Left Termination proof of /tmp/tmpzLARnq/reminder-ioi.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 87 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 46 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 183 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 31 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇔, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔, 0 ms)

        ↳25 PiDP

        ↳26 PiDPToQDPProof (⇒, 0 ms)

        ↳27 QDP

        ↳28 QDPSizeChangeProof (⇔, 0 ms)

        ↳29 YES

    ↳30 PiDP

        ↳31 UsableRulesProof (⇔, 0 ms)

        ↳32 PiDP

        ↳33 PiDPToQDPProof (⇒, 0 ms)

        ↳34 QDP

        ↳35 QDPOrderProof (⇔, 37 ms)

        ↳36 QDP

        ↳37 DependencyGraphProof (⇔, 0 ms)

        ↳38 TRUE

    ↳39 PiDP

        ↳40 UsableRulesProof (⇔, 0 ms)

        ↳41 PiDP

        ↳42 PiDPToQDPProof (⇒, 0 ms)

        ↳43 QDP

        ↳44 QDPSizeChangeProof (⇔, 0 ms)

        ↳45 YES

(0) Obligation:

Clauses:

rem(X, Y, R) :- ','(notZero(Y), ','(sub(X, Y, Z), rem(Z, Y, R))).
rem(X, Y, X) :- ','(notZero(Y), geq(X, Y)).
sub(s(X), s(Y), Z) :- sub(X, Y, Z).
sub(X, 0, X).
notZero(s(X)).
geq(s(X), s(Y)) :- geq(X, Y).
geq(X, 0).

Query: rem(g,a,g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: rem(T1, T2, T3)\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: rem(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | rem(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(notZero(T10), ','(sub(T7, T10, X7), rem(X7, T10, T9))) | rem(T7, T2, T9), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT9 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(notZero(T10), ','(sub(T7, T10, X7), rem(X7, T10, T9))), SCOPE: 2, CLAUSE: 4 | ?_2 | rem(T7, T2, T9), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT9 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(notZero(T10), ','(sub(T7, T10, X7), rem(X7, T10, T9))), SCOPE: 2, CLAUSE: 4\n\nT7 is ground\nT9 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ?_2 | rem(T7, T2, T9), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(sub(T7, s(T16), X7), rem(X7, s(T16), T9))\n\nT7 is ground\nT9 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: sub(T7, s(T16), X7)\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: rem(T19, s(T20), T9)\n\nT9 is ground\nT20 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: sub(T7, s(T16), X7), SCOPE: 3, CLAUSE: 2 | sub(T7, s(T16), X7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: sub(T7, s(T16), X7), SCOPE: 3, CLAUSE: 2\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: sub(T7, s(T16), X7), SCOPE: 3, CLAUSE: 3\n\nT7 is ground\nX7 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: sub(T31, T33, X40)\n\nT31 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: sub(T31, T33, X40), SCOPE: 4, CLAUSE: 2 | sub(T31, T33, X40), SCOPE: 4, CLAUSE: 3\n\nT31 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: sub(T31, T33, X40), SCOPE: 4, CLAUSE: 2\n\nT31 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: sub(T31, T33, X40), SCOPE: 4, CLAUSE: 3\n\nT31 is ground\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: sub(T44, T46, X64)\n\nT44 is ground\nX64 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: rem(T7, T2, T9), SCOPE: 1, CLAUSE: 1\n\nT7 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(notZero(T62), geq(T60, T62))\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(notZero(T62), geq(T60, T62)), SCOPE: 5, CLAUSE: 4\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: geq(T60, s(T68))\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: geq(T60, s(T68)), SCOPE: 6, CLAUSE: 5 | geq(T60, s(T68)), SCOPE: 6, CLAUSE: 6\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: geq(T60, s(T68)), SCOPE: 6, CLAUSE: 5\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: geq(T60, s(T68)), SCOPE: 6, CLAUSE: 6\n\nT60 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: geq(T79, T81)\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: geq(T79, T81), SCOPE: 7, CLAUSE: 5 | geq(T79, T81), SCOPE: 7, CLAUSE: 6\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: geq(T79, T81), SCOPE: 7, CLAUSE: 5\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: geq(T79, T81), SCOPE: 7, CLAUSE: 6\n\nT79 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: geq(T92, T94)\n\nT92 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 45 [arrowhead = none ];
45 -> 2;

