Left Termination proof of /tmp/tmpBEhCnF/maximum-ffb.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 86 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 33 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 0 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 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(a,a,g)

(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\nT3 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\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: less(T8, T6) | max(T1, T2, T6), 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\nT3 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(T1, T2, T6), 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(T1, T2, T6), 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(T1, T2, T6), 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(T1, T2, T6), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: less(T47, s(T46))\n\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: less(T54, s(T53))\n\nT53 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T53) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T54, s(T53)), SCOPE: 4, CLAUSE: 2 | less(T54, s(T53)), SCOPE: 4, CLAUSE: 3\n\nT53 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T53) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: less(T54, s(T53)), SCOPE: 4, CLAUSE: 2\n\nT53 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T53) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: less(T54, s(T53)), SCOPE: 4, CLAUSE: 3\n\nT53 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T53) !=? max(X4, X5, X4)", 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: less(T66, T65)\n\nT65 is ground\nX4 is free\nX5 is free\nmax(T1, T2, T65) !=? max(X4, X5, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 36 [arrowhead = none ];
36 -> 2;

37 [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 -> 37 [arrowhead = none ];
37 -> 3;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 38 [arrowhead = none ];
38 -> 4;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 39 [arrowhead = none ];
39 -> 5;

40 [label="EVAL with clause\nmax(X46, X47, X47) :- less(X46, s(X47)).\nand substitutionT1 -> T54,\nX46 -> T54,\nT2 -> T53,\nX47 -> T53,\nT3 -> T53,\nT52 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 40 [arrowhead = none ];
40 -> 26;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 41 [arrowhead = none ];
41 -> 27;

42 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 43 [arrowhead = none ];
43 -> 7;

44 [label="EVAL with clause\nless(0, s(X10)).\nand substitutionT8 -> 0,\nX10 -> T13,\nT6 -> s(T13)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 44 [arrowhead = none ];
44 -> 8;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 45 [arrowhead = none ];
45 -> 9;

46 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 46 [arrowhead = none ];
46 -> 11;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
7 -> 47 [arrowhead = none ];
47 -> 12;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 48 [arrowhead = none ];
48 -> 10;

49 [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 -> 49 [arrowhead = none ];
49 -> 13;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 50 [arrowhead = none ];
50 -> 14;

51 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 51 [arrowhead = none ];
51 -> 23;

52 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 52 [arrowhead = none ];
52 -> 15;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 53 [arrowhead = none ];
53 -> 16;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
15 -> 54 [arrowhead = none ];
54 -> 17;

55 [label="EVAL with clause\nless(0, s(X27)).\nand substitutionT24 -> 0,\nX27 -> T31,\nT23 -> s(T31)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 55 [arrowhead = none ];
55 -> 18;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 56 [arrowhead = none ];
56 -> 19;

57 [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 -> 57 [arrowhead = none ];
57 -> 21;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 58 [arrowhead = none ];
58 -> 22;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 59 [arrowhead = none ];
59 -> 20;

60 [label="INSTANCE with matching:\nT24 -> T38\nT23 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
21 -> 60 [arrowhead = none , style=dashed];
60 -> 13[style=dashed];

61 [label="EVAL with clause\nmax(X40, X41, X41) :- less(X40, s(X41)).\nand substitutionT1 -> T47,\nX40 -> T47,\nT2 -> T46,\nX41 -> T46,\nT6 -> T46,\nT45 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 61 [arrowhead = none ];
61 -> 24;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 62 [arrowhead = none ];
62 -> 25;

63 [label="INSTANCE with matching:\nT24 -> T47\nT23 -> s(T46)", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 63 [arrowhead = none , style=dashed];
63 -> 13[style=dashed];

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 64 [arrowhead = none ];
64 -> 28;

65 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 65 [arrowhead = none ];
65 -> 29;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 66 [arrowhead = none ];
66 -> 30;

67 [label="EVAL with clause\nless(0, s(X52)).\nand substitutionT54 -> 0,\nT53 -> T59,\nX52 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 67 [arrowhead = none ];
67 -> 31;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 68 [arrowhead = none ];
68 -> 32;

69 [label="EVAL with clause\nless(s(X57), s(X58)) :- less(X57, X58).\nand substitutionX57 -> T66,\nT54 -> s(T66),\nT53 -> T65,\nX58 -> T65,\nT64 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 69 [arrowhead = none ];
69 -> 34;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 70 [arrowhead = none ];
70 -> 35;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 71 [arrowhead = none ];
71 -> 33;

72 [label="INSTANCE with matching:\nT24 -> T66\nT23 -> T65", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 72 [arrowhead = none , style=dashed];
72 -> 13[style=dashed];

}

(2) Obligation:

Clauses:

lessA(0, s(T31)).
lessA(s(T38), s(T37)) :- lessA(T38, T37).
maxB(s(T13), 0, s(T13)).
maxB(s(T23), s(T24), s(T23)) :- lessA(T24, T23).
maxB(T47, T46, T46) :- lessA(T47, s(T46)).
maxB(0, T59, T59).
maxB(s(T66), T65, T65) :- lessA(T66, T65).

Query: maxB(a,a,g)

(3) PrologToPiTRSProof (SOUND transformation)

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

maxB_in_aag(s(T13), 0, s(T13)) → maxB_out_aag(s(T13), 0, s(T13))
maxB_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, lessA_out_ag(T24, T23)) → maxB_out_aag(s(T23), s(T24), s(T23))
maxB_in_aag(T47, T46, T46) → U3_aag(T47, T46, lessA_in_ag(T47, s(T46)))
U3_aag(T47, T46, lessA_out_ag(T47, s(T46))) → maxB_out_aag(T47, T46, T46)
maxB_in_aag(0, T59, T59) → maxB_out_aag(0, T59, T59)
maxB_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, lessA_in_ag(T66, T65))
U4_aag(T66, T65, lessA_out_ag(T66, T65)) → maxB_out_aag(s(T66), T65, T65)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

maxB_in_aag(s(T13), 0, s(T13)) → maxB_out_aag(s(T13), 0, s(T13))
maxB_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, lessA_out_ag(T24, T23)) → maxB_out_aag(s(T23), s(T24), s(T23))
maxB_in_aag(T47, T46, T46) → U3_aag(T47, T46, lessA_in_ag(T47, s(T46)))
U3_aag(T47, T46, lessA_out_ag(T47, s(T46))) → maxB_out_aag(T47, T46, T46)
maxB_in_aag(0, T59, T59) → maxB_out_aag(0, T59, T59)
maxB_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, lessA_in_ag(T66, T65))
U4_aag(T66, T65, lessA_out_ag(T66, T65)) → maxB_out_aag(s(T66), T65, T65)

(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:

MAXB_IN_AAG(s(T23), s(T24), s(T23)) → U2_AAG(T23, T24, lessA_in_ag(T24, T23))
MAXB_IN_AAG(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)
MAXB_IN_AAG(T47, T46, T46) → U3_AAG(T47, T46, lessA_in_ag(T47, s(T46)))
MAXB_IN_AAG(T47, T46, T46) → LESSA_IN_AG(T47, s(T46))
MAXB_IN_AAG(s(T66), T65, T65) → U4_AAG(T66, T65, lessA_in_ag(T66, T65))
MAXB_IN_AAG(s(T66), T65, T65) → LESSA_IN_AG(T66, T65)

The TRS R consists of the following rules:

maxB_in_aag(s(T13), 0, s(T13)) → maxB_out_aag(s(T13), 0, s(T13))
maxB_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, lessA_out_ag(T24, T23)) → maxB_out_aag(s(T23), s(T24), s(T23))
maxB_in_aag(T47, T46, T46) → U3_aag(T47, T46, lessA_in_ag(T47, s(T46)))
U3_aag(T47, T46, lessA_out_ag(T47, s(T46))) → maxB_out_aag(T47, T46, T46)
maxB_in_aag(0, T59, T59) → maxB_out_aag(0, T59, T59)
maxB_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, lessA_in_ag(T66, T65))
U4_aag(T66, T65, lessA_out_ag(T66, T65)) → maxB_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
maxB_in_aag(x1, x2, x3) = maxB_in_aag(x3)
s(x1) = s(x1)
maxB_out_aag(x1, x2, x3) = maxB_out_aag(x1, x2)
U2_aag(x1, x2, x3) = U2_aag(x1, 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)
U3_aag(x1, x2, x3) = U3_aag(x2, x3)
U4_aag(x1, x2, x3) = U4_aag(x2, x3)
MAXB_IN_AAG(x1, x2, x3) = MAXB_IN_AAG(x3)
U2_AAG(x1, x2, x3) = U2_AAG(x1, x3)
LESSA_IN_AG(x1, x2) = LESSA_IN_AG(x2)
U1_AG(x1, x2, x3) = U1_AG(x3)
U3_AAG(x1, x2, x3) = U3_AAG(x2, x3)
U4_AAG(x1, x2, x3) = U4_AAG(x2, x3)

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

(6) Obligation:

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

MAXB_IN_AAG(s(T23), s(T24), s(T23)) → U2_AAG(T23, T24, lessA_in_ag(T24, T23))
MAXB_IN_AAG(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)
MAXB_IN_AAG(T47, T46, T46) → U3_AAG(T47, T46, lessA_in_ag(T47, s(T46)))
MAXB_IN_AAG(T47, T46, T46) → LESSA_IN_AG(T47, s(T46))
MAXB_IN_AAG(s(T66), T65, T65) → U4_AAG(T66, T65, lessA_in_ag(T66, T65))
MAXB_IN_AAG(s(T66), T65, T65) → LESSA_IN_AG(T66, T65)

The TRS R consists of the following rules:

maxB_in_aag(s(T13), 0, s(T13)) → maxB_out_aag(s(T13), 0, s(T13))
maxB_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, lessA_out_ag(T24, T23)) → maxB_out_aag(s(T23), s(T24), s(T23))
maxB_in_aag(T47, T46, T46) → U3_aag(T47, T46, lessA_in_ag(T47, s(T46)))
U3_aag(T47, T46, lessA_out_ag(T47, s(T46))) → maxB_out_aag(T47, T46, T46)
maxB_in_aag(0, T59, T59) → maxB_out_aag(0, T59, T59)
maxB_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, lessA_in_ag(T66, T65))
U4_aag(T66, T65, lessA_out_ag(T66, T65)) → maxB_out_aag(s(T66), T65, T65)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 7 less nodes.

(8) 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:

maxB_in_aag(s(T13), 0, s(T13)) → maxB_out_aag(s(T13), 0, s(T13))
maxB_in_aag(s(T23), s(T24), s(T23)) → U2_aag(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))
U2_aag(T23, T24, lessA_out_ag(T24, T23)) → maxB_out_aag(s(T23), s(T24), s(T23))
maxB_in_aag(T47, T46, T46) → U3_aag(T47, T46, lessA_in_ag(T47, s(T46)))
U3_aag(T47, T46, lessA_out_ag(T47, s(T46))) → maxB_out_aag(T47, T46, T46)
maxB_in_aag(0, T59, T59) → maxB_out_aag(0, T59, T59)
maxB_in_aag(s(T66), T65, T65) → U4_aag(T66, T65, lessA_in_ag(T66, T65))
U4_aag(T66, T65, lessA_out_ag(T66, T65)) → maxB_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
maxB_in_aag(x1, x2, x3) = maxB_in_aag(x3)
s(x1) = s(x1)
maxB_out_aag(x1, x2, x3) = maxB_out_aag(x1, x2)
U2_aag(x1, x2, x3) = U2_aag(x1, 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)
U3_aag(x1, x2, x3) = U3_aag(x2, x3)
U4_aag(x1, x2, x3) = U4_aag(x2, x3)
LESSA_IN_AG(x1, x2) = LESSA_IN_AG(x2)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) 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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) 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.

(13) 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) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES