Left Termination proof of /tmp/tmp8iSTeC/maximum-bff.pl

Left Termination of the query pattern
max(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 (⇒, 0 ms)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔, 0 ms)

        ↳22 YES

(0) Obligation:

Clauses:

max(X, Y, X) :- less(Y, X).
max(X, Y, Y) :- less(X, s(Y)).
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).

Query: max(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: max(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: max(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | max(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: less(T8, T6) | max(T6, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: max(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: less(T8, T6), SCOPE: 2, CLAUSE: 2 | less(T8, T6), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T6, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: less(T8, T6), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: less(T8, T6), SCOPE: 2, CLAUSE: 3 | ?_2 | max(T6, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: less(T8, T6), SCOPE: 2, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ?_2 | max(T6, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: less(T24, T23)\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: less(T24, T23), SCOPE: 3, CLAUSE: 2 | less(T24, T23), SCOPE: 3, CLAUSE: 3\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: less(T24, T23), SCOPE: 3, CLAUSE: 2\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: less(T24, T23), SCOPE: 3, CLAUSE: 3\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: less(T38, T37)\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: max(T6, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(T45, s(T47))\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(T45, s(T47)), SCOPE: 4, CLAUSE: 2 | less(T45, s(T47)), SCOPE: 4, CLAUSE: 3\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: less(T45, s(T47)), SCOPE: 4, CLAUSE: 2\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T45, s(T47)), SCOPE: 4, CLAUSE: 3\n\nT45 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: less(T59, T61)\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: less(T59, T61), SCOPE: 5, CLAUSE: 2 | less(T59, T61), SCOPE: 5, CLAUSE: 3\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: less(T59, T61), SCOPE: 5, CLAUSE: 2\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: less(T59, T61), SCOPE: 5, CLAUSE: 3\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(T73, T75)\n\nT73 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: less(T80, s(T82))\n\nT80 is ground\nX4 is free\nX5 is free\nmax(T80, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: less(T80, s(T82)), SCOPE: 6, CLAUSE: 2 | less(T80, s(T82)), SCOPE: 6, CLAUSE: 3\n\nT80 is ground\nX4 is free\nX5 is free\nmax(T80, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T80, s(T82)), SCOPE: 6, CLAUSE: 2\n\nT80 is ground\nX4 is free\nX5 is free\nmax(T80, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: less(T80, s(T82)), SCOPE: 6, CLAUSE: 3\n\nT80 is ground\nX4 is free\nX5 is free\nmax(T80, T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: less(T92, T94)\n\nT92 is ground\nX4 is free\nX5 is free\nmax(s(T92), T2, T3) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 52 [arrowhead = none ];
52 -> 2;

53 [label="EVAL with clause\nmax(X4, X5, X4) :- less(X5, X4).\nand substitutionT1 -> T6,\nX4 -> T6,\nT2 -> T8,\nX5 -> T8,\nT3 -> T6,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 53 [arrowhead = none ];
53 -> 3;

54 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 54 [arrowhead = none ];
54 -> 4;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 55 [arrowhead = none ];
55 -> 5;

56 [label="EVAL with clause\nmax(X72, X73, X73) :- less(X72, s(X73)).\nand substitutionT1 -> T80,\nX72 -> T80,\nT2 -> T82,\nX73 -> T82,\nT3 -> T82,\nT81 -> T82", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 56 [arrowhead = none ];
56 -> 42;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 57 [arrowhead = none ];
57 -> 43;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 58 [arrowhead = none ];
58 -> 6;

59 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 59 [arrowhead = none ];
59 -> 7;

60 [label="EVAL with clause\nless(0, s(X10)).\nand substitutionT8 -> 0,\nX10 -> T13,\nT6 -> s(T13)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 60 [arrowhead = none ];
60 -> 8;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 61 [arrowhead = none ];
61 -> 9;

62 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 62 [arrowhead = none ];
62 -> 11;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 64 [arrowhead = none ];
64 -> 10;

65 [label="EVAL with clause\nless(s(X19), s(X20)) :- less(X19, X20).\nand substitutionX19 -> T24,\nT8 -> s(T24),\nX20 -> T23,\nT6 -> s(T23),\nT22 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 65 [arrowhead = none ];
65 -> 13;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 66 [arrowhead = none ];
66 -> 14;

67 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 67 [arrowhead = none ];
67 -> 23;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 68 [arrowhead = none ];
68 -> 15;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 69 [arrowhead = none ];
69 -> 16;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 70 [arrowhead = none ];
70 -> 17;

71 [label="EVAL with clause\nless(0, s(X27)).\nand substitutionT24 -> 0,\nX27 -> T31,\nT23 -> s(T31)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 71 [arrowhead = none ];
71 -> 18;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 72 [arrowhead = none ];
72 -> 19;

73 [label="EVAL with clause\nless(s(X32), s(X33)) :- less(X32, X33).\nand substitutionX32 -> T38,\nT24 -> s(T38),\nX33 -> T37,\nT23 -> s(T37),\nT36 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 73 [arrowhead = none ];
73 -> 21;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 74 [arrowhead = none ];
74 -> 22;

75 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 75 [arrowhead = none ];
75 -> 20;

76 [label="INSTANCE with matching:\nT24 -> T38\nT23 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 76 [arrowhead = none , style=dashed];
76 -> 13[style=dashed];

77 [label="EVAL with clause\nmax(X40, X41, X41) :- less(X40, s(X41)).\nand substitutionT6 -> T45,\nX40 -> T45,\nT2 -> T47,\nX41 -> T47,\nT3 -> T47,\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 77 [arrowhead = none ];
77 -> 24;

78 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 78 [arrowhead = none ];
78 -> 25;

79 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 79 [arrowhead = none ];
79 -> 26;

80 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 80 [arrowhead = none ];
80 -> 27;

81 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
26 -> 81 [arrowhead = none ];
81 -> 28;

82 [label="EVAL with clause\nless(0, s(X48)).\nand substitutionT45 -> 0,\nT47 -> T54,\nX48 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 82 [arrowhead = none ];
82 -> 29;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 83 [arrowhead = none ];
83 -> 30;

84 [label="EVAL with clause\nless(s(X53), s(X54)) :- less(X53, X54).\nand substitutionX53 -> T59,\nT45 -> s(T59),\nT47 -> T61,\nX54 -> T61,\nT60 -> T61", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 84 [arrowhead = none ];
84 -> 32;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 85 [arrowhead = none ];
85 -> 33;

86 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 86 [arrowhead = none ];
86 -> 31;

87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 87 [arrowhead = none ];
87 -> 34;

88 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 88 [arrowhead = none ];
88 -> 35;

89 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
34 -> 89 [arrowhead = none ];
89 -> 36;

90 [label="EVAL with clause\nless(0, s(X61)).\nand substitutionT59 -> 0,\nX61 -> T68,\nT61 -> s(T68)", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 90 [arrowhead = none ];
90 -> 37;

91 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 91 [arrowhead = none ];
91 -> 38;

92 [label="EVAL with clause\nless(s(X66), s(X67)) :- less(X66, X67).\nand substitutionX66 -> T73,\nT59 -> s(T73),\nX67 -> T75,\nT61 -> s(T75),\nT74 -> T75", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 92 [arrowhead = none ];
92 -> 40;

93 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 93 [arrowhead = none ];
93 -> 41;

94 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 94 [arrowhead = none ];
94 -> 39;

95 [label="INSTANCE with matching:\nT59 -> T73\nT61 -> T75", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 95 [arrowhead = none , style=dashed];
95 -> 32[style=dashed];

96 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
42 -> 96 [arrowhead = none ];
96 -> 44;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 97 [arrowhead = none ];
97 -> 45;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 98 [arrowhead = none ];
98 -> 46;

99 [label="EVAL with clause\nless(0, s(X78)).\nand substitutionT80 -> 0,\nT82 -> T87,\nX78 -> T87", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 99 [arrowhead = none ];
99 -> 47;

100 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 100 [arrowhead = none ];
100 -> 48;

101 [label="EVAL with clause\nless(s(X83), s(X84)) :- less(X83, X84).\nand substitutionX83 -> T92,\nT80 -> s(T92),\nT82 -> T94,\nX84 -> T94,\nT93 -> T94", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 101 [arrowhead = none ];
101 -> 50;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 102 [arrowhead = none ];
102 -> 51;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 103 [arrowhead = none ];
103 -> 49;

104 [label="INSTANCE with matching:\nT59 -> T92\nT61 -> T94", fontsize=14, style = filled, fillcolor = lightgrey];
50 -> 104 [arrowhead = none , style=dashed];
104 -> 32[style=dashed];

}

(2) Obligation:

Clauses:

lessA(0, s(T31)).
lessA(s(T38), s(T37)) :- lessA(T38, T37).
lessB(0, s(T68)).
lessB(s(T73), s(T75)) :- lessB(T73, T75).
maxC(s(T13), 0, s(T13)).
maxC(s(T23), s(T24), s(T23)) :- lessA(T24, T23).
maxC(0, T54, T54).
maxC(s(T59), T61, T61) :- lessB(T59, T61).
maxC(0, T87, T87).
maxC(s(T92), T94, T94) :- lessB(T92, T94).

Query: maxC(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:
maxC_in: (b,f,f)
lessA_in: (f,b)
lessB_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

maxC_in_gaa(s(T13), 0, s(T13)) → maxC_out_gaa(s(T13), 0, s(T13))
maxC_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, lessA_in_ag(T24, T23))
lessA_in_ag(0, s(T31)) → lessA_out_ag(0, s(T31))
lessA_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, lessA_in_ag(T38, T37))
U1_ag(T38, T37, lessA_out_ag(T38, T37)) → lessA_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, lessA_out_ag(T24, T23)) → maxC_out_gaa(s(T23), s(T24), s(T23))
maxC_in_gaa(0, T54, T54) → maxC_out_gaa(0, T54, T54)
maxC_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, lessB_in_ga(T59, T61))
lessB_in_ga(0, s(T68)) → lessB_out_ga(0, s(T68))
lessB_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, lessB_in_ga(T73, T75))
U2_ga(T73, T75, lessB_out_ga(T73, T75)) → lessB_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, lessB_out_ga(T59, T61)) → maxC_out_gaa(s(T59), T61, T61)
maxC_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, lessB_in_ga(T92, T94))
U5_gaa(T92, T94, lessB_out_ga(T92, T94)) → maxC_out_gaa(s(T92), T94, T94)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

maxC_in_gaa(s(T13), 0, s(T13)) → maxC_out_gaa(s(T13), 0, s(T13))
maxC_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, lessA_in_ag(T24, T23))
lessA_in_ag(0, s(T31)) → lessA_out_ag(0, s(T31))
lessA_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, lessA_in_ag(T38, T37))
U1_ag(T38, T37, lessA_out_ag(T38, T37)) → lessA_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, lessA_out_ag(T24, T23)) → maxC_out_gaa(s(T23), s(T24), s(T23))
maxC_in_gaa(0, T54, T54) → maxC_out_gaa(0, T54, T54)
maxC_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, lessB_in_ga(T59, T61))
lessB_in_ga(0, s(T68)) → lessB_out_ga(0, s(T68))
lessB_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, lessB_in_ga(T73, T75))
U2_ga(T73, T75, lessB_out_ga(T73, T75)) → lessB_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, lessB_out_ga(T59, T61)) → maxC_out_gaa(s(T59), T61, T61)
maxC_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, lessB_in_ga(T92, T94))
U5_gaa(T92, T94, lessB_out_ga(T92, T94)) → maxC_out_gaa(s(T92), T94, T94)

(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:

MAXC_IN_GAA(s(T23), s(T24), s(T23)) → U3_GAA(T23, T24, lessA_in_ag(T24, T23))
MAXC_IN_GAA(s(T23), s(T24), s(T23)) → LESSA_IN_AG(T24, T23)
LESSA_IN_AG(s(T38), s(T37)) → U1_AG(T38, T37, lessA_in_ag(T38, T37))
LESSA_IN_AG(s(T38), s(T37)) → LESSA_IN_AG(T38, T37)
MAXC_IN_GAA(s(T59), T61, T61) → U4_GAA(T59, T61, lessB_in_ga(T59, T61))
MAXC_IN_GAA(s(T59), T61, T61) → LESSB_IN_GA(T59, T61)
LESSB_IN_GA(s(T73), s(T75)) → U2_GA(T73, T75, lessB_in_ga(T73, T75))
LESSB_IN_GA(s(T73), s(T75)) → LESSB_IN_GA(T73, T75)
MAXC_IN_GAA(s(T92), T94, T94) → U5_GAA(T92, T94, lessB_in_ga(T92, T94))

The TRS R consists of the following rules:

maxC_in_gaa(s(T13), 0, s(T13)) → maxC_out_gaa(s(T13), 0, s(T13))
maxC_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, lessA_in_ag(T24, T23))
lessA_in_ag(0, s(T31)) → lessA_out_ag(0, s(T31))
lessA_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, lessA_in_ag(T38, T37))
U1_ag(T38, T37, lessA_out_ag(T38, T37)) → lessA_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, lessA_out_ag(T24, T23)) → maxC_out_gaa(s(T23), s(T24), s(T23))
maxC_in_gaa(0, T54, T54) → maxC_out_gaa(0, T54, T54)
maxC_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, lessB_in_ga(T59, T61))
lessB_in_ga(0, s(T68)) → lessB_out_ga(0, s(T68))
lessB_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, lessB_in_ga(T73, T75))
U2_ga(T73, T75, lessB_out_ga(T73, T75)) → lessB_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, lessB_out_ga(T59, T61)) → maxC_out_gaa(s(T59), T61, T61)
maxC_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, lessB_in_ga(T92, T94))
U5_gaa(T92, T94, lessB_out_ga(T92, T94)) → maxC_out_gaa(s(T92), T94, T94)

The argument filtering Pi contains the following mapping:
maxC_in_gaa(x1, x2, x3) = maxC_in_gaa(x1)
s(x1) = s(x1)
maxC_out_gaa(x1, x2, x3) = maxC_out_gaa
U3_gaa(x1, x2, x3) = U3_gaa(x3)
lessA_in_ag(x1, x2) = lessA_in_ag(x2)
lessA_out_ag(x1, x2) = lessA_out_ag(x1)
U1_ag(x1, x2, x3) = U1_ag(x3)
0 = 0
U4_gaa(x1, x2, x3) = U4_gaa(x3)
lessB_in_ga(x1, x2) = lessB_in_ga(x1)
lessB_out_ga(x1, x2) = lessB_out_ga
U2_ga(x1, x2, x3) = U2_ga(x3)
U5_gaa(x1, x2, x3) = U5_gaa(x3)
MAXC_IN_GAA(x1, x2, x3) = MAXC_IN_GAA(x1)
U3_GAA(x1, x2, x3) = U3_GAA(x3)
LESSA_IN_AG(x1, x2) = LESSA_IN_AG(x2)
U1_AG(x1, x2, x3) = U1_AG(x3)
U4_GAA(x1, x2, x3) = U4_GAA(x3)
LESSB_IN_GA(x1, x2) = LESSB_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x3)
U5_GAA(x1, x2, x3) = U5_GAA(x3)

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

(6) Obligation:

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

MAXC_IN_GAA(s(T23), s(T24), s(T23)) → U3_GAA(T23, T24, lessA_in_ag(T24, T23))
MAXC_IN_GAA(s(T23), s(T24), s(T23)) → LESSA_IN_AG(T24, T23)
LESSA_IN_AG(s(T38), s(T37)) → U1_AG(T38, T37, lessA_in_ag(T38, T37))
LESSA_IN_AG(s(T38), s(T37)) → LESSA_IN_AG(T38, T37)
MAXC_IN_GAA(s(T59), T61, T61) → U4_GAA(T59, T61, lessB_in_ga(T59, T61))
MAXC_IN_GAA(s(T59), T61, T61) → LESSB_IN_GA(T59, T61)
LESSB_IN_GA(s(T73), s(T75)) → U2_GA(T73, T75, lessB_in_ga(T73, T75))
LESSB_IN_GA(s(T73), s(T75)) → LESSB_IN_GA(T73, T75)
MAXC_IN_GAA(s(T92), T94, T94) → U5_GAA(T92, T94, lessB_in_ga(T92, T94))

The TRS R consists of the following rules:

maxC_in_gaa(s(T13), 0, s(T13)) → maxC_out_gaa(s(T13), 0, s(T13))
maxC_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, lessA_in_ag(T24, T23))
lessA_in_ag(0, s(T31)) → lessA_out_ag(0, s(T31))
lessA_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, lessA_in_ag(T38, T37))
U1_ag(T38, T37, lessA_out_ag(T38, T37)) → lessA_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, lessA_out_ag(T24, T23)) → maxC_out_gaa(s(T23), s(T24), s(T23))
maxC_in_gaa(0, T54, T54) → maxC_out_gaa(0, T54, T54)
maxC_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, lessB_in_ga(T59, T61))
lessB_in_ga(0, s(T68)) → lessB_out_ga(0, s(T68))
lessB_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, lessB_in_ga(T73, T75))
U2_ga(T73, T75, lessB_out_ga(T73, T75)) → lessB_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, lessB_out_ga(T59, T61)) → maxC_out_gaa(s(T59), T61, T61)
maxC_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, lessB_in_ga(T92, T94))
U5_gaa(T92, T94, lessB_out_ga(T92, T94)) → maxC_out_gaa(s(T92), T94, T94)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 7 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LESSB_IN_GA(s(T73), s(T75)) → LESSB_IN_GA(T73, T75)

The TRS R consists of the following rules:

maxC_in_gaa(s(T13), 0, s(T13)) → maxC_out_gaa(s(T13), 0, s(T13))
maxC_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, lessA_in_ag(T24, T23))
lessA_in_ag(0, s(T31)) → lessA_out_ag(0, s(T31))
lessA_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, lessA_in_ag(T38, T37))
U1_ag(T38, T37, lessA_out_ag(T38, T37)) → lessA_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, lessA_out_ag(T24, T23)) → maxC_out_gaa(s(T23), s(T24), s(T23))
maxC_in_gaa(0, T54, T54) → maxC_out_gaa(0, T54, T54)
maxC_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, lessB_in_ga(T59, T61))
lessB_in_ga(0, s(T68)) → lessB_out_ga(0, s(T68))
lessB_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, lessB_in_ga(T73, T75))
U2_ga(T73, T75, lessB_out_ga(T73, T75)) → lessB_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, lessB_out_ga(T59, T61)) → maxC_out_gaa(s(T59), T61, T61)
maxC_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, lessB_in_ga(T92, T94))
U5_gaa(T92, T94, lessB_out_ga(T92, T94)) → maxC_out_gaa(s(T92), T94, T94)

(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:

LESSB_IN_GA(s(T73), s(T75)) → LESSB_IN_GA(T73, T75)

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

LESSB_IN_GA(s(T73)) → LESSB_IN_GA(T73)

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:

LESSB_IN_GA(s(T73)) → LESSB_IN_GA(T73)
The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

LESSA_IN_AG(s(T38), s(T37)) → LESSA_IN_AG(T38, T37)

The TRS R consists of the following rules:

maxC_in_gaa(s(T13), 0, s(T13)) → maxC_out_gaa(s(T13), 0, s(T13))
maxC_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, lessA_in_ag(T24, T23))
lessA_in_ag(0, s(T31)) → lessA_out_ag(0, s(T31))
lessA_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, lessA_in_ag(T38, T37))
U1_ag(T38, T37, lessA_out_ag(T38, T37)) → lessA_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, lessA_out_ag(T24, T23)) → maxC_out_gaa(s(T23), s(T24), s(T23))
maxC_in_gaa(0, T54, T54) → maxC_out_gaa(0, T54, T54)
maxC_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, lessB_in_ga(T59, T61))
lessB_in_ga(0, s(T68)) → lessB_out_ga(0, s(T68))
lessB_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, lessB_in_ga(T73, T75))
U2_ga(T73, T75, lessB_out_ga(T73, T75)) → lessB_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, lessB_out_ga(T59, T61)) → maxC_out_gaa(s(T59), T61, T61)
maxC_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, lessB_in_ga(T92, T94))
U5_gaa(T92, T94, lessB_out_ga(T92, T94)) → maxC_out_gaa(s(T92), T94, T94)

(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:

LESSA_IN_AG(s(T38), s(T37)) → LESSA_IN_AG(T38, T37)

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

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:

LESSA_IN_AG(s(T37)) → LESSA_IN_AG(T37)

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:

LESSA_IN_AG(s(T37)) → LESSA_IN_AG(T37)
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 (SOUND transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) YES