Left Termination proof of /tmp/tmp7uaEGX/maximum-bff.pl

Left Termination of the query pattern max_in_3(g, a, a) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPiTRSProof (⇐)

↳4 PiTRS

↳5 DependencyPairsProof (⇔)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇐)

        ↳13 QDP

        ↳14 QDPSizeChangeProof (⇔)

        ↳15 YES

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

        ↳18 PiDP

        ↳19 PiDPToQDPProof (⇐)

        ↳20 QDP

        ↳21 QDPSizeChangeProof (⇔)

        ↳22 YES

(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).

Queries:

max(g,a,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

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\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", 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", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T80, s(T82)), SCOPE: 6, CLAUSE: 2\n\nT80 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: less(T80, s(T82)), SCOPE: 6, CLAUSE: 3\n\nT80 is ground", 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", 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 substitution\nT1 -> 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 = lightblue];
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(X88, X89, X89) :- less(X88, s(X89)).\nand substitution\nT1 -> T80\nX88 -> T80\nT2 -> T82\nX89 -> 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 = lightblue];
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(X12)).\nand substitution\nT8 -> 0\nX12 -> 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 = lightblue];
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(X22), s(X23)) :- less(X22, X23).\nand substitution\nX22 -> T24\nT8 -> s(T24)\nX23 -> 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 = lightblue];
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(X32)).\nand substitution\nT24 -> 0\nX32 -> 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 = lightblue];
16 -> 72 [arrowhead = none ];
72 -> 19;

73 [label="EVAL with clause\nless(s(X38), s(X39)) :- less(X38, X39).\nand substitution\nX38 -> T38\nT24 -> s(T38)\nX39 -> 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 = lightblue];
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(X48, X49, X49) :- less(X48, s(X49)).\nand substitution\nT6 -> T45\nX48 -> T45\nT2 -> T47\nX49 -> 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 = lightblue];
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(X58)).\nand substitution\nT45 -> 0\nT47 -> T54\nX58 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 82 [arrowhead = none ];
82 -> 29;

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

84 [label="EVAL with clause\nless(s(X64), s(X65)) :- less(X64, X65).\nand substitution\nX64 -> T59\nT45 -> s(T59)\nT47 -> T61\nX65 -> T61\nT60 -> T61", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 84 [arrowhead = none ];
84 -> 32;

85 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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(X74)).\nand substitution\nT59 -> 0\nX74 -> 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 = lightblue];
35 -> 91 [arrowhead = none ];
91 -> 38;

92 [label="EVAL with clause\nless(s(X80), s(X81)) :- less(X80, X81).\nand substitution\nX80 -> T73\nT59 -> s(T73)\nX81 -> 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 = lightblue];
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(X96)).\nand substitution\nT80 -> 0\nT82 -> T87\nX96 -> T87", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 99 [arrowhead = none ];
99 -> 47;

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

101 [label="EVAL with clause\nless(s(X102), s(X103)) :- less(X102, X103).\nand substitution\nX102 -> T92\nT80 -> s(T92)\nT82 -> T94\nX103 -> T94\nT93 -> T94", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 101 [arrowhead = none ];
101 -> 50;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
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:

less13(0, s(T31)).
less13(s(T38), s(T37)) :- less13(T38, T37).
less32(0, s(T68)).
less32(s(T73), s(T75)) :- less32(T73, T75).
max1(s(T13), 0, s(T13)).
max1(s(T23), s(T24), s(T23)) :- less13(T24, T23).
max1(0, T54, T54).
max1(s(T59), T61, T61) :- less32(T59, T61).
max1(0, T87, T87).
max1(s(T92), T94, T94) :- less32(T92, T94).

Queries:

max1(g,a,a).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
max1_in: (b,f,f)
less13_in: (f,b)
less32_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_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:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_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:

MAX1_IN_GAA(s(T23), s(T24), s(T23)) → U3_GAA(T23, T24, less13_in_ag(T24, T23))
MAX1_IN_GAA(s(T23), s(T24), s(T23)) → LESS13_IN_AG(T24, T23)
LESS13_IN_AG(s(T38), s(T37)) → U1_AG(T38, T37, less13_in_ag(T38, T37))
LESS13_IN_AG(s(T38), s(T37)) → LESS13_IN_AG(T38, T37)
MAX1_IN_GAA(s(T59), T61, T61) → U4_GAA(T59, T61, less32_in_ga(T59, T61))
MAX1_IN_GAA(s(T59), T61, T61) → LESS32_IN_GA(T59, T61)
LESS32_IN_GA(s(T73), s(T75)) → U2_GA(T73, T75, less32_in_ga(T73, T75))
LESS32_IN_GA(s(T73), s(T75)) → LESS32_IN_GA(T73, T75)
MAX1_IN_GAA(s(T92), T94, T94) → U5_GAA(T92, T94, less32_in_ga(T92, T94))

The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_out_gaa(s(T92), T94, T94)

