nT5 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(bool(T53), dis(T5)), SCOPE: 5, CLAUSE: 5\n\nT5 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: dis(T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: dis(T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: dis(or(T4, T5)), SCOPE: 1, CLAUSE: 1\n\nT4 is ground\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: con(or(T64, T65))\n\nT64 is ground\nT65 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: con(T67)\n\nT67 is ground\nX3 is free\nX4 is free\ndis(T67) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: con(T67), SCOPE: 6, CLAUSE: 2 | con(T67), SCOPE: 6, CLAUSE: 3\n\nT67 is ground\nX3 is free\nX4 is free\ndis(T67) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: con(T67), SCOPE: 6, CLAUSE: 2\n\nT67 is ground\nX3 is free\nX4 is free\ndis(T67) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: con(T67), SCOPE: 6, CLAUSE: 3\n\nT67 is ground\nX3 is free\nX4 is free\ndis(T67) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(dis(T76), con(T77))\n\nT76 is ground\nT77 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: bool(T82)\n\nT82 is ground\nX3 is free\nX4 is free\ndis(T82) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: bool(T82), SCOPE: 7, CLAUSE: 4 | bool(T82), SCOPE: 7, CLAUSE: 5\n\nT82 is ground\nX3 is free\nX4 is free\ndis(T82) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: bool(T82), SCOPE: 7, CLAUSE: 4\n\nT82 is ground\nX3 is free\nX4 is free\ndis(T82) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: bool(T82), SCOPE: 7, CLAUSE: 5\n\nT82 is ground\nX3 is free\nX4 is free\ndis(T82) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 59 [arrowhead = none ];
59 -> 2;

60 [label="EVAL with clause\ndis(or(X3, X4)) :- ','(con(X3), dis(X4)).\nand substitutionX3 -> T4,\nX4 -> T5,\nT1 -> or(T4, T5)", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 60 [arrowhead = none ];
60 -> 3;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 61 [arrowhead = none ];
61 -> 4;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 62 [arrowhead = none ];
62 -> 5;

63 [label="ONLY EVAL with clause\ndis(X67) :- con(X67).\nand substitutionT1 -> T67,\nX67 -> T67", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 63 [arrowhead = none ];
63 -> 43;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 64 [arrowhead = none ];
64 -> 6;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 65 [arrowhead = none ];
65 -> 7;

66 [label="EVAL with clause\ncon(and(X13, X14)) :- ','(dis(X13), con(X14)).\nand substitutionX13 -> T14,\nX14 -> T15,\nT4 -> and(T14, T15)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 66 [arrowhead = none ];
66 -> 8;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 67 [arrowhead = none ];
67 -> 9;

68 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 68 [arrowhead = none ];
68 -> 31;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 69 [arrowhead = none ];
69 -> 32;

70 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
8 -> 70 [arrowhead = none ];
70 -> 10;

71 [label="SPLIT 2\nnew knowledge:\nT14 is ground", fontsize=14, style = filled, fillcolor = violet];
8 -> 71 [arrowhead = none ];
71 -> 11;

72 [label="INSTANCE with matching:\nT1 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
10 -> 72 [arrowhead = none , style=dashed];
72 -> 1[style=dashed];

73 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
11 -> 73 [arrowhead = none ];
73 -> 12;

74 [label="SPLIT 2\nnew knowledge:\nT15 is ground", fontsize=14, style = filled, fillcolor = violet];
11 -> 74 [arrowhead = none ];
74 -> 13;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 75 [arrowhead = none ];
75 -> 14;

76 [label="INSTANCE with matching:\nT1 -> T5", fontsize=14, style = filled, fillcolor = lightgrey];
13 -> 76 [arrowhead = none , style=dashed];
76 -> 1[style=dashed];

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 77 [arrowhead = none ];
77 -> 15;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
14 -> 78 [arrowhead = none ];
78 -> 16;

79 [label="EVAL with clause\ncon(and(X31, X32)) :- ','(dis(X31), con(X32)).\nand substitutionX31 -> T32,\nX32 -> T33,\nT15 -> and(T32, T33)", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 79 [arrowhead = none ];
79 -> 17;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 80 [arrowhead = none ];
80 -> 18;

81 [label="ONLY EVAL with clause\ncon(X41) :- bool(X41).\nand substitutionT15 -> T42,\nX41 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 81 [arrowhead = none ];
81 -> 21;

82 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 82 [arrowhead = none ];
82 -> 19;

83 [label="SPLIT 2\nnew knowledge:\nT32 is ground", fontsize=14, style = filled, fillcolor = violet];
17 -> 83 [arrowhead = none ];
83 -> 20;

84 [label="INSTANCE with matching:\nT1 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 84 [arrowhead = none , style=dashed];
84 -> 1[style=dashed];

85 [label="INSTANCE with matching:\nT15 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 85 [arrowhead = none , style=dashed];
85 -> 12[style=dashed];