46 [label="ONLY EVAL with clause\nrem(X4, X5, X6) :- ','(notZero(X5), ','(sub(X4, X5, X7), rem(X7, X5, X6))).\nand substitutionT1 -> T7,\nX4 -> T7,\nT2 -> T10,\nX5 -> T10,\nT3 -> T9,\nX6 -> T9,\nT8 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 46 [arrowhead = none ];
46 -> 3;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 47 [arrowhead = none ];
47 -> 4;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 48 [arrowhead = none ];
48 -> 5;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
4 -> 49 [arrowhead = none ];
49 -> 6;

50 [label="EVAL with clause\nnotZero(s(X12)).\nand substitutionX12 -> T16,\nT10 -> s(T16),\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 50 [arrowhead = none ];
50 -> 7;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 51 [arrowhead = none ];
51 -> 8;

52 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 52 [arrowhead = none ];
52 -> 25;

53 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
7 -> 53 [arrowhead = none ];
53 -> 9;

54 [label="SPLIT 2\nnew knowledge:\nT7 is ground\nT20 is ground\nT19 is ground\nreplacements:X7 -> T19,\nT16 -> T20", fontsize=14, style = filled, fillcolor = violet];
7 -> 54 [arrowhead = none ];
54 -> 10;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 55 [arrowhead = none ];
55 -> 11;

56 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> s(T20)\nT3 -> T9", fontsize=14, style = filled, fillcolor = lightgrey];
10 -> 56 [arrowhead = none , style=dashed];
56 -> 1[style=dashed];

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 57 [arrowhead = none ];
57 -> 12;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
11 -> 58 [arrowhead = none ];
58 -> 13;

59 [label="EVAL with clause\nsub(s(X37), s(X38), X39) :- sub(X37, X38, X39).\nand substitutionX37 -> T31,\nT7 -> s(T31),\nT16 -> T33,\nX38 -> T33,\nX7 -> X40,\nX39 -> X40,\nT32 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 59 [arrowhead = none ];
59 -> 14;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 60 [arrowhead = none ];
60 -> 15;

61 [label="BACKTRACK\nfor clause: sub(X, 0, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 61 [arrowhead = none ];
61 -> 24;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 62 [arrowhead = none ];
62 -> 16;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 63 [arrowhead = none ];
63 -> 17;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 64 [arrowhead = none ];
64 -> 18;

65 [label="EVAL with clause\nsub(s(X61), s(X62), X63) :- sub(X61, X62, X63).\nand substitutionX61 -> T44,\nT31 -> s(T44),\nX62 -> T46,\nT33 -> s(T46),\nX40 -> X64,\nX63 -> X64,\nT45 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 65 [arrowhead = none ];
65 -> 19;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 66 [arrowhead = none ];
66 -> 20;

67 [label="EVAL with clause\nsub(X71, 0, X71).\nand substitutionT31 -> T51,\nX71 -> T51,\nT33 -> 0,\nX40 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 67 [arrowhead = none ];
67 -> 21;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 68 [arrowhead = none ];
68 -> 22;

69 [label="INSTANCE with matching:\nT31 -> T44\nT33 -> T46\nX40 -> X64", fontsize=14, style = filled, fillcolor = lightgrey];
19 -> 69 [arrowhead = none , style=dashed];
69 -> 14[style=dashed];

70 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 70 [arrowhead = none ];
70 -> 23;

71 [label="EVAL with clause\nrem(X81, X82, X81) :- ','(notZero(X82), geq(X81, X82)).\nand substitutionT7 -> T60,\nX81 -> T60,\nT2 -> T62,\nX82 -> T62,\nT9 -> T60,\nT61 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 71 [arrowhead = none ];
71 -> 26;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 72 [arrowhead = none ];
72 -> 27;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 73 [arrowhead = none ];
73 -> 28;

74 [label="EVAL with clause\nnotZero(s(X87)).\nand substitutionX87 -> T68,\nT62 -> s(T68),\nT67 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 74 [arrowhead = none ];
74 -> 29;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 75 [arrowhead = none ];
75 -> 30;

76 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 76 [arrowhead = none ];
76 -> 31;

77 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 77 [arrowhead = none ];
77 -> 32;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
31 -> 78 [arrowhead = none ];
78 -> 33;

79 [label="EVAL with clause\ngeq(s(X98), s(X99)) :- geq(X98, X99).\nand substitutionX98 -> T79,\nT60 -> s(T79),\nT68 -> T81,\nX99 -> T81,\nT80 -> T81", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 79 [arrowhead = none ];
79 -> 34;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 80 [arrowhead = none ];
80 -> 35;

81 [label="BACKTRACK\nfor clause: geq(X, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 81 [arrowhead = none ];
81 -> 44;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
34 -> 82 [arrowhead = none ];
82 -> 36;

83 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 83 [arrowhead = none ];
83 -> 37;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 84 [arrowhead = none ];
84 -> 38;

85 [label="EVAL with clause\ngeq(s(X110), s(X111)) :- geq(X110, X111).\nand substitutionX110 -> T92,\nT79 -> s(T92),\nX111 -> T94,\nT81 -> s(T94),\nT93 -> T94", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 85 [arrowhead = none ];
85 -> 39;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 86 [arrowhead = none ];
86 -> 40;

87 [label="EVAL with clause\ngeq(X116, 0).\nand substitutionT79 -> T99,\nX116 -> T99,\nT81 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 87 [arrowhead = none ];
87 -> 41;

88 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 88 [arrowhead = none ];
88 -> 42;

89 [label="INSTANCE with matching:\nT79 -> T92\nT81 -> T94", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 89 [arrowhead = none , style=dashed];
89 -> 34[style=dashed];

90 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 90 [arrowhead = none ];
90 -> 43;

}

(2) Obligation:

Clauses:

subA(s(T44), s(T46), X64) :- subA(T44, T46, X64).
subA(T51, 0, T51).
geqB(s(T92), s(T94)) :- geqB(T92, T94).
geqB(T99, 0).
subC(s(T31), T33, X40) :- subA(T31, T33, X40).
remD(T7, s(T16), T9) :- subC(T7, T16, X7).
remD(T7, s(T20), T9) :- ','(subC(T7, T20, T19), remD(T19, s(T20), T9)).
remD(s(T79), s(T81), s(T79)) :- geqB(T79, T81).

Query: remD(g,a,g)

(3) PrologToPiTRSProof (SOUND transformation)

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

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

(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:

REMD_IN_GAG(T7, s(T16), T9) → U4_GAG(T7, T16, T9, subC_in_gaa(T7, T16, X7))
REMD_IN_GAG(T7, s(T16), T9) → SUBC_IN_GAA(T7, T16, X7)
SUBC_IN_GAA(s(T31), T33, X40) → U3_GAA(T31, T33, X40, subA_in_gaa(T31, T33, X40))
SUBC_IN_GAA(s(T31), T33, X40) → SUBA_IN_GAA(T31, T33, X40)
SUBA_IN_GAA(s(T44), s(T46), X64) → U1_GAA(T44, T46, X64, subA_in_gaa(T44, T46, X64))
SUBA_IN_GAA(s(T44), s(T46), X64) → SUBA_IN_GAA(T44, T46, X64)
REMD_IN_GAG(T7, s(T20), T9) → U5_GAG(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_GAG(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_GAG(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
U5_GAG(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → REMD_IN_GGG(T19, s(T20), T9)
REMD_IN_GGG(T7, s(T16), T9) → U4_GGG(T7, T16, T9, subC_in_gga(T7, T16, X7))
REMD_IN_GGG(T7, s(T16), T9) → SUBC_IN_GGA(T7, T16, X7)
SUBC_IN_GGA(s(T31), T33, X40) → U3_GGA(T31, T33, X40, subA_in_gga(T31, T33, X40))
SUBC_IN_GGA(s(T31), T33, X40) → SUBA_IN_GGA(T31, T33, X40)
SUBA_IN_GGA(s(T44), s(T46), X64) → U1_GGA(T44, T46, X64, subA_in_gga(T44, T46, X64))
SUBA_IN_GGA(s(T44), s(T46), X64) → SUBA_IN_GGA(T44, T46, X64)
REMD_IN_GGG(T7, s(T20), T9) → U5_GGG(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_GGG(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_GGG(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
U5_GGG(T7, T20, T9, subC_out_gga(T7, T20, T19)) → REMD_IN_GGG(T19, s(T20), T9)
REMD_IN_GGG(s(T79), s(T81), s(T79)) → U7_GGG(T79, T81, geqB_in_gg(T79, T81))
REMD_IN_GGG(s(T79), s(T81), s(T79)) → GEQB_IN_GG(T79, T81)
GEQB_IN_GG(s(T92), s(T94)) → U2_GG(T92, T94, geqB_in_gg(T92, T94))
GEQB_IN_GG(s(T92), s(T94)) → GEQB_IN_GG(T92, T94)
REMD_IN_GAG(s(T79), s(T81), s(T79)) → U7_GAG(T79, T81, geqB_in_ga(T79, T81))
REMD_IN_GAG(s(T79), s(T81), s(T79)) → GEQB_IN_GA(T79, T81)
GEQB_IN_GA(s(T92), s(T94)) → U2_GA(T92, T94, geqB_in_ga(T92, T94))
GEQB_IN_GA(s(T92), s(T94)) → GEQB_IN_GA(T92, T94)

The TRS R consists of the following rules:

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

The argument filtering Pi contains the following mapping:
remD_in_gag(x1, x2, x3) = remD_in_gag(x1, x3)
U4_gag(x1, x2, x3, x4) = U4_gag(x4)
subC_in_gaa(x1, x2, x3) = subC_in_gaa(x1)
s(x1) = s(x1)
U3_gaa(x1, x2, x3, x4) = U3_gaa(x4)
subA_in_gaa(x1, x2, x3) = subA_in_gaa(x1)
U1_gaa(x1, x2, x3, x4) = U1_gaa(x4)
subA_out_gaa(x1, x2, x3) = subA_out_gaa(x2, x3)
subC_out_gaa(x1, x2, x3) = subC_out_gaa(x2, x3)
remD_out_gag(x1, x2, x3) = remD_out_gag(x2)
U5_gag(x1, x2, x3, x4) = U5_gag(x3, x4)
U6_gag(x1, x2, x3, x4) = U6_gag(x2, x4)
remD_in_ggg(x1, x2, x3) = remD_in_ggg(x1, x2, x3)
U4_ggg(x1, x2, x3, x4) = U4_ggg(x4)
subC_in_gga(x1, x2, x3) = subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
subA_in_gga(x1, x2, x3) = subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
0 = 0
subA_out_gga(x1, x2, x3) = subA_out_gga(x3)
subC_out_gga(x1, x2, x3) = subC_out_gga(x3)
remD_out_ggg(x1, x2, x3) = remD_out_ggg
U5_ggg(x1, x2, x3, x4) = U5_ggg(x2, x3, x4)
U6_ggg(x1, x2, x3, x4) = U6_ggg(x4)
U7_ggg(x1, x2, x3) = U7_ggg(x3)
geqB_in_gg(x1, x2) = geqB_in_gg(x1, x2)
U2_gg(x1, x2, x3) = U2_gg(x3)
geqB_out_gg(x1, x2) = geqB_out_gg
U7_gag(x1, x2, x3) = U7_gag(x3)
geqB_in_ga(x1, x2) = geqB_in_ga(x1)
U2_ga(x1, x2, x3) = U2_ga(x3)
geqB_out_ga(x1, x2) = geqB_out_ga(x2)
REMD_IN_GAG(x1, x2, x3) = REMD_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4) = U4_GAG(x4)
SUBC_IN_GAA(x1, x2, x3) = SUBC_IN_GAA(x1)
U3_GAA(x1, x2, x3, x4) = U3_GAA(x4)
SUBA_IN_GAA(x1, x2, x3) = SUBA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4) = U1_GAA(x4)
U5_GAG(x1, x2, x3, x4) = U5_GAG(x3, x4)
U6_GAG(x1, x2, x3, x4) = U6_GAG(x2, x4)
REMD_IN_GGG(x1, x2, x3) = REMD_IN_GGG(x1, x2, x3)
U4_GGG(x1, x2, x3, x4) = U4_GGG(x4)
SUBC_IN_GGA(x1, x2, x3) = SUBC_IN_GGA(x1, x2)
U3_GGA(x1, x2, x3, x4) = U3_GGA(x4)
SUBA_IN_GGA(x1, x2, x3) = SUBA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
U5_GGG(x1, x2, x3, x4) = U5_GGG(x2, x3, x4)
U6_GGG(x1, x2, x3, x4) = U6_GGG(x4)
U7_GGG(x1, x2, x3) = U7_GGG(x3)
GEQB_IN_GG(x1, x2) = GEQB_IN_GG(x1, x2)
U2_GG(x1, x2, x3) = U2_GG(x3)
U7_GAG(x1, x2, x3) = U7_GAG(x3)
GEQB_IN_GA(x1, x2) = GEQB_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x3)

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

(6) Obligation:

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

REMD_IN_GAG(T7, s(T16), T9) → U4_GAG(T7, T16, T9, subC_in_gaa(T7, T16, X7))
REMD_IN_GAG(T7, s(T16), T9) → SUBC_IN_GAA(T7, T16, X7)
SUBC_IN_GAA(s(T31), T33, X40) → U3_GAA(T31, T33, X40, subA_in_gaa(T31, T33, X40))
SUBC_IN_GAA(s(T31), T33, X40) → SUBA_IN_GAA(T31, T33, X40)
SUBA_IN_GAA(s(T44), s(T46), X64) → U1_GAA(T44, T46, X64, subA_in_gaa(T44, T46, X64))
SUBA_IN_GAA(s(T44), s(T46), X64) → SUBA_IN_GAA(T44, T46, X64)
REMD_IN_GAG(T7, s(T20), T9) → U5_GAG(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_GAG(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_GAG(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
U5_GAG(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → REMD_IN_GGG(T19, s(T20), T9)
REMD_IN_GGG(T7, s(T16), T9) → U4_GGG(T7, T16, T9, subC_in_gga(T7, T16, X7))
REMD_IN_GGG(T7, s(T16), T9) → SUBC_IN_GGA(T7, T16, X7)
SUBC_IN_GGA(s(T31), T33, X40) → U3_GGA(T31, T33, X40, subA_in_gga(T31, T33, X40))
SUBC_IN_GGA(s(T31), T33, X40) → SUBA_IN_GGA(T31, T33, X40)
SUBA_IN_GGA(s(T44), s(T46), X64) → U1_GGA(T44, T46, X64, subA_in_gga(T44, T46, X64))
SUBA_IN_GGA(s(T44), s(T46), X64) → SUBA_IN_GGA(T44, T46, X64)
REMD_IN_GGG(T7, s(T20), T9) → U5_GGG(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_GGG(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_GGG(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
U5_GGG(T7, T20, T9, subC_out_gga(T7, T20, T19)) → REMD_IN_GGG(T19, s(T20), T9)
REMD_IN_GGG(s(T79), s(T81), s(T79)) → U7_GGG(T79, T81, geqB_in_gg(T79, T81))
REMD_IN_GGG(s(T79), s(T81), s(T79)) → GEQB_IN_GG(T79, T81)
GEQB_IN_GG(s(T92), s(T94)) → U2_GG(T92, T94, geqB_in_gg(T92, T94))
GEQB_IN_GG(s(T92), s(T94)) → GEQB_IN_GG(T92, T94)
REMD_IN_GAG(s(T79), s(T81), s(T79)) → U7_GAG(T79, T81, geqB_in_ga(T79, T81))
REMD_IN_GAG(s(T79), s(T81), s(T79)) → GEQB_IN_GA(T79, T81)
GEQB_IN_GA(s(T92), s(T94)) → U2_GA(T92, T94, geqB_in_ga(T92, T94))
GEQB_IN_GA(s(T92), s(T94)) → GEQB_IN_GA(T92, T94)

The TRS R consists of the following rules:

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 5 SCCs with 20 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

GEQB_IN_GA(s(T92), s(T94)) → GEQB_IN_GA(T92, T94)

The TRS R consists of the following rules:

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

(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:

GEQB_IN_GA(s(T92), s(T94)) → GEQB_IN_GA(T92, T94)

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

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

GEQB_IN_GA(s(T92)) → GEQB_IN_GA(T92)

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:

GEQB_IN_GA(s(T92)) → GEQB_IN_GA(T92)
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

GEQB_IN_GG(s(T92), s(T94)) → GEQB_IN_GG(T92, T94)

The TRS R consists of the following rules:

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

(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:

GEQB_IN_GG(s(T92), s(T94)) → GEQB_IN_GG(T92, T94)

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

(19) PiDPToQDPProof (EQUIVALENT 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:

GEQB_IN_GG(s(T92), s(T94)) → GEQB_IN_GG(T92, T94)

R is empty.
Q is empty.
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:

GEQB_IN_GG(s(T92), s(T94)) → GEQB_IN_GG(T92, T94)
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:

SUBA_IN_GGA(s(T44), s(T46), X64) → SUBA_IN_GGA(T44, T46, X64)

The TRS R consists of the following rules:

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

(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:

SUBA_IN_GGA(s(T44), s(T46), X64) → SUBA_IN_GGA(T44, T46, X64)

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

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:

SUBA_IN_GGA(s(T44), s(T46)) → SUBA_IN_GGA(T44, T46)

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

(28) 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:

SUBA_IN_GGA(s(T44), s(T46)) → SUBA_IN_GGA(T44, T46)
The graph contains the following edges 1 > 1, 2 > 2

(29) YES

(30) Obligation:

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

REMD_IN_GGG(T7, s(T20), T9) → U5_GGG(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_GGG(T7, T20, T9, subC_out_gga(T7, T20, T19)) → REMD_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

(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:

REMD_IN_GGG(T7, s(T20), T9) → U5_GGG(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_GGG(T7, T20, T9, subC_out_gga(T7, T20, T19)) → REMD_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)

The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
subC_in_gga(x1, x2, x3) = subC_in_gga(x1, x2)
U3_gga(x1, x2, x3, x4) = U3_gga(x4)
subA_in_gga(x1, x2, x3) = subA_in_gga(x1, x2)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
0 = 0
subA_out_gga(x1, x2, x3) = subA_out_gga(x3)
subC_out_gga(x1, x2, x3) = subC_out_gga(x3)
REMD_IN_GGG(x1, x2, x3) = REMD_IN_GGG(x1, x2, x3)
U5_GGG(x1, x2, x3, x4) = U5_GGG(x2, x3, x4)

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:

REMD_IN_GGG(T7, s(T20), T9) → U5_GGG(T20, T9, subC_in_gga(T7, T20))
U5_GGG(T20, T9, subC_out_gga(T19)) → REMD_IN_GGG(T19, s(T20), T9)

The TRS R consists of the following rules:

subC_in_gga(s(T31), T33) → U3_gga(subA_in_gga(T31, T33))
U3_gga(subA_out_gga(X40)) → subC_out_gga(X40)
subA_in_gga(s(T44), s(T46)) → U1_gga(subA_in_gga(T44, T46))
subA_in_gga(T51, 0) → subA_out_gga(T51)
U1_gga(subA_out_gga(X64)) → subA_out_gga(X64)

The set Q consists of the following terms:

subC_in_gga(x0, x1)
U3_gga(x0)
subA_in_gga(x0, x1)
U1_gga(x0)

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

(35) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

U5_GGG(T20, T9, subC_out_gga(T19)) → REMD_IN_GGG(T19, s(T20), T9)

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

POL(0) = 0
POL(REMD_IN_GGG(x₁, x₂, x₃)) = 1 + x₁
POL(U1_gga(x₁)) = x₁
POL(U3_gga(x₁)) = x₁
POL(U5_GGG(x₁, x₂, x₃)) = 1 + x₃
POL(s(x₁)) = 1 + x₁
POL(subA_in_gga(x₁, x₂)) = 1 + x₁
POL(subA_out_gga(x₁)) = 1 + x₁
POL(subC_in_gga(x₁, x₂)) = x₁
POL(subC_out_gga(x₁)) = 1 + x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

subC_in_gga(s(T31), T33) → U3_gga(subA_in_gga(T31, T33))
subA_in_gga(s(T44), s(T46)) → U1_gga(subA_in_gga(T44, T46))
subA_in_gga(T51, 0) → subA_out_gga(T51)
U3_gga(subA_out_gga(X40)) → subC_out_gga(X40)
U1_gga(subA_out_gga(X64)) → subA_out_gga(X64)

(36) Obligation:

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

REMD_IN_GGG(T7, s(T20), T9) → U5_GGG(T20, T9, subC_in_gga(T7, T20))

The TRS R consists of the following rules:

subC_in_gga(s(T31), T33) → U3_gga(subA_in_gga(T31, T33))
U3_gga(subA_out_gga(X40)) → subC_out_gga(X40)
subA_in_gga(s(T44), s(T46)) → U1_gga(subA_in_gga(T44, T46))
subA_in_gga(T51, 0) → subA_out_gga(T51)
U1_gga(subA_out_gga(X64)) → subA_out_gga(X64)

The set Q consists of the following terms:

subC_in_gga(x0, x1)
U3_gga(x0)
subA_in_gga(x0, x1)
U1_gga(x0)

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

(37) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(38) TRUE

(39) Obligation:

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

SUBA_IN_GAA(s(T44), s(T46), X64) → SUBA_IN_GAA(T44, T46, X64)

The TRS R consists of the following rules:

remD_in_gag(T7, s(T16), T9) → U4_gag(T7, T16, T9, subC_in_gaa(T7, T16, X7))
subC_in_gaa(s(T31), T33, X40) → U3_gaa(T31, T33, X40, subA_in_gaa(T31, T33, X40))
subA_in_gaa(s(T44), s(T46), X64) → U1_gaa(T44, T46, X64, subA_in_gaa(T44, T46, X64))
subA_in_gaa(T51, 0, T51) → subA_out_gaa(T51, 0, T51)
U1_gaa(T44, T46, X64, subA_out_gaa(T44, T46, X64)) → subA_out_gaa(s(T44), s(T46), X64)
U3_gaa(T31, T33, X40, subA_out_gaa(T31, T33, X40)) → subC_out_gaa(s(T31), T33, X40)
U4_gag(T7, T16, T9, subC_out_gaa(T7, T16, X7)) → remD_out_gag(T7, s(T16), T9)
remD_in_gag(T7, s(T20), T9) → U5_gag(T7, T20, T9, subC_in_gaa(T7, T20, T19))
U5_gag(T7, T20, T9, subC_out_gaa(T7, T20, T19)) → U6_gag(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(T7, s(T16), T9) → U4_ggg(T7, T16, T9, subC_in_gga(T7, T16, X7))
subC_in_gga(s(T31), T33, X40) → U3_gga(T31, T33, X40, subA_in_gga(T31, T33, X40))
subA_in_gga(s(T44), s(T46), X64) → U1_gga(T44, T46, X64, subA_in_gga(T44, T46, X64))
subA_in_gga(T51, 0, T51) → subA_out_gga(T51, 0, T51)
U1_gga(T44, T46, X64, subA_out_gga(T44, T46, X64)) → subA_out_gga(s(T44), s(T46), X64)
U3_gga(T31, T33, X40, subA_out_gga(T31, T33, X40)) → subC_out_gga(s(T31), T33, X40)
U4_ggg(T7, T16, T9, subC_out_gga(T7, T16, X7)) → remD_out_ggg(T7, s(T16), T9)
remD_in_ggg(T7, s(T20), T9) → U5_ggg(T7, T20, T9, subC_in_gga(T7, T20, T19))
U5_ggg(T7, T20, T9, subC_out_gga(T7, T20, T19)) → U6_ggg(T7, T20, T9, remD_in_ggg(T19, s(T20), T9))
remD_in_ggg(s(T79), s(T81), s(T79)) → U7_ggg(T79, T81, geqB_in_gg(T79, T81))
geqB_in_gg(s(T92), s(T94)) → U2_gg(T92, T94, geqB_in_gg(T92, T94))
geqB_in_gg(T99, 0) → geqB_out_gg(T99, 0)
U2_gg(T92, T94, geqB_out_gg(T92, T94)) → geqB_out_gg(s(T92), s(T94))
U7_ggg(T79, T81, geqB_out_gg(T79, T81)) → remD_out_ggg(s(T79), s(T81), s(T79))
U6_ggg(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_ggg(T7, s(T20), T9)
U6_gag(T7, T20, T9, remD_out_ggg(T19, s(T20), T9)) → remD_out_gag(T7, s(T20), T9)
remD_in_gag(s(T79), s(T81), s(T79)) → U7_gag(T79, T81, geqB_in_ga(T79, T81))
geqB_in_ga(s(T92), s(T94)) → U2_ga(T92, T94, geqB_in_ga(T92, T94))
geqB_in_ga(T99, 0) → geqB_out_ga(T99, 0)
U2_ga(T92, T94, geqB_out_ga(T92, T94)) → geqB_out_ga(s(T92), s(T94))
U7_gag(T79, T81, geqB_out_ga(T79, T81)) → remD_out_gag(s(T79), s(T81), s(T79))

(40) UsableRulesProof (EQUIVALENT transformation)

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

(41) Obligation:

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

SUBA_IN_GAA(s(T44), s(T46), X64) → SUBA_IN_GAA(T44, T46, X64)

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

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

(42) PiDPToQDPProof (SOUND transformation)

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

(43) Obligation:

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

SUBA_IN_GAA(s(T44)) → SUBA_IN_GAA(T44)

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

(44) 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:

SUBA_IN_GAA(s(T44)) → SUBA_IN_GAA(T44)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (SOUND transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) YES

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) YES

(30) Obligation:

(31) UsableRulesProof (EQUIVALENT transformation)

(32) Obligation:

(33) PiDPToQDPProof (SOUND transformation)

(34) Obligation:

(35) QDPOrderProof (EQUIVALENT transformation)

(36) Obligation:

(37) DependencyGraphProof (EQUIVALENT transformation)

(38) TRUE

(39) Obligation:

(40) UsableRulesProof (EQUIVALENT transformation)

(41) Obligation:

(42) PiDPToQDPProof (SOUND transformation)

(43) Obligation:

(44) QDPSizeChangeProof (EQUIVALENT transformation)

(45) YES