Left Termination proof of /tmp/tmpUGk48d/slowsort-fb.pl

Left Termination of the query pattern ss_in_2(a, g) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

    ↳3 PrologToPiTRSProof (⇐)

        ↳4 PiTRS

        ↳5 DependencyPairsProof (⇔)

        ↳6 PiDP

        ↳7 DependencyGraphProof (⇔)

        ↳8 AND

            ↳9 PiDP

                ↳10 UsableRulesProof (⇔)

                ↳11 PiDP

                ↳12 PiDPToQDPProof (⇔)

                ↳13 QDP

                ↳14 QDPSizeChangeProof (⇔)

                ↳15 YES

            ↳16 PiDP

                ↳17 UsableRulesProof (⇔)

                ↳18 PiDP

                ↳19 PiDPToQDPProof (⇐)

                ↳20 QDP

                ↳21 QDPSizeChangeProof (⇔)

                ↳22 YES

            ↳23 PiDP

                ↳24 UsableRulesProof (⇔)

                ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                ↳27 QDP

                ↳28 NonTerminationProof (⇔)

                ↳29 NO

            ↳30 PiDP

                ↳31 UsableRulesProof (⇔)

                ↳32 PiDP

                ↳33 PiDPToQDPProof (⇐)

                ↳34 QDP

                ↳35 NonTerminationProof (⇔)

                ↳36 NO

            ↳37 PiDP

                ↳38 UsableRulesProof (⇔)

                ↳39 PiDP

                ↳40 PiDPToQDPProof (⇐)

                ↳41 QDP

                ↳42 QDPSizeChangeProof (⇔)

                ↳43 YES

    ↳44 PrologToPiTRSProof (⇐)

        ↳45 PiTRS

        ↳46 DependencyPairsProof (⇔)

        ↳47 PiDP

        ↳48 DependencyGraphProof (⇔)

        ↳49 AND

            ↳50 PiDP

                ↳51 UsableRulesProof (⇔)

                ↳52 PiDP

                ↳53 PiDPToQDPProof (⇔)

                ↳54 QDP

                ↳55 QDPSizeChangeProof (⇔)

                ↳56 YES

            ↳57 PiDP

                ↳58 UsableRulesProof (⇔)

                ↳59 PiDP

                ↳60 PiDPToQDPProof (⇔)

                ↳61 QDP

                ↳62 QDPSizeChangeProof (⇔)

                ↳63 YES

            ↳64 PiDP

                ↳65 UsableRulesProof (⇔)

                ↳66 PiDP

                ↳67 PiDPToQDPProof (⇐)

                ↳68 QDP

                ↳69 NonTerminationProof (⇔)

                ↳70 NO

            ↳71 PiDP

                ↳72 UsableRulesProof (⇔)

                ↳73 PiDP

                ↳74 PiDPToQDPProof (⇐)

                ↳75 QDP

                ↳76 NonTerminationProof (⇔)

                ↳77 NO

            ↳78 PiDP

                ↳79 UsableRulesProof (⇔)

                ↳80 PiDP

                ↳81 PiDPToQDPProof (⇐)

                ↳82 QDP

                ↳83 QDPSizeChangeProof (⇔)

                ↳84 YES

(0) Obligation:

Clauses:

ss(Xs, Ys) :- ','(perm(Xs, Ys), ordered(Ys)).
perm([], []).
perm(Xs, .(X, Ys)) :- ','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).
app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).
ordered([]).
ordered(.(X1, [])).
ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))).
less(0, s(X2)).
less(s(X), s(Y)) :- less(X, Y).

Queries:

ss(a,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: ss(T1, T2)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: ss(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(perm(T7, T6), ordered(T6))\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(perm(T7, T6), ordered(T6)), SCOPE: 2, CLAUSE: 1 | ','(perm(T7, T6), ordered(T6)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(perm(T7, T6), ordered(T6)), SCOPE: 2, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(perm(T7, T6), ordered(T6)), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ordered([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ordered([]), SCOPE: 3, CLAUSE: 5 | ordered([]), SCOPE: 3, CLAUSE: 6 | ordered([]), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ordered([]), SCOPE: 3, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ordered([]), SCOPE: 3, CLAUSE: 6 | ordered([]), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ordered([]), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(app(X23, .(T15, X24), T17), ','(app(X23, X24, X25), ','(perm(X25, T16), ordered(.(T15, T16)))))\n\nT15 is ground\nT16 is ground\nX23 is free; \nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: app(X23, .(T15, X24), T17)\n\nT15 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(app(T20, T21, X25), ','(perm(X25, T16), ordered(.(T15, T16))))\n\nT15 is ground\nT16 is ground\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(X23, .(T15, X24), T17), SCOPE: 4, CLAUSE: 3 | app(X23, .(T15, X24), T17), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(X23, .(T15, X24), T17), SCOPE: 4, CLAUSE: 3\n\nT15 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(X23, .(T15, X24), T17), SCOPE: 4, CLAUSE: 4\n\nT15 is ground\nX23 is free; \nX24 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: app(X67, .(T33, X68), T35)\n\nT33 is ground\nX67 is free; \nX68 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app(T20, T21, X25)\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(perm(T40, T16), ordered(.(T15, T16)))\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T20, T21, X25), SCOPE: 5, CLAUSE: 3 | app(T20, T21, X25), SCOPE: 5, CLAUSE: 4\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: app(T20, T21, X25), SCOPE: 5, CLAUSE: 3\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T20, T21, X25), SCOPE: 5, CLAUSE: 4\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T57, T58, X101)\n\nX101 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: perm(T40, T16)\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ordered(.(T15, T16))\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: perm(T40, T16), SCOPE: 6, CLAUSE: 1 | perm(T40, T16), SCOPE: 6, CLAUSE: 2\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: perm(T40, T16), SCOPE: 6, CLAUSE: 1\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: perm(T40, T16), SCOPE: 6, CLAUSE: 2\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(app(X120, .(T68, X121), T70), ','(app(X120, X121, X122), perm(X122, T69)))\n\nT68 is ground\nT69 is ground\nX120 is free; \nX121 is free; \nX122 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: app(X120, .(T68, X121), T70)\n\nT68 is ground\nX120 is free; \nX121 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: ','(app(T73, T74, X122), perm(X122, T69))\n\nT69 is ground\nX122 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: app(T73, T74, X122)\n\nX122 is free", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: perm(T79, T69)\n\nT69 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: ordered(.(T15, T16)), SCOPE: 7, CLAUSE: 5 | ordered(.(T15, T16)), SCOPE: 7, CLAUSE: 6 | ordered(.(T15, T16)), SCOPE: 7, CLAUSE: 7\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ordered(.(T15, T16)), SCOPE: 7, CLAUSE: 6 | ordered(.(T15, T16)), SCOPE: 7, CLAUSE: 7\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ordered(.(T15, T16)), SCOPE: 7, CLAUSE: 6\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ordered(.(T15, T16)), SCOPE: 7, CLAUSE: 7\n\nT15 is ground\nT16 is ground", 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: ','(less(T95, s(T96)), ordered(.(T96, T97)))\n\nT95 is ground\nT96 is ground\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: less(T95, s(T96))\n\nT95 is ground\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ordered(.(T96, T97))\n\nT96 is ground\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: less(T95, s(T96)), SCOPE: 8, CLAUSE: 8 | less(T95, s(T96)), SCOPE: 8, CLAUSE: 9\n\nT95 is ground\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: less(T95, s(T96)), SCOPE: 8, CLAUSE: 8\n\nT95 is ground\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: less(T95, s(T96)), SCOPE: 8, CLAUSE: 9\n\nT95 is ground\nT96 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: less(T111, T112)\n\nT111 is ground\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: less(T111, T112), SCOPE: 9, CLAUSE: 8 | less(T111, T112), SCOPE: 9, CLAUSE: 9\n\nT111 is ground\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: less(T111, T112), SCOPE: 9, CLAUSE: 8\n\nT111 is ground\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: less(T111, T112), SCOPE: 9, CLAUSE: 9\n\nT111 is ground\nT112 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: less(T124, T125)\n\nT124 is ground\nT125 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 79 [arrowhead = none ];
79 -> 2;