86 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 86 [arrowhead = none ];
86 -> 22;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 87 [arrowhead = none ];
87 -> 23;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 88 [arrowhead = none ];
88 -> 24;

89 [label="EVAL with clause\nbool(0).\nand substitutionT42 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 89 [arrowhead = none ];
89 -> 25;

90 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 90 [arrowhead = none ];
90 -> 26;

91 [label="EVAL with clause\nbool(1).\nand substitutionT42 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 91 [arrowhead = none ];
91 -> 28;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 92 [arrowhead = none ];
92 -> 29;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 93 [arrowhead = none ];
93 -> 27;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 94 [arrowhead = none ];
94 -> 30;

95 [label="ONLY EVAL with clause\ncon(X52) :- bool(X52).\nand substitutionT4 -> T53,\nX52 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 95 [arrowhead = none ];
95 -> 33;

96 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 96 [arrowhead = none ];
96 -> 41;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 97 [arrowhead = none ];
97 -> 34;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 98 [arrowhead = none ];
98 -> 35;

99 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 99 [arrowhead = none ];
99 -> 36;

100 [label="EVAL with clause\nbool(0).\nand substitutionT53 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 100 [arrowhead = none ];
100 -> 37;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 101 [arrowhead = none ];
101 -> 38;

102 [label="EVAL with clause\nbool(1).\nand substitutionT53 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 102 [arrowhead = none ];
102 -> 39;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 103 [arrowhead = none ];
103 -> 40;

104 [label="INSTANCE with matching:\nT1 -> T5", fontsize=14, style = filled, fillcolor = lightgrey];
37 -> 104 [arrowhead = none , style=dashed];
104 -> 1[style=dashed];

105 [label="INSTANCE with matching:\nT1 -> T5", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 105 [arrowhead = none , style=dashed];
105 -> 1[style=dashed];

106 [label="ONLY EVAL with clause\ndis(X63) :- con(X63).\nand substitutionT4 -> T64,\nT5 -> T65,\nX63 -> or(T64, T65)", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 106 [arrowhead = none ];
106 -> 42;

107 [label="INSTANCE with matching:\nT15 -> or(T64, T65)", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 107 [arrowhead = none , style=dashed];
107 -> 12[style=dashed];

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 108 [arrowhead = none ];
108 -> 44;

109 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 109 [arrowhead = none ];
109 -> 45;

110 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 110 [arrowhead = none ];
110 -> 46;

111 [label="EVAL with clause\ncon(and(X76, X77)) :- ','(dis(X76), con(X77)).\nand substitutionX76 -> T76,\nX77 -> T77,\nT67 -> and(T76, T77)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 111 [arrowhead = none ];
111 -> 47;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 112 [arrowhead = none ];
112 -> 48;

113 [label="ONLY EVAL with clause\ncon(X82) :- bool(X82).\nand substitutionT67 -> T82,\nX82 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 113 [arrowhead = none ];
113 -> 49;

114 [label="INSTANCE with matching:\nT32 -> T76\nT33 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
47 -> 114 [arrowhead = none , style=dashed];
114 -> 17[style=dashed];

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
49 -> 115 [arrowhead = none ];
115 -> 50;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 116 [arrowhead = none ];
116 -> 51;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
50 -> 117 [arrowhead = none ];
117 -> 52;

118 [label="EVAL with clause\nbool(0).\nand substitutionT82 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 118 [arrowhead = none ];
118 -> 53;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 119 [arrowhead = none ];
119 -> 54;

120 [label="EVAL with clause\nbool(1).\nand substitutionT82 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 120 [arrowhead = none ];
120 -> 56;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 121 [arrowhead = none ];
121 -> 57;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 122 [arrowhead = none ];
122 -> 55;

123 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 123 [arrowhead = none ];
123 -> 58;

}

(2) Obligation:

Clauses:

conA(and(T32, T33)) :- pB(T32, T33).
conA(0).
conA(1).
pB(T32, T33) :- disC(T32).
pB(T32, T33) :- ','(disC(T32), conA(T33)).
disC(or(and(T14, T15), T5)) :- disC(T14).
disC(or(and(T14, T15), T5)) :- ','(disC(T14), conA(T15)).
disC(or(and(T14, T15), T5)) :- ','(disC(T14), ','(conA(T15), disC(T5))).
disC(or(0, T5)) :- disC(T5).
disC(or(1, T5)) :- disC(T5).
disC(or(T64, T65)) :- conA(or(T64, T65)).
disC(and(T76, T77)) :- pB(T76, T77).
disC(0).
disC(1).

Query: disC(g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
disC_in: (b)
conA_in: (b)
pB_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

disC_in_g(or(and(T14, T15), T5)) → U4_g(T14, T15, T5, disC_in_g(T14))
disC_in_g(or(0, T5)) → U7_g(T5, disC_in_g(T5))
disC_in_g(or(1, T5)) → U8_g(T5, disC_in_g(T5))
disC_in_g(or(T64, T65)) → U9_g(T64, T65, conA_in_g(or(T64, T65)))
conA_in_g(and(T32, T33)) → U1_g(T32, T33, pB_in_gg(T32, T33))
pB_in_gg(T32, T33) → U2_gg(T32, T33, disC_in_g(T32))
disC_in_g(and(T76, T77)) → U10_g(T76, T77, pB_in_gg(T76, T77))
U10_g(T76, T77, pB_out_gg(T76, T77)) → disC_out_g(and(T76, T77))
disC_in_g(0) → disC_out_g(0)
disC_in_g(1) → disC_out_g(1)
U2_gg(T32, T33, disC_out_g(T32)) → pB_out_gg(T32, T33)
U2_gg(T32, T33, disC_out_g(T32)) → U3_gg(T32, T33, conA_in_g(T33))
conA_in_g(0) → conA_out_g(0)
conA_in_g(1) → conA_out_g(1)
U3_gg(T32, T33, conA_out_g(T33)) → pB_out_gg(T32, T33)
U1_g(T32, T33, pB_out_gg(T32, T33)) → conA_out_g(and(T32, T33))
U9_g(T64, T65, conA_out_g(or(T64, T65))) → disC_out_g(or(T64, T65))
U8_g(T5, disC_out_g(T5)) → disC_out_g(or(1, T5))
U7_g(T5, disC_out_g(T5)) → disC_out_g(or(0, T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → disC_out_g(or(and(T14, T15), T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → U5_g(T14, T15, T5, conA_in_g(T15))
U5_g(T14, T15, T5, conA_out_g(T15)) → disC_out_g(or(and(T14, T15), T5))
U5_g(T14, T15, T5, conA_out_g(T15)) → U6_g(T14, T15, T5, disC_in_g(T5))
U6_g(T14, T15, T5, disC_out_g(T5)) → disC_out_g(or(and(T14, T15), T5))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

disC_in_g(or(and(T14, T15), T5)) → U4_g(T14, T15, T5, disC_in_g(T14))
disC_in_g(or(0, T5)) → U7_g(T5, disC_in_g(T5))
disC_in_g(or(1, T5)) → U8_g(T5, disC_in_g(T5))
disC_in_g(or(T64, T65)) → U9_g(T64, T65, conA_in_g(or(T64, T65)))
conA_in_g(and(T32, T33)) → U1_g(T32, T33, pB_in_gg(T32, T33))
pB_in_gg(T32, T33) → U2_gg(T32, T33, disC_in_g(T32))
disC_in_g(and(T76, T77)) → U10_g(T76, T77, pB_in_gg(T76, T77))
U10_g(T76, T77, pB_out_gg(T76, T77)) → disC_out_g(and(T76, T77))
disC_in_g(0) → disC_out_g(0)
disC_in_g(1) → disC_out_g(1)
U2_gg(T32, T33, disC_out_g(T32)) → pB_out_gg(T32, T33)
U2_gg(T32, T33, disC_out_g(T32)) → U3_gg(T32, T33, conA_in_g(T33))
conA_in_g(0) → conA_out_g(0)
conA_in_g(1) → conA_out_g(1)
U3_gg(T32, T33, conA_out_g(T33)) → pB_out_gg(T32, T33)
U1_g(T32, T33, pB_out_gg(T32, T33)) → conA_out_g(and(T32, T33))
U9_g(T64, T65, conA_out_g(or(T64, T65))) → disC_out_g(or(T64, T65))
U8_g(T5, disC_out_g(T5)) → disC_out_g(or(1, T5))
U7_g(T5, disC_out_g(T5)) → disC_out_g(or(0, T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → disC_out_g(or(and(T14, T15), T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → U5_g(T14, T15, T5, conA_in_g(T15))
U5_g(T14, T15, T5, conA_out_g(T15)) → disC_out_g(or(and(T14, T15), T5))
U5_g(T14, T15, T5, conA_out_g(T15)) → U6_g(T14, T15, T5, disC_in_g(T5))
U6_g(T14, T15, T5, disC_out_g(T5)) → disC_out_g(or(and(T14, T15), T5))

(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:

