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

(0) Obligation:

Clauses:

p(0).
p(s(X)) :- ','(q(X), ','(!, r)).
p(X) :- r.
q(0).
q(s(X)) :- ','(p(X), ','(!, r)).
q(X) :- r.
r.

Query: p(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: p(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: p(T1), SCOPE: 1, CLAUSE: 0 | p(T1), SCOPE: 1, CLAUSE: 1 | p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | p(0), SCOPE: 1, CLAUSE: 1 | p(0), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: p(T1), SCOPE: 1, CLAUSE: 1 | p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\np(T1) !=? p(0)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: p(0), SCOPE: 1, CLAUSE: 1 | p(0), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: p(0), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: r, SCOPE: 2, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(q(T3), ','(!_1, r)) | p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: p(T1), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX5 is free\np(T1) !=? p(0)\np(T1) !=? p(s(X5))", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 3 | ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 4 | ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 5 | ?_3 | p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(!_1, r) | ','(q(0), ','(!_1, r)), SCOPE: 3, CLAUSE: 4 | ','(q(0), ','(!_1, r)), SCOPE: 3, CLAUSE: 5 | ?_3 | p(s(0)), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 4 | ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 5 | ?_3 | p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: r, SCOPE: 4, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 4 | ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 5\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ?_3 | p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 4\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: ','(q(T3), ','(!_1, r)), SCOPE: 3, CLAUSE: 5\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="B: ','(','(p(T10), ','(!_3, r)), ','(!_1, r))\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: p(T10)\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="C: ','(','(!_3, r), ','(!_1, r))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(r, ','(!_1, r))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: ','(r, ','(!_1, r)), SCOPE: 5, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(!_1, r)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: r, SCOPE: 6, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: ','(r, ','(!_1, r))\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(r, ','(!_1, r)), SCOPE: 7, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: ','(!_1, r)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: r, SCOPE: 8, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: p(s(T3)), SCOPE: 1, CLAUSE: 2\n\nT3 is ground\nq(T3) !=? q(0)", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="117: r, SCOPE: 9, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
119 [label="119: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
121 [label="121: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
123 [label="123: r\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
125 [label="125: r, SCOPE: 10, CLAUSE: 6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
130 [label="130: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
131 [label="131: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
132 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 132 [arrowhead = none ];
132 -> 6;

133 [label="EVAL with clause\np(0).\nand substitutionT1 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 133 [arrowhead = none ];
133 -> 8;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 134 [arrowhead = none ];
134 -> 11;

135 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 135 [arrowhead = none ];
135 -> 14;

136 [label="EVAL with clause\np(s(X5)) :- ','(q(X5), ','(!_1, r)).\nand substitutionX5 -> T3,\nT1 -> s(T3)", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 136 [arrowhead = none ];
136 -> 32;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 137 [arrowhead = none ];
137 -> 34;

138 [label="BACKTRACK\nfor clause: p(s(X)) :- ','(q(X), ','(!, r))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 138 [arrowhead = none ];
138 -> 16;

139 [label="ONLY EVAL with clause\np(X3) :- r.\nand substitutionX3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 139 [arrowhead = none ];
139 -> 20;

140 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 140 [arrowhead = none ];
140 -> 24;

141 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 141 [arrowhead = none ];
141 -> 28;

142 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 142 [arrowhead = none ];
142 -> 30;

143 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 143 [arrowhead = none ];
143 -> 38;

144 [label="ONLY EVAL with clause\np(X20) :- r.\nand substitutionT1 -> T18,\nX20 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 144 [arrowhead = none ];
144 -> 123;

145 [label="EVAL with clause\nq(0).\nand substitutionT3 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 145 [arrowhead = none ];
145 -> 40;

146 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 146 [arrowhead = none ];
146 -> 42;

147 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 147 [arrowhead = none ];
147 -> 44;

148 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 148 [arrowhead = none ];
148 -> 54;

149 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 149 [arrowhead = none ];
149 -> 56;

150 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
44 -> 150 [arrowhead = none ];
150 -> 47;

151 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 151 [arrowhead = none ];
151 -> 49;

152 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 152 [arrowhead = none ];
152 -> 51;

153 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 153 [arrowhead = none ];
153 -> 58;

154 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
54 -> 154 [arrowhead = none ];
154 -> 60;

155 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
56 -> 155 [arrowhead = none ];
155 -> 113;

156 [label="EVAL with clause\nq(s(X12)) :- ','(p(X12), ','(!_3, r)).\nand substitutionX12 -> T10,\nT3 -> s(T10)", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 156 [arrowhead = none ];
156 -> 62;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 157 [arrowhead = none ];
157 -> 64;

158 [label="ONLY EVAL with clause\nq(X15) :- r.\nand substitutionT3 -> T13,\nX15 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 158 [arrowhead = none ];
158 -> 97;

159 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
62 -> 159 [arrowhead = none ];
159 -> 69;

160 [label="SPLIT 2\nnew knowledge:\nT10 is ground", fontsize=14, style = filled, fillcolor = violet];
62 -> 160 [arrowhead = none ];
160 -> 70;

161 [label="INSTANCE with matching:\nT1 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
69 -> 161 [arrowhead = none , style=dashed];
161 -> 2[style=dashed];

