Left Termination proof of /tmp/tmptTDao0/divremain.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 83 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 40 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 15 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 0 ms)

        ↳20 QDP

        ↳21 QDPOrderProof (⇔, 0 ms)

        ↳22 QDP

        ↳23 DependencyGraphProof (⇔, 0 ms)

        ↳24 TRUE

(0) Obligation:

Clauses:

div(X1, 0, X2, X3) :- ','(!, failure(a)).
div(0, X4, Z, R) :- ','(!, ','(eq(Z, 0), eq(R, 0))).
div(X, Y, s(Z), R) :- ','(minus(X, Y, U), ','(!, div(U, Y, Z, R))).
div(X, X5, X6, X).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).
failure(b).
eq(X, X).

Query: div(g,g,a,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: div(T1, T2, T3, T4)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 0 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, failure(a)) | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T8, 0, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 1 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX10 is free\nX11 is free\nX12 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: failure(a), SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(!_1, ','(eq(T17, 0), eq(T18, 0))) | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 2 | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT14 is ground\nX10 is free\nX11 is free\nX12 is free\ndiv(0, T14, T3, T4) !=? div(X10, 0, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 2 | div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(eq(T17, 0), eq(T18, 0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(eq(T17, 0), eq(T18, 0)), SCOPE: 3, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: eq(T22, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: eq(T22, 0), SCOPE: 4, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))) | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX29 is free\nX30 is free\nX31 is free\nX32 is free\ndiv(T1, T2, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T1, T2, T3, T4) !=? div(0, X16, X17, X18)\ndiv(T1, T2, T3, T4) !=? div(X29, X30, s(X31), X32)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 4 | ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5 | ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5 | ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(minus(T30, T31, X33), ','(!_1, div(X33, T31, T34, T35))), SCOPE: 5, CLAUSE: 5\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\nX33 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ?_5 | div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(minus(T45, T46, X54), ','(!_1, div(X54, s(T46), T34, T35)))\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: minus(T45, T46, X54)\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(!_1, div(T49, s(T46), T34, T35))\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: minus(T45, T46, X54), SCOPE: 6, CLAUSE: 4 | minus(T45, T46, X54), SCOPE: 6, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: minus(T45, T46, X54), SCOPE: 6, CLAUSE: 4\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: minus(T45, T46, X54), SCOPE: 6, CLAUSE: 5\n\nT45 is ground\nT46 is ground\nX54 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: minus(T61, T62, X75)\n\nT61 is ground\nT62 is ground\nX75 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: div(T49, s(T46), T34, T35)\n\nT46 is ground\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: div(T30, T31, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT30 is ground\nT31 is ground\nX10 is free\nX11 is free\nX12 is free\nX16 is free\nX17 is free\nX18 is free\ndiv(T30, T31, T3, T4) !=? div(X10, 0, X11, X12)\ndiv(T30, T31, T3, T4) !=? div(0, X16, X17, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: 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="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 44 [arrowhead = none ];
44 -> 2;

45 [label="EVAL with clause\ndiv(X10, 0, X11, X12) :- ','(!_1, failure(a)).\nand substitutionT1 -> T8,\nX10 -> T8,\nT2 -> 0,\nT3 -> T9,\nX11 -> T9,\nT4 -> T10,\nX12 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 45 [arrowhead = none ];
45 -> 3;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 46 [arrowhead = none ];
46 -> 4;

47 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 47 [arrowhead = none ];
47 -> 5;

48 [label="EVAL with clause\ndiv(0, X16, X17, X18) :- ','(!_1, ','(eq(X17, 0), eq(X18, 0))).\nand substitutionT1 -> 0,\nT2 -> T14,\nX16 -> T14,\nT3 -> T17,\nX17 -> T17,\nT4 -> T18,\nX18 -> T18,\nT15 -> T17,\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 48 [arrowhead = none ];
48 -> 8;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 49 [arrowhead = none ];
49 -> 9;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 50 [arrowhead = none ];
50 -> 6;

51 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 51 [arrowhead = none ];
51 -> 7;

52 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 52 [arrowhead = none ];
52 -> 10;

53 [label="EVAL with clause\ndiv(X29, X30, s(X31), X32) :- ','(minus(X29, X30, X33), ','(!_1, div(X33, X30, X31, X32))).\nand substitutionT1 -> T30,\nX29 -> T30,\nT2 -> T31,\nX30 -> T31,\nX31 -> T34,\nT3 -> s(T34),\nT4 -> T35,\nX32 -> T35,\nT32 -> T34,\nT33 -> T35", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 54 [arrowhead = none ];
54 -> 19;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 55 [arrowhead = none ];
55 -> 11;

56 [label="EVAL with clause\neq(X21, X21).\nand substitutionT17 -> 0,\nX21 -> 0,\nT21 -> 0,\nT18 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 56 [arrowhead = none ];
56 -> 12;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 57 [arrowhead = none ];
57 -> 13;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 58 [arrowhead = none ];
58 -> 14;

59 [label="EVAL with clause\neq(X24, X24).\nand substitutionT22 -> 0,\nX24 -> 0,\nT25 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 59 [arrowhead = none ];
59 -> 15;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 60 [arrowhead = none ];
60 -> 16;

61 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 61 [arrowhead = none ];
61 -> 17;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 62 [arrowhead = none ];
62 -> 20;

63 [label="EVAL with clause\ndiv(X93, X94, X95, X93).\nand substitutionT1 -> T78,\nX93 -> T78,\nT2 -> T79,\nX94 -> T79,\nT3 -> T80,\nX95 -> T80,\nT4 -> T78", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 63 [arrowhead = none ];
63 -> 41;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 64 [arrowhead = none ];
64 -> 42;

65 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T30, T31, T3, T4), div(X10, 0, X11, X12))", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 65 [arrowhead = none ];
65 -> 21;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 66 [arrowhead = none ];
66 -> 22;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
21 -> 67 [arrowhead = none ];
67 -> 23;

68 [label="EVAL with clause\nminus(s(X51), s(X52), X53) :- minus(X51, X52, X53).\nand substitutionX51 -> T45,\nT30 -> s(T45),\nX52 -> T46,\nT31 -> s(T46),\nX33 -> X54,\nX53 -> X54", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 68 [arrowhead = none ];
68 -> 24;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 69 [arrowhead = none ];
69 -> 25;

70 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 70 [arrowhead = none ];
70 -> 37;

71 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 71 [arrowhead = none ];
71 -> 26;

72 [label="SPLIT 2\nnew knowledge:\nT45 is ground\nT46 is ground\nT49 is ground\nreplacements:X54 -> T49", fontsize=14, style = filled, fillcolor = violet];
24 -> 72 [arrowhead = none ];
72 -> 27;

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

74 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 74 [arrowhead = none ];
74 -> 36;

75 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 75 [arrowhead = none ];
75 -> 29;

76 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 76 [arrowhead = none ];
76 -> 30;

77 [label="EVAL with clause\nminus(X63, 0, X63).\nand substitutionT45 -> T56,\nX63 -> T56,\nT46 -> 0,\nX54 -> T56", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 77 [arrowhead = none ];
77 -> 31;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 78 [arrowhead = none ];
78 -> 32;

79 [label="EVAL with clause\nminus(s(X72), s(X73), X74) :- minus(X72, X73, X74).\nand substitutionX72 -> T61,\nT45 -> s(T61),\nX73 -> T62,\nT46 -> s(T62),\nX54 -> X75,\nX74 -> X75", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 79 [arrowhead = none ];
79 -> 34;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 80 [arrowhead = none ];
80 -> 35;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 81 [arrowhead = none ];
81 -> 33;

82 [label="INSTANCE with matching:\nT45 -> T61\nT46 -> T62\nX54 -> X75", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 82 [arrowhead = none , style=dashed];
82 -> 26[style=dashed];

83 [label="INSTANCE with matching:\nT1 -> T49\nT2 -> s(T46)\nT3 -> T34\nT4 -> T35", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 83 [arrowhead = none , style=dashed];
83 -> 1[style=dashed];

84 [label="EVAL with clause\ndiv(X87, X88, X89, X87).\nand substitutionT30 -> T72,\nX87 -> T72,\nT31 -> T73,\nX88 -> T73,\nT3 -> T74,\nX89 -> T74,\nT4 -> T72", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 84 [arrowhead = none ];
84 -> 38;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 85 [arrowhead = none ];
85 -> 39;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 86 [arrowhead = none ];
86 -> 40;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 87 [arrowhead = none ];
87 -> 43;

}

(2) Obligation:

Clauses:

minusA(T56, 0, T56).
minusA(s(T61), s(T62), X75) :- minusA(T61, T62, X75).
divB(0, T14, 0, 0).
divB(s(T45), s(T46), s(T34), T35) :- minusA(T45, T46, X54).
divB(s(T45), s(T46), s(T34), T35) :- ','(minusA(T45, T46, T49), divB(T49, s(T46), T34, T35)).
divB(T72, T73, T74, T72).
divB(T78, T79, T80, T78).

Query: divB(g,g,a,a)

(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:
divB_in: (b,b,f,f)
minusA_in: (b,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

divB_in_ggaa(0, T14, 0, 0) → divB_out_ggaa(0, T14, 0, 0)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U2_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, X54))
minusA_in_gga(T56, 0, T56) → minusA_out_gga(T56, 0, T56)
minusA_in_gga(s(T61), s(T62), X75) → U1_gga(T61, T62, X75, minusA_in_gga(T61, T62, X75))
U1_gga(T61, T62, X75, minusA_out_gga(T61, T62, X75)) → minusA_out_gga(s(T61), s(T62), X75)
U2_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, X54)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U3_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → U4_ggaa(T45, T46, T34, T35, divB_in_ggaa(T49, s(T46), T34, T35))
divB_in_ggaa(T72, T73, T74, T72) → divB_out_ggaa(T72, T73, T74, T72)
U4_ggaa(T45, T46, T34, T35, divB_out_ggaa(T49, s(T46), T34, T35)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

divB_in_ggaa(0, T14, 0, 0) → divB_out_ggaa(0, T14, 0, 0)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U2_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, X54))
minusA_in_gga(T56, 0, T56) → minusA_out_gga(T56, 0, T56)
minusA_in_gga(s(T61), s(T62), X75) → U1_gga(T61, T62, X75, minusA_in_gga(T61, T62, X75))
U1_gga(T61, T62, X75, minusA_out_gga(T61, T62, X75)) → minusA_out_gga(s(T61), s(T62), X75)
U2_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, X54)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U3_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → U4_ggaa(T45, T46, T34, T35, divB_in_ggaa(T49, s(T46), T34, T35))
divB_in_ggaa(T72, T73, T74, T72) → divB_out_ggaa(T72, T73, T74, T72)
U4_ggaa(T45, T46, T34, T35, divB_out_ggaa(T49, s(T46), T34, T35)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)

(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:

DIVB_IN_GGAA(s(T45), s(T46), s(T34), T35) → U2_GGAA(T45, T46, T34, T35, minusA_in_gga(T45, T46, X54))
DIVB_IN_GGAA(s(T45), s(T46), s(T34), T35) → MINUSA_IN_GGA(T45, T46, X54)
MINUSA_IN_GGA(s(T61), s(T62), X75) → U1_GGA(T61, T62, X75, minusA_in_gga(T61, T62, X75))
MINUSA_IN_GGA(s(T61), s(T62), X75) → MINUSA_IN_GGA(T61, T62, X75)
DIVB_IN_GGAA(s(T45), s(T46), s(T34), T35) → U3_GGAA(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_GGAA(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → U4_GGAA(T45, T46, T34, T35, divB_in_ggaa(T49, s(T46), T34, T35))
U3_GGAA(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → DIVB_IN_GGAA(T49, s(T46), T34, T35)

The TRS R consists of the following rules:

divB_in_ggaa(0, T14, 0, 0) → divB_out_ggaa(0, T14, 0, 0)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U2_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, X54))
minusA_in_gga(T56, 0, T56) → minusA_out_gga(T56, 0, T56)
minusA_in_gga(s(T61), s(T62), X75) → U1_gga(T61, T62, X75, minusA_in_gga(T61, T62, X75))
U1_gga(T61, T62, X75, minusA_out_gga(T61, T62, X75)) → minusA_out_gga(s(T61), s(T62), X75)
U2_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, X54)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U3_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → U4_ggaa(T45, T46, T34, T35, divB_in_ggaa(T49, s(T46), T34, T35))
divB_in_ggaa(T72, T73, T74, T72) → divB_out_ggaa(T72, T73, T74, T72)
U4_ggaa(T45, T46, T34, T35, divB_out_ggaa(T49, s(T46), T34, T35)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)

The argument filtering Pi contains the following mapping:
divB_in_ggaa(x1, x2, x3, x4) = divB_in_ggaa(x1, x2)
0 = 0
divB_out_ggaa(x1, x2, x3, x4) = divB_out_ggaa
s(x1) = s(x1)
U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x5)
minusA_in_gga(x1, x2, x3) = minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3) = minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U3_ggaa(x1, x2, x3, x4, x5) = U3_ggaa(x2, x5)
U4_ggaa(x1, x2, x3, x4, x5) = U4_ggaa(x5)
DIVB_IN_GGAA(x1, x2, x3, x4) = DIVB_IN_GGAA(x1, x2)
U2_GGAA(x1, x2, x3, x4, x5) = U2_GGAA(x5)
MINUSA_IN_GGA(x1, x2, x3) = MINUSA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4) = U1_GGA(x4)
U3_GGAA(x1, x2, x3, x4, x5) = U3_GGAA(x2, x5)
U4_GGAA(x1, x2, x3, x4, x5) = U4_GGAA(x5)

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

(6) Obligation:

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

DIVB_IN_GGAA(s(T45), s(T46), s(T34), T35) → U2_GGAA(T45, T46, T34, T35, minusA_in_gga(T45, T46, X54))
DIVB_IN_GGAA(s(T45), s(T46), s(T34), T35) → MINUSA_IN_GGA(T45, T46, X54)
MINUSA_IN_GGA(s(T61), s(T62), X75) → U1_GGA(T61, T62, X75, minusA_in_gga(T61, T62, X75))
MINUSA_IN_GGA(s(T61), s(T62), X75) → MINUSA_IN_GGA(T61, T62, X75)
DIVB_IN_GGAA(s(T45), s(T46), s(T34), T35) → U3_GGAA(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_GGAA(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → U4_GGAA(T45, T46, T34, T35, divB_in_ggaa(T49, s(T46), T34, T35))
U3_GGAA(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → DIVB_IN_GGAA(T49, s(T46), T34, T35)

The TRS R consists of the following rules:

divB_in_ggaa(0, T14, 0, 0) → divB_out_ggaa(0, T14, 0, 0)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U2_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, X54))
minusA_in_gga(T56, 0, T56) → minusA_out_gga(T56, 0, T56)
minusA_in_gga(s(T61), s(T62), X75) → U1_gga(T61, T62, X75, minusA_in_gga(T61, T62, X75))
U1_gga(T61, T62, X75, minusA_out_gga(T61, T62, X75)) → minusA_out_gga(s(T61), s(T62), X75)
U2_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, X54)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U3_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → U4_ggaa(T45, T46, T34, T35, divB_in_ggaa(T49, s(T46), T34, T35))
divB_in_ggaa(T72, T73, T74, T72) → divB_out_ggaa(T72, T73, T74, T72)
U4_ggaa(T45, T46, T34, T35, divB_out_ggaa(T49, s(T46), T34, T35)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MINUSA_IN_GGA(s(T61), s(T62), X75) → MINUSA_IN_GGA(T61, T62, X75)

The TRS R consists of the following rules:

divB_in_ggaa(0, T14, 0, 0) → divB_out_ggaa(0, T14, 0, 0)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U2_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, X54))
minusA_in_gga(T56, 0, T56) → minusA_out_gga(T56, 0, T56)
minusA_in_gga(s(T61), s(T62), X75) → U1_gga(T61, T62, X75, minusA_in_gga(T61, T62, X75))
U1_gga(T61, T62, X75, minusA_out_gga(T61, T62, X75)) → minusA_out_gga(s(T61), s(T62), X75)
U2_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, X54)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U3_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → U4_ggaa(T45, T46, T34, T35, divB_in_ggaa(T49, s(T46), T34, T35))
divB_in_ggaa(T72, T73, T74, T72) → divB_out_ggaa(T72, T73, T74, T72)
U4_ggaa(T45, T46, T34, T35, divB_out_ggaa(T49, s(T46), T34, T35)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)

The argument filtering Pi contains the following mapping:
divB_in_ggaa(x1, x2, x3, x4) = divB_in_ggaa(x1, x2)
0 = 0
divB_out_ggaa(x1, x2, x3, x4) = divB_out_ggaa
s(x1) = s(x1)
U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x5)
minusA_in_gga(x1, x2, x3) = minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3) = minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U3_ggaa(x1, x2, x3, x4, x5) = U3_ggaa(x2, x5)
U4_ggaa(x1, x2, x3, x4, x5) = U4_ggaa(x5)
MINUSA_IN_GGA(x1, x2, x3) = MINUSA_IN_GGA(x1, x2)

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

(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:

MINUSA_IN_GGA(s(T61), s(T62), X75) → MINUSA_IN_GGA(T61, T62, X75)

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

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:

MINUSA_IN_GGA(s(T61), s(T62)) → MINUSA_IN_GGA(T61, T62)

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:

MINUSA_IN_GGA(s(T61), s(T62)) → MINUSA_IN_GGA(T61, T62)
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:

DIVB_IN_GGAA(s(T45), s(T46), s(T34), T35) → U3_GGAA(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_GGAA(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → DIVB_IN_GGAA(T49, s(T46), T34, T35)

The TRS R consists of the following rules:

divB_in_ggaa(0, T14, 0, 0) → divB_out_ggaa(0, T14, 0, 0)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U2_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, X54))
minusA_in_gga(T56, 0, T56) → minusA_out_gga(T56, 0, T56)
minusA_in_gga(s(T61), s(T62), X75) → U1_gga(T61, T62, X75, minusA_in_gga(T61, T62, X75))
U1_gga(T61, T62, X75, minusA_out_gga(T61, T62, X75)) → minusA_out_gga(s(T61), s(T62), X75)
U2_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, X54)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)
divB_in_ggaa(s(T45), s(T46), s(T34), T35) → U3_ggaa(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_ggaa(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → U4_ggaa(T45, T46, T34, T35, divB_in_ggaa(T49, s(T46), T34, T35))
divB_in_ggaa(T72, T73, T74, T72) → divB_out_ggaa(T72, T73, T74, T72)
U4_ggaa(T45, T46, T34, T35, divB_out_ggaa(T49, s(T46), T34, T35)) → divB_out_ggaa(s(T45), s(T46), s(T34), T35)

The argument filtering Pi contains the following mapping:
divB_in_ggaa(x1, x2, x3, x4) = divB_in_ggaa(x1, x2)
0 = 0
divB_out_ggaa(x1, x2, x3, x4) = divB_out_ggaa
s(x1) = s(x1)
U2_ggaa(x1, x2, x3, x4, x5) = U2_ggaa(x5)
minusA_in_gga(x1, x2, x3) = minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3) = minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
U3_ggaa(x1, x2, x3, x4, x5) = U3_ggaa(x2, x5)
U4_ggaa(x1, x2, x3, x4, x5) = U4_ggaa(x5)
DIVB_IN_GGAA(x1, x2, x3, x4) = DIVB_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5) = U3_GGAA(x2, x5)

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

(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:

DIVB_IN_GGAA(s(T45), s(T46), s(T34), T35) → U3_GGAA(T45, T46, T34, T35, minusA_in_gga(T45, T46, T49))
U3_GGAA(T45, T46, T34, T35, minusA_out_gga(T45, T46, T49)) → DIVB_IN_GGAA(T49, s(T46), T34, T35)

The TRS R consists of the following rules:

minusA_in_gga(T56, 0, T56) → minusA_out_gga(T56, 0, T56)
minusA_in_gga(s(T61), s(T62), X75) → U1_gga(T61, T62, X75, minusA_in_gga(T61, T62, X75))
U1_gga(T61, T62, X75, minusA_out_gga(T61, T62, X75)) → minusA_out_gga(s(T61), s(T62), X75)

The argument filtering Pi contains the following mapping:
0 = 0
s(x1) = s(x1)
minusA_in_gga(x1, x2, x3) = minusA_in_gga(x1, x2)
minusA_out_gga(x1, x2, x3) = minusA_out_gga(x3)
U1_gga(x1, x2, x3, x4) = U1_gga(x4)
DIVB_IN_GGAA(x1, x2, x3, x4) = DIVB_IN_GGAA(x1, x2)
U3_GGAA(x1, x2, x3, x4, x5) = U3_GGAA(x2, x5)

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:

DIVB_IN_GGAA(s(T45), s(T46)) → U3_GGAA(T46, minusA_in_gga(T45, T46))
U3_GGAA(T46, minusA_out_gga(T49)) → DIVB_IN_GGAA(T49, s(T46))

The TRS R consists of the following rules:

minusA_in_gga(T56, 0) → minusA_out_gga(T56)
minusA_in_gga(s(T61), s(T62)) → U1_gga(minusA_in_gga(T61, T62))
U1_gga(minusA_out_gga(X75)) → minusA_out_gga(X75)

The set Q consists of the following terms:

minusA_in_gga(x0, x1)
U1_gga(x0)

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

(21) QDPOrderProof (EQUIVALENT transformation)

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

The following pairs can be oriented strictly and are deleted.

DIVB_IN_GGAA(s(T45), s(T46)) → U3_GGAA(T46, minusA_in_gga(T45, T46))

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

POL(0) = 0
POL(DIVB_IN_GGAA(x₁, x₂)) = x₁
POL(U1_gga(x₁)) = x₁
POL(U3_GGAA(x₁, x₂)) = x₂
POL(minusA_in_gga(x₁, x₂)) = x₁
POL(minusA_out_gga(x₁)) = x₁
POL(s(x₁)) = 1 + x₁

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

minusA_in_gga(T56, 0) → minusA_out_gga(T56)
minusA_in_gga(s(T61), s(T62)) → U1_gga(minusA_in_gga(T61, T62))
U1_gga(minusA_out_gga(X75)) → minusA_out_gga(X75)

(22) Obligation:

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

U3_GGAA(T46, minusA_out_gga(T49)) → DIVB_IN_GGAA(T49, s(T46))

The TRS R consists of the following rules:

minusA_in_gga(T56, 0) → minusA_out_gga(T56)
minusA_in_gga(s(T61), s(T62)) → U1_gga(minusA_in_gga(T61, T62))
U1_gga(minusA_out_gga(X75)) → minusA_out_gga(X75)

The set Q consists of the following terms:

minusA_in_gga(x0, x1)
U1_gga(x0)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(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 (SOUND transformation)

(20) Obligation:

(21) QDPOrderProof (EQUIVALENT transformation)

(22) Obligation:

(23) DependencyGraphProof (EQUIVALENT transformation)

(24) TRUE