DISC_IN_G(or(and(T14, T15), T5)) → U4_G(T14, T15, T5, disC_in_g(T14))
DISC_IN_G(or(and(T14, T15), T5)) → DISC_IN_G(T14)
DISC_IN_G(or(0, T5)) → U7_G(T5, disC_in_g(T5))
DISC_IN_G(or(0, T5)) → DISC_IN_G(T5)
DISC_IN_G(or(1, T5)) → U8_G(T5, disC_in_g(T5))
DISC_IN_G(or(1, T5)) → DISC_IN_G(T5)
DISC_IN_G(or(T64, T65)) → U9_G(T64, T65, conA_in_g(or(T64, T65)))
DISC_IN_G(or(T64, T65)) → CONA_IN_G(or(T64, T65))
CONA_IN_G(and(T32, T33)) → U1_G(T32, T33, pB_in_gg(T32, T33))
CONA_IN_G(and(T32, T33)) → PB_IN_GG(T32, T33)
PB_IN_GG(T32, T33) → U2_GG(T32, T33, disC_in_g(T32))
PB_IN_GG(T32, T33) → DISC_IN_G(T32)
DISC_IN_G(and(T76, T77)) → U10_G(T76, T77, pB_in_gg(T76, T77))
DISC_IN_G(and(T76, T77)) → PB_IN_GG(T76, T77)
U2_GG(T32, T33, disC_out_g(T32)) → U3_GG(T32, T33, conA_in_g(T33))
U2_GG(T32, T33, disC_out_g(T32)) → CONA_IN_G(T33)
U4_G(T14, T15, T5, disC_out_g(T14)) → U5_G(T14, T15, T5, conA_in_g(T15))
U4_G(T14, T15, T5, disC_out_g(T14)) → CONA_IN_G(T15)
U5_G(T14, T15, T5, conA_out_g(T15)) → U6_G(T14, T15, T5, disC_in_g(T5))
U5_G(T14, T15, T5, conA_out_g(T15)) → DISC_IN_G(T5)

The TRS R consists of the following rules:

disC_in_g(or(and(T14, T15), T5)) → U4_g(T14, T15, T5, disC_in_g(T14))
disC_in_g(or(0, T5)) → U7_g(T5, disC_in_g(T5))
disC_in_g(or(1, T5)) → U8_g(T5, disC_in_g(T5))
disC_in_g(or(T64, T65)) → U9_g(T64, T65, conA_in_g(or(T64, T65)))
conA_in_g(and(T32, T33)) → U1_g(T32, T33, pB_in_gg(T32, T33))
pB_in_gg(T32, T33) → U2_gg(T32, T33, disC_in_g(T32))
disC_in_g(and(T76, T77)) → U10_g(T76, T77, pB_in_gg(T76, T77))
U10_g(T76, T77, pB_out_gg(T76, T77)) → disC_out_g(and(T76, T77))
disC_in_g(0) → disC_out_g(0)
disC_in_g(1) → disC_out_g(1)
U2_gg(T32, T33, disC_out_g(T32)) → pB_out_gg(T32, T33)
U2_gg(T32, T33, disC_out_g(T32)) → U3_gg(T32, T33, conA_in_g(T33))
conA_in_g(0) → conA_out_g(0)
conA_in_g(1) → conA_out_g(1)
U3_gg(T32, T33, conA_out_g(T33)) → pB_out_gg(T32, T33)
U1_g(T32, T33, pB_out_gg(T32, T33)) → conA_out_g(and(T32, T33))
U9_g(T64, T65, conA_out_g(or(T64, T65))) → disC_out_g(or(T64, T65))
U8_g(T5, disC_out_g(T5)) → disC_out_g(or(1, T5))
U7_g(T5, disC_out_g(T5)) → disC_out_g(or(0, T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → disC_out_g(or(and(T14, T15), T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → U5_g(T14, T15, T5, conA_in_g(T15))
U5_g(T14, T15, T5, conA_out_g(T15)) → disC_out_g(or(and(T14, T15), T5))
U5_g(T14, T15, T5, conA_out_g(T15)) → U6_g(T14, T15, T5, disC_in_g(T5))
U6_g(T14, T15, T5, disC_out_g(T5)) → disC_out_g(or(and(T14, T15), T5))

The argument filtering Pi contains the following mapping:
disC_in_g(x1) = disC_in_g(x1)
or(x1, x2) = or(x1, x2)
and(x1, x2) = and(x1, x2)
U4_g(x1, x2, x3, x4) = U4_g(x2, x3, x4)
0 = 0
U7_g(x1, x2) = U7_g(x2)
1 = 1
U8_g(x1, x2) = U8_g(x2)
U9_g(x1, x2, x3) = U9_g(x3)
conA_in_g(x1) = conA_in_g(x1)
U1_g(x1, x2, x3) = U1_g(x3)
pB_in_gg(x1, x2) = pB_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x2, x3)
U10_g(x1, x2, x3) = U10_g(x3)
pB_out_gg(x1, x2) = pB_out_gg
disC_out_g(x1) = disC_out_g
U3_gg(x1, x2, x3) = U3_gg(x3)
conA_out_g(x1) = conA_out_g
U5_g(x1, x2, x3, x4) = U5_g(x3, x4)
U6_g(x1, x2, x3, x4) = U6_g(x4)
DISC_IN_G(x1) = DISC_IN_G(x1)
U4_G(x1, x2, x3, x4) = U4_G(x2, x3, x4)
U7_G(x1, x2) = U7_G(x2)
U8_G(x1, x2) = U8_G(x2)
U9_G(x1, x2, x3) = U9_G(x3)
CONA_IN_G(x1) = CONA_IN_G(x1)
U1_G(x1, x2, x3) = U1_G(x3)
PB_IN_GG(x1, x2) = PB_IN_GG(x1, x2)
U2_GG(x1, x2, x3) = U2_GG(x2, x3)
U10_G(x1, x2, x3) = U10_G(x3)
U3_GG(x1, x2, x3) = U3_GG(x3)
U5_G(x1, x2, x3, x4) = U5_G(x3, x4)
U6_G(x1, x2, x3, x4) = U6_G(x4)

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

(6) Obligation:

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

DISC_IN_G(or(and(T14, T15), T5)) → U4_G(T14, T15, T5, disC_in_g(T14))
DISC_IN_G(or(and(T14, T15), T5)) → DISC_IN_G(T14)
DISC_IN_G(or(0, T5)) → U7_G(T5, disC_in_g(T5))
DISC_IN_G(or(0, T5)) → DISC_IN_G(T5)
DISC_IN_G(or(1, T5)) → U8_G(T5, disC_in_g(T5))
DISC_IN_G(or(1, T5)) → DISC_IN_G(T5)
DISC_IN_G(or(T64, T65)) → U9_G(T64, T65, conA_in_g(or(T64, T65)))
DISC_IN_G(or(T64, T65)) → CONA_IN_G(or(T64, T65))
CONA_IN_G(and(T32, T33)) → U1_G(T32, T33, pB_in_gg(T32, T33))
CONA_IN_G(and(T32, T33)) → PB_IN_GG(T32, T33)
PB_IN_GG(T32, T33) → U2_GG(T32, T33, disC_in_g(T32))
PB_IN_GG(T32, T33) → DISC_IN_G(T32)
DISC_IN_G(and(T76, T77)) → U10_G(T76, T77, pB_in_gg(T76, T77))
DISC_IN_G(and(T76, T77)) → PB_IN_GG(T76, T77)
U2_GG(T32, T33, disC_out_g(T32)) → U3_GG(T32, T33, conA_in_g(T33))
U2_GG(T32, T33, disC_out_g(T32)) → CONA_IN_G(T33)
U4_G(T14, T15, T5, disC_out_g(T14)) → U5_G(T14, T15, T5, conA_in_g(T15))
U4_G(T14, T15, T5, disC_out_g(T14)) → CONA_IN_G(T15)
U5_G(T14, T15, T5, conA_out_g(T15)) → U6_G(T14, T15, T5, disC_in_g(T5))
U5_G(T14, T15, T5, conA_out_g(T15)) → DISC_IN_G(T5)

The TRS R consists of the following rules:

disC_in_g(or(and(T14, T15), T5)) → U4_g(T14, T15, T5, disC_in_g(T14))
disC_in_g(or(0, T5)) → U7_g(T5, disC_in_g(T5))
disC_in_g(or(1, T5)) → U8_g(T5, disC_in_g(T5))
disC_in_g(or(T64, T65)) → U9_g(T64, T65, conA_in_g(or(T64, T65)))
conA_in_g(and(T32, T33)) → U1_g(T32, T33, pB_in_gg(T32, T33))
pB_in_gg(T32, T33) → U2_gg(T32, T33, disC_in_g(T32))
disC_in_g(and(T76, T77)) → U10_g(T76, T77, pB_in_gg(T76, T77))
U10_g(T76, T77, pB_out_gg(T76, T77)) → disC_out_g(and(T76, T77))
disC_in_g(0) → disC_out_g(0)
disC_in_g(1) → disC_out_g(1)
U2_gg(T32, T33, disC_out_g(T32)) → pB_out_gg(T32, T33)
U2_gg(T32, T33, disC_out_g(T32)) → U3_gg(T32, T33, conA_in_g(T33))
conA_in_g(0) → conA_out_g(0)
conA_in_g(1) → conA_out_g(1)
U3_gg(T32, T33, conA_out_g(T33)) → pB_out_gg(T32, T33)
U1_g(T32, T33, pB_out_gg(T32, T33)) → conA_out_g(and(T32, T33))
U9_g(T64, T65, conA_out_g(or(T64, T65))) → disC_out_g(or(T64, T65))
U8_g(T5, disC_out_g(T5)) → disC_out_g(or(1, T5))
U7_g(T5, disC_out_g(T5)) → disC_out_g(or(0, T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → disC_out_g(or(and(T14, T15), T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → U5_g(T14, T15, T5, conA_in_g(T15))
U5_g(T14, T15, T5, conA_out_g(T15)) → disC_out_g(or(and(T14, T15), T5))
U5_g(T14, T15, T5, conA_out_g(T15)) → U6_g(T14, T15, T5, disC_in_g(T5))
U6_g(T14, T15, T5, disC_out_g(T5)) → disC_out_g(or(and(T14, T15), T5))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 8 less nodes.

