Left Termination proof of /tmp/tmpOOOF6z/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 PrologToPiTRSViaGraphTransformerProof (⇒, 109 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 11 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 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) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: max(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: 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];
10 [label="10: less(T8, T6) | max(T1, T2, T6), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: 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];
16 [label="16: 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];
18 [label="18: less(T8, T6), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: 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];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: less(T8, T6), SCOPE: 2, CLAUSE: 3\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ?_2 | max(T1, T2, T6), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="B: less(T24, T23)\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: less(T24, T23), SCOPE: 3, CLAUSE: 2 | less(T24, T23), SCOPE: 3, CLAUSE: 3\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: less(T24, T23), SCOPE: 3, CLAUSE: 2\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: less(T24, T23), SCOPE: 3, CLAUSE: 3\n\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: less(T38, T37)\n\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: max(T1, T2, T6), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: less(T47, s(T46))\n\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: 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];
78 [label="78: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: 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];
80 [label="80: 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];
81 [label="81: 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];
82 [label="82: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: 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];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 87 [arrowhead = none ];
87 -> 6;

88 [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];
6 -> 88 [arrowhead = none ];
88 -> 10;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 89 [arrowhead = none ];
89 -> 13;

90 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 90 [arrowhead = none ];
90 -> 16;

91 [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];
13 -> 91 [arrowhead = none ];
91 -> 77;

92 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 92 [arrowhead = none ];
92 -> 78;

93 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 93 [arrowhead = none ];
93 -> 18;

94 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
16 -> 94 [arrowhead = none ];
94 -> 20;

95 [label="EVAL with clause\nless(0, s(X10)).\nand substitutionT8 -> 0,\nX10 -> T13,\nT6 -> s(T13)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 95 [arrowhead = none ];
95 -> 22;

96 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 96 [arrowhead = none ];
96 -> 24;

97 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 97 [arrowhead = none ];
97 -> 28;

98 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
20 -> 98 [arrowhead = none ];
98 -> 30;

99 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 99 [arrowhead = none ];
99 -> 26;

100 [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];
28 -> 100 [arrowhead = none ];
100 -> 35;

101 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 101 [arrowhead = none ];
101 -> 37;

102 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 102 [arrowhead = none ];
102 -> 64;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
35 -> 103 [arrowhead = none ];
103 -> 44;

104 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 104 [arrowhead = none ];
104 -> 47;

105 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
44 -> 105 [arrowhead = none ];
105 -> 48;

106 [label="EVAL with clause\nless(0, s(X27)).\nand substitutionT24 -> 0,\nX27 -> T31,\nT23 -> s(T31)", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 106 [arrowhead = none ];
106 -> 54;

107 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 107 [arrowhead = none ];
107 -> 55;

108 [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];
48 -> 108 [arrowhead = none ];
108 -> 60;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 109 [arrowhead = none ];
109 -> 61;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 110 [arrowhead = none ];
110 -> 56;

111 [label="INSTANCE with matching:\nT24 -> T38\nT23 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
60 -> 111 [arrowhead = none , style=dashed];
111 -> 35[style=dashed];

112 [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];
64 -> 112 [arrowhead = none ];
112 -> 67;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 113 [arrowhead = none ];
113 -> 69;

114 [label="INSTANCE with matching:\nT24 -> T47\nT23 -> s(T46)", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 114 [arrowhead = none , style=dashed];
114 -> 35[style=dashed];

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
77 -> 115 [arrowhead = none ];
115 -> 79;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
79 -> 116 [arrowhead = none ];
116 -> 80;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
79 -> 117 [arrowhead = none ];
117 -> 81;

118 [label="EVAL with clause\nless(0, s(X52)).\nand substitutionT54 -> 0,\nT53 -> T59,\nX52 -> T59", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 118 [arrowhead = none ];
118 -> 82;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 119 [arrowhead = none ];
119 -> 83;

120 [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];
81 -> 120 [arrowhead = none ];
120 -> 85;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 121 [arrowhead = none ];
121 -> 86;

122 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 122 [arrowhead = none ];
122 -> 84;

123 [label="INSTANCE with matching:\nT24 -> T66\nT23 -> T65", fontsize=14, style = filled, fillcolor = lightgrey];
85 -> 123 [arrowhead = none , style=dashed];
123 -> 35[style=dashed];

}

(2) Obligation:

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

maxA_in_aag(s(T13), 0, s(T13)) → maxA_out_aag(s(T13), 0, s(T13))
maxA_in_aag(s(T23), s(T24), s(T23)) → U1_aag(T23, T24, lessB_in_ag(T24, T23))
lessB_in_ag(0, s(T31)) → lessB_out_ag(0, s(T31))
lessB_in_ag(s(T38), s(T37)) → U4_ag(T38, T37, lessB_in_ag(T38, T37))
U4_ag(T38, T37, lessB_out_ag(T38, T37)) → lessB_out_ag(s(T38), s(T37))
U1_aag(T23, T24, lessB_out_ag(T24, T23)) → maxA_out_aag(s(T23), s(T24), s(T23))
maxA_in_aag(T47, T46, T46) → U2_aag(T47, T46, lessB_in_ag(T47, s(T46)))
U2_aag(T47, T46, lessB_out_ag(T47, s(T46))) → maxA_out_aag(T47, T46, T46)
maxA_in_aag(0, T59, T59) → maxA_out_aag(0, T59, T59)
maxA_in_aag(s(T66), T65, T65) → U3_aag(T66, T65, lessB_in_ag(T66, T65))
U3_aag(T66, T65, lessB_out_ag(T66, T65)) → maxA_out_aag(s(T66), T65, T65)

(3) 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:

MAXA_IN_AAG(s(T23), s(T24), s(T23)) → U1_AAG(T23, T24, lessB_in_ag(T24, T23))
MAXA_IN_AAG(s(T23), s(T24), s(T23)) → LESSB_IN_AG(T24, T23)
LESSB_IN_AG(s(T38), s(T37)) → U4_AG(T38, T37, lessB_in_ag(T38, T37))
LESSB_IN_AG(s(T38), s(T37)) → LESSB_IN_AG(T38, T37)
MAXA_IN_AAG(T47, T46, T46) → U2_AAG(T47, T46, lessB_in_ag(T47, s(T46)))
MAXA_IN_AAG(T47, T46, T46) → LESSB_IN_AG(T47, s(T46))
MAXA_IN_AAG(s(T66), T65, T65) → U3_AAG(T66, T65, lessB_in_ag(T66, T65))
MAXA_IN_AAG(s(T66), T65, T65) → LESSB_IN_AG(T66, T65)

The TRS R consists of the following rules:

maxA_in_aag(s(T13), 0, s(T13)) → maxA_out_aag(s(T13), 0, s(T13))
maxA_in_aag(s(T23), s(T24), s(T23)) → U1_aag(T23, T24, lessB_in_ag(T24, T23))
lessB_in_ag(0, s(T31)) → lessB_out_ag(0, s(T31))
lessB_in_ag(s(T38), s(T37)) → U4_ag(T38, T37, lessB_in_ag(T38, T37))
U4_ag(T38, T37, lessB_out_ag(T38, T37)) → lessB_out_ag(s(T38), s(T37))
U1_aag(T23, T24, lessB_out_ag(T24, T23)) → maxA_out_aag(s(T23), s(T24), s(T23))
maxA_in_aag(T47, T46, T46) → U2_aag(T47, T46, lessB_in_ag(T47, s(T46)))
U2_aag(T47, T46, lessB_out_ag(T47, s(T46))) → maxA_out_aag(T47, T46, T46)
maxA_in_aag(0, T59, T59) → maxA_out_aag(0, T59, T59)
maxA_in_aag(s(T66), T65, T65) → U3_aag(T66, T65, lessB_in_ag(T66, T65))
U3_aag(T66, T65, lessB_out_ag(T66, T65)) → maxA_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
maxA_in_aag(x1, x2, x3) = maxA_in_aag(x3)
s(x1) = s(x1)
maxA_out_aag(x1, x2, x3) = maxA_out_aag(x1, x2, x3)
U1_aag(x1, x2, x3) = U1_aag(x1, x3)
lessB_in_ag(x1, x2) = lessB_in_ag(x2)
lessB_out_ag(x1, x2) = lessB_out_ag(x1, x2)
U4_ag(x1, x2, x3) = U4_ag(x2, x3)
U2_aag(x1, x2, x3) = U2_aag(x2, x3)
U3_aag(x1, x2, x3) = U3_aag(x2, x3)
MAXA_IN_AAG(x1, x2, x3) = MAXA_IN_AAG(x3)
U1_AAG(x1, x2, x3) = U1_AAG(x1, x3)
LESSB_IN_AG(x1, x2) = LESSB_IN_AG(x2)
U4_AG(x1, x2, x3) = U4_AG(x2, x3)
U2_AAG(x1, x2, x3) = U2_AAG(x2, x3)
U3_AAG(x1, x2, x3) = U3_AAG(x2, x3)

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

(4) Obligation:

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

MAXA_IN_AAG(s(T23), s(T24), s(T23)) → U1_AAG(T23, T24, lessB_in_ag(T24, T23))
MAXA_IN_AAG(s(T23), s(T24), s(T23)) → LESSB_IN_AG(T24, T23)
LESSB_IN_AG(s(T38), s(T37)) → U4_AG(T38, T37, lessB_in_ag(T38, T37))
LESSB_IN_AG(s(T38), s(T37)) → LESSB_IN_AG(T38, T37)
MAXA_IN_AAG(T47, T46, T46) → U2_AAG(T47, T46, lessB_in_ag(T47, s(T46)))
MAXA_IN_AAG(T47, T46, T46) → LESSB_IN_AG(T47, s(T46))
MAXA_IN_AAG(s(T66), T65, T65) → U3_AAG(T66, T65, lessB_in_ag(T66, T65))
MAXA_IN_AAG(s(T66), T65, T65) → LESSB_IN_AG(T66, T65)

The TRS R consists of the following rules:

maxA_in_aag(s(T13), 0, s(T13)) → maxA_out_aag(s(T13), 0, s(T13))
maxA_in_aag(s(T23), s(T24), s(T23)) → U1_aag(T23, T24, lessB_in_ag(T24, T23))
lessB_in_ag(0, s(T31)) → lessB_out_ag(0, s(T31))
lessB_in_ag(s(T38), s(T37)) → U4_ag(T38, T37, lessB_in_ag(T38, T37))
U4_ag(T38, T37, lessB_out_ag(T38, T37)) → lessB_out_ag(s(T38), s(T37))
U1_aag(T23, T24, lessB_out_ag(T24, T23)) → maxA_out_aag(s(T23), s(T24), s(T23))
maxA_in_aag(T47, T46, T46) → U2_aag(T47, T46, lessB_in_ag(T47, s(T46)))
U2_aag(T47, T46, lessB_out_ag(T47, s(T46))) → maxA_out_aag(T47, T46, T46)
maxA_in_aag(0, T59, T59) → maxA_out_aag(0, T59, T59)
maxA_in_aag(s(T66), T65, T65) → U3_aag(T66, T65, lessB_in_ag(T66, T65))
U3_aag(T66, T65, lessB_out_ag(T66, T65)) → maxA_out_aag(s(T66), T65, T65)

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

LESSB_IN_AG(s(T38), s(T37)) → LESSB_IN_AG(T38, T37)

The TRS R consists of the following rules:

maxA_in_aag(s(T13), 0, s(T13)) → maxA_out_aag(s(T13), 0, s(T13))
maxA_in_aag(s(T23), s(T24), s(T23)) → U1_aag(T23, T24, lessB_in_ag(T24, T23))
lessB_in_ag(0, s(T31)) → lessB_out_ag(0, s(T31))
lessB_in_ag(s(T38), s(T37)) → U4_ag(T38, T37, lessB_in_ag(T38, T37))
U4_ag(T38, T37, lessB_out_ag(T38, T37)) → lessB_out_ag(s(T38), s(T37))
U1_aag(T23, T24, lessB_out_ag(T24, T23)) → maxA_out_aag(s(T23), s(T24), s(T23))
maxA_in_aag(T47, T46, T46) → U2_aag(T47, T46, lessB_in_ag(T47, s(T46)))
U2_aag(T47, T46, lessB_out_ag(T47, s(T46))) → maxA_out_aag(T47, T46, T46)
maxA_in_aag(0, T59, T59) → maxA_out_aag(0, T59, T59)
maxA_in_aag(s(T66), T65, T65) → U3_aag(T66, T65, lessB_in_ag(T66, T65))
U3_aag(T66, T65, lessB_out_ag(T66, T65)) → maxA_out_aag(s(T66), T65, T65)

The argument filtering Pi contains the following mapping:
maxA_in_aag(x1, x2, x3) = maxA_in_aag(x3)
s(x1) = s(x1)
maxA_out_aag(x1, x2, x3) = maxA_out_aag(x1, x2, x3)
U1_aag(x1, x2, x3) = U1_aag(x1, x3)
lessB_in_ag(x1, x2) = lessB_in_ag(x2)
lessB_out_ag(x1, x2) = lessB_out_ag(x1, x2)
U4_ag(x1, x2, x3) = U4_ag(x2, x3)
U2_aag(x1, x2, x3) = U2_aag(x2, x3)
U3_aag(x1, x2, x3) = U3_aag(x2, x3)
LESSB_IN_AG(x1, x2) = LESSB_IN_AG(x2)

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

(7) UsableRulesProof (EQUIVALENT transformation)

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

(8) Obligation:

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

LESSB_IN_AG(s(T38), s(T37)) → LESSB_IN_AG(T38, T37)

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

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

(9) PiDPToQDPProof (SOUND transformation)

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

(10) Obligation:

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

LESSB_IN_AG(s(T37)) → LESSB_IN_AG(T37)

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

(11) 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_AG(s(T37)) → LESSB_IN_AG(T37)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) PiDPToQDPProof (SOUND transformation)

(10) Obligation:

(11) QDPSizeChangeProof (EQUIVALENT transformation)

(12) YES