80 [label="ONLY EVAL with clause\nss(X5, X6) :- ','(perm(X5, X6), ordered(X6)).\nand substitution\nT1 -> T7\nX5 -> T7\nT2 -> T6\nX6 -> T6\nT5 -> T7", fontsize=14, style = filled, fillcolor = white];
2 -> 80 [arrowhead = none ];
80 -> 3;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 81 [arrowhead = none ];
81 -> 4;

82 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 82 [arrowhead = none ];
82 -> 5;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 83 [arrowhead = none ];
83 -> 6;

84 [label="EVAL with clause\nperm([], []).\nand substitution\nT7 -> []\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 84 [arrowhead = none ];
84 -> 7;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 85 [arrowhead = none ];
85 -> 8;

86 [label="EVAL with clause\nperm(X20, .(X21, X22)) :- ','(app(X23, .(X21, X24), X20), ','(app(X23, X24, X25), perm(X25, X22))).\nand substitution\nT7 -> T17\nX20 -> T17\nX21 -> T15\nX22 -> T16\nT6 -> .(T15, T16)\nT14 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 86 [arrowhead = none ];
86 -> 16;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 87 [arrowhead = none ];
87 -> 17;

88 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 88 [arrowhead = none ];
88 -> 9;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 89 [arrowhead = none ];
89 -> 10;

90 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
9 -> 90 [arrowhead = none ];
90 -> 11;

91 [label="ONLY EVAL with clause\nordered([]).\nand substitution", fontsize=14, style = filled, fillcolor = white];
10 -> 91 [arrowhead = none ];
91 -> 12;

92 [label="BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification", fontsize=14, style = filled, fillcolor = white];
11 -> 92 [arrowhead = none ];
92 -> 14;

93 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 93 [arrowhead = none ];
93 -> 13;

94 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = white];
14 -> 94 [arrowhead = none ];
94 -> 15;

95 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
16 -> 95 [arrowhead = none ];
95 -> 18;

96 [label="SPLIT 2\nreplacements:\nX23 -> T20\nX24 -> T21", fontsize=14, style = filled, fillcolor = violet];
16 -> 96 [arrowhead = none ];
96 -> 19;

97 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 97 [arrowhead = none ];
97 -> 20;

98 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
19 -> 98 [arrowhead = none ];
98 -> 28;

99 [label="SPLIT 2\nreplacements:\nX25 -> T40", fontsize=14, style = filled, fillcolor = violet];
19 -> 99 [arrowhead = none ];
99 -> 29;

100 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 100 [arrowhead = none ];
100 -> 21;

101 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 101 [arrowhead = none ];
101 -> 22;

102 [label="EVAL with clause\napp([], X45, X45).\nand substitution\nX23 -> []\nT15 -> T28\nX24 -> X46\nX45 -> .(T28, X46)\nT17 -> .(T28, X46)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 102 [arrowhead = none ];
102 -> 23;

103 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 103 [arrowhead = none ];
103 -> 24;

104 [label="EVAL with clause\napp(.(X62, X63), X64, .(X62, X65)) :- app(X63, X64, X65).\nand substitution\nX62 -> X66\nX63 -> X67\nX23 -> .(X66, X67)\nT15 -> T33\nX24 -> X68\nX64 -> .(T33, X68)\nX65 -> T35\nT17 -> .(X66, T35)\nT34 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 104 [arrowhead = none ];
104 -> 26;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 105 [arrowhead = none ];
105 -> 27;

106 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 106 [arrowhead = none ];
106 -> 25;

107 [label="INSTANCE with matching:\nX23 -> X67\nT15 -> T33\nX24 -> X68\nT17 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
26 -> 107 [arrowhead = none , style=dashed];
107 -> 18[style=dashed];

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 108 [arrowhead = none ];
108 -> 30;

109 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
29 -> 109 [arrowhead = none ];
109 -> 38;

110 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
29 -> 110 [arrowhead = none ];
110 -> 39;

111 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 111 [arrowhead = none ];
111 -> 31;

112 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 112 [arrowhead = none ];
112 -> 32;

113 [label="EVAL with clause\napp([], X85, X85).\nand substitution\nT20 -> []\nT21 -> T47\nX85 -> T47\nX25 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 113 [arrowhead = none ];
113 -> 33;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 114 [arrowhead = none ];
114 -> 34;

115 [label="EVAL with clause\napp(.(X97, X98), X99, .(X97, X100)) :- app(X98, X99, X100).\nand substitution\nX97 -> T54\nX98 -> T57\nT20 -> .(T54, T57)\nT21 -> T58\nX99 -> T58\nX100 -> X101\nX25 -> .(T54, X101)\nT55 -> T57\nT56 -> T58", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 115 [arrowhead = none ];
115 -> 36;

116 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 116 [arrowhead = none ];
116 -> 37;

117 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 117 [arrowhead = none ];
117 -> 35;

118 [label="INSTANCE with matching:\nT20 -> T57\nT21 -> T58\nX25 -> X101", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 118 [arrowhead = none , style=dashed];
118 -> 28[style=dashed];

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
38 -> 119 [arrowhead = none ];
119 -> 40;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 120 [arrowhead = none ];
120 -> 52;

121 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 121 [arrowhead = none ];
121 -> 41;

122 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
40 -> 122 [arrowhead = none ];
122 -> 42;

123 [label="EVAL with clause\nperm([], []).\nand substitution\nT40 -> []\nT16 -> []", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 123 [arrowhead = none ];
123 -> 43;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 124 [arrowhead = none ];
124 -> 44;

125 [label="EVAL with clause\nperm(X117, .(X118, X119)) :- ','(app(X120, .(X118, X121), X117), ','(app(X120, X121, X122), perm(X122, X119))).\nand substitution\nT40 -> T70\nX117 -> T70\nX118 -> T68\nX119 -> T69\nT16 -> .(T68, T69)\nT67 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 125 [arrowhead = none ];
125 -> 46;

126 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 126 [arrowhead = none ];
126 -> 47;

127 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 127 [arrowhead = none ];
127 -> 45;

128 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
46 -> 128 [arrowhead = none ];
128 -> 48;

129 [label="SPLIT 2\nreplacements:\nX120 -> T73\nX121 -> T74", fontsize=14, style = filled, fillcolor = violet];
46 -> 129 [arrowhead = none ];
129 -> 49;

130 [label="INSTANCE with matching:\nX23 -> X120\nT15 -> T68\nX24 -> X121\nT17 -> T70", fontsize=14, style = filled, fillcolor = lightgrey];
48 -> 130 [arrowhead = none , style=dashed];
130 -> 18[style=dashed];

131 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
49 -> 131 [arrowhead = none ];
131 -> 50;

132 [label="SPLIT 2\nreplacements:\nX122 -> T79", fontsize=14, style = filled, fillcolor = violet];
49 -> 132 [arrowhead = none ];
132 -> 51;

133 [label="INSTANCE with matching:\nT20 -> T73\nT21 -> T74\nX25 -> X122", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 133 [arrowhead = none , style=dashed];
133 -> 28[style=dashed];

134 [label="INSTANCE with matching:\nT40 -> T79\nT16 -> T69", fontsize=14, style = filled, fillcolor = lightgrey];
51 -> 134 [arrowhead = none , style=dashed];
134 -> 38[style=dashed];

135 [label="BACKTRACK\nfor clause: ordered([])because of non-unification", fontsize=14, style = filled, fillcolor = white];
52 -> 135 [arrowhead = none ];
135 -> 53;

136 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 136 [arrowhead = none ];
136 -> 54;

137 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
53 -> 137 [arrowhead = none ];
137 -> 55;

138 [label="EVAL with clause\nordered(.(X144, [])).\nand substitution\nT15 -> T88\nX144 -> T88\nT16 -> []", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 138 [arrowhead = none ];
138 -> 56;

139 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 139 [arrowhead = none ];
139 -> 57;

140 [label="EVAL with clause\nordered(.(X152, .(X153, X154))) :- ','(less(X152, s(X153)), ordered(.(X153, X154))).\nand substitution\nT15 -> T95\nX152 -> T95\nX153 -> T96\nX154 -> T97\nT16 -> .(T96, T97)", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 140 [arrowhead = none ];
140 -> 59;

141 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 141 [arrowhead = none ];
141 -> 60;

142 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 142 [arrowhead = none ];
142 -> 58;

143 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
59 -> 143 [arrowhead = none ];
143 -> 61;

144 [label="SPLIT 2", fontsize=14, style = filled, fillcolor = violet];
59 -> 144 [arrowhead = none ];
144 -> 62;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
61 -> 145 [arrowhead = none ];
145 -> 63;

146 [label="INSTANCE with matching:\nT15 -> T96\nT16 -> T97", fontsize=14, style = filled, fillcolor = lightgrey];
62 -> 146 [arrowhead = none , style=dashed];
146 -> 39[style=dashed];

147 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 147 [arrowhead = none ];
147 -> 64;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
63 -> 148 [arrowhead = none ];
148 -> 65;

149 [label="EVAL with clause\nless(0, s(X166)).\nand substitution\nT95 -> 0\nT96 -> T106\nX166 -> T106", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 149 [arrowhead = none ];
149 -> 66;

150 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 150 [arrowhead = none ];
150 -> 67;

151 [label="EVAL with clause\nless(s(X172), s(X173)) :- less(X172, X173).\nand substitution\nX172 -> T111\nT95 -> s(T111)\nT96 -> T112\nX173 -> T112", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 151 [arrowhead = none ];
151 -> 69;

152 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 152 [arrowhead = none ];
152 -> 70;

153 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 153 [arrowhead = none ];
153 -> 68;

154 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
69 -> 154 [arrowhead = none ];
154 -> 71;

155 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 155 [arrowhead = none ];
155 -> 72;

156 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
71 -> 156 [arrowhead = none ];
156 -> 73;

157 [label="EVAL with clause\nless(0, s(X182)).\nand substitution\nT111 -> 0\nX182 -> T119\nT112 -> s(T119)", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 157 [arrowhead = none ];
157 -> 74;

158 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
72 -> 158 [arrowhead = none ];
158 -> 75;

159 [label="EVAL with clause\nless(s(X188), s(X189)) :- less(X188, X189).\nand substitution\nX188 -> T124\nT111 -> s(T124)\nX189 -> T125\nT112 -> s(T125)", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 159 [arrowhead = none ];
159 -> 77;

160 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
73 -> 160 [arrowhead = none ];
160 -> 78;

161 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 161 [arrowhead = none ];
161 -> 76;

162 [label="INSTANCE with matching:\nT111 -> T124\nT112 -> T125", fontsize=14, style = filled, fillcolor = lightgrey];
77 -> 162 [arrowhead = none , style=dashed];
162 -> 69[style=dashed];

}

(2) Obligation:

Clauses:

app18([], T28, X46, .(T28, X46)).
app18(.(X66, X67), T33, X68, .(X66, T35)) :- app18(X67, T33, X68, T35).
app28([], T47, T47).
app28(.(T54, T57), T58, .(T54, X101)) :- app28(T57, T58, X101).
perm38([], []).
perm38(T70, .(T68, T69)) :- app18(X120, T68, X121, T70).
perm38(T70, .(T68, T69)) :- ','(app18(T73, T68, T74, T70), app28(T73, T74, X122)).
perm38(T70, .(T68, T69)) :- ','(app18(T73, T68, T74, T70), ','(app28(T73, T74, T79), perm38(T79, T69))).
ordered39(T88, []).
ordered39(T95, .(T96, T97)) :- less61(T95, T96).
ordered39(T95, .(T96, T97)) :- ','(less61(T95, T96), ordered39(T96, T97)).
less69(0, s(T119)).
less69(s(T124), s(T125)) :- less69(T124, T125).
less61(0, T106).
less61(s(T111), T112) :- less69(T111, T112).
ss1([], []).
ss1(T17, .(T15, T16)) :- app18(X23, T15, X24, T17).
ss1(T17, .(T15, T16)) :- ','(app18(T20, T15, T21, T17), app28(T20, T21, X25)).
ss1(T17, .(T15, T16)) :- ','(app18(T20, T15, T21, T17), ','(app28(T20, T21, T40), perm38(T40, T16))).
ss1(T17, .(T15, T16)) :- ','(app18(T20, T15, T21, T17), ','(app28(T20, T21, T40), ','(perm38(T40, T16), ordered39(T15, T16)))).

Queries:

ss1(a,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
ss1_in: (f,b)
app18_in: (f,b,f,f)
app28_in: (f,f,f)
perm38_in: (f,b)
ordered39_in: (b,b)
less61_in: (b,b)
less69_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(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:

SS1_IN_AG(T17, .(T15, T16)) → U12_AG(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
SS1_IN_AG(T17, .(T15, T16)) → APP18_IN_AGAA(X23, T15, X24, T17)
APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → U1_AGAA(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → APP18_IN_AGAA(X67, T33, X68, T35)
SS1_IN_AG(T17, .(T15, T16)) → U13_AG(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_AG(T17, T15, T16, app28_in_aaa(T20, T21, X25))
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → APP28_IN_AAA(T20, T21, X25)
APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → U2_AAA(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → APP28_IN_AAA(T57, T58, X101)
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_AG(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_AG(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_AG(T17, T15, T16, perm38_in_ag(T40, T16))
U15_AG(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → PERM38_IN_AG(T40, T16)
PERM38_IN_AG(T70, .(T68, T69)) → U3_AG(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
PERM38_IN_AG(T70, .(T68, T69)) → APP18_IN_AGAA(X120, T68, X121, T70)
PERM38_IN_AG(T70, .(T68, T69)) → U4_AG(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_AG(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → APP28_IN_AAA(T73, T74, X122)
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_AG(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_AG(T70, T68, T69, perm38_in_ag(T79, T69))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → PERM38_IN_AG(T79, T69)
U16_AG(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_AG(T17, T15, T16, ordered39_in_gg(T15, T16))
U16_AG(T17, T15, T16, perm38_out_ag(T40, T16)) → ORDERED39_IN_GG(T15, T16)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))
ORDERED39_IN_GG(T95, .(T96, T97)) → LESS61_IN_GG(T95, T96)
LESS61_IN_GG(s(T111), T112) → U11_GG(T111, T112, less69_in_gg(T111, T112))
LESS61_IN_GG(s(T111), T112) → LESS69_IN_GG(T111, T112)
LESS69_IN_GG(s(T124), s(T125)) → U10_GG(T124, T125, less69_in_gg(T124, T125))
LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → U9_GG(T95, T96, T97, ordered39_in_gg(T96, T97))
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

The argument filtering Pi contains the following mapping:
ss1_in_ag(x1, x2) = ss1_in_ag(x2)
[] = []
ss1_out_ag(x1, x2) = ss1_out_ag
.(x1, x2) = .(x1, x2)
U12_ag(x1, x2, x3, x4) = U12_ag(x4)
app18_in_agaa(x1, x2, x3, x4) = app18_in_agaa(x2)
app18_out_agaa(x1, x2, x3, x4) = app18_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6) = U1_agaa(x6)
U13_ag(x1, x2, x3, x4) = U13_ag(x2, x3, x4)
U14_ag(x1, x2, x3, x4) = U14_ag(x4)
app28_in_aaa(x1, x2, x3) = app28_in_aaa
app28_out_aaa(x1, x2, x3) = app28_out_aaa
U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5)
U15_ag(x1, x2, x3, x4) = U15_ag(x2, x3, x4)
U16_ag(x1, x2, x3, x4) = U16_ag(x2, x3, x4)
perm38_in_ag(x1, x2) = perm38_in_ag(x2)
perm38_out_ag(x1, x2) = perm38_out_ag
U3_ag(x1, x2, x3, x4) = U3_ag(x4)
U4_ag(x1, x2, x3, x4) = U4_ag(x3, x4)
U5_ag(x1, x2, x3, x4) = U5_ag(x4)
U6_ag(x1, x2, x3, x4) = U6_ag(x3, x4)
U7_ag(x1, x2, x3, x4) = U7_ag(x4)
U17_ag(x1, x2, x3, x4) = U17_ag(x4)
ordered39_in_gg(x1, x2) = ordered39_in_gg(x1, x2)
ordered39_out_gg(x1, x2) = ordered39_out_gg
U8_gg(x1, x2, x3, x4) = U8_gg(x2, x3, x4)
less61_in_gg(x1, x2) = less61_in_gg(x1, x2)
0 = 0
less61_out_gg(x1, x2) = less61_out_gg
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x3)
less69_in_gg(x1, x2) = less69_in_gg(x1, x2)
less69_out_gg(x1, x2) = less69_out_gg
U10_gg(x1, x2, x3) = U10_gg(x3)
U9_gg(x1, x2, x3, x4) = U9_gg(x4)
SS1_IN_AG(x1, x2) = SS1_IN_AG(x2)
U12_AG(x1, x2, x3, x4) = U12_AG(x4)
APP18_IN_AGAA(x1, x2, x3, x4) = APP18_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6) = U1_AGAA(x6)
U13_AG(x1, x2, x3, x4) = U13_AG(x2, x3, x4)
U14_AG(x1, x2, x3, x4) = U14_AG(x4)
APP28_IN_AAA(x1, x2, x3) = APP28_IN_AAA
U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5)
U15_AG(x1, x2, x3, x4) = U15_AG(x2, x3, x4)
U16_AG(x1, x2, x3, x4) = U16_AG(x2, x3, x4)
PERM38_IN_AG(x1, x2) = PERM38_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x4)
U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4)
U5_AG(x1, x2, x3, x4) = U5_AG(x4)
U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4)
U7_AG(x1, x2, x3, x4) = U7_AG(x4)
U17_AG(x1, x2, x3, x4) = U17_AG(x4)
ORDERED39_IN_GG(x1, x2) = ORDERED39_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4) = U8_GG(x2, x3, x4)
LESS61_IN_GG(x1, x2) = LESS61_IN_GG(x1, x2)
U11_GG(x1, x2, x3) = U11_GG(x3)
LESS69_IN_GG(x1, x2) = LESS69_IN_GG(x1, x2)
U10_GG(x1, x2, x3) = U10_GG(x3)
U9_GG(x1, x2, x3, x4) = U9_GG(x4)

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

(6) Obligation:

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

SS1_IN_AG(T17, .(T15, T16)) → U12_AG(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
SS1_IN_AG(T17, .(T15, T16)) → APP18_IN_AGAA(X23, T15, X24, T17)
APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → U1_AGAA(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → APP18_IN_AGAA(X67, T33, X68, T35)
SS1_IN_AG(T17, .(T15, T16)) → U13_AG(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_AG(T17, T15, T16, app28_in_aaa(T20, T21, X25))
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → APP28_IN_AAA(T20, T21, X25)
APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → U2_AAA(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → APP28_IN_AAA(T57, T58, X101)
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_AG(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_AG(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_AG(T17, T15, T16, perm38_in_ag(T40, T16))
U15_AG(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → PERM38_IN_AG(T40, T16)
PERM38_IN_AG(T70, .(T68, T69)) → U3_AG(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
PERM38_IN_AG(T70, .(T68, T69)) → APP18_IN_AGAA(X120, T68, X121, T70)
PERM38_IN_AG(T70, .(T68, T69)) → U4_AG(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_AG(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → APP28_IN_AAA(T73, T74, X122)
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_AG(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_AG(T70, T68, T69, perm38_in_ag(T79, T69))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → PERM38_IN_AG(T79, T69)
U16_AG(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_AG(T17, T15, T16, ordered39_in_gg(T15, T16))
U16_AG(T17, T15, T16, perm38_out_ag(T40, T16)) → ORDERED39_IN_GG(T15, T16)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))
ORDERED39_IN_GG(T95, .(T96, T97)) → LESS61_IN_GG(T95, T96)
LESS61_IN_GG(s(T111), T112) → U11_GG(T111, T112, less69_in_gg(T111, T112))
LESS61_IN_GG(s(T111), T112) → LESS69_IN_GG(T111, T112)
LESS69_IN_GG(s(T124), s(T125)) → U10_GG(T124, T125, less69_in_gg(T124, T125))
LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → U9_GG(T95, T96, T97, ordered39_in_gg(T96, T97))
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 22 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)

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

(14) QDPSizeChangeProof (EQUIVALENT transformation)

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

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

LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)
The graph contains the following edges 1 > 1, 2 > 2

(15) YES

(16) Obligation:

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

U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))

The TRS R consists of the following rules:

less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))

The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
less61_in_gg(x1, x2) = less61_in_gg(x1, x2)
0 = 0
less61_out_gg(x1, x2) = less61_out_gg
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x3)
less69_in_gg(x1, x2) = less69_in_gg(x1, x2)
less69_out_gg(x1, x2) = less69_out_gg
U10_gg(x1, x2, x3) = U10_gg(x3)
ORDERED39_IN_GG(x1, x2) = ORDERED39_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4) = U8_GG(x2, x3, x4)

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

(19) PiDPToQDPProof (SOUND transformation)

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

(20) Obligation:

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

U8_GG(T96, T97, less61_out_gg) → ORDERED39_IN_GG(T96, T97)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T96, T97, less61_in_gg(T95, T96))

The TRS R consists of the following rules:

less61_in_gg(0, T106) → less61_out_gg
less61_in_gg(s(T111), T112) → U11_gg(less69_in_gg(T111, T112))
U11_gg(less69_out_gg) → less61_out_gg
less69_in_gg(0, s(T119)) → less69_out_gg
less69_in_gg(s(T124), s(T125)) → U10_gg(less69_in_gg(T124, T125))
U10_gg(less69_out_gg) → less69_out_gg

The set Q consists of the following terms:

less61_in_gg(x0, x1)
U11_gg(x0)
less69_in_gg(x0, x1)
U10_gg(x0)

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

(21) 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:

ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T96, T97, less61_in_gg(T95, T96))
The graph contains the following edges 2 > 1, 2 > 2
U8_GG(T96, T97, less61_out_gg) → ORDERED39_IN_GG(T96, T97)
The graph contains the following edges 1 >= 1, 2 >= 2

(22) YES

(23) Obligation:

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

APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → APP28_IN_AAA(T57, T58, X101)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → APP28_IN_AAA(T57, T58, X101)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP28_IN_AAA(x1, x2, x3) = APP28_IN_AAA

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

APP28_IN_AAA → APP28_IN_AAA

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

(28) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APP28_IN_AAA evaluates to t =APP28_IN_AAA

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

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP28_IN_AAA to APP28_IN_AAA.

(29) NO

(30) Obligation:

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

APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → APP18_IN_AGAA(X67, T33, X68, T35)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(31) UsableRulesProof (EQUIVALENT transformation)

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

(32) Obligation:

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

APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → APP18_IN_AGAA(X67, T33, X68, T35)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP18_IN_AGAA(x1, x2, x3, x4) = APP18_IN_AGAA(x2)

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

(33) PiDPToQDPProof (SOUND transformation)

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

(34) Obligation:

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

APP18_IN_AGAA(T33) → APP18_IN_AGAA(T33)

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

(35) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

s = APP18_IN_AGAA(T33) evaluates to t =APP18_IN_AGAA(T33)

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

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP18_IN_AGAA(T33) to APP18_IN_AGAA(T33).

(36) NO

(37) Obligation:

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

PERM38_IN_AG(T70, .(T68, T69)) → U4_AG(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_AG(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → PERM38_IN_AG(T79, T69)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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

PERM38_IN_AG(T70, .(T68, T69)) → U4_AG(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_AG(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → PERM38_IN_AG(T79, T69)

The TRS R consists of the following rules:

app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
app18_in_agaa(x1, x2, x3, x4) = app18_in_agaa(x2)
app18_out_agaa(x1, x2, x3, x4) = app18_out_agaa
U1_agaa(x1, x2, x3, x4, x5, x6) = U1_agaa(x6)
app28_in_aaa(x1, x2, x3) = app28_in_aaa
app28_out_aaa(x1, x2, x3) = app28_out_aaa
U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5)
PERM38_IN_AG(x1, x2) = PERM38_IN_AG(x2)
U4_AG(x1, x2, x3, x4) = U4_AG(x3, x4)
U6_AG(x1, x2, x3, x4) = U6_AG(x3, x4)

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

(40) PiDPToQDPProof (SOUND transformation)

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

(41) Obligation:

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

PERM38_IN_AG(.(T68, T69)) → U4_AG(T69, app18_in_agaa(T68))
U4_AG(T69, app18_out_agaa) → U6_AG(T69, app28_in_aaa)
U6_AG(T69, app28_out_aaa) → PERM38_IN_AG(T69)

The TRS R consists of the following rules:

app18_in_agaa(T28) → app18_out_agaa
app18_in_agaa(T33) → U1_agaa(app18_in_agaa(T33))
app28_in_aaa → app28_out_aaa
app28_in_aaa → U2_aaa(app28_in_aaa)
U1_agaa(app18_out_agaa) → app18_out_agaa
U2_aaa(app28_out_aaa) → app28_out_aaa

The set Q consists of the following terms:

app18_in_agaa(x0)
app28_in_aaa
U1_agaa(x0)
U2_aaa(x0)

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

(42) 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:

U4_AG(T69, app18_out_agaa) → U6_AG(T69, app28_in_aaa)
The graph contains the following edges 1 >= 1
U6_AG(T69, app28_out_aaa) → PERM38_IN_AG(T69)
The graph contains the following edges 1 >= 1
PERM38_IN_AG(.(T68, T69)) → U4_AG(T69, app18_in_agaa(T68))
The graph contains the following edges 1 > 1

(43) YES

(44) PrologToPiTRSProof (SOUND transformation)

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(45) Obligation:

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

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(46) 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:

SS1_IN_AG(T17, .(T15, T16)) → U12_AG(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
SS1_IN_AG(T17, .(T15, T16)) → APP18_IN_AGAA(X23, T15, X24, T17)
APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → U1_AGAA(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → APP18_IN_AGAA(X67, T33, X68, T35)
SS1_IN_AG(T17, .(T15, T16)) → U13_AG(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_AG(T17, T15, T16, app28_in_aaa(T20, T21, X25))
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → APP28_IN_AAA(T20, T21, X25)
APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → U2_AAA(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → APP28_IN_AAA(T57, T58, X101)
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_AG(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_AG(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_AG(T17, T15, T16, perm38_in_ag(T40, T16))
U15_AG(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → PERM38_IN_AG(T40, T16)
PERM38_IN_AG(T70, .(T68, T69)) → U3_AG(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
PERM38_IN_AG(T70, .(T68, T69)) → APP18_IN_AGAA(X120, T68, X121, T70)
PERM38_IN_AG(T70, .(T68, T69)) → U4_AG(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_AG(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → APP28_IN_AAA(T73, T74, X122)
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_AG(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_AG(T70, T68, T69, perm38_in_ag(T79, T69))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → PERM38_IN_AG(T79, T69)
U16_AG(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_AG(T17, T15, T16, ordered39_in_gg(T15, T16))
U16_AG(T17, T15, T16, perm38_out_ag(T40, T16)) → ORDERED39_IN_GG(T15, T16)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))
ORDERED39_IN_GG(T95, .(T96, T97)) → LESS61_IN_GG(T95, T96)
LESS61_IN_GG(s(T111), T112) → U11_GG(T111, T112, less69_in_gg(T111, T112))
LESS61_IN_GG(s(T111), T112) → LESS69_IN_GG(T111, T112)
LESS69_IN_GG(s(T124), s(T125)) → U10_GG(T124, T125, less69_in_gg(T124, T125))
LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → U9_GG(T95, T96, T97, ordered39_in_gg(T96, T97))
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

The argument filtering Pi contains the following mapping:
ss1_in_ag(x1, x2) = ss1_in_ag(x2)
[] = []
ss1_out_ag(x1, x2) = ss1_out_ag(x2)
.(x1, x2) = .(x1, x2)
U12_ag(x1, x2, x3, x4) = U12_ag(x2, x3, x4)
app18_in_agaa(x1, x2, x3, x4) = app18_in_agaa(x2)
app18_out_agaa(x1, x2, x3, x4) = app18_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6) = U1_agaa(x3, x6)
U13_ag(x1, x2, x3, x4) = U13_ag(x2, x3, x4)
U14_ag(x1, x2, x3, x4) = U14_ag(x2, x3, x4)
app28_in_aaa(x1, x2, x3) = app28_in_aaa
app28_out_aaa(x1, x2, x3) = app28_out_aaa
U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5)
U15_ag(x1, x2, x3, x4) = U15_ag(x2, x3, x4)
U16_ag(x1, x2, x3, x4) = U16_ag(x2, x3, x4)
perm38_in_ag(x1, x2) = perm38_in_ag(x2)
perm38_out_ag(x1, x2) = perm38_out_ag(x2)
U3_ag(x1, x2, x3, x4) = U3_ag(x2, x3, x4)
U4_ag(x1, x2, x3, x4) = U4_ag(x2, x3, x4)
U5_ag(x1, x2, x3, x4) = U5_ag(x2, x3, x4)
U6_ag(x1, x2, x3, x4) = U6_ag(x2, x3, x4)
U7_ag(x1, x2, x3, x4) = U7_ag(x2, x3, x4)
U17_ag(x1, x2, x3, x4) = U17_ag(x2, x3, x4)
ordered39_in_gg(x1, x2) = ordered39_in_gg(x1, x2)
ordered39_out_gg(x1, x2) = ordered39_out_gg(x1, x2)
U8_gg(x1, x2, x3, x4) = U8_gg(x1, x2, x3, x4)
less61_in_gg(x1, x2) = less61_in_gg(x1, x2)
0 = 0
less61_out_gg(x1, x2) = less61_out_gg(x1, x2)
s(x1) = s(x1)
U11_gg(x1, x2, x3) = U11_gg(x1, x2, x3)
less69_in_gg(x1, x2) = less69_in_gg(x1, x2)
less69_out_gg(x1, x2) = less69_out_gg(x1, x2)
U10_gg(x1, x2, x3) = U10_gg(x1, x2, x3)
U9_gg(x1, x2, x3, x4) = U9_gg(x1, x2, x3, x4)
SS1_IN_AG(x1, x2) = SS1_IN_AG(x2)
U12_AG(x1, x2, x3, x4) = U12_AG(x2, x3, x4)
APP18_IN_AGAA(x1, x2, x3, x4) = APP18_IN_AGAA(x2)
U1_AGAA(x1, x2, x3, x4, x5, x6) = U1_AGAA(x3, x6)
U13_AG(x1, x2, x3, x4) = U13_AG(x2, x3, x4)
U14_AG(x1, x2, x3, x4) = U14_AG(x2, x3, x4)
APP28_IN_AAA(x1, x2, x3) = APP28_IN_AAA
U2_AAA(x1, x2, x3, x4, x5) = U2_AAA(x5)
U15_AG(x1, x2, x3, x4) = U15_AG(x2, x3, x4)
U16_AG(x1, x2, x3, x4) = U16_AG(x2, x3, x4)
PERM38_IN_AG(x1, x2) = PERM38_IN_AG(x2)
U3_AG(x1, x2, x3, x4) = U3_AG(x2, x3, x4)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4)
U5_AG(x1, x2, x3, x4) = U5_AG(x2, x3, x4)
U6_AG(x1, x2, x3, x4) = U6_AG(x2, x3, x4)
U7_AG(x1, x2, x3, x4) = U7_AG(x2, x3, x4)
U17_AG(x1, x2, x3, x4) = U17_AG(x2, x3, x4)
ORDERED39_IN_GG(x1, x2) = ORDERED39_IN_GG(x1, x2)
U8_GG(x1, x2, x3, x4) = U8_GG(x1, x2, x3, x4)
LESS61_IN_GG(x1, x2) = LESS61_IN_GG(x1, x2)
U11_GG(x1, x2, x3) = U11_GG(x1, x2, x3)
LESS69_IN_GG(x1, x2) = LESS69_IN_GG(x1, x2)
U10_GG(x1, x2, x3) = U10_GG(x1, x2, x3)
U9_GG(x1, x2, x3, x4) = U9_GG(x1, x2, x3, x4)

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

(47) Obligation:

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

SS1_IN_AG(T17, .(T15, T16)) → U12_AG(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
SS1_IN_AG(T17, .(T15, T16)) → APP18_IN_AGAA(X23, T15, X24, T17)
APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → U1_AGAA(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → APP18_IN_AGAA(X67, T33, X68, T35)
SS1_IN_AG(T17, .(T15, T16)) → U13_AG(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_AG(T17, T15, T16, app28_in_aaa(T20, T21, X25))
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → APP28_IN_AAA(T20, T21, X25)
APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → U2_AAA(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → APP28_IN_AAA(T57, T58, X101)
U13_AG(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_AG(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_AG(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_AG(T17, T15, T16, perm38_in_ag(T40, T16))
U15_AG(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → PERM38_IN_AG(T40, T16)
PERM38_IN_AG(T70, .(T68, T69)) → U3_AG(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
PERM38_IN_AG(T70, .(T68, T69)) → APP18_IN_AGAA(X120, T68, X121, T70)
PERM38_IN_AG(T70, .(T68, T69)) → U4_AG(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_AG(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → APP28_IN_AAA(T73, T74, X122)
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_AG(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_AG(T70, T68, T69, perm38_in_ag(T79, T69))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → PERM38_IN_AG(T79, T69)
U16_AG(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_AG(T17, T15, T16, ordered39_in_gg(T15, T16))
U16_AG(T17, T15, T16, perm38_out_ag(T40, T16)) → ORDERED39_IN_GG(T15, T16)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))
ORDERED39_IN_GG(T95, .(T96, T97)) → LESS61_IN_GG(T95, T96)
LESS61_IN_GG(s(T111), T112) → U11_GG(T111, T112, less69_in_gg(T111, T112))
LESS61_IN_GG(s(T111), T112) → LESS69_IN_GG(T111, T112)
LESS69_IN_GG(s(T124), s(T125)) → U10_GG(T124, T125, less69_in_gg(T124, T125))
LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → U9_GG(T95, T96, T97, ordered39_in_gg(T96, T97))
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(48) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 22 less nodes.

(49) Complex Obligation (AND)

(50) Obligation:

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

LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(51) UsableRulesProof (EQUIVALENT transformation)

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

(52) Obligation:

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

LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)

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

(53) PiDPToQDPProof (EQUIVALENT transformation)

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

(54) Obligation:

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

LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)

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

(55) 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:

LESS69_IN_GG(s(T124), s(T125)) → LESS69_IN_GG(T124, T125)
The graph contains the following edges 1 > 1, 2 > 2

(56) YES

(57) Obligation:

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

U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(58) UsableRulesProof (EQUIVALENT transformation)

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

(59) Obligation:

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

U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))

The TRS R consists of the following rules:

less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))

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

(60) PiDPToQDPProof (EQUIVALENT transformation)

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

(61) Obligation:

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

U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)
ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))

The TRS R consists of the following rules:

less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))

The set Q consists of the following terms:

less61_in_gg(x0, x1)
U11_gg(x0, x1, x2)
less69_in_gg(x0, x1)
U10_gg(x0, x1, x2)

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

(62) 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:

ORDERED39_IN_GG(T95, .(T96, T97)) → U8_GG(T95, T96, T97, less61_in_gg(T95, T96))
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3
U8_GG(T95, T96, T97, less61_out_gg(T95, T96)) → ORDERED39_IN_GG(T96, T97)
The graph contains the following edges 2 >= 1, 4 > 1, 3 >= 2

(63) YES

(64) Obligation:

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

APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → APP28_IN_AAA(T57, T58, X101)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(65) UsableRulesProof (EQUIVALENT transformation)

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

(66) Obligation:

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

APP28_IN_AAA(.(T54, T57), T58, .(T54, X101)) → APP28_IN_AAA(T57, T58, X101)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP28_IN_AAA(x1, x2, x3) = APP28_IN_AAA

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

(67) PiDPToQDPProof (SOUND transformation)

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

(68) Obligation:

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

APP28_IN_AAA → APP28_IN_AAA

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

(69) NonTerminationProof (EQUIVALENT transformation)

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP28_IN_AAA to APP28_IN_AAA.

(70) NO

(71) Obligation:

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

APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → APP18_IN_AGAA(X67, T33, X68, T35)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(72) UsableRulesProof (EQUIVALENT transformation)

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

(73) Obligation:

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

APP18_IN_AGAA(.(X66, X67), T33, X68, .(X66, T35)) → APP18_IN_AGAA(X67, T33, X68, T35)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APP18_IN_AGAA(x1, x2, x3, x4) = APP18_IN_AGAA(x2)

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

(74) PiDPToQDPProof (SOUND transformation)

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

(75) Obligation:

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

APP18_IN_AGAA(T33) → APP18_IN_AGAA(T33)

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

(76) NonTerminationProof (EQUIVALENT transformation)

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from APP18_IN_AGAA(T33) to APP18_IN_AGAA(T33).

(77) NO

(78) Obligation:

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

PERM38_IN_AG(T70, .(T68, T69)) → U4_AG(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_AG(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → PERM38_IN_AG(T79, T69)

The TRS R consists of the following rules:

ss1_in_ag([], []) → ss1_out_ag([], [])
ss1_in_ag(T17, .(T15, T16)) → U12_ag(T17, T15, T16, app18_in_agaa(X23, T15, X24, T17))
app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U12_ag(T17, T15, T16, app18_out_agaa(X23, T15, X24, T17)) → ss1_out_ag(T17, .(T15, T16))
ss1_in_ag(T17, .(T15, T16)) → U13_ag(T17, T15, T16, app18_in_agaa(T20, T15, T21, T17))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U14_ag(T17, T15, T16, app28_in_aaa(T20, T21, X25))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))
U14_ag(T17, T15, T16, app28_out_aaa(T20, T21, X25)) → ss1_out_ag(T17, .(T15, T16))
U13_ag(T17, T15, T16, app18_out_agaa(T20, T15, T21, T17)) → U15_ag(T17, T15, T16, app28_in_aaa(T20, T21, T40))
U15_ag(T17, T15, T16, app28_out_aaa(T20, T21, T40)) → U16_ag(T17, T15, T16, perm38_in_ag(T40, T16))
perm38_in_ag([], []) → perm38_out_ag([], [])
perm38_in_ag(T70, .(T68, T69)) → U3_ag(T70, T68, T69, app18_in_agaa(X120, T68, X121, T70))
U3_ag(T70, T68, T69, app18_out_agaa(X120, T68, X121, T70)) → perm38_out_ag(T70, .(T68, T69))
perm38_in_ag(T70, .(T68, T69)) → U4_ag(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U5_ag(T70, T68, T69, app28_in_aaa(T73, T74, X122))
U5_ag(T70, T68, T69, app28_out_aaa(T73, T74, X122)) → perm38_out_ag(T70, .(T68, T69))
U4_ag(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_ag(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_ag(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → U7_ag(T70, T68, T69, perm38_in_ag(T79, T69))
U7_ag(T70, T68, T69, perm38_out_ag(T79, T69)) → perm38_out_ag(T70, .(T68, T69))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → ss1_out_ag(T17, .(T15, T16))
U16_ag(T17, T15, T16, perm38_out_ag(T40, T16)) → U17_ag(T17, T15, T16, ordered39_in_gg(T15, T16))
ordered39_in_gg(T88, []) → ordered39_out_gg(T88, [])
ordered39_in_gg(T95, .(T96, T97)) → U8_gg(T95, T96, T97, less61_in_gg(T95, T96))
less61_in_gg(0, T106) → less61_out_gg(0, T106)
less61_in_gg(s(T111), T112) → U11_gg(T111, T112, less69_in_gg(T111, T112))
less69_in_gg(0, s(T119)) → less69_out_gg(0, s(T119))
less69_in_gg(s(T124), s(T125)) → U10_gg(T124, T125, less69_in_gg(T124, T125))
U10_gg(T124, T125, less69_out_gg(T124, T125)) → less69_out_gg(s(T124), s(T125))
U11_gg(T111, T112, less69_out_gg(T111, T112)) → less61_out_gg(s(T111), T112)
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → ordered39_out_gg(T95, .(T96, T97))
U8_gg(T95, T96, T97, less61_out_gg(T95, T96)) → U9_gg(T95, T96, T97, ordered39_in_gg(T96, T97))
U9_gg(T95, T96, T97, ordered39_out_gg(T96, T97)) → ordered39_out_gg(T95, .(T96, T97))
U17_ag(T17, T15, T16, ordered39_out_gg(T15, T16)) → ss1_out_ag(T17, .(T15, T16))

(79) UsableRulesProof (EQUIVALENT transformation)

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

(80) Obligation:

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

PERM38_IN_AG(T70, .(T68, T69)) → U4_AG(T70, T68, T69, app18_in_agaa(T73, T68, T74, T70))
U4_AG(T70, T68, T69, app18_out_agaa(T73, T68, T74, T70)) → U6_AG(T70, T68, T69, app28_in_aaa(T73, T74, T79))
U6_AG(T70, T68, T69, app28_out_aaa(T73, T74, T79)) → PERM38_IN_AG(T79, T69)

The TRS R consists of the following rules:

app18_in_agaa([], T28, X46, .(T28, X46)) → app18_out_agaa([], T28, X46, .(T28, X46))
app18_in_agaa(.(X66, X67), T33, X68, .(X66, T35)) → U1_agaa(X66, X67, T33, X68, T35, app18_in_agaa(X67, T33, X68, T35))
app28_in_aaa([], T47, T47) → app28_out_aaa([], T47, T47)
app28_in_aaa(.(T54, T57), T58, .(T54, X101)) → U2_aaa(T54, T57, T58, X101, app28_in_aaa(T57, T58, X101))
U1_agaa(X66, X67, T33, X68, T35, app18_out_agaa(X67, T33, X68, T35)) → app18_out_agaa(.(X66, X67), T33, X68, .(X66, T35))
U2_aaa(T54, T57, T58, X101, app28_out_aaa(T57, T58, X101)) → app28_out_aaa(.(T54, T57), T58, .(T54, X101))

The argument filtering Pi contains the following mapping:
[] = []
.(x1, x2) = .(x1, x2)
app18_in_agaa(x1, x2, x3, x4) = app18_in_agaa(x2)
app18_out_agaa(x1, x2, x3, x4) = app18_out_agaa(x2)
U1_agaa(x1, x2, x3, x4, x5, x6) = U1_agaa(x3, x6)
app28_in_aaa(x1, x2, x3) = app28_in_aaa
app28_out_aaa(x1, x2, x3) = app28_out_aaa
U2_aaa(x1, x2, x3, x4, x5) = U2_aaa(x5)
PERM38_IN_AG(x1, x2) = PERM38_IN_AG(x2)
U4_AG(x1, x2, x3, x4) = U4_AG(x2, x3, x4)
U6_AG(x1, x2, x3, x4) = U6_AG(x2, x3, x4)

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

(81) PiDPToQDPProof (SOUND transformation)

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

(82) Obligation:

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

PERM38_IN_AG(.(T68, T69)) → U4_AG(T68, T69, app18_in_agaa(T68))
U4_AG(T68, T69, app18_out_agaa(T68)) → U6_AG(T68, T69, app28_in_aaa)
U6_AG(T68, T69, app28_out_aaa) → PERM38_IN_AG(T69)

The TRS R consists of the following rules:

app18_in_agaa(T28) → app18_out_agaa(T28)
app18_in_agaa(T33) → U1_agaa(T33, app18_in_agaa(T33))
app28_in_aaa → app28_out_aaa
app28_in_aaa → U2_aaa(app28_in_aaa)
U1_agaa(T33, app18_out_agaa(T33)) → app18_out_agaa(T33)
U2_aaa(app28_out_aaa) → app28_out_aaa

The set Q consists of the following terms:

app18_in_agaa(x0)
app28_in_aaa
U1_agaa(x0, x1)
U2_aaa(x0)

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

(83) 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:

U4_AG(T68, T69, app18_out_agaa(T68)) → U6_AG(T68, T69, app28_in_aaa)
The graph contains the following edges 1 >= 1, 3 > 1, 2 >= 2
U6_AG(T68, T69, app28_out_aaa) → PERM38_IN_AG(T69)
The graph contains the following edges 2 >= 1
PERM38_IN_AG(.(T68, T69)) → U4_AG(T68, T69, app18_in_agaa(T68))
The graph contains the following edges 1 > 1, 1 > 2