(8) Obligation:

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

U4_G(T14, T15, T5, disC_out_g(T14)) → U5_G(T14, T15, T5, conA_in_g(T15))
U5_G(T14, T15, T5, conA_out_g(T15)) → DISC_IN_G(T5)
DISC_IN_G(or(and(T14, T15), T5)) → U4_G(T14, T15, T5, disC_in_g(T14))
U4_G(T14, T15, T5, disC_out_g(T14)) → CONA_IN_G(T15)
CONA_IN_G(and(T32, T33)) → PB_IN_GG(T32, T33)
PB_IN_GG(T32, T33) → U2_GG(T32, T33, disC_in_g(T32))
U2_GG(T32, T33, disC_out_g(T32)) → CONA_IN_G(T33)
PB_IN_GG(T32, T33) → DISC_IN_G(T32)
DISC_IN_G(or(and(T14, T15), T5)) → DISC_IN_G(T14)
DISC_IN_G(or(0, T5)) → DISC_IN_G(T5)
DISC_IN_G(or(1, T5)) → DISC_IN_G(T5)
DISC_IN_G(and(T76, T77)) → PB_IN_GG(T76, T77)

The TRS R consists of the following rules:

disC_in_g(or(and(T14, T15), T5)) → U4_g(T14, T15, T5, disC_in_g(T14))
disC_in_g(or(0, T5)) → U7_g(T5, disC_in_g(T5))
disC_in_g(or(1, T5)) → U8_g(T5, disC_in_g(T5))
disC_in_g(or(T64, T65)) → U9_g(T64, T65, conA_in_g(or(T64, T65)))
conA_in_g(and(T32, T33)) → U1_g(T32, T33, pB_in_gg(T32, T33))
pB_in_gg(T32, T33) → U2_gg(T32, T33, disC_in_g(T32))
disC_in_g(and(T76, T77)) → U10_g(T76, T77, pB_in_gg(T76, T77))
U10_g(T76, T77, pB_out_gg(T76, T77)) → disC_out_g(and(T76, T77))
disC_in_g(0) → disC_out_g(0)
disC_in_g(1) → disC_out_g(1)
U2_gg(T32, T33, disC_out_g(T32)) → pB_out_gg(T32, T33)
U2_gg(T32, T33, disC_out_g(T32)) → U3_gg(T32, T33, conA_in_g(T33))
conA_in_g(0) → conA_out_g(0)
conA_in_g(1) → conA_out_g(1)
U3_gg(T32, T33, conA_out_g(T33)) → pB_out_gg(T32, T33)
U1_g(T32, T33, pB_out_gg(T32, T33)) → conA_out_g(and(T32, T33))
U9_g(T64, T65, conA_out_g(or(T64, T65))) → disC_out_g(or(T64, T65))
U8_g(T5, disC_out_g(T5)) → disC_out_g(or(1, T5))
U7_g(T5, disC_out_g(T5)) → disC_out_g(or(0, T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → disC_out_g(or(and(T14, T15), T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → U5_g(T14, T15, T5, conA_in_g(T15))
U5_g(T14, T15, T5, conA_out_g(T15)) → disC_out_g(or(and(T14, T15), T5))
U5_g(T14, T15, T5, conA_out_g(T15)) → U6_g(T14, T15, T5, disC_in_g(T5))
U6_g(T14, T15, T5, disC_out_g(T5)) → disC_out_g(or(and(T14, T15), T5))

The argument filtering Pi contains the following mapping:
disC_in_g(x1) = disC_in_g(x1)
or(x1, x2) = or(x1, x2)
and(x1, x2) = and(x1, x2)
U4_g(x1, x2, x3, x4) = U4_g(x2, x3, x4)
0 = 0
U7_g(x1, x2) = U7_g(x2)
1 = 1
U8_g(x1, x2) = U8_g(x2)
U9_g(x1, x2, x3) = U9_g(x3)
conA_in_g(x1) = conA_in_g(x1)
U1_g(x1, x2, x3) = U1_g(x3)
pB_in_gg(x1, x2) = pB_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x2, x3)
U10_g(x1, x2, x3) = U10_g(x3)
pB_out_gg(x1, x2) = pB_out_gg
disC_out_g(x1) = disC_out_g
U3_gg(x1, x2, x3) = U3_gg(x3)
conA_out_g(x1) = conA_out_g
U5_g(x1, x2, x3, x4) = U5_g(x3, x4)
U6_g(x1, x2, x3, x4) = U6_g(x4)
DISC_IN_G(x1) = DISC_IN_G(x1)
U4_G(x1, x2, x3, x4) = U4_G(x2, x3, x4)
CONA_IN_G(x1) = CONA_IN_G(x1)
PB_IN_GG(x1, x2) = PB_IN_GG(x1, x2)
U2_GG(x1, x2, x3) = U2_GG(x2, x3)
U5_G(x1, x2, x3, x4) = U5_G(x3, x4)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

