Left Termination of the query pattern dis_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(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).

Queries:

dis(g).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

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\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(T66, T67))\n\nT66 is ground\nT67 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: con(T71)\n\nT71 is ground\ndis(T71) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: con(T71), SCOPE: 6, CLAUSE: 2 | con(T71), SCOPE: 6, CLAUSE: 3\n\nT71 is ground\ndis(T71) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: con(T71), SCOPE: 6, CLAUSE: 2\n\nT71 is ground\ndis(T71) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: con(T71), SCOPE: 6, CLAUSE: 3\n\nT71 is ground\ndis(T71) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: ','(dis(T80), con(T81))\n\nT80 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: bool(T86)\n\nT86 is ground\ndis(T86) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: bool(T86), SCOPE: 7, CLAUSE: 4 | bool(T86), SCOPE: 7, CLAUSE: 5\n\nT86 is ground\ndis(T86) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: bool(T86), SCOPE: 7, CLAUSE: 4\n\nT86 is ground\ndis(T86) !=? dis(or(X3, X4))", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: bool(T86), SCOPE: 7, CLAUSE: 5\n\nT86 is ground\ndis(T86) !=? 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 substitution\nX3 -> 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 = lightblue];
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(X75) :- con(X75).\nand substitution\nT1 -> T71\nX75 -> T71", fontsize=14, style = filled, fillcolor = white];
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(X15, X16)) :- ','(dis(X15), con(X16)).\nand substitution\nX15 -> T14\nX16 -> 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 = lightblue];
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", 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", 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(X35, X36)) :- ','(dis(X35), con(X36)).\nand substitution\nX35 -> T32\nX36 -> 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 = lightblue];
15 -> 80 [arrowhead = none ];
80 -> 18;

81 [label="ONLY EVAL with clause\ncon(X47) :- bool(X47).\nand substitution\nT15 -> T42\nX47 -> T42", fontsize=14, style = filled, fillcolor = white];
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", 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 substitution\nT42 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 89 [arrowhead = none ];
89 -> 25;

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

91 [label="EVAL with clause\nbool(1).\nand substitution\nT42 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 91 [arrowhead = none ];
91 -> 28;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X58) :- bool(X58).\nand substitution\nT4 -> T53\nX58 -> T53", fontsize=14, style = filled, fillcolor = white];
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 substitution\nT53 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 100 [arrowhead = none ];
100 -> 37;

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

102 [label="EVAL with clause\nbool(1).\nand substitution\nT53 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 102 [arrowhead = none ];
102 -> 39;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X70) :- con(X70).\nand substitution\nT4 -> T66\nT5 -> T67\nX70 -> or(T66, T67)", fontsize=14, style = filled, fillcolor = white];
41 -> 106 [arrowhead = none ];
106 -> 42;

107 [label="INSTANCE with matching:\nT15 -> or(T66, T67)", 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(X84, X85)) :- ','(dis(X84), con(X85)).\nand substitution\nX84 -> T80\nX85 -> T81\nT71 -> and(T80, T81)", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 111 [arrowhead = none ];
111 -> 47;

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

113 [label="ONLY EVAL with clause\ncon(X92) :- bool(X92).\nand substitution\nT71 -> T86\nX92 -> T86", fontsize=14, style = filled, fillcolor = white];
46 -> 113 [arrowhead = none ];
113 -> 49;

114 [label="INSTANCE with matching:\nT32 -> T80\nT33 -> T81", 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 substitution\nT86 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 118 [arrowhead = none ];
118 -> 53;

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

120 [label="EVAL with clause\nbool(1).\nand substitution\nT86 -> 1", fontsize=14, style = filled, fillcolor = lightblue];
52 -> 120 [arrowhead = none ];
120 -> 56;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:

Triples:

con12(and(T32, T33)) :- p17(T32, T33).
p17(T32, T33) :- dis1(T32).
p17(T32, T33) :- ','(disc1(T32), con12(T33)).
dis1(or(and(T14, T15), T5)) :- dis1(T14).
dis1(or(and(T14, T15), T5)) :- ','(disc1(T14), con12(T15)).
dis1(or(and(T14, T15), T5)) :- ','(disc1(T14), ','(conc12(T15), dis1(T5))).
dis1(or(0, T5)) :- dis1(T5).
dis1(or(1, T5)) :- dis1(T5).
dis1(or(T66, T67)) :- con12(or(T66, T67)).
dis1(and(T80, T81)) :- p17(T80, T81).

Clauses:

disc1(or(and(T14, T15), T5)) :- ','(disc1(T14), ','(conc12(T15), disc1(T5))).
disc1(or(0, T5)) :- disc1(T5).
disc1(or(1, T5)) :- disc1(T5).
disc1(or(T66, T67)) :- conc12(or(T66, T67)).
disc1(and(T80, T81)) :- qc17(T80, T81).
disc1(0).
disc1(1).
conc12(and(T32, T33)) :- qc17(T32, T33).
conc12(0).
conc12(1).
qc17(T32, T33) :- ','(disc1(T32), conc12(T33)).

Afs:

dis1(x1)  =  dis1(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
dis1_in: (b)
disc1_in: (b)
conc12_in: (b)
qc17_in: (b,b)
con12_in: (b)
p17_in: (b,b)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

DIS1_IN_G(or(and(T14, T15), T5)) → U5_G(T14, T15, T5, dis1_in_g(T14))
DIS1_IN_G(or(and(T14, T15), T5)) → DIS1_IN_G(T14)
DIS1_IN_G(or(and(T14, T15), T5)) → U6_G(T14, T15, T5, disc1_in_g(T14))
U6_G(T14, T15, T5, disc1_out_g(T14)) → U7_G(T14, T15, T5, con12_in_g(T15))
U6_G(T14, T15, T5, disc1_out_g(T14)) → CON12_IN_G(T15)
CON12_IN_G(and(T32, T33)) → U1_G(T32, T33, p17_in_gg(T32, T33))
CON12_IN_G(and(T32, T33)) → P17_IN_GG(T32, T33)
P17_IN_GG(T32, T33) → U2_GG(T32, T33, dis1_in_g(T32))
P17_IN_GG(T32, T33) → DIS1_IN_G(T32)
DIS1_IN_G(or(0, T5)) → U10_G(T5, dis1_in_g(T5))
DIS1_IN_G(or(0, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(or(1, T5)) → U11_G(T5, dis1_in_g(T5))
DIS1_IN_G(or(1, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(or(T66, T67)) → U12_G(T66, T67, con12_in_g(or(T66, T67)))
DIS1_IN_G(or(T66, T67)) → CON12_IN_G(or(T66, T67))
DIS1_IN_G(and(T80, T81)) → U13_G(T80, T81, p17_in_gg(T80, T81))
DIS1_IN_G(and(T80, T81)) → P17_IN_GG(T80, T81)
P17_IN_GG(T32, T33) → U3_GG(T32, T33, disc1_in_g(T32))
U3_GG(T32, T33, disc1_out_g(T32)) → U4_GG(T32, T33, con12_in_g(T33))
U3_GG(T32, T33, disc1_out_g(T32)) → CON12_IN_G(T33)
U6_G(T14, T15, T5, disc1_out_g(T14)) → U8_G(T14, T15, T5, conc12_in_g(T15))
U8_G(T14, T15, T5, conc12_out_g(T15)) → U9_G(T14, T15, T5, dis1_in_g(T5))
U8_G(T14, T15, T5, conc12_out_g(T15)) → DIS1_IN_G(T5)

The TRS R consists of the following rules:

disc1_in_g(or(and(T14, T15), T5)) → U15_g(T14, T15, T5, disc1_in_g(T14))
disc1_in_g(or(0, T5)) → U18_g(T5, disc1_in_g(T5))
disc1_in_g(or(1, T5)) → U19_g(T5, disc1_in_g(T5))
disc1_in_g(or(T66, T67)) → U20_g(T66, T67, conc12_in_g(or(T66, T67)))
conc12_in_g(and(T32, T33)) → U22_g(T32, T33, qc17_in_gg(T32, T33))
qc17_in_gg(T32, T33) → U23_gg(T32, T33, disc1_in_g(T32))
disc1_in_g(and(T80, T81)) → U21_g(T80, T81, qc17_in_gg(T80, T81))
U21_g(T80, T81, qc17_out_gg(T80, T81)) → disc1_out_g(and(T80, T81))
disc1_in_g(0) → disc1_out_g(0)
disc1_in_g(1) → disc1_out_g(1)
U23_gg(T32, T33, disc1_out_g(T32)) → U24_gg(T32, T33, conc12_in_g(T33))
conc12_in_g(0) → conc12_out_g(0)
conc12_in_g(1) → conc12_out_g(1)
U24_gg(T32, T33, conc12_out_g(T33)) → qc17_out_gg(T32, T33)
U22_g(T32, T33, qc17_out_gg(T32, T33)) → conc12_out_g(and(T32, T33))
U20_g(T66, T67, conc12_out_g(or(T66, T67))) → disc1_out_g(or(T66, T67))
U19_g(T5, disc1_out_g(T5)) → disc1_out_g(or(1, T5))
U18_g(T5, disc1_out_g(T5)) → disc1_out_g(or(0, T5))
U15_g(T14, T15, T5, disc1_out_g(T14)) → U16_g(T14, T15, T5, conc12_in_g(T15))
U16_g(T14, T15, T5, conc12_out_g(T15)) → U17_g(T14, T15, T5, disc1_in_g(T5))
U17_g(T14, T15, T5, disc1_out_g(T5)) → disc1_out_g(or(and(T14, T15), T5))

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

DIS1_IN_G(or(and(T14, T15), T5)) → U5_G(T14, T15, T5, dis1_in_g(T14))
DIS1_IN_G(or(and(T14, T15), T5)) → DIS1_IN_G(T14)
DIS1_IN_G(or(and(T14, T15), T5)) → U6_G(T14, T15, T5, disc1_in_g(T14))
U6_G(T14, T15, T5, disc1_out_g(T14)) → U7_G(T14, T15, T5, con12_in_g(T15))
U6_G(T14, T15, T5, disc1_out_g(T14)) → CON12_IN_G(T15)
CON12_IN_G(and(T32, T33)) → U1_G(T32, T33, p17_in_gg(T32, T33))
CON12_IN_G(and(T32, T33)) → P17_IN_GG(T32, T33)
P17_IN_GG(T32, T33) → U2_GG(T32, T33, dis1_in_g(T32))
P17_IN_GG(T32, T33) → DIS1_IN_G(T32)
DIS1_IN_G(or(0, T5)) → U10_G(T5, dis1_in_g(T5))
DIS1_IN_G(or(0, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(or(1, T5)) → U11_G(T5, dis1_in_g(T5))
DIS1_IN_G(or(1, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(or(T66, T67)) → U12_G(T66, T67, con12_in_g(or(T66, T67)))
DIS1_IN_G(or(T66, T67)) → CON12_IN_G(or(T66, T67))
DIS1_IN_G(and(T80, T81)) → U13_G(T80, T81, p17_in_gg(T80, T81))
DIS1_IN_G(and(T80, T81)) → P17_IN_GG(T80, T81)
P17_IN_GG(T32, T33) → U3_GG(T32, T33, disc1_in_g(T32))
U3_GG(T32, T33, disc1_out_g(T32)) → U4_GG(T32, T33, con12_in_g(T33))
U3_GG(T32, T33, disc1_out_g(T32)) → CON12_IN_G(T33)
U6_G(T14, T15, T5, disc1_out_g(T14)) → U8_G(T14, T15, T5, conc12_in_g(T15))
U8_G(T14, T15, T5, conc12_out_g(T15)) → U9_G(T14, T15, T5, dis1_in_g(T5))
U8_G(T14, T15, T5, conc12_out_g(T15)) → DIS1_IN_G(T5)

The TRS R consists of the following rules:

disc1_in_g(or(and(T14, T15), T5)) → U15_g(T14, T15, T5, disc1_in_g(T14))
disc1_in_g(or(0, T5)) → U18_g(T5, disc1_in_g(T5))
disc1_in_g(or(1, T5)) → U19_g(T5, disc1_in_g(T5))
disc1_in_g(or(T66, T67)) → U20_g(T66, T67, conc12_in_g(or(T66, T67)))
conc12_in_g(and(T32, T33)) → U22_g(T32, T33, qc17_in_gg(T32, T33))
qc17_in_gg(T32, T33) → U23_gg(T32, T33, disc1_in_g(T32))
disc1_in_g(and(T80, T81)) → U21_g(T80, T81, qc17_in_gg(T80, T81))
U21_g(T80, T81, qc17_out_gg(T80, T81)) → disc1_out_g(and(T80, T81))
disc1_in_g(0) → disc1_out_g(0)
disc1_in_g(1) → disc1_out_g(1)
U23_gg(T32, T33, disc1_out_g(T32)) → U24_gg(T32, T33, conc12_in_g(T33))
conc12_in_g(0) → conc12_out_g(0)
conc12_in_g(1) → conc12_out_g(1)
U24_gg(T32, T33, conc12_out_g(T33)) → qc17_out_gg(T32, T33)
U22_g(T32, T33, qc17_out_gg(T32, T33)) → conc12_out_g(and(T32, T33))
U20_g(T66, T67, conc12_out_g(or(T66, T67))) → disc1_out_g(or(T66, T67))
U19_g(T5, disc1_out_g(T5)) → disc1_out_g(or(1, T5))
U18_g(T5, disc1_out_g(T5)) → disc1_out_g(or(0, T5))
U15_g(T14, T15, T5, disc1_out_g(T14)) → U16_g(T14, T15, T5, conc12_in_g(T15))
U16_g(T14, T15, T5, conc12_out_g(T15)) → U17_g(T14, T15, T5, disc1_in_g(T5))
U17_g(T14, T15, T5, disc1_out_g(T5)) → disc1_out_g(or(and(T14, T15), T5))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 11 less nodes.

(6) Obligation:

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

DIS1_IN_G(or(and(T14, T15), T5)) → U6_G(T14, T15, T5, disc1_in_g(T14))
U6_G(T14, T15, T5, disc1_out_g(T14)) → CON12_IN_G(T15)
CON12_IN_G(and(T32, T33)) → P17_IN_GG(T32, T33)
P17_IN_GG(T32, T33) → DIS1_IN_G(T32)
DIS1_IN_G(or(and(T14, T15), T5)) → DIS1_IN_G(T14)
DIS1_IN_G(or(0, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(or(1, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(and(T80, T81)) → P17_IN_GG(T80, T81)
P17_IN_GG(T32, T33) → U3_GG(T32, T33, disc1_in_g(T32))
U3_GG(T32, T33, disc1_out_g(T32)) → CON12_IN_G(T33)
U6_G(T14, T15, T5, disc1_out_g(T14)) → U8_G(T14, T15, T5, conc12_in_g(T15))
U8_G(T14, T15, T5, conc12_out_g(T15)) → DIS1_IN_G(T5)

The TRS R consists of the following rules:

disc1_in_g(or(and(T14, T15), T5)) → U15_g(T14, T15, T5, disc1_in_g(T14))
disc1_in_g(or(0, T5)) → U18_g(T5, disc1_in_g(T5))
disc1_in_g(or(1, T5)) → U19_g(T5, disc1_in_g(T5))
disc1_in_g(or(T66, T67)) → U20_g(T66, T67, conc12_in_g(or(T66, T67)))
conc12_in_g(and(T32, T33)) → U22_g(T32, T33, qc17_in_gg(T32, T33))
qc17_in_gg(T32, T33) → U23_gg(T32, T33, disc1_in_g(T32))
disc1_in_g(and(T80, T81)) → U21_g(T80, T81, qc17_in_gg(T80, T81))
U21_g(T80, T81, qc17_out_gg(T80, T81)) → disc1_out_g(and(T80, T81))
disc1_in_g(0) → disc1_out_g(0)
disc1_in_g(1) → disc1_out_g(1)
U23_gg(T32, T33, disc1_out_g(T32)) → U24_gg(T32, T33, conc12_in_g(T33))
conc12_in_g(0) → conc12_out_g(0)
conc12_in_g(1) → conc12_out_g(1)
U24_gg(T32, T33, conc12_out_g(T33)) → qc17_out_gg(T32, T33)
U22_g(T32, T33, qc17_out_gg(T32, T33)) → conc12_out_g(and(T32, T33))
U20_g(T66, T67, conc12_out_g(or(T66, T67))) → disc1_out_g(or(T66, T67))
U19_g(T5, disc1_out_g(T5)) → disc1_out_g(or(1, T5))
U18_g(T5, disc1_out_g(T5)) → disc1_out_g(or(0, T5))
U15_g(T14, T15, T5, disc1_out_g(T14)) → U16_g(T14, T15, T5, conc12_in_g(T15))
U16_g(T14, T15, T5, conc12_out_g(T15)) → U17_g(T14, T15, T5, disc1_in_g(T5))
U17_g(T14, T15, T5, disc1_out_g(T5)) → disc1_out_g(or(and(T14, T15), T5))

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

DIS1_IN_G(or(and(T14, T15), T5)) → U6_G(T14, T15, T5, disc1_in_g(T14))
U6_G(T14, T15, T5, disc1_out_g(T14)) → CON12_IN_G(T15)
CON12_IN_G(and(T32, T33)) → P17_IN_GG(T32, T33)
P17_IN_GG(T32, T33) → DIS1_IN_G(T32)
DIS1_IN_G(or(and(T14, T15), T5)) → DIS1_IN_G(T14)
DIS1_IN_G(or(0, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(or(1, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(and(T80, T81)) → P17_IN_GG(T80, T81)
P17_IN_GG(T32, T33) → U3_GG(T32, T33, disc1_in_g(T32))
U3_GG(T32, T33, disc1_out_g(T32)) → CON12_IN_G(T33)
U6_G(T14, T15, T5, disc1_out_g(T14)) → U8_G(T14, T15, T5, conc12_in_g(T15))
U8_G(T14, T15, T5, conc12_out_g(T15)) → DIS1_IN_G(T5)

The TRS R consists of the following rules:

disc1_in_g(or(and(T14, T15), T5)) → U15_g(T14, T15, T5, disc1_in_g(T14))
disc1_in_g(or(0, T5)) → U18_g(T5, disc1_in_g(T5))
disc1_in_g(or(1, T5)) → U19_g(T5, disc1_in_g(T5))
disc1_in_g(or(T66, T67)) → U20_g(T66, T67, conc12_in_g(or(T66, T67)))
disc1_in_g(and(T80, T81)) → U21_g(T80, T81, qc17_in_gg(T80, T81))
disc1_in_g(0) → disc1_out_g(0)
disc1_in_g(1) → disc1_out_g(1)
conc12_in_g(and(T32, T33)) → U22_g(T32, T33, qc17_in_gg(T32, T33))
conc12_in_g(0) → conc12_out_g(0)
conc12_in_g(1) → conc12_out_g(1)
U15_g(T14, T15, T5, disc1_out_g(T14)) → U16_g(T14, T15, T5, conc12_in_g(T15))
U18_g(T5, disc1_out_g(T5)) → disc1_out_g(or(0, T5))
U19_g(T5, disc1_out_g(T5)) → disc1_out_g(or(1, T5))
U21_g(T80, T81, qc17_out_gg(T80, T81)) → disc1_out_g(and(T80, T81))
U22_g(T32, T33, qc17_out_gg(T32, T33)) → conc12_out_g(and(T32, T33))
U16_g(T14, T15, T5, conc12_out_g(T15)) → U17_g(T14, T15, T5, disc1_in_g(T5))
qc17_in_gg(T32, T33) → U23_gg(T32, T33, disc1_in_g(T32))
U17_g(T14, T15, T5, disc1_out_g(T5)) → disc1_out_g(or(and(T14, T15), T5))
U23_gg(T32, T33, disc1_out_g(T32)) → U24_gg(T32, T33, conc12_in_g(T33))
U24_gg(T32, T33, conc12_out_g(T33)) → qc17_out_gg(T32, T33)

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

(9) PiDPToQDPProof (EQUIVALENT transformation)

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

(10) Obligation:

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

DIS1_IN_G(or(and(T14, T15), T5)) → U6_G(T14, T15, T5, disc1_in_g(T14))
U6_G(T14, T15, T5, disc1_out_g(T14)) → CON12_IN_G(T15)
CON12_IN_G(and(T32, T33)) → P17_IN_GG(T32, T33)
P17_IN_GG(T32, T33) → DIS1_IN_G(T32)
DIS1_IN_G(or(and(T14, T15), T5)) → DIS1_IN_G(T14)
DIS1_IN_G(or(0, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(or(1, T5)) → DIS1_IN_G(T5)
DIS1_IN_G(and(T80, T81)) → P17_IN_GG(T80, T81)
P17_IN_GG(T32, T33) → U3_GG(T32, T33, disc1_in_g(T32))
U3_GG(T32, T33, disc1_out_g(T32)) → CON12_IN_G(T33)
U6_G(T14, T15, T5, disc1_out_g(T14)) → U8_G(T14, T15, T5, conc12_in_g(T15))
U8_G(T14, T15, T5, conc12_out_g(T15)) → DIS1_IN_G(T5)

The TRS R consists of the following rules:

disc1_in_g(or(and(T14, T15), T5)) → U15_g(T14, T15, T5, disc1_in_g(T14))
disc1_in_g(or(0, T5)) → U18_g(T5, disc1_in_g(T5))
disc1_in_g(or(1, T5)) → U19_g(T5, disc1_in_g(T5))
disc1_in_g(or(T66, T67)) → U20_g(T66, T67, conc12_in_g(or(T66, T67)))
disc1_in_g(and(T80, T81)) → U21_g(T80, T81, qc17_in_gg(T80, T81))
disc1_in_g(0) → disc1_out_g(0)
disc1_in_g(1) → disc1_out_g(1)
conc12_in_g(and(T32, T33)) → U22_g(T32, T33, qc17_in_gg(T32, T33))
conc12_in_g(0) → conc12_out_g(0)
conc12_in_g(1) → conc12_out_g(1)
U15_g(T14, T15, T5, disc1_out_g(T14)) → U16_g(T14, T15, T5, conc12_in_g(T15))
U18_g(T5, disc1_out_g(T5)) → disc1_out_g(or(0, T5))
U19_g(T5, disc1_out_g(T5)) → disc1_out_g(or(1, T5))
U21_g(T80, T81, qc17_out_gg(T80, T81)) → disc1_out_g(and(T80, T81))
U22_g(T32, T33, qc17_out_gg(T32, T33)) → conc12_out_g(and(T32, T33))
U16_g(T14, T15, T5, conc12_out_g(T15)) → U17_g(T14, T15, T5, disc1_in_g(T5))
qc17_in_gg(T32, T33) → U23_gg(T32, T33, disc1_in_g(T32))
U17_g(T14, T15, T5, disc1_out_g(T5)) → disc1_out_g(or(and(T14, T15), T5))
U23_gg(T32, T33, disc1_out_g(T32)) → U24_gg(T32, T33, conc12_in_g(T33))
U24_gg(T32, T33, conc12_out_g(T33)) → qc17_out_gg(T32, T33)

The set Q consists of the following terms:

disc1_in_g(x0)
conc12_in_g(x0)
U15_g(x0, x1, x2, x3)
U18_g(x0, x1)
U19_g(x0, x1)
U21_g(x0, x1, x2)
U22_g(x0, x1, x2)
U16_g(x0, x1, x2, x3)
qc17_in_gg(x0, x1)
U17_g(x0, x1, x2, x3)
U23_gg(x0, x1, x2)
U24_gg(x0, x1, x2)

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

(11) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

• CON12_IN_G(and(T32, T33)) → P17_IN_GG(T32, T33)
  The graph contains the following edges 1 > 1, 1 > 2

• DIS1_IN_G(or(and(T14, T15), T5)) → U6_G(T14, T15, T5, disc1_in_g(T14))
  The graph contains the following edges 1 > 1, 1 > 2, 1 > 3

• U6_G(T14, T15, T5, disc1_out_g(T14)) → CON12_IN_G(T15)
  The graph contains the following edges 2 >= 1

• U6_G(T14, T15, T5, disc1_out_g(T14)) → U8_G(T14, T15, T5, conc12_in_g(T15))
  The graph contains the following edges 1 >= 1, 4 > 1, 2 >= 2, 3 >= 3

• P17_IN_GG(T32, T33) → DIS1_IN_G(T32)
  The graph contains the following edges 1 >= 1

• P17_IN_GG(T32, T33) → U3_GG(T32, T33, disc1_in_g(T32))
  The graph contains the following edges 1 >= 1, 2 >= 2

• U3_GG(T32, T33, disc1_out_g(T32)) → CON12_IN_G(T33)
  The graph contains the following edges 2 >= 1

• DIS1_IN_G(and(T80, T81)) → P17_IN_GG(T80, T81)
  The graph contains the following edges 1 > 1, 1 > 2

• U8_G(T14, T15, T5, conc12_out_g(T15)) → DIS1_IN_G(T5)
  The graph contains the following edges 3 >= 1

• DIS1_IN_G(or(and(T14, T15), T5)) → DIS1_IN_G(T14)
  The graph contains the following edges 1 > 1

• DIS1_IN_G(or(0, T5)) → DIS1_IN_G(T5)
  The graph contains the following edges 1 > 1

• DIS1_IN_G(or(1, T5)) → DIS1_IN_G(T5)
  The graph contains the following edges 1 > 1

(12) YES