Left Termination proof of /tmp/tmp0m9b9C/ifdiv.pl

Left Termination of the query pattern div_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇐)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇐)

↳4 PiDP

↳5 DependencyGraphProof (⇔)

↳6 AND

    ↳7 PiDP

        ↳8 UsableRulesProof (⇔)

        ↳9 PiDP

        ↳10 PiDPToQDPProof (⇐)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔)

        ↳13 YES

    ↳14 PiDP

        ↳15 UsableRulesProof (⇔)

        ↳16 PiDP

        ↳17 PiDPToQDPProof (⇐)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔)

        ↳20 YES

    ↳21 PiDP

        ↳22 PiDPToQDPProof (⇐)

        ↳23 QDP

        ↳24 Narrowing (⇐)

        ↳25 QDP

        ↳26 Narrowing (⇐)

        ↳27 QDP

        ↳28 UsableRulesProof (⇔)

        ↳29 QDP

        ↳30 QReductionProof (⇔)

        ↳31 QDP

        ↳32 Instantiation (⇔)

        ↳33 QDP

        ↳34 DependencyGraphProof (⇔)

        ↳35 AND

            ↳36 QDP

                ↳37 Instantiation (⇔)

                ↳38 QDP

                ↳39 Instantiation (⇔)

                ↳40 QDP

                ↳41 ForwardInstantiation (⇔)

                ↳42 QDP

                ↳43 QDPOrderProof (⇔)

                ↳44 QDP

                ↳45 DependencyGraphProof (⇔)

                ↳46 TRUE

            ↳47 QDP

                ↳48 UsableRulesProof (⇔)

                ↳49 QDP

                ↳50 QReductionProof (⇔)

                ↳51 QDP

                ↳52 NonTerminationProof (⇔)

                ↳53 NO

(0) Obligation:

Clauses:

le(0, Y, true).
le(s(X), 0, false).
le(s(X), s(Y), B) :- le(X, Y, B).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
div(X, s(Y), Z) :- ','(le(s(Y), X, B), if(B, X, s(Y), Z)).
if(false, X, s(Y), 0).
if(true, X, s(Y), s(Z)) :- ','(minus(X, Y, U), div(U, s(Y), Z)).

Queries:

div(g,g,a).

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: div(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: div(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(le(s(T8), T7, X7), if(X7, T7, s(T8), T10))\n\nT7 is ground\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(le(s(T8), T7, X7), if(X7, T7, s(T8), T10)), SCOPE: 2, CLAUSE: 0 | ','(le(s(T8), T7, X7), if(X7, T7, s(T8), T10)), SCOPE: 2, CLAUSE: 1 | ','(le(s(T8), T7, X7), if(X7, T7, s(T8), T10)), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(le(s(T8), T7, X7), if(X7, T7, s(T8), T10)), SCOPE: 2, CLAUSE: 1 | ','(le(s(T8), T7, X7), if(X7, T7, s(T8), T10)), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(le(s(T8), T7, X7), if(X7, T7, s(T8), T10)), SCOPE: 2, CLAUSE: 1\n\nT7 is ground\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(le(s(T8), T7, X7), if(X7, T7, s(T8), T10)), SCOPE: 2, CLAUSE: 2\n\nT7 is ground\nT8 is ground\nX7 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: if(false, 0, s(T15), T10)\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: if(false, 0, s(T15), T10), SCOPE: 3, CLAUSE: 6 | if(false, 0, s(T15), T10), SCOPE: 3, CLAUSE: 7\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: if(false, 0, s(T15), T10), SCOPE: 3, CLAUSE: 6\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: if(false, 0, s(T15), T10), SCOPE: 3, CLAUSE: 7\n\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(le(T27, T28, X48), if(X48, s(T28), s(T27), T10))\n\nT27 is ground\nT28 is ground\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: le(T27, T28, X48)\n\nT27 is ground\nT28 is ground\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: if(T31, s(T28), s(T27), T10)\n\nT27 is ground\nT28 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: le(T27, T28, X48), SCOPE: 4, CLAUSE: 0 | le(T27, T28, X48), SCOPE: 4, CLAUSE: 1 | le(T27, T28, X48), SCOPE: 4, CLAUSE: 2\n\nT27 is ground\nT28 is ground\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: le(T27, T28, X48), SCOPE: 4, CLAUSE: 0\n\nT27 is ground\nT28 is ground\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: le(T27, T28, X48), SCOPE: 4, CLAUSE: 1 | le(T27, T28, X48), SCOPE: 4, CLAUSE: 2\n\nT27 is ground\nT28 is ground\nX48 is free; ", 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: le(T27, T28, X48), SCOPE: 4, CLAUSE: 1\n\nT27 is ground\nT28 is ground\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: le(T27, T28, X48), SCOPE: 4, CLAUSE: 2\n\nT27 is ground\nT28 is ground\nX48 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: le(T48, T49, X79)\n\nT48 is ground\nT49 is ground\nX79 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: if(T31, s(T28), s(T27), T10), SCOPE: 5, CLAUSE: 6 | if(T31, s(T28), s(T27), T10), SCOPE: 5, CLAUSE: 7\n\nT27 is ground\nT28 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: if(T31, s(T28), s(T27), T10), SCOPE: 5, CLAUSE: 6\n\nT27 is ground\nT28 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: if(T31, s(T28), s(T27), T10), SCOPE: 5, CLAUSE: 7\n\nT27 is ground\nT28 is ground\nT31 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(minus(s(T72), T73, X111), div(X111, s(T73), T75))\n\nT72 is ground\nT73 is ground\nX111 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: minus(s(T72), T73, X111)\n\nT72 is ground\nT73 is ground\nX111 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: div(T78, s(T73), T75)\n\nT73 is ground\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: minus(s(T72), T73, X111), SCOPE: 6, CLAUSE: 3 | minus(s(T72), T73, X111), SCOPE: 6, CLAUSE: 4\n\nT72 is ground\nT73 is ground\nX111 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: minus(s(T72), T73, X111), SCOPE: 6, CLAUSE: 3\n\nT72 is ground\nT73 is ground\nX111 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: minus(s(T72), T73, X111), SCOPE: 6, CLAUSE: 4\n\nT72 is ground\nT73 is ground\nX111 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: minus(T90, T91, X136)\n\nT90 is ground\nT91 is ground\nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: minus(T90, T91, X136), SCOPE: 7, CLAUSE: 3 | minus(T90, T91, X136), SCOPE: 7, CLAUSE: 4\n\nT90 is ground\nT91 is ground\nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: minus(T90, T91, X136), SCOPE: 7, CLAUSE: 3\n\nT90 is ground\nT91 is ground\nX136 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: minus(T90, T91, X136), SCOPE: 7, CLAUSE: 4\n\nT90 is ground\nT91 is ground\nX136 is free; ", 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="59: minus(T103, T104, X159)\n\nT103 is ground\nT104 is ground\nX159 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 61 [arrowhead = none ];
61 -> 2;