162 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
70 -> 162 [arrowhead = none ];
162 -> 75;

163 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
75 -> 163 [arrowhead = none ];
163 -> 78;

164 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 164 [arrowhead = none ];
164 -> 82;

165 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
82 -> 165 [arrowhead = none ];
165 -> 85;

166 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 166 [arrowhead = none ];
166 -> 88;

167 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 167 [arrowhead = none ];
167 -> 90;

168 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 168 [arrowhead = none ];
168 -> 92;

169 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
97 -> 169 [arrowhead = none ];
169 -> 101;

170 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
101 -> 170 [arrowhead = none ];
170 -> 104;

171 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
104 -> 171 [arrowhead = none ];
171 -> 106;

172 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
106 -> 172 [arrowhead = none ];
172 -> 107;

173 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 173 [arrowhead = none ];
173 -> 111;

174 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
111 -> 174 [arrowhead = none ];
174 -> 112;

175 [label="ONLY EVAL with clause\np(X18) :- r.\nand substitutionT3 -> T16,\nX18 -> s(T16)", fontsize=14, style = filled, fillcolor = lightblue];
113 -> 175 [arrowhead = none ];
175 -> 115;

176 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
115 -> 176 [arrowhead = none ];
176 -> 117;

177 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
117 -> 177 [arrowhead = none ];
177 -> 119;

178 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
119 -> 178 [arrowhead = none ];
178 -> 121;

179 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
123 -> 179 [arrowhead = none ];
179 -> 125;

180 [label="ONLY EVAL with clause\nr.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
125 -> 180 [arrowhead = none ];
180 -> 130;

181 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
130 -> 181 [arrowhead = none ];
181 -> 131;

}

(2) Obligation:

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

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pB_in_g(T10))
pB_in_g(T10) → U2_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U2_g(T10, pA_out_g(T10)) → U3_g(T10, pC_in_)
pC_in_ → pC_out_
U3_g(T10, pC_out_) → pB_out_g(T10)
U1_g(T10, pB_out_g(T10)) → pA_out_g(s(s(T10)))

Pi is empty.

(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:

PA_IN_G(s(s(T10))) → U1_G(T10, pB_in_g(T10))
PA_IN_G(s(s(T10))) → PB_IN_G(T10)
PB_IN_G(T10) → U2_G(T10, pA_in_g(T10))
PB_IN_G(T10) → PA_IN_G(T10)
U2_G(T10, pA_out_g(T10)) → U3_G(T10, pC_in_)
U2_G(T10, pA_out_g(T10)) → PC_IN_

The TRS R consists of the following rules:

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pB_in_g(T10))
pB_in_g(T10) → U2_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U2_g(T10, pA_out_g(T10)) → U3_g(T10, pC_in_)
pC_in_ → pC_out_
U3_g(T10, pC_out_) → pB_out_g(T10)
U1_g(T10, pB_out_g(T10)) → pA_out_g(s(s(T10)))

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

(4) Obligation:

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

PA_IN_G(s(s(T10))) → U1_G(T10, pB_in_g(T10))
PA_IN_G(s(s(T10))) → PB_IN_G(T10)
PB_IN_G(T10) → U2_G(T10, pA_in_g(T10))
PB_IN_G(T10) → PA_IN_G(T10)
U2_G(T10, pA_out_g(T10)) → U3_G(T10, pC_in_)
U2_G(T10, pA_out_g(T10)) → PC_IN_

The TRS R consists of the following rules:

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pB_in_g(T10))
pB_in_g(T10) → U2_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U2_g(T10, pA_out_g(T10)) → U3_g(T10, pC_in_)
pC_in_ → pC_out_
U3_g(T10, pC_out_) → pB_out_g(T10)
U1_g(T10, pB_out_g(T10)) → pA_out_g(s(s(T10)))

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(6) Obligation:

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

PA_IN_G(s(s(T10))) → PB_IN_G(T10)
PB_IN_G(T10) → PA_IN_G(T10)

The TRS R consists of the following rules:

pA_in_g(0) → pA_out_g(0)
pA_in_g(s(0)) → pA_out_g(s(0))
pA_in_g(s(s(T10))) → U1_g(T10, pB_in_g(T10))
pB_in_g(T10) → U2_g(T10, pA_in_g(T10))
pA_in_g(s(T13)) → pA_out_g(s(T13))
pA_in_g(T18) → pA_out_g(T18)
U2_g(T10, pA_out_g(T10)) → U3_g(T10, pC_in_)
pC_in_ → pC_out_
U3_g(T10, pC_out_) → pB_out_g(T10)
U1_g(T10, pB_out_g(T10)) → pA_out_g(s(s(T10)))

Pi is empty.
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:

PA_IN_G(s(s(T10))) → PB_IN_G(T10)
PB_IN_G(T10) → PA_IN_G(T10)

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

(9) PiDPToQDPProof (EQUIVALENT 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:

PA_IN_G(s(s(T10))) → PB_IN_G(T10)
PB_IN_G(T10) → PA_IN_G(T10)

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:

• PB_IN_G(T10) → PA_IN_G(T10)
  The graph contains the following edges 1 >= 1

• PA_IN_G(s(s(T10))) → PB_IN_G(T10)
  The graph contains the following edges 1 > 1

(12) YES