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

(0) Obligation:

Clauses:

less(0, Y) :- ','(!, =(Y, s(X2))).
less(X, Y) :- ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1))).
p(0, 0).
p(s(X), X).
=(X, X).

Query: less(g,a)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

less(0, Y) :- ','(!, user_defined_=(Y, s(X2))).
less(X, Y) :- ','(p(X, X1), ','(p(Y, Y1), less(X1, Y1))).
p(0, 0).
p(s(X), X).
user_defined_=(X, X).

Query: less(g,a)

(3) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: less(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: less(T1, T2), SCOPE: 1, CLAUSE: 0 | less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(!_1, user_defined_=(T5, s(X5))) | less(0, T2), SCOPE: 1, CLAUSE: 1\n\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: less(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX4 is free\nless(T1, T2) !=? less(0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: user_defined_=(T5, s(X5))\n\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: user_defined_=(T5, s(X5)), SCOPE: 2, CLAUSE: 4\n\nX5 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: 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];
29 [label="29: ','(p(T14, X13), ','(p(T16, X14), less(X13, X14)))\n\nT14 is ground\nX4 is free\nX13 is free\nX14 is free\nless(T14, T2) !=? less(0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(p(T14, X13), ','(p(T16, X14), less(X13, X14))), SCOPE: 3, CLAUSE: 2 | ','(p(T14, X13), ','(p(T16, X14), less(X13, X14))), SCOPE: 3, CLAUSE: 3\n\nT14 is ground\nX4 is free\nX13 is free\nX14 is free\nless(T14, T2) !=? less(0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T14, X13), ','(p(T16, X14), less(X13, X14))), SCOPE: 3, CLAUSE: 3\n\nT14 is ground\nX4 is free\nX13 is free\nX14 is free\nless(T14, T2) !=? less(0, X4)", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(p(T16, X14), less(T19, X14))\n\nT19 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(p(T16, X14), less(T19, X14)), SCOPE: 4, CLAUSE: 2 | ','(p(T16, X14), less(T19, X14)), SCOPE: 4, CLAUSE: 3\n\nT19 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(p(T16, X14), less(T19, X14)), SCOPE: 4, CLAUSE: 2\n\nT19 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: ','(p(T16, X14), less(T19, X14)), SCOPE: 4, CLAUSE: 3\n\nT19 is ground\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="B: less(T19, 0)\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: less(T19, 0), SCOPE: 5, CLAUSE: 0 | less(T19, 0), SCOPE: 5, CLAUSE: 1\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: ','(!_5, user_defined_=(0, s(X27))) | less(0, 0), SCOPE: 5, CLAUSE: 1\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: less(T19, 0), SCOPE: 5, CLAUSE: 1\n\nT19 is ground\nX26 is free\nless(T19, 0) !=? less(0, X26)", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: user_defined_=(0, s(X27))\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: user_defined_=(0, s(X27)), SCOPE: 6, CLAUSE: 4\n\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(p(T22, X40), ','(p(0, X41), less(X40, X41)))\n\nT22 is ground\nX26 is free\nX40 is free\nX41 is free\nless(T22, 0) !=? less(0, X26)", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ','(p(T22, X40), ','(p(0, X41), less(X40, X41))), SCOPE: 7, CLAUSE: 2 | ','(p(T22, X40), ','(p(0, X41), less(X40, X41))), SCOPE: 7, CLAUSE: 3\n\nT22 is ground\nX26 is free\nX40 is free\nX41 is free\nless(T22, 0) !=? less(0, X26)", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(p(T22, X40), ','(p(0, X41), less(X40, X41))), SCOPE: 7, CLAUSE: 3\n\nT22 is ground\nX26 is free\nX40 is free\nX41 is free\nless(T22, 0) !=? less(0, X26)", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(p(0, X41), less(T25, X41))\n\nT25 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: ','(p(0, X41), less(T25, X41)), SCOPE: 8, CLAUSE: 2 | ','(p(0, X41), less(T25, X41)), SCOPE: 8, CLAUSE: 3\n\nT25 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: ','(p(0, X41), less(T25, X41)), SCOPE: 8, CLAUSE: 2\n\nT25 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: ','(p(0, X41), less(T25, X41)), SCOPE: 8, CLAUSE: 3\n\nT25 is ground\nX41 is free", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: less(T25, 0)\n\nT25 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: less(T19, T29)\n\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 102 [arrowhead = none ];
102 -> 6;

103 [label="EVAL with clause\nless(0, X4) :- ','(!_1, user_defined_=(X4, s(X5))).\nand substitutionT1 -> 0,\nT2 -> T5,\nX4 -> T5,\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 103 [arrowhead = none ];
103 -> 9;

104 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 104 [arrowhead = none ];
104 -> 11;

105 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 105 [arrowhead = none ];
105 -> 15;

106 [label="ONLY EVAL with clause\nless(X11, X12) :- ','(p(X11, X13), ','(p(X12, X14), less(X13, X14))).\nand substitutionT1 -> T14,\nX11 -> T14,\nT2 -> T16,\nX12 -> T16,\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 106 [arrowhead = none ];
106 -> 29;

107 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 107 [arrowhead = none ];
107 -> 17;

108 [label="EVAL with clause\nuser_defined_=(X8, X8).\nand substitutionT5 -> s(T11),\nX8 -> s(T11),\nX5 -> T11,\nT10 -> s(T11)", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 108 [arrowhead = none ];
108 -> 20;

109 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 109 [arrowhead = none ];
109 -> 24;

110 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 110 [arrowhead = none ];
110 -> 26;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 111 [arrowhead = none ];
111 -> 32;

112 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T14, T2), less(0, X4))", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 112 [arrowhead = none ];
112 -> 34;

113 [label="EVAL with clause\np(s(X17), X17).\nand substitutionX17 -> T19,\nT14 -> s(T19),\nX13 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 113 [arrowhead = none ];
113 -> 37;

114 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 114 [arrowhead = none ];
114 -> 40;

115 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
37 -> 115 [arrowhead = none ];
115 -> 43;

116 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 116 [arrowhead = none ];
116 -> 46;

117 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
43 -> 117 [arrowhead = none ];
117 -> 48;

118 [label="EVAL with clause\np(0, 0).\nand substitutionT16 -> 0,\nX14 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
46 -> 118 [arrowhead = none ];
118 -> 52;

119 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 119 [arrowhead = none ];
119 -> 54;

120 [label="EVAL with clause\np(s(X60), X60).\nand substitutionX60 -> T29,\nT16 -> s(T29),\nX14 -> T29,\nT28 -> T29", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 120 [arrowhead = none ];
120 -> 100;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 121 [arrowhead = none ];
121 -> 101;

122 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
52 -> 122 [arrowhead = none ];
122 -> 60;

123 [label="EVAL with clause\nless(0, X26) :- ','(!_5, user_defined_=(X26, s(X27))).\nand substitutionT19 -> 0,\nX26 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 123 [arrowhead = none ];
123 -> 61;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 124 [arrowhead = none ];
124 -> 62;

125 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 125 [arrowhead = none ];
125 -> 64;

126 [label="ONLY EVAL with clause\nless(X38, X39) :- ','(p(X38, X40), ','(p(X39, X41), less(X40, X41))).\nand substitutionT19 -> T22,\nX38 -> T22,\nX39 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
62 -> 126 [arrowhead = none ];
126 -> 78;

127 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
64 -> 127 [arrowhead = none ];
127 -> 66;

128 [label="BACKTRACK\nfor clause: user_defined_=(X, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
66 -> 128 [arrowhead = none ];
128 -> 70;

129 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
78 -> 129 [arrowhead = none ];
129 -> 80;

130 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (less(T22, 0), less(0, X26))", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 130 [arrowhead = none ];
130 -> 82;

131 [label="EVAL with clause\np(s(X47), X47).\nand substitutionX47 -> T25,\nT22 -> s(T25),\nX40 -> T25", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 131 [arrowhead = none ];
131 -> 85;

132 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 132 [arrowhead = none ];
132 -> 86;

133 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 133 [arrowhead = none ];
133 -> 89;

134 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
89 -> 134 [arrowhead = none ];
134 -> 91;

135 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
89 -> 135 [arrowhead = none ];
135 -> 92;

136 [label="ONLY EVAL with clause\np(0, 0).\nand substitutionX41 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 136 [arrowhead = none ];
136 -> 93;

137 [label="BACKTRACK\nfor clause: p(s(X), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 137 [arrowhead = none ];
137 -> 95;

138 [label="INSTANCE with matching:\nT19 -> T25", fontsize=14, style = filled, fillcolor = lightgrey];
93 -> 138 [arrowhead = none , style=dashed];
138 -> 52[style=dashed];

139 [label="INSTANCE with matching:\nT1 -> T19\nT2 -> T29", fontsize=14, style = filled, fillcolor = lightgrey];
100 -> 139 [arrowhead = none , style=dashed];
139 -> 2[style=dashed];

}

(4) Obligation:

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

lessA_in_ga(0, s(T11)) → lessA_out_ga(0, s(T11))
lessA_in_ga(s(T19), 0) → U1_ga(T19, lessB_in_g(T19))
lessB_in_g(s(T25)) → U3_g(T25, lessB_in_g(T25))
U3_g(T25, lessB_out_g(T25)) → lessB_out_g(s(T25))
U1_ga(T19, lessB_out_g(T19)) → lessA_out_ga(s(T19), 0)
lessA_in_ga(s(T19), s(T29)) → U2_ga(T19, T29, lessA_in_ga(T19, T29))
U2_ga(T19, T29, lessA_out_ga(T19, T29)) → lessA_out_ga(s(T19), s(T29))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)

(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:

LESSA_IN_GA(s(T19), 0) → U1_GA(T19, lessB_in_g(T19))
LESSA_IN_GA(s(T19), 0) → LESSB_IN_G(T19)
LESSB_IN_G(s(T25)) → U3_G(T25, lessB_in_g(T25))
LESSB_IN_G(s(T25)) → LESSB_IN_G(T25)
LESSA_IN_GA(s(T19), s(T29)) → U2_GA(T19, T29, lessA_in_ga(T19, T29))
LESSA_IN_GA(s(T19), s(T29)) → LESSA_IN_GA(T19, T29)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T11)) → lessA_out_ga(0, s(T11))
lessA_in_ga(s(T19), 0) → U1_ga(T19, lessB_in_g(T19))
lessB_in_g(s(T25)) → U3_g(T25, lessB_in_g(T25))
U3_g(T25, lessB_out_g(T25)) → lessB_out_g(s(T25))
U1_ga(T19, lessB_out_g(T19)) → lessA_out_ga(s(T19), 0)
lessA_in_ga(s(T19), s(T29)) → U2_ga(T19, T29, lessA_in_ga(T19, T29))
U2_ga(T19, T29, lessA_out_ga(T19, T29)) → lessA_out_ga(s(T19), s(T29))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
LESSA_IN_GA(x1, x2)  =  LESSA_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x1, x2)
LESSB_IN_G(x1)  =  LESSB_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(6) Obligation:

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

LESSA_IN_GA(s(T19), 0) → U1_GA(T19, lessB_in_g(T19))
LESSA_IN_GA(s(T19), 0) → LESSB_IN_G(T19)
LESSB_IN_G(s(T25)) → U3_G(T25, lessB_in_g(T25))
LESSB_IN_G(s(T25)) → LESSB_IN_G(T25)
LESSA_IN_GA(s(T19), s(T29)) → U2_GA(T19, T29, lessA_in_ga(T19, T29))
LESSA_IN_GA(s(T19), s(T29)) → LESSA_IN_GA(T19, T29)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T11)) → lessA_out_ga(0, s(T11))
lessA_in_ga(s(T19), 0) → U1_ga(T19, lessB_in_g(T19))
lessB_in_g(s(T25)) → U3_g(T25, lessB_in_g(T25))
U3_g(T25, lessB_out_g(T25)) → lessB_out_g(s(T25))
U1_ga(T19, lessB_out_g(T19)) → lessA_out_ga(s(T19), 0)
lessA_in_ga(s(T19), s(T29)) → U2_ga(T19, T29, lessA_in_ga(T19, T29))
U2_ga(T19, T29, lessA_out_ga(T19, T29)) → lessA_out_ga(s(T19), s(T29))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
LESSA_IN_GA(x1, x2)  =  LESSA_IN_GA(x1)
U1_GA(x1, x2)  =  U1_GA(x1, x2)
LESSB_IN_G(x1)  =  LESSB_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x1, x2)
U2_GA(x1, x2, x3)  =  U2_GA(x1, x3)

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

(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:

LESSB_IN_G(s(T25)) → LESSB_IN_G(T25)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T11)) → lessA_out_ga(0, s(T11))
lessA_in_ga(s(T19), 0) → U1_ga(T19, lessB_in_g(T19))
lessB_in_g(s(T25)) → U3_g(T25, lessB_in_g(T25))
U3_g(T25, lessB_out_g(T25)) → lessB_out_g(s(T25))
U1_ga(T19, lessB_out_g(T19)) → lessA_out_ga(s(T19), 0)
lessA_in_ga(s(T19), s(T29)) → U2_ga(T19, T29, lessA_in_ga(T19, T29))
U2_ga(T19, T29, lessA_out_ga(T19, T29)) → lessA_out_ga(s(T19), s(T29))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
LESSB_IN_G(x1)  =  LESSB_IN_G(x1)

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:

LESSB_IN_G(s(T25)) → LESSB_IN_G(T25)

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

(12) PiDPToQDPProof (EQUIVALENT 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_G(s(T25)) → LESSB_IN_G(T25)

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_G(s(T25)) → LESSB_IN_G(T25)
  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_GA(s(T19), s(T29)) → LESSA_IN_GA(T19, T29)

The TRS R consists of the following rules:

lessA_in_ga(0, s(T11)) → lessA_out_ga(0, s(T11))
lessA_in_ga(s(T19), 0) → U1_ga(T19, lessB_in_g(T19))
lessB_in_g(s(T25)) → U3_g(T25, lessB_in_g(T25))
U3_g(T25, lessB_out_g(T25)) → lessB_out_g(s(T25))
U1_ga(T19, lessB_out_g(T19)) → lessA_out_ga(s(T19), 0)
lessA_in_ga(s(T19), s(T29)) → U2_ga(T19, T29, lessA_in_ga(T19, T29))
U2_ga(T19, T29, lessA_out_ga(T19, T29)) → lessA_out_ga(s(T19), s(T29))

The argument filtering Pi contains the following mapping:
lessA_in_ga(x1, x2)  =  lessA_in_ga(x1)
0  =  0
lessA_out_ga(x1, x2)  =  lessA_out_ga(x1)
s(x1)  =  s(x1)
U1_ga(x1, x2)  =  U1_ga(x1, x2)
lessB_in_g(x1)  =  lessB_in_g(x1)
U3_g(x1, x2)  =  U3_g(x1, x2)
lessB_out_g(x1)  =  lessB_out_g(x1)
U2_ga(x1, x2, x3)  =  U2_ga(x1, x3)
LESSA_IN_GA(x1, x2)  =  LESSA_IN_GA(x1)

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:

LESSA_IN_GA(s(T19), s(T29)) → LESSA_IN_GA(T19, T29)

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

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_GA(s(T19)) → LESSA_IN_GA(T19)

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_GA(s(T19)) → LESSA_IN_GA(T19)
  The graph contains the following edges 1 > 1

(22) YES