62 [label="EVAL with clause\ndiv(X4, s(X5), X6) :- ','(le(s(X5), X4, X7), if(X7, X4, s(X5), X6)).\nand substitution\nT1 -> T7\nX4 -> T7\nX5 -> T8\nT2 -> s(T8)\nT3 -> T10\nX6 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 62 [arrowhead = none ];
62 -> 3;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 63 [arrowhead = none ];
63 -> 4;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 64 [arrowhead = none ];
64 -> 5;

65 [label="BACKTRACK\nfor clause: le(0, Y, true)because of non-unification", fontsize=14, style = filled, fillcolor = white];
5 -> 65 [arrowhead = none ];
65 -> 6;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 66 [arrowhead = none ];
66 -> 7;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
6 -> 67 [arrowhead = none ];
67 -> 8;

68 [label="EVAL with clause\nle(s(X16), 0, false).\nand substitution\nT8 -> T15\nX16 -> T15\nT7 -> 0\nX7 -> false", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 68 [arrowhead = none ];
68 -> 9;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 69 [arrowhead = none ];
69 -> 10;

70 [label="EVAL with clause\nle(s(X45), s(X46), X47) :- le(X45, X46, X47).\nand substitution\nT8 -> T27\nX45 -> T27\nX46 -> T28\nT7 -> s(T28)\nX7 -> X48\nX47 -> X48", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 70 [arrowhead = none ];
70 -> 18;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 71 [arrowhead = none ];
71 -> 19;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 72 [arrowhead = none ];
72 -> 11;

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

74 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 74 [arrowhead = none ];
74 -> 13;

75 [label="EVAL with clause\nif(false, X30, s(X31), 0).\nand substitution\nX30 -> 0\nT15 -> T22\nX31 -> T22\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 75 [arrowhead = none ];
75 -> 14;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 76 [arrowhead = none ];
76 -> 15;

77 [label="BACKTRACK\nfor clause: if(true, X, s(Y), s(Z)) :- ','(minus(X, Y, U), div(U, s(Y), Z))because of non-unification", fontsize=14, style = filled, fillcolor = white];
13 -> 77 [arrowhead = none ];
77 -> 17;

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

79 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
18 -> 79 [arrowhead = none ];
79 -> 20;

80 [label="SPLIT 2\nnew knowledge:\nT31 is ground\nreplacements:\nX48 -> T31", fontsize=14, style = filled, fillcolor = violet];
18 -> 80 [arrowhead = none ];
80 -> 21;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 81 [arrowhead = none ];
81 -> 22;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 82 [arrowhead = none ];
82 -> 35;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 83 [arrowhead = none ];
83 -> 23;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 84 [arrowhead = none ];
84 -> 24;

85 [label="EVAL with clause\nle(0, X60, true).\nand substitution\nT27 -> 0\nT28 -> T38\nX60 -> T38\nX48 -> true", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 85 [arrowhead = none ];
85 -> 25;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 86 [arrowhead = none ];
86 -> 26;

87 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 87 [arrowhead = none ];
87 -> 28;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
24 -> 88 [arrowhead = none ];
88 -> 29;

89 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 89 [arrowhead = none ];
89 -> 27;

90 [label="EVAL with clause\nle(s(X66), 0, false).\nand substitution\nX66 -> T43\nT27 -> s(T43)\nT28 -> 0\nX48 -> false", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 90 [arrowhead = none ];
90 -> 30;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 91 [arrowhead = none ];
91 -> 31;

92 [label="EVAL with clause\nle(s(X76), s(X77), X78) :- le(X76, X77, X78).\nand substitution\nX76 -> T48\nT27 -> s(T48)\nX77 -> T49\nT28 -> s(T49)\nX48 -> X79\nX78 -> X79", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 92 [arrowhead = none ];
92 -> 33;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 93 [arrowhead = none ];
93 -> 34;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 94 [arrowhead = none ];
94 -> 32;

95 [label="INSTANCE with matching:\nT27 -> T48\nT28 -> T49\nX48 -> X79", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 95 [arrowhead = none , style=dashed];
95 -> 20[style=dashed];

96 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 96 [arrowhead = none ];
96 -> 36;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
35 -> 97 [arrowhead = none ];
97 -> 37;

98 [label="EVAL with clause\nif(false, X97, s(X98), 0).\nand substitution\nT31 -> false\nT28 -> T64\nX97 -> s(T64)\nT27 -> T65\nX98 -> T65\nT10 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 98 [arrowhead = none ];
98 -> 38;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 99 [arrowhead = none ];
99 -> 39;

100 [label="EVAL with clause\nif(true, X108, s(X109), s(X110)) :- ','(minus(X108, X109, X111), div(X111, s(X109), X110)).\nand substitution\nT31 -> true\nT28 -> T72\nX108 -> s(T72)\nT27 -> T73\nX109 -> T73\nX110 -> T75\nT10 -> s(T75)\nT74 -> T75", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 100 [arrowhead = none ];
100 -> 41;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 101 [arrowhead = none ];
101 -> 42;

102 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 102 [arrowhead = none ];
102 -> 40;

103 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
41 -> 103 [arrowhead = none ];
103 -> 43;

104 [label="SPLIT 2\nnew knowledge:\nT78 is ground\nreplacements:\nX111 -> T78", fontsize=14, style = filled, fillcolor = violet];
41 -> 104 [arrowhead = none ];
104 -> 44;

105 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 105 [arrowhead = none ];
105 -> 45;

106 [label="INSTANCE with matching:\nT1 -> T78\nT2 -> s(T73)\nT3 -> T75", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 106 [arrowhead = none , style=dashed];
106 -> 1[style=dashed];

107 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 107 [arrowhead = none ];
107 -> 46;

108 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 108 [arrowhead = none ];
108 -> 47;

109 [label="EVAL with clause\nminus(X123, 0, X123).\nand substitution\nT72 -> T85\nX123 -> s(T85)\nT73 -> 0\nX111 -> s(T85)", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 109 [arrowhead = none ];
109 -> 48;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 110 [arrowhead = none ];
110 -> 49;

111 [label="EVAL with clause\nminus(s(X133), s(X134), X135) :- minus(X133, X134, X135).\nand substitution\nT72 -> T90\nX133 -> T90\nX134 -> T91\nT73 -> s(T91)\nX111 -> X136\nX135 -> X136", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 111 [arrowhead = none ];
111 -> 51;

112 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 112 [arrowhead = none ];
112 -> 52;

113 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 113 [arrowhead = none ];
113 -> 50;

114 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
51 -> 114 [arrowhead = none ];
114 -> 53;

115 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 115 [arrowhead = none ];
115 -> 54;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 116 [arrowhead = none ];
116 -> 55;

117 [label="EVAL with clause\nminus(X146, 0, X146).\nand substitution\nT90 -> T98\nX146 -> T98\nT91 -> 0\nX136 -> T98", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 117 [arrowhead = none ];
117 -> 56;

118 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 118 [arrowhead = none ];
118 -> 57;

119 [label="EVAL with clause\nminus(s(X156), s(X157), X158) :- minus(X156, X157, X158).\nand substitution\nX156 -> T103\nT90 -> s(T103)\nX157 -> T104\nT91 -> s(T104)\nX136 -> X159\nX158 -> X159", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 119 [arrowhead = none ];
119 -> 59;

120 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 120 [arrowhead = none ];
120 -> 60;

121 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 121 [arrowhead = none ];
121 -> 58;

122 [label="INSTANCE with matching:\nT90 -> T103\nT91 -> T104\nX136 -> X159", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 122 [arrowhead = none , style=dashed];
122 -> 51[style=dashed];

}

(2) Obligation:

Triples:

le20(s(T48), s(T49), X79) :- le20(T48, T49, X79).
minus51(s(T103), s(T104), X159) :- minus51(T103, T104, X159).
div1(s(T28), s(T27), T10) :- le20(T27, T28, X48).
div1(s(T90), s(s(T91)), s(T75)) :- ','(lec20(s(T91), T90, true), minus51(T90, T91, X136)).
div1(s(T72), s(T73), s(T75)) :- ','(lec20(T73, T72, true), ','(minusc43(T72, T73, T78), div1(T78, s(T73), T75))).

Clauses:

lec20(0, T38, true).
lec20(s(T43), 0, false).
lec20(s(T48), s(T49), X79) :- lec20(T48, T49, X79).
divc1(0, s(T22), 0).
divc1(s(T64), s(T65), 0) :- lec20(T65, T64, false).
divc1(s(T72), s(T73), s(T75)) :- ','(lec20(T73, T72, true), ','(minusc43(T72, T73, T78), divc1(T78, s(T73), T75))).
minusc51(T98, 0, T98).
minusc51(s(T103), s(T104), X159) :- minusc51(T103, T104, X159).
minusc43(T85, 0, s(T85)).
minusc43(T90, s(T91), X136) :- minusc51(T90, T91, X136).