U4_G(T14, T15, T5, disC_out_g(T14)) → U5_G(T14, T15, T5, conA_in_g(T15))
U5_G(T14, T15, T5, conA_out_g(T15)) → DISC_IN_G(T5)
DISC_IN_G(or(and(T14, T15), T5)) → U4_G(T14, T15, T5, disC_in_g(T14))
U4_G(T14, T15, T5, disC_out_g(T14)) → CONA_IN_G(T15)
CONA_IN_G(and(T32, T33)) → PB_IN_GG(T32, T33)
PB_IN_GG(T32, T33) → U2_GG(T32, T33, disC_in_g(T32))
U2_GG(T32, T33, disC_out_g(T32)) → CONA_IN_G(T33)
PB_IN_GG(T32, T33) → DISC_IN_G(T32)
DISC_IN_G(or(and(T14, T15), T5)) → DISC_IN_G(T14)
DISC_IN_G(or(0, T5)) → DISC_IN_G(T5)
DISC_IN_G(or(1, T5)) → DISC_IN_G(T5)
DISC_IN_G(and(T76, T77)) → PB_IN_GG(T76, T77)

The TRS R consists of the following rules:

conA_in_g(and(T32, T33)) → U1_g(T32, T33, pB_in_gg(T32, T33))
conA_in_g(0) → conA_out_g(0)
conA_in_g(1) → conA_out_g(1)
disC_in_g(or(and(T14, T15), T5)) → U4_g(T14, T15, T5, disC_in_g(T14))
disC_in_g(or(0, T5)) → U7_g(T5, disC_in_g(T5))
disC_in_g(or(1, T5)) → U8_g(T5, disC_in_g(T5))
disC_in_g(or(T64, T65)) → U9_g(T64, T65, conA_in_g(or(T64, T65)))
disC_in_g(and(T76, T77)) → U10_g(T76, T77, pB_in_gg(T76, T77))
disC_in_g(0) → disC_out_g(0)
disC_in_g(1) → disC_out_g(1)
U1_g(T32, T33, pB_out_gg(T32, T33)) → conA_out_g(and(T32, T33))
U4_g(T14, T15, T5, disC_out_g(T14)) → disC_out_g(or(and(T14, T15), T5))
U4_g(T14, T15, T5, disC_out_g(T14)) → U5_g(T14, T15, T5, conA_in_g(T15))
U7_g(T5, disC_out_g(T5)) → disC_out_g(or(0, T5))
U8_g(T5, disC_out_g(T5)) → disC_out_g(or(1, T5))
U10_g(T76, T77, pB_out_gg(T76, T77)) → disC_out_g(and(T76, T77))
pB_in_gg(T32, T33) → U2_gg(T32, T33, disC_in_g(T32))
U5_g(T14, T15, T5, conA_out_g(T15)) → disC_out_g(or(and(T14, T15), T5))
U5_g(T14, T15, T5, conA_out_g(T15)) → U6_g(T14, T15, T5, disC_in_g(T5))
U2_gg(T32, T33, disC_out_g(T32)) → pB_out_gg(T32, T33)
U2_gg(T32, T33, disC_out_g(T32)) → U3_gg(T32, T33, conA_in_g(T33))
U6_g(T14, T15, T5, disC_out_g(T5)) → disC_out_g(or(and(T14, T15), T5))
U3_gg(T32, T33, conA_out_g(T33)) → pB_out_gg(T32, T33)

The argument filtering Pi contains the following mapping:
disC_in_g(x1) = disC_in_g(x1)
or(x1, x2) = or(x1, x2)
and(x1, x2) = and(x1, x2)
U4_g(x1, x2, x3, x4) = U4_g(x2, x3, x4)
0 = 0
U7_g(x1, x2) = U7_g(x2)
1 = 1
U8_g(x1, x2) = U8_g(x2)
U9_g(x1, x2, x3) = U9_g(x3)
conA_in_g(x1) = conA_in_g(x1)
U1_g(x1, x2, x3) = U1_g(x3)
pB_in_gg(x1, x2) = pB_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x2, x3)
U10_g(x1, x2, x3) = U10_g(x3)
pB_out_gg(x1, x2) = pB_out_gg
disC_out_g(x1) = disC_out_g
U3_gg(x1, x2, x3) = U3_gg(x3)
conA_out_g(x1) = conA_out_g
U5_g(x1, x2, x3, x4) = U5_g(x3, x4)
U6_g(x1, x2, x3, x4) = U6_g(x4)
DISC_IN_G(x1) = DISC_IN_G(x1)
U4_G(x1, x2, x3, x4) = U4_G(x2, x3, x4)
CONA_IN_G(x1) = CONA_IN_G(x1)
PB_IN_GG(x1, x2) = PB_IN_GG(x1, x2)
U2_GG(x1, x2, x3) = U2_GG(x2, x3)
U5_G(x1, x2, x3, x4) = U5_G(x3, x4)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

U4_G(T15, T5, disC_out_g) → U5_G(T5, conA_in_g(T15))
U5_G(T5, conA_out_g) → DISC_IN_G(T5)
DISC_IN_G(or(and(T14, T15), T5)) → U4_G(T15, T5, disC_in_g(T14))
U4_G(T15, T5, disC_out_g) → CONA_IN_G(T15)
CONA_IN_G(and(T32, T33)) → PB_IN_GG(T32, T33)
PB_IN_GG(T32, T33) → U2_GG(T33, disC_in_g(T32))
U2_GG(T33, disC_out_g) → CONA_IN_G(T33)
PB_IN_GG(T32, T33) → DISC_IN_G(T32)
DISC_IN_G(or(and(T14, T15), T5)) → DISC_IN_G(T14)
DISC_IN_G(or(0, T5)) → DISC_IN_G(T5)
DISC_IN_G(or(1, T5)) → DISC_IN_G(T5)
DISC_IN_G(and(T76, T77)) → PB_IN_GG(T76, T77)

The TRS R consists of the following rules:

conA_in_g(and(T32, T33)) → U1_g(pB_in_gg(T32, T33))
conA_in_g(0) → conA_out_g
conA_in_g(1) → conA_out_g
disC_in_g(or(and(T14, T15), T5)) → U4_g(T15, T5, disC_in_g(T14))
disC_in_g(or(0, T5)) → U7_g(disC_in_g(T5))
disC_in_g(or(1, T5)) → U8_g(disC_in_g(T5))
disC_in_g(or(T64, T65)) → U9_g(conA_in_g(or(T64, T65)))
disC_in_g(and(T76, T77)) → U10_g(pB_in_gg(T76, T77))
disC_in_g(0) → disC_out_g
disC_in_g(1) → disC_out_g
U1_g(pB_out_gg) → conA_out_g
U4_g(T15, T5, disC_out_g) → disC_out_g
U4_g(T15, T5, disC_out_g) → U5_g(T5, conA_in_g(T15))
U7_g(disC_out_g) → disC_out_g
U8_g(disC_out_g) → disC_out_g
U10_g(pB_out_gg) → disC_out_g
pB_in_gg(T32, T33) → U2_gg(T33, disC_in_g(T32))
U5_g(T5, conA_out_g) → disC_out_g
U5_g(T5, conA_out_g) → U6_g(disC_in_g(T5))
U2_gg(T33, disC_out_g) → pB_out_gg
U2_gg(T33, disC_out_g) → U3_gg(conA_in_g(T33))
U6_g(disC_out_g) → disC_out_g
U3_gg(conA_out_g) → pB_out_gg

The set Q consists of the following terms:

conA_in_g(x0)
disC_in_g(x0)
U1_g(x0)
U4_g(x0, x1, x2)
U7_g(x0)
U8_g(x0)
U10_g(x0)
pB_in_gg(x0, x1)
U5_g(x0, x1)
U2_gg(x0, x1)
U6_g(x0)
U3_gg(x0)

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

(13) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

U5_G(T5, conA_out_g) → DISC_IN_G(T5)
The graph contains the following edges 1 >= 1
DISC_IN_G(or(and(T14, T15), T5)) → U4_G(T15, T5, disC_in_g(T14))
The graph contains the following edges 1 > 1, 1 > 2
U4_G(T15, T5, disC_out_g) → U5_G(T5, conA_in_g(T15))
The graph contains the following edges 2 >= 1
DISC_IN_G(and(T76, T77)) → PB_IN_GG(T76, T77)
The graph contains the following edges 1 > 1, 1 > 2
U4_G(T15, T5, disC_out_g) → CONA_IN_G(T15)
The graph contains the following edges 1 >= 1
PB_IN_GG(T32, T33) → DISC_IN_G(T32)
The graph contains the following edges 1 >= 1
CONA_IN_G(and(T32, T33)) → PB_IN_GG(T32, T33)
The graph contains the following edges 1 > 1, 1 > 2
PB_IN_GG(T32, T33) → U2_GG(T33, disC_in_g(T32))
The graph contains the following edges 2 >= 1
U2_GG(T33, disC_out_g) → CONA_IN_G(T33)
The graph contains the following edges 1 >= 1
DISC_IN_G(or(and(T14, T15), T5)) → DISC_IN_G(T14)
The graph contains the following edges 1 > 1
DISC_IN_G(or(0, T5)) → DISC_IN_G(T5)
The graph contains the following edges 1 > 1
DISC_IN_G(or(1, T5)) → DISC_IN_G(T5)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES