Left Termination proof of /tmp/tmptvtW1R/divrempredef.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 (⇒, 89 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 10 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔, 0 ms)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇒, 0 ms)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔, 0 ms)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔, 0 ms)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇒, 4 ms)

        ↳20 QDP

        ↳21 QDPOrderProof (⇔, 18 ms)

        ↳22 QDP

        ↳23 DependencyGraphProof (⇔, 0 ms)

        ↳24 TRUE

(0) Obligation:

Clauses:

div(X, 0, Z, R) :- ','(!, fail).
div(0, Y, Z, R) :- ','(!, ','(=(Z, 0), =(R, 0))).
div(X, Y, s(Z), R) :- ','(minus(X, Y, U), ','(!, div(U, Y, Z, R))).
div(X, Y, 0, X).
minus(X, 0, X).
minus(s(X), s(Y), Z) :- minus(X, Y, Z).

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, fail) | 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\nX4 is free\nX5 is free\nX6 is free\ndiv(T1, T2, T3, T4) !=? div(X4, 0, X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: fail\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_1, ','(=(T17, 0), =(T18, 0))) | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 2 | div(0, T14, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT14 is ground\nX4 is free\nX5 is free\nX6 is free\ndiv(0, T14, T3, T4) !=? div(X4, 0, X5, X6)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: 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\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\ndiv(T1, T2, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T1, T2, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(=(T17, 0), =(T18, 0))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: =(T19, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))) | div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX21 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: div(T1, T2, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nT2 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX17 is free\nX18 is free\nX19 is free\nX20 is free\ndiv(T1, T2, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T1, T2, T3, T4) !=? div(0, X10, X11, X12)\ndiv(T1, T2, T3, T4) !=? div(X17, X18, s(X19), X20)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))), SCOPE: 2, CLAUSE: 4 | ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))), SCOPE: 2, CLAUSE: 5 | ?_2 | div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX21 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))), SCOPE: 2, CLAUSE: 5 | ?_2 | div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX21 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(minus(T24, T25, X21), ','(!_1, div(X21, T25, T28, T29))), SCOPE: 2, CLAUSE: 5\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\nX21 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ?_2 | div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(minus(T39, T40, X42), ','(!_1, div(X42, s(T40), T28, T29)))\n\nT39 is ground\nT40 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: minus(T39, T40, X42)\n\nT39 is ground\nT40 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(!_1, div(T43, s(T40), T28, T29))\n\nT40 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: minus(T39, T40, X42), SCOPE: 3, CLAUSE: 4 | minus(T39, T40, X42), SCOPE: 3, CLAUSE: 5\n\nT39 is ground\nT40 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: minus(T39, T40, X42), SCOPE: 3, CLAUSE: 4\n\nT39 is ground\nT40 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: minus(T39, T40, X42), SCOPE: 3, CLAUSE: 5\n\nT39 is ground\nT40 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: minus(T55, T56, X63)\n\nT55 is ground\nT56 is ground\nX63 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: div(T43, s(T40), T28, T29)\n\nT40 is ground\nT43 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: div(T24, T25, T3, T4), SCOPE: 1, CLAUSE: 3\n\nT24 is ground\nT25 is ground\nX4 is free\nX5 is free\nX6 is free\nX10 is free\nX11 is free\nX12 is free\ndiv(T24, T25, T3, T4) !=? div(X4, 0, X5, X6)\ndiv(T24, T25, T3, T4) !=? div(0, X10, X11, X12)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", 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="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 41 [arrowhead = none ];
41 -> 2;

42 [label="EVAL with clause\ndiv(X4, 0, X5, X6) :- ','(!_1, fail).\nand substitutionT1 -> T8,\nX4 -> T8,\nT2 -> 0,\nT3 -> T9,\nX5 -> T9,\nT4 -> T10,\nX6 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 42 [arrowhead = none ];
42 -> 3;

43 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 43 [arrowhead = none ];
43 -> 4;

44 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 44 [arrowhead = none ];
44 -> 5;

45 [label="EVAL with clause\ndiv(0, X10, X11, X12) :- ','(!_1, ','(=(X11, 0), =(X12, 0))).\nand substitutionT1 -> 0,\nT2 -> T14,\nX10 -> T14,\nT3 -> T17,\nX11 -> T17,\nT4 -> T18,\nX12 -> T18,\nT15 -> T17,\nT16 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 45 [arrowhead = none ];
45 -> 7;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 46 [arrowhead = none ];
46 -> 8;

47 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 47 [arrowhead = none ];
47 -> 6;

48 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 48 [arrowhead = none ];
48 -> 9;

49 [label="EVAL with clause\ndiv(X17, X18, s(X19), X20) :- ','(minus(X17, X18, X21), ','(!_1, div(X21, X18, X19, X20))).\nand substitutionT1 -> T24,\nX17 -> T24,\nT2 -> T25,\nX18 -> T25,\nX19 -> T28,\nT3 -> s(T28),\nT4 -> T29,\nX20 -> T29,\nT26 -> T28,\nT27 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 49 [arrowhead = none ];
49 -> 15;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 50 [arrowhead = none ];
50 -> 16;

51 [label="UNIFY CASE with substitutionT17 -> 0,\nT18 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 51 [arrowhead = none ];
51 -> 10;

52 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 52 [arrowhead = none ];
52 -> 11;

53 [label="UNIFY CASE with substitutionT19 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 53 [arrowhead = none ];
53 -> 12;

54 [label="UNIFY-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 54 [arrowhead = none ];
54 -> 13;

55 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 55 [arrowhead = none ];
55 -> 14;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 56 [arrowhead = none ];
56 -> 17;

57 [label="EVAL with clause\ndiv(X77, X78, 0, X77).\nand substitutionT1 -> T68,\nX77 -> T68,\nT2 -> T69,\nX78 -> T69,\nT3 -> 0,\nT4 -> T68", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 57 [arrowhead = none ];
57 -> 38;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 58 [arrowhead = none ];
58 -> 39;

59 [label="BACKTRACK\nfor clause: minus(X, 0, X)\nwith clash: (div(T24, T25, T3, T4), div(X4, 0, X5, X6))", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 59 [arrowhead = none ];
59 -> 18;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 60 [arrowhead = none ];
60 -> 19;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 61 [arrowhead = none ];
61 -> 20;

62 [label="EVAL with clause\nminus(s(X39), s(X40), X41) :- minus(X39, X40, X41).\nand substitutionX39 -> T39,\nT24 -> s(T39),\nX40 -> T40,\nT25 -> s(T40),\nX21 -> X42,\nX41 -> X42", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 62 [arrowhead = none ];
62 -> 21;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 63 [arrowhead = none ];
63 -> 22;

64 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 64 [arrowhead = none ];
64 -> 34;

65 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 65 [arrowhead = none ];
65 -> 23;

66 [label="SPLIT 2\nnew knowledge:\nT39 is ground\nT40 is ground\nT43 is ground\nreplacements:X42 -> T43", fontsize=14, style = filled, fillcolor = violet];
21 -> 66 [arrowhead = none ];
66 -> 24;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 67 [arrowhead = none ];
67 -> 25;

68 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 69 [arrowhead = none ];
69 -> 26;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 70 [arrowhead = none ];
70 -> 27;

71 [label="EVAL with clause\nminus(X51, 0, X51).\nand substitutionT39 -> T50,\nX51 -> T50,\nT40 -> 0,\nX42 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 71 [arrowhead = none ];
71 -> 28;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 72 [arrowhead = none ];
72 -> 29;

73 [label="EVAL with clause\nminus(s(X60), s(X61), X62) :- minus(X60, X61, X62).\nand substitutionX60 -> T55,\nT39 -> s(T55),\nX61 -> T56,\nT40 -> s(T56),\nX42 -> X63,\nX62 -> X63", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 73 [arrowhead = none ];
73 -> 31;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 74 [arrowhead = none ];
74 -> 32;

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

76 [label="INSTANCE with matching:\nT39 -> T55\nT40 -> T56\nX42 -> X63", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 76 [arrowhead = none , style=dashed];
76 -> 23[style=dashed];

77 [label="INSTANCE with matching:\nT1 -> T43\nT2 -> s(T40)\nT3 -> T28\nT4 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 77 [arrowhead = none , style=dashed];
77 -> 1[style=dashed];

78 [label="EVAL with clause\ndiv(X73, X74, 0, X73).\nand substitutionT24 -> T64,\nX73 -> T64,\nT25 -> T65,\nX74 -> T65,\nT3 -> 0,\nT4 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 78 [arrowhead = none ];
78 -> 35;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 79 [arrowhead = none ];
79 -> 36;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 80 [arrowhead = none ];
80 -> 37;

81 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 81 [arrowhead = none ];
81 -> 40;

}

(2) Obligation:

Clauses:

minusA(T50, 0, T50).
minusA(s(T55), s(T56), X63) :- minusA(T55, T56, X63).
divB(0, T14, 0, 0).
divB(s(T39), s(T40), s(T28), T29) :- minusA(T39, T40, X42).
divB(s(T39), s(T40), s(T28), T29) :- ','(minusA(T39, T40, T43), divB(T43, s(T40), T28, T29)).
divB(T64, T65, 0, T64).
divB(T68, T69, 0, T68).

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(T39), s(T40), s(T28), T29) → U2_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, X42))
minusA_in_gga(T50, 0, T50) → minusA_out_gga(T50, 0, T50)
minusA_in_gga(s(T55), s(T56), X63) → U1_gga(T55, T56, X63, minusA_in_gga(T55, T56, X63))
U1_gga(T55, T56, X63, minusA_out_gga(T55, T56, X63)) → minusA_out_gga(s(T55), s(T56), X63)
U2_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, X42)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)
divB_in_ggaa(s(T39), s(T40), s(T28), T29) → U3_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → U4_ggaa(T39, T40, T28, T29, divB_in_ggaa(T43, s(T40), T28, T29))
divB_in_ggaa(T64, T65, 0, T64) → divB_out_ggaa(T64, T65, 0, T64)
U4_ggaa(T39, T40, T28, T29, divB_out_ggaa(T43, s(T40), T28, T29)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)

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(T39), s(T40), s(T28), T29) → U2_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, X42))
minusA_in_gga(T50, 0, T50) → minusA_out_gga(T50, 0, T50)
minusA_in_gga(s(T55), s(T56), X63) → U1_gga(T55, T56, X63, minusA_in_gga(T55, T56, X63))
U1_gga(T55, T56, X63, minusA_out_gga(T55, T56, X63)) → minusA_out_gga(s(T55), s(T56), X63)
U2_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, X42)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)
divB_in_ggaa(s(T39), s(T40), s(T28), T29) → U3_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → U4_ggaa(T39, T40, T28, T29, divB_in_ggaa(T43, s(T40), T28, T29))
divB_in_ggaa(T64, T65, 0, T64) → divB_out_ggaa(T64, T65, 0, T64)
U4_ggaa(T39, T40, T28, T29, divB_out_ggaa(T43, s(T40), T28, T29)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)

(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(T39), s(T40), s(T28), T29) → U2_GGAA(T39, T40, T28, T29, minusA_in_gga(T39, T40, X42))
DIVB_IN_GGAA(s(T39), s(T40), s(T28), T29) → MINUSA_IN_GGA(T39, T40, X42)
MINUSA_IN_GGA(s(T55), s(T56), X63) → U1_GGA(T55, T56, X63, minusA_in_gga(T55, T56, X63))
MINUSA_IN_GGA(s(T55), s(T56), X63) → MINUSA_IN_GGA(T55, T56, X63)
DIVB_IN_GGAA(s(T39), s(T40), s(T28), T29) → U3_GGAA(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_GGAA(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → U4_GGAA(T39, T40, T28, T29, divB_in_ggaa(T43, s(T40), T28, T29))
U3_GGAA(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → DIVB_IN_GGAA(T43, s(T40), T28, T29)

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(T39), s(T40), s(T28), T29) → U2_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, X42))
minusA_in_gga(T50, 0, T50) → minusA_out_gga(T50, 0, T50)
minusA_in_gga(s(T55), s(T56), X63) → U1_gga(T55, T56, X63, minusA_in_gga(T55, T56, X63))
U1_gga(T55, T56, X63, minusA_out_gga(T55, T56, X63)) → minusA_out_gga(s(T55), s(T56), X63)
U2_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, X42)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)
divB_in_ggaa(s(T39), s(T40), s(T28), T29) → U3_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → U4_ggaa(T39, T40, T28, T29, divB_in_ggaa(T43, s(T40), T28, T29))
divB_in_ggaa(T64, T65, 0, T64) → divB_out_ggaa(T64, T65, 0, T64)
U4_ggaa(T39, T40, T28, T29, divB_out_ggaa(T43, s(T40), T28, T29)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)

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(T39), s(T40), s(T28), T29) → U2_GGAA(T39, T40, T28, T29, minusA_in_gga(T39, T40, X42))
DIVB_IN_GGAA(s(T39), s(T40), s(T28), T29) → MINUSA_IN_GGA(T39, T40, X42)
MINUSA_IN_GGA(s(T55), s(T56), X63) → U1_GGA(T55, T56, X63, minusA_in_gga(T55, T56, X63))
MINUSA_IN_GGA(s(T55), s(T56), X63) → MINUSA_IN_GGA(T55, T56, X63)
DIVB_IN_GGAA(s(T39), s(T40), s(T28), T29) → U3_GGAA(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_GGAA(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → U4_GGAA(T39, T40, T28, T29, divB_in_ggaa(T43, s(T40), T28, T29))
U3_GGAA(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → DIVB_IN_GGAA(T43, s(T40), T28, T29)

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(T39), s(T40), s(T28), T29) → U2_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, X42))
minusA_in_gga(T50, 0, T50) → minusA_out_gga(T50, 0, T50)
minusA_in_gga(s(T55), s(T56), X63) → U1_gga(T55, T56, X63, minusA_in_gga(T55, T56, X63))
U1_gga(T55, T56, X63, minusA_out_gga(T55, T56, X63)) → minusA_out_gga(s(T55), s(T56), X63)
U2_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, X42)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)
divB_in_ggaa(s(T39), s(T40), s(T28), T29) → U3_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → U4_ggaa(T39, T40, T28, T29, divB_in_ggaa(T43, s(T40), T28, T29))
divB_in_ggaa(T64, T65, 0, T64) → divB_out_ggaa(T64, T65, 0, T64)
U4_ggaa(T39, T40, T28, T29, divB_out_ggaa(T43, s(T40), T28, T29)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)

(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(T55), s(T56), X63) → MINUSA_IN_GGA(T55, T56, X63)

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(T39), s(T40), s(T28), T29) → U2_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, X42))
minusA_in_gga(T50, 0, T50) → minusA_out_gga(T50, 0, T50)
minusA_in_gga(s(T55), s(T56), X63) → U1_gga(T55, T56, X63, minusA_in_gga(T55, T56, X63))
U1_gga(T55, T56, X63, minusA_out_gga(T55, T56, X63)) → minusA_out_gga(s(T55), s(T56), X63)
U2_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, X42)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)
divB_in_ggaa(s(T39), s(T40), s(T28), T29) → U3_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → U4_ggaa(T39, T40, T28, T29, divB_in_ggaa(T43, s(T40), T28, T29))
divB_in_ggaa(T64, T65, 0, T64) → divB_out_ggaa(T64, T65, 0, T64)
U4_ggaa(T39, T40, T28, T29, divB_out_ggaa(T43, s(T40), T28, T29)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)

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(T55), s(T56), X63) → MINUSA_IN_GGA(T55, T56, X63)

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(T55), s(T56)) → MINUSA_IN_GGA(T55, T56)

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(T55), s(T56)) → MINUSA_IN_GGA(T55, T56)
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(T39), s(T40), s(T28), T29) → U3_GGAA(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_GGAA(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → DIVB_IN_GGAA(T43, s(T40), T28, T29)

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(T39), s(T40), s(T28), T29) → U2_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, X42))
minusA_in_gga(T50, 0, T50) → minusA_out_gga(T50, 0, T50)
minusA_in_gga(s(T55), s(T56), X63) → U1_gga(T55, T56, X63, minusA_in_gga(T55, T56, X63))
U1_gga(T55, T56, X63, minusA_out_gga(T55, T56, X63)) → minusA_out_gga(s(T55), s(T56), X63)
U2_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, X42)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)
divB_in_ggaa(s(T39), s(T40), s(T28), T29) → U3_ggaa(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_ggaa(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → U4_ggaa(T39, T40, T28, T29, divB_in_ggaa(T43, s(T40), T28, T29))
divB_in_ggaa(T64, T65, 0, T64) → divB_out_ggaa(T64, T65, 0, T64)
U4_ggaa(T39, T40, T28, T29, divB_out_ggaa(T43, s(T40), T28, T29)) → divB_out_ggaa(s(T39), s(T40), s(T28), T29)

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(T39), s(T40), s(T28), T29) → U3_GGAA(T39, T40, T28, T29, minusA_in_gga(T39, T40, T43))
U3_GGAA(T39, T40, T28, T29, minusA_out_gga(T39, T40, T43)) → DIVB_IN_GGAA(T43, s(T40), T28, T29)

The TRS R consists of the following rules:

minusA_in_gga(T50, 0, T50) → minusA_out_gga(T50, 0, T50)
minusA_in_gga(s(T55), s(T56), X63) → U1_gga(T55, T56, X63, minusA_in_gga(T55, T56, X63))
U1_gga(T55, T56, X63, minusA_out_gga(T55, T56, X63)) → minusA_out_gga(s(T55), s(T56), X63)

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(T39), s(T40)) → U3_GGAA(T40, minusA_in_gga(T39, T40))
U3_GGAA(T40, minusA_out_gga(T43)) → DIVB_IN_GGAA(T43, s(T40))

The TRS R consists of the following rules:

minusA_in_gga(T50, 0) → minusA_out_gga(T50)
minusA_in_gga(s(T55), s(T56)) → U1_gga(minusA_in_gga(T55, T56))
U1_gga(minusA_out_gga(X63)) → minusA_out_gga(X63)

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(T39), s(T40)) → U3_GGAA(T40, minusA_in_gga(T39, T40))

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(T50, 0) → minusA_out_gga(T50)
minusA_in_gga(s(T55), s(T56)) → U1_gga(minusA_in_gga(T55, T56))
U1_gga(minusA_out_gga(X63)) → minusA_out_gga(X63)

(22) Obligation:

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

U3_GGAA(T40, minusA_out_gga(T43)) → DIVB_IN_GGAA(T43, s(T40))

The TRS R consists of the following rules:

minusA_in_gga(T50, 0) → minusA_out_gga(T50)
minusA_in_gga(s(T55), s(T56)) → U1_gga(minusA_in_gga(T55, T56))
U1_gga(minusA_out_gga(X63)) → minusA_out_gga(X63)

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