Afs:

div1(x1, x2, x3) = div1(x1, x2)

(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:
div1_in: (b,b,f)
le20_in: (b,b,f)
lec20_in: (b,b,b)
minus51_in: (b,b,f)
minusc43_in: (b,b,f)
minusc51_in: (b,b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

DIV1_IN_GGA(s(T28), s(T27), T10) → U3_GGA(T28, T27, T10, le20_in_gga(T27, T28, X48))
DIV1_IN_GGA(s(T28), s(T27), T10) → LE20_IN_GGA(T27, T28, X48)
LE20_IN_GGA(s(T48), s(T49), X79) → U1_GGA(T48, T49, X79, le20_in_gga(T48, T49, X79))
LE20_IN_GGA(s(T48), s(T49), X79) → LE20_IN_GGA(T48, T49, X79)
DIV1_IN_GGA(s(T90), s(s(T91)), s(T75)) → U4_GGA(T90, T91, T75, lec20_in_ggg(s(T91), T90, true))
U4_GGA(T90, T91, T75, lec20_out_ggg(s(T91), T90, true)) → U5_GGA(T90, T91, T75, minus51_in_gga(T90, T91, X136))
U4_GGA(T90, T91, T75, lec20_out_ggg(s(T91), T90, true)) → MINUS51_IN_GGA(T90, T91, X136)
MINUS51_IN_GGA(s(T103), s(T104), X159) → U2_GGA(T103, T104, X159, minus51_in_gga(T103, T104, X159))
MINUS51_IN_GGA(s(T103), s(T104), X159) → MINUS51_IN_GGA(T103, T104, X159)
DIV1_IN_GGA(s(T72), s(T73), s(T75)) → U6_GGA(T72, T73, T75, lec20_in_ggg(T73, T72, true))
U6_GGA(T72, T73, T75, lec20_out_ggg(T73, T72, true)) → U7_GGA(T72, T73, T75, minusc43_in_gga(T72, T73, T78))
U7_GGA(T72, T73, T75, minusc43_out_gga(T72, T73, T78)) → U8_GGA(T72, T73, T75, div1_in_gga(T78, s(T73), T75))
U7_GGA(T72, T73, T75, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73), T75)

The TRS R consists of the following rules:

lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
minusc43_in_gga(T85, 0, s(T85)) → minusc43_out_gga(T85, 0, s(T85))
minusc43_in_gga(T90, s(T91), X136) → U16_gga(T90, T91, X136, minusc51_in_gga(T90, T91, X136))
minusc51_in_gga(T98, 0, T98) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104), X159) → U15_gga(T103, T104, X159, minusc51_in_gga(T103, T104, X159))
U15_gga(T103, T104, X159, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
U16_gga(T90, T91, X136, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)

The argument filtering Pi contains the following mapping:
div1_in_gga(x1, x2, x3) = div1_in_gga(x1, x2)
s(x1) = s(x1)
le20_in_gga(x1, x2, x3) = le20_in_gga(x1, x2)
lec20_in_ggg(x1, x2, x3) = lec20_in_ggg(x1, x2, x3)
0 = 0
true = true
lec20_out_ggg(x1, x2, x3) = lec20_out_ggg(x1, x2, x3)
false = false
U10_ggg(x1, x2, x3, x4) = U10_ggg(x1, x2, x3, x4)
minus51_in_gga(x1, x2, x3) = minus51_in_gga(x1, x2)
minusc43_in_gga(x1, x2, x3) = minusc43_in_gga(x1, x2)
minusc43_out_gga(x1, x2, x3) = minusc43_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4) = U16_gga(x1, x2, x4)
minusc51_in_gga(x1, x2, x3) = minusc51_in_gga(x1, x2)
minusc51_out_gga(x1, x2, x3) = minusc51_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3, x4) = U15_gga(x1, x2, x4)
DIV1_IN_GGA(x1, x2, x3) = DIV1_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x1, x2, x4)
LE20_IN_GGA(x1, x2, x3) = LE20_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x1, x2, x4)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
U5_GGA(x1, x2, x3, x4) = U5_GGA(x1, x2, x4)
MINUS51_IN_GGA(x1, x2, x3) = MINUS51_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4)

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:

DIV1_IN_GGA(s(T28), s(T27), T10) → U3_GGA(T28, T27, T10, le20_in_gga(T27, T28, X48))
DIV1_IN_GGA(s(T28), s(T27), T10) → LE20_IN_GGA(T27, T28, X48)
LE20_IN_GGA(s(T48), s(T49), X79) → U1_GGA(T48, T49, X79, le20_in_gga(T48, T49, X79))
LE20_IN_GGA(s(T48), s(T49), X79) → LE20_IN_GGA(T48, T49, X79)
DIV1_IN_GGA(s(T90), s(s(T91)), s(T75)) → U4_GGA(T90, T91, T75, lec20_in_ggg(s(T91), T90, true))
U4_GGA(T90, T91, T75, lec20_out_ggg(s(T91), T90, true)) → U5_GGA(T90, T91, T75, minus51_in_gga(T90, T91, X136))
U4_GGA(T90, T91, T75, lec20_out_ggg(s(T91), T90, true)) → MINUS51_IN_GGA(T90, T91, X136)
MINUS51_IN_GGA(s(T103), s(T104), X159) → U2_GGA(T103, T104, X159, minus51_in_gga(T103, T104, X159))
MINUS51_IN_GGA(s(T103), s(T104), X159) → MINUS51_IN_GGA(T103, T104, X159)
DIV1_IN_GGA(s(T72), s(T73), s(T75)) → U6_GGA(T72, T73, T75, lec20_in_ggg(T73, T72, true))
U6_GGA(T72, T73, T75, lec20_out_ggg(T73, T72, true)) → U7_GGA(T72, T73, T75, minusc43_in_gga(T72, T73, T78))
U7_GGA(T72, T73, T75, minusc43_out_gga(T72, T73, T78)) → U8_GGA(T72, T73, T75, div1_in_gga(T78, s(T73), T75))
U7_GGA(T72, T73, T75, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73), T75)

The TRS R consists of the following rules:

lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
minusc43_in_gga(T85, 0, s(T85)) → minusc43_out_gga(T85, 0, s(T85))
minusc43_in_gga(T90, s(T91), X136) → U16_gga(T90, T91, X136, minusc51_in_gga(T90, T91, X136))
minusc51_in_gga(T98, 0, T98) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104), X159) → U15_gga(T103, T104, X159, minusc51_in_gga(T103, T104, X159))
U15_gga(T103, T104, X159, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
U16_gga(T90, T91, X136, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 8 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

MINUS51_IN_GGA(s(T103), s(T104), X159) → MINUS51_IN_GGA(T103, T104, X159)

The TRS R consists of the following rules:

lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
minusc43_in_gga(T85, 0, s(T85)) → minusc43_out_gga(T85, 0, s(T85))
minusc43_in_gga(T90, s(T91), X136) → U16_gga(T90, T91, X136, minusc51_in_gga(T90, T91, X136))
minusc51_in_gga(T98, 0, T98) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104), X159) → U15_gga(T103, T104, X159, minusc51_in_gga(T103, T104, X159))
U15_gga(T103, T104, X159, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
U16_gga(T90, T91, X136, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
lec20_in_ggg(x1, x2, x3) = lec20_in_ggg(x1, x2, x3)
0 = 0
true = true
lec20_out_ggg(x1, x2, x3) = lec20_out_ggg(x1, x2, x3)
false = false
U10_ggg(x1, x2, x3, x4) = U10_ggg(x1, x2, x3, x4)
minusc43_in_gga(x1, x2, x3) = minusc43_in_gga(x1, x2)
minusc43_out_gga(x1, x2, x3) = minusc43_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4) = U16_gga(x1, x2, x4)
minusc51_in_gga(x1, x2, x3) = minusc51_in_gga(x1, x2)
minusc51_out_gga(x1, x2, x3) = minusc51_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3, x4) = U15_gga(x1, x2, x4)
MINUS51_IN_GGA(x1, x2, x3) = MINUS51_IN_GGA(x1, x2)

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

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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

MINUS51_IN_GGA(s(T103), s(T104), X159) → MINUS51_IN_GGA(T103, T104, X159)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
MINUS51_IN_GGA(x1, x2, x3) = MINUS51_IN_GGA(x1, x2)

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

(10) PiDPToQDPProof (SOUND transformation)

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

(11) Obligation:

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

MINUS51_IN_GGA(s(T103), s(T104)) → MINUS51_IN_GGA(T103, T104)

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

(12) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

MINUS51_IN_GGA(s(T103), s(T104)) → MINUS51_IN_GGA(T103, T104)
The graph contains the following edges 1 > 1, 2 > 2

(13) YES

(14) Obligation:

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

LE20_IN_GGA(s(T48), s(T49), X79) → LE20_IN_GGA(T48, T49, X79)

The TRS R consists of the following rules:

lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
minusc43_in_gga(T85, 0, s(T85)) → minusc43_out_gga(T85, 0, s(T85))
minusc43_in_gga(T90, s(T91), X136) → U16_gga(T90, T91, X136, minusc51_in_gga(T90, T91, X136))
minusc51_in_gga(T98, 0, T98) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104), X159) → U15_gga(T103, T104, X159, minusc51_in_gga(T103, T104, X159))
U15_gga(T103, T104, X159, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
U16_gga(T90, T91, X136, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
lec20_in_ggg(x1, x2, x3) = lec20_in_ggg(x1, x2, x3)
0 = 0
true = true
lec20_out_ggg(x1, x2, x3) = lec20_out_ggg(x1, x2, x3)
false = false
U10_ggg(x1, x2, x3, x4) = U10_ggg(x1, x2, x3, x4)
minusc43_in_gga(x1, x2, x3) = minusc43_in_gga(x1, x2)
minusc43_out_gga(x1, x2, x3) = minusc43_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4) = U16_gga(x1, x2, x4)
minusc51_in_gga(x1, x2, x3) = minusc51_in_gga(x1, x2)
minusc51_out_gga(x1, x2, x3) = minusc51_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3, x4) = U15_gga(x1, x2, x4)
LE20_IN_GGA(x1, x2, x3) = LE20_IN_GGA(x1, x2)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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

LE20_IN_GGA(s(T48), s(T49), X79) → LE20_IN_GGA(T48, T49, X79)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LE20_IN_GGA(x1, x2, x3) = LE20_IN_GGA(x1, x2)

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

(17) PiDPToQDPProof (SOUND transformation)

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

(18) Obligation:

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

LE20_IN_GGA(s(T48), s(T49)) → LE20_IN_GGA(T48, T49)

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

(19) 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:

LE20_IN_GGA(s(T48), s(T49)) → LE20_IN_GGA(T48, T49)
The graph contains the following edges 1 > 1, 2 > 2

(20) YES

(21) Obligation:

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

DIV1_IN_GGA(s(T72), s(T73), s(T75)) → U6_GGA(T72, T73, T75, lec20_in_ggg(T73, T72, true))
U6_GGA(T72, T73, T75, lec20_out_ggg(T73, T72, true)) → U7_GGA(T72, T73, T75, minusc43_in_gga(T72, T73, T78))
U7_GGA(T72, T73, T75, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73), T75)

The TRS R consists of the following rules:

lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
minusc43_in_gga(T85, 0, s(T85)) → minusc43_out_gga(T85, 0, s(T85))
minusc43_in_gga(T90, s(T91), X136) → U16_gga(T90, T91, X136, minusc51_in_gga(T90, T91, X136))
minusc51_in_gga(T98, 0, T98) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104), X159) → U15_gga(T103, T104, X159, minusc51_in_gga(T103, T104, X159))
U15_gga(T103, T104, X159, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
U16_gga(T90, T91, X136, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
lec20_in_ggg(x1, x2, x3) = lec20_in_ggg(x1, x2, x3)
0 = 0
true = true
lec20_out_ggg(x1, x2, x3) = lec20_out_ggg(x1, x2, x3)
false = false
U10_ggg(x1, x2, x3, x4) = U10_ggg(x1, x2, x3, x4)
minusc43_in_gga(x1, x2, x3) = minusc43_in_gga(x1, x2)
minusc43_out_gga(x1, x2, x3) = minusc43_out_gga(x1, x2, x3)
U16_gga(x1, x2, x3, x4) = U16_gga(x1, x2, x4)
minusc51_in_gga(x1, x2, x3) = minusc51_in_gga(x1, x2)
minusc51_out_gga(x1, x2, x3) = minusc51_out_gga(x1, x2, x3)
U15_gga(x1, x2, x3, x4) = U15_gga(x1, x2, x4)
DIV1_IN_GGA(x1, x2, x3) = DIV1_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4) = U7_GGA(x1, x2, x4)

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

(22) PiDPToQDPProof (SOUND transformation)

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

(23) Obligation:

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

DIV1_IN_GGA(s(T72), s(T73)) → U6_GGA(T72, T73, lec20_in_ggg(T73, T72, true))
U6_GGA(T72, T73, lec20_out_ggg(T73, T72, true)) → U7_GGA(T72, T73, minusc43_in_gga(T72, T73))
U7_GGA(T72, T73, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73))

The TRS R consists of the following rules:

lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
minusc43_in_gga(T85, 0) → minusc43_out_gga(T85, 0, s(T85))
minusc43_in_gga(T90, s(T91)) → U16_gga(T90, T91, minusc51_in_gga(T90, T91))
minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc43_in_gga(x0, x1)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(24) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule DIV1_IN_GGA(s(T72), s(T73)) → U6_GGA(T72, T73, lec20_in_ggg(T73, T72, true)) at position [2] we obtained the following new rules [LPAR04]:

DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))
DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))

(25) Obligation:

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

U6_GGA(T72, T73, lec20_out_ggg(T73, T72, true)) → U7_GGA(T72, T73, minusc43_in_gga(T72, T73))
U7_GGA(T72, T73, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73))
DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))
DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))

The TRS R consists of the following rules:

lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
minusc43_in_gga(T85, 0) → minusc43_out_gga(T85, 0, s(T85))
minusc43_in_gga(T90, s(T91)) → U16_gga(T90, T91, minusc51_in_gga(T90, T91))
minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc43_in_gga(x0, x1)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(26) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule U6_GGA(T72, T73, lec20_out_ggg(T73, T72, true)) → U7_GGA(T72, T73, minusc43_in_gga(T72, T73)) at position [2] we obtained the following new rules [LPAR04]:

U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))
U6_GGA(x0, s(x1), lec20_out_ggg(s(x1), x0, true)) → U7_GGA(x0, s(x1), U16_gga(x0, x1, minusc51_in_gga(x0, x1)))

(27) Obligation:

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

U7_GGA(T72, T73, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73))
DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))
DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))
U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))
U6_GGA(x0, s(x1), lec20_out_ggg(s(x1), x0, true)) → U7_GGA(x0, s(x1), U16_gga(x0, x1, minusc51_in_gga(x0, x1)))

The TRS R consists of the following rules:

lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
minusc43_in_gga(T85, 0) → minusc43_out_gga(T85, 0, s(T85))
minusc43_in_gga(T90, s(T91)) → U16_gga(T90, T91, minusc51_in_gga(T90, T91))
minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc43_in_gga(x0, x1)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(28) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(29) Obligation:

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

U7_GGA(T72, T73, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73))
DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))
DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))
U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))
U6_GGA(x0, s(x1), lec20_out_ggg(s(x1), x0, true)) → U7_GGA(x0, s(x1), U16_gga(x0, x1, minusc51_in_gga(x0, x1)))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc43_in_gga(x0, x1)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(30) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

minusc43_in_gga(x0, x1)

(31) Obligation:

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

U7_GGA(T72, T73, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73))
DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))
DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))
U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))
U6_GGA(x0, s(x1), lec20_out_ggg(s(x1), x0, true)) → U7_GGA(x0, s(x1), U16_gga(x0, x1, minusc51_in_gga(x0, x1)))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(32) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_GGA(T72, T73, minusc43_out_gga(T72, T73, T78)) → DIV1_IN_GGA(T78, s(T73)) we obtained the following new rules [LPAR04]:

U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0))) → DIV1_IN_GGA(s(z0), s(0))
U7_GGA(z0, s(z1), minusc43_out_gga(z0, s(z1), x2)) → DIV1_IN_GGA(x2, s(s(z1)))

(33) Obligation:

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

DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))
DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))
U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))
U6_GGA(x0, s(x1), lec20_out_ggg(s(x1), x0, true)) → U7_GGA(x0, s(x1), U16_gga(x0, x1, minusc51_in_gga(x0, x1)))
U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0))) → DIV1_IN_GGA(s(z0), s(0))
U7_GGA(z0, s(z1), minusc43_out_gga(z0, s(z1), x2)) → DIV1_IN_GGA(x2, s(s(z1)))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(34) DependencyGraphProof (EQUIVALENT transformation)

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

(35) Complex Obligation (AND)

(36) Obligation:

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

U6_GGA(x0, s(x1), lec20_out_ggg(s(x1), x0, true)) → U7_GGA(x0, s(x1), U16_gga(x0, x1, minusc51_in_gga(x0, x1)))
U7_GGA(z0, s(z1), minusc43_out_gga(z0, s(z1), x2)) → DIV1_IN_GGA(x2, s(s(z1)))
DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(37) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GGA(x0, s(x1), lec20_out_ggg(s(x1), x0, true)) → U7_GGA(x0, s(x1), U16_gga(x0, x1, minusc51_in_gga(x0, x1))) we obtained the following new rules [LPAR04]:

U6_GGA(s(z0), s(z1), lec20_out_ggg(s(z1), s(z0), true)) → U7_GGA(s(z0), s(z1), U16_gga(s(z0), z1, minusc51_in_gga(s(z0), z1)))

(38) Obligation:

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

U7_GGA(z0, s(z1), minusc43_out_gga(z0, s(z1), x2)) → DIV1_IN_GGA(x2, s(s(z1)))
DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))
U6_GGA(s(z0), s(z1), lec20_out_ggg(s(z1), s(z0), true)) → U7_GGA(s(z0), s(z1), U16_gga(s(z0), z1, minusc51_in_gga(s(z0), z1)))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(39) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U7_GGA(z0, s(z1), minusc43_out_gga(z0, s(z1), x2)) → DIV1_IN_GGA(x2, s(s(z1))) we obtained the following new rules [LPAR04]:

U7_GGA(s(z0), s(z1), minusc43_out_gga(s(z0), s(z1), x2)) → DIV1_IN_GGA(x2, s(s(z1)))

(40) Obligation:

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

DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))
U6_GGA(s(z0), s(z1), lec20_out_ggg(s(z1), s(z0), true)) → U7_GGA(s(z0), s(z1), U16_gga(s(z0), z1, minusc51_in_gga(s(z0), z1)))
U7_GGA(s(z0), s(z1), minusc43_out_gga(s(z0), s(z1), x2)) → DIV1_IN_GGA(x2, s(s(z1)))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(41) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule U7_GGA(s(z0), s(z1), minusc43_out_gga(s(z0), s(z1), x2)) → DIV1_IN_GGA(x2, s(s(z1))) we obtained the following new rules [LPAR04]:

U7_GGA(s(x0), s(x1), minusc43_out_gga(s(x0), s(x1), s(s(y_0)))) → DIV1_IN_GGA(s(s(y_0)), s(s(x1)))

(42) Obligation:

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

DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))
U6_GGA(s(z0), s(z1), lec20_out_ggg(s(z1), s(z0), true)) → U7_GGA(s(z0), s(z1), U16_gga(s(z0), z1, minusc51_in_gga(s(z0), z1)))
U7_GGA(s(x0), s(x1), minusc43_out_gga(s(x0), s(x1), s(s(y_0)))) → DIV1_IN_GGA(s(s(y_0)), s(s(x1)))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(43) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U6_GGA(s(z0), s(z1), lec20_out_ggg(s(z1), s(z0), true)) → U7_GGA(s(z0), s(z1), U16_gga(s(z0), z1, minusc51_in_gga(s(z0), z1)))

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

POL(0) = 0
POL(DIV1_IN_GGA(x₁, x₂)) = x₁
POL(U10_ggg(x₁, x₂, x₃, x₄)) = 0
POL(U15_gga(x₁, x₂, x₃)) = x₃
POL(U16_gga(x₁, x₂, x₃)) = x₃
POL(U6_GGA(x₁, x₂, x₃)) = 1 + x₁
POL(U7_GGA(x₁, x₂, x₃)) = x₃
POL(false) = 0
POL(lec20_in_ggg(x₁, x₂, x₃)) = 0
POL(lec20_out_ggg(x₁, x₂, x₃)) = 0
POL(minusc43_out_gga(x₁, x₂, x₃)) = x₃
POL(minusc51_in_gga(x₁, x₂)) = x₁
POL(minusc51_out_gga(x₁, x₂, x₃)) = x₃
POL(s(x₁)) = 1 + x₁
POL(true) = 0

The following usable rules [FROCOS05] were oriented:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)

(44) Obligation:

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

DIV1_IN_GGA(s(s(x1)), s(s(x0))) → U6_GGA(s(x1), s(x0), U10_ggg(x0, x1, true, lec20_in_ggg(x0, x1, true)))
U7_GGA(s(x0), s(x1), minusc43_out_gga(s(x0), s(x1), s(s(y_0)))) → DIV1_IN_GGA(s(s(y_0)), s(s(x1)))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(45) DependencyGraphProof (EQUIVALENT transformation)

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

(46) TRUE

(47) Obligation:

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

U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))
U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0))) → DIV1_IN_GGA(s(z0), s(0))
DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))

The TRS R consists of the following rules:

minusc51_in_gga(T98, 0) → minusc51_out_gga(T98, 0, T98)
minusc51_in_gga(s(T103), s(T104)) → U15_gga(T103, T104, minusc51_in_gga(T103, T104))
U16_gga(T90, T91, minusc51_out_gga(T90, T91, X136)) → minusc43_out_gga(T90, s(T91), X136)
U15_gga(T103, T104, minusc51_out_gga(T103, T104, X159)) → minusc51_out_gga(s(T103), s(T104), X159)
lec20_in_ggg(0, T38, true) → lec20_out_ggg(0, T38, true)
lec20_in_ggg(s(T48), s(T49), X79) → U10_ggg(T48, T49, X79, lec20_in_ggg(T48, T49, X79))
U10_ggg(T48, T49, X79, lec20_out_ggg(T48, T49, X79)) → lec20_out_ggg(s(T48), s(T49), X79)
lec20_in_ggg(s(T43), 0, false) → lec20_out_ggg(s(T43), 0, false)

The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(48) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(49) Obligation:

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

U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))
U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0))) → DIV1_IN_GGA(s(z0), s(0))
DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))

R is empty.
The set Q consists of the following terms:

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

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

(50) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

lec20_in_ggg(x0, x1, x2)
U10_ggg(x0, x1, x2, x3)
minusc51_in_gga(x0, x1)
U15_gga(x0, x1, x2)
U16_gga(x0, x1, x2)

(51) Obligation:

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

U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))
U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0))) → DIV1_IN_GGA(s(z0), s(0))
DIV1_IN_GGA(s(x0), s(0)) → U6_GGA(x0, 0, lec20_out_ggg(0, x0, true))

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

(52) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0))) evaluates to t =U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0)))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0))) → DIV1_IN_GGA(s(z0), s(0))
with rule U7_GGA(z0', 0, minusc43_out_gga(z0', 0, s(z0'))) → DIV1_IN_GGA(s(z0'), s(0)) at position [] and matcher [z0' / z0]

DIV1_IN_GGA(s(z0), s(0)) → U6_GGA(z0, 0, lec20_out_ggg(0, z0, true))
with rule DIV1_IN_GGA(s(x0'), s(0)) → U6_GGA(x0', 0, lec20_out_ggg(0, x0', true)) at position [] and matcher [x0' / z0]

U6_GGA(z0, 0, lec20_out_ggg(0, z0, true)) → U7_GGA(z0, 0, minusc43_out_gga(z0, 0, s(z0)))
with rule U6_GGA(x0, 0, lec20_out_ggg(0, x0, true)) → U7_GGA(x0, 0, minusc43_out_gga(x0, 0, s(x0)))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(0) Obligation:

(1) PrologToDTProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) TriplesToPiDPProof (SOUND transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) PiDPToQDPProof (SOUND transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) YES

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) PiDPToQDPProof (SOUND transformation)

(18) Obligation:

(19) QDPSizeChangeProof (EQUIVALENT transformation)

(20) YES

(21) Obligation:

(22) PiDPToQDPProof (SOUND transformation)

(23) Obligation:

(24) Narrowing (SOUND transformation)

(25) Obligation:

(26) Narrowing (SOUND transformation)

(27) Obligation:

(28) UsableRulesProof (EQUIVALENT transformation)

(29) Obligation:

(30) QReductionProof (EQUIVALENT transformation)

(31) Obligation:

(32) Instantiation (EQUIVALENT transformation)

(33) Obligation:

(34) DependencyGraphProof (EQUIVALENT transformation)

(35) Complex Obligation (AND)

(36) Obligation:

(37) Instantiation (EQUIVALENT transformation)

(38) Obligation:

(39) Instantiation (EQUIVALENT transformation)

(40) Obligation:

(41) ForwardInstantiation (EQUIVALENT transformation)

(42) Obligation:

(43) QDPOrderProof (EQUIVALENT transformation)

(44) Obligation:

(45) DependencyGraphProof (EQUIVALENT transformation)

(46) TRUE

(47) Obligation:

(48) UsableRulesProof (EQUIVALENT transformation)

(49) Obligation:

(50) QReductionProof (EQUIVALENT transformation)

(51) Obligation:

(52) NonTerminationProof (EQUIVALENT transformation)

(53) NO