The argument filtering Pi contains the following mapping:
max1_in_gaa(x1, x2, x3) = max1_in_gaa(x1)
s(x1) = s(x1)
max1_out_gaa(x1, x2, x3) = max1_out_gaa(x1)
U3_gaa(x1, x2, x3) = U3_gaa(x1, x3)
less13_in_ag(x1, x2) = less13_in_ag(x2)
less13_out_ag(x1, x2) = less13_out_ag(x1, x2)
U1_ag(x1, x2, x3) = U1_ag(x2, x3)
0 = 0
U4_gaa(x1, x2, x3) = U4_gaa(x1, x3)
less32_in_ga(x1, x2) = less32_in_ga(x1)
less32_out_ga(x1, x2) = less32_out_ga(x1)
U2_ga(x1, x2, x3) = U2_ga(x1, x3)
U5_gaa(x1, x2, x3) = U5_gaa(x1, x3)
MAX1_IN_GAA(x1, x2, x3) = MAX1_IN_GAA(x1)
U3_GAA(x1, x2, x3) = U3_GAA(x1, x3)
LESS13_IN_AG(x1, x2) = LESS13_IN_AG(x2)
U1_AG(x1, x2, x3) = U1_AG(x2, x3)
U4_GAA(x1, x2, x3) = U4_GAA(x1, x3)
LESS32_IN_GA(x1, x2) = LESS32_IN_GA(x1)
U2_GA(x1, x2, x3) = U2_GA(x1, x3)
U5_GAA(x1, x2, x3) = U5_GAA(x1, x3)

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

(6) Obligation:

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

MAX1_IN_GAA(s(T23), s(T24), s(T23)) → U3_GAA(T23, T24, less13_in_ag(T24, T23))
MAX1_IN_GAA(s(T23), s(T24), s(T23)) → LESS13_IN_AG(T24, T23)
LESS13_IN_AG(s(T38), s(T37)) → U1_AG(T38, T37, less13_in_ag(T38, T37))
LESS13_IN_AG(s(T38), s(T37)) → LESS13_IN_AG(T38, T37)
MAX1_IN_GAA(s(T59), T61, T61) → U4_GAA(T59, T61, less32_in_ga(T59, T61))
MAX1_IN_GAA(s(T59), T61, T61) → LESS32_IN_GA(T59, T61)
LESS32_IN_GA(s(T73), s(T75)) → U2_GA(T73, T75, less32_in_ga(T73, T75))
LESS32_IN_GA(s(T73), s(T75)) → LESS32_IN_GA(T73, T75)
MAX1_IN_GAA(s(T92), T94, T94) → U5_GAA(T92, T94, less32_in_ga(T92, T94))

The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_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:

LESS32_IN_GA(s(T73), s(T75)) → LESS32_IN_GA(T73, T75)

The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_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:

LESS32_IN_GA(s(T73), s(T75)) → LESS32_IN_GA(T73, T75)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LESS32_IN_GA(x1, x2) = LESS32_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:

LESS32_IN_GA(s(T73)) → LESS32_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:

LESS32_IN_GA(s(T73)) → LESS32_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:

LESS13_IN_AG(s(T38), s(T37)) → LESS13_IN_AG(T38, T37)

The TRS R consists of the following rules:

max1_in_gaa(s(T13), 0, s(T13)) → max1_out_gaa(s(T13), 0, s(T13))
max1_in_gaa(s(T23), s(T24), s(T23)) → U3_gaa(T23, T24, less13_in_ag(T24, T23))
less13_in_ag(0, s(T31)) → less13_out_ag(0, s(T31))
less13_in_ag(s(T38), s(T37)) → U1_ag(T38, T37, less13_in_ag(T38, T37))
U1_ag(T38, T37, less13_out_ag(T38, T37)) → less13_out_ag(s(T38), s(T37))
U3_gaa(T23, T24, less13_out_ag(T24, T23)) → max1_out_gaa(s(T23), s(T24), s(T23))
max1_in_gaa(0, T54, T54) → max1_out_gaa(0, T54, T54)
max1_in_gaa(s(T59), T61, T61) → U4_gaa(T59, T61, less32_in_ga(T59, T61))
less32_in_ga(0, s(T68)) → less32_out_ga(0, s(T68))
less32_in_ga(s(T73), s(T75)) → U2_ga(T73, T75, less32_in_ga(T73, T75))
U2_ga(T73, T75, less32_out_ga(T73, T75)) → less32_out_ga(s(T73), s(T75))
U4_gaa(T59, T61, less32_out_ga(T59, T61)) → max1_out_gaa(s(T59), T61, T61)
max1_in_gaa(s(T92), T94, T94) → U5_gaa(T92, T94, less32_in_ga(T92, T94))
U5_gaa(T92, T94, less32_out_ga(T92, T94)) → max1_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:

LESS13_IN_AG(s(T38), s(T37)) → LESS13_IN_AG(T38, T37)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1) = s(x1)
LESS13_IN_AG(x1, x2) = LESS13_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:

LESS13_IN_AG(s(T37)) → LESS13_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:

LESS13_IN_AG(s(T37)) → LESS13_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