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

(0) Obligation:

Clauses:

h(X) :- ','(f(X), g(X)).
f(c(0, X1)) :- !.
f(c(X, Y)) :- ','(p(X, P), f(c(P, s(Y)))).
g(c(X2, 0)) :- !.
g(c(X, Y)) :- ','(p(Y, P), g(c(s(X), P))).
p(0, 0).
p(s(X), X).

Query: h(g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: h(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: h(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(f(T3), g(T3))\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: ','(f(T3), g(T3)), SCOPE: 2, CLAUSE: 1 | ','(f(T3), g(T3)), SCOPE: 2, CLAUSE: 2\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(!_2, g(c(0, T6))) | ','(f(c(0, T6)), g(c(0, T6))), SCOPE: 2, CLAUSE: 2\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(f(T3), g(T3)), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX7 is free\nf(T3) !=? f(c(0, X7))", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: g(c(0, T6))\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: g(c(0, T6)), SCOPE: 3, CLAUSE: 3 | g(c(0, T6)), SCOPE: 3, CLAUSE: 4\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: !_3 | g(c(0, 0)), SCOPE: 3, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: g(c(0, T6)), SCOPE: 3, CLAUSE: 4\n\nT6 is ground\nX11 is free\ng(c(0, T6)) !=? g(c(X11, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T9, X20), g(c(s(0), X20)))\n\nT9 is ground\nX11 is free\nX20 is free\ng(c(0, T9)) !=? g(c(X11, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T9, X20), g(c(s(0), X20))), SCOPE: 4, CLAUSE: 5 | ','(p(T9, X20), g(c(s(0), X20))), SCOPE: 4, CLAUSE: 6\n\nT9 is ground\nX11 is free\nX20 is free\ng(c(0, T9)) !=? g(c(X11, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T9, X20), g(c(s(0), X20))), SCOPE: 4, CLAUSE: 6\n\nT9 is ground\nX11 is free\nX20 is free\ng(c(0, T9)) !=? g(c(X11, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: g(c(s(0), T12))\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: g(c(s(0), T12)), SCOPE: 5, CLAUSE: 3 | g(c(s(0), T12)), SCOPE: 5, CLAUSE: 4\n\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: !_5 | g(c(s(0), 0)), SCOPE: 5, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: g(c(s(0), T12)), SCOPE: 5, CLAUSE: 4\n\nT12 is ground\nX28 is free\ng(c(s(0), T12)) !=? g(c(X28, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(p(T15, X37), g(c(s(s(0)), X37)))\n\nT15 is ground\nX28 is free\nX37 is free\ng(c(s(0), T15)) !=? g(c(X28, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(p(T15, X37), g(c(s(s(0)), X37))), SCOPE: 6, CLAUSE: 5 | ','(p(T15, X37), g(c(s(s(0)), X37))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX28 is free\nX37 is free\ng(c(s(0), T15)) !=? g(c(X28, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(p(T15, X37), g(c(s(s(0)), X37))), SCOPE: 6, CLAUSE: 6\n\nT15 is ground\nX28 is free\nX37 is free\ng(c(s(0), T15)) !=? g(c(X28, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: g(c(s(s(0)), T18))\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: g(c(s(s(0)), T18)), SCOPE: 7, CLAUSE: 3 | g(c(s(s(0)), T18)), SCOPE: 7, CLAUSE: 4\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: !_7 | g(c(s(s(0)), 0)), SCOPE: 7, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: g(c(s(s(0)), T18)), SCOPE: 7, CLAUSE: 4\n\nT18 is ground\nX45 is free\ng(c(s(s(0)), T18)) !=? g(c(X45, 0))", 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: ','(p(T21, X54), g(c(s(s(s(0))), X54)))\n\nT21 is ground\nX45 is free\nX54 is free\ng(c(s(s(0)), T21)) !=? g(c(X45, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(p(T21, X54), g(c(s(s(s(0))), X54))), SCOPE: 8, CLAUSE: 5 | ','(p(T21, X54), g(c(s(s(s(0))), X54))), SCOPE: 8, CLAUSE: 6\n\nT21 is ground\nX45 is free\nX54 is free\ng(c(s(s(0)), T21)) !=? g(c(X45, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(p(T21, X54), g(c(s(s(s(0))), X54))), SCOPE: 8, CLAUSE: 6\n\nT21 is ground\nX45 is free\nX54 is free\ng(c(s(s(0)), T21)) !=? g(c(X45, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: g(c(s(s(s(0))), T24))\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: g(c(s(s(s(0))), T24)), SCOPE: 9, CLAUSE: 3 | g(c(s(s(s(0))), T24)), SCOPE: 9, CLAUSE: 4\n\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: !_9 | g(c(s(s(s(0))), 0)), SCOPE: 9, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: g(c(s(s(s(0))), T24)), SCOPE: 9, CLAUSE: 4\n\nT24 is ground\nX62 is free\ng(c(s(s(s(0))), T24)) !=? g(c(X62, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ','(p(T27, X71), g(c(s(s(s(s(0)))), X71)))\n\nT27 is ground\nX62 is free\nX71 is free\ng(c(s(s(s(0))), T27)) !=? g(c(X62, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(p(T27, X71), g(c(s(s(s(s(0)))), X71))), SCOPE: 10, CLAUSE: 5 | ','(p(T27, X71), g(c(s(s(s(s(0)))), X71))), SCOPE: 10, CLAUSE: 6\n\nT27 is ground\nX62 is free\nX71 is free\ng(c(s(s(s(0))), T27)) !=? g(c(X62, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: ','(p(T27, X71), g(c(s(s(s(s(0)))), X71))), SCOPE: 10, CLAUSE: 6\n\nT27 is ground\nX62 is free\nX71 is free\ng(c(s(s(s(0))), T27)) !=? g(c(X62, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: g(c(s(s(s(s(0)))), T30))\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: g(c(s(s(s(s(0)))), T30)), SCOPE: 11, CLAUSE: 3 | g(c(s(s(s(s(0)))), T30)), SCOPE: 11, CLAUSE: 4\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: !_11 | g(c(s(s(s(s(0)))), 0)), SCOPE: 11, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: g(c(s(s(s(s(0)))), T30)), SCOPE: 11, CLAUSE: 4\n\nT30 is ground\nX79 is free\ng(c(s(s(s(s(0)))), T30)) !=? g(c(X79, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: ','(p(T33, X88), g(c(s(s(s(s(s(0))))), X88)))\n\nT33 is ground\nX79 is free\nX88 is free\ng(c(s(s(s(s(0)))), T33)) !=? g(c(X79, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(p(T33, X88), g(c(s(s(s(s(s(0))))), X88))), SCOPE: 12, CLAUSE: 5 | ','(p(T33, X88), g(c(s(s(s(s(s(0))))), X88))), SCOPE: 12, CLAUSE: 6\n\nT33 is ground\nX79 is free\nX88 is free\ng(c(s(s(s(s(0)))), T33)) !=? g(c(X79, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: ','(p(T33, X88), g(c(s(s(s(s(s(0))))), X88))), SCOPE: 12, CLAUSE: 6\n\nT33 is ground\nX79 is free\nX88 is free\ng(c(s(s(s(s(0)))), T33)) !=? g(c(X79, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: g(c(s(s(s(s(s(0))))), T36))\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: g(c(s(s(s(s(s(0))))), T36)), SCOPE: 13, CLAUSE: 3 | g(c(s(s(s(s(s(0))))), T36)), SCOPE: 13, CLAUSE: 4\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: !_13 | g(c(s(s(s(s(s(0))))), 0)), SCOPE: 13, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: g(c(s(s(s(s(s(0))))), T36)), SCOPE: 13, CLAUSE: 4\n\nT36 is ground\nX96 is free\ng(c(s(s(s(s(s(0))))), T36)) !=? g(c(X96, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: ','(p(T39, X105), g(c(s(s(s(s(s(s(0)))))), X105)))\n\nT39 is ground\nX96 is free\nX105 is free\ng(c(s(s(s(s(s(0))))), T39)) !=? g(c(X96, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: ','(p(T39, X105), g(c(s(s(s(s(s(s(0)))))), X105))), SCOPE: 14, CLAUSE: 5 | ','(p(T39, X105), g(c(s(s(s(s(s(s(0)))))), X105))), SCOPE: 14, CLAUSE: 6\n\nT39 is ground\nX96 is free\nX105 is free\ng(c(s(s(s(s(s(0))))), T39)) !=? g(c(X96, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(p(T39, X105), g(c(s(s(s(s(s(s(0)))))), X105))), SCOPE: 14, CLAUSE: 6\n\nT39 is ground\nX96 is free\nX105 is free\ng(c(s(s(s(s(s(0))))), T39)) !=? g(c(X96, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: g(c(s(s(s(s(s(s(0)))))), T42))\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: g(c(s(s(s(s(s(s(0)))))), T42)), SCOPE: 15, CLAUSE: 3 | g(c(s(s(s(s(s(s(0)))))), T42)), SCOPE: 15, CLAUSE: 4\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="69: !_15 | g(c(s(s(s(s(s(s(0)))))), 0)), SCOPE: 15, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
70 [label="70: g(c(s(s(s(s(s(s(0)))))), T42)), SCOPE: 15, CLAUSE: 4\n\nT42 is ground\nX113 is free\ng(c(s(s(s(s(s(s(0)))))), T42)) !=? g(c(X113, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
72 [label="72: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
73 [label="73: ','(p(T45, X122), g(c(s(s(s(s(s(s(s(0))))))), X122)))\n\nT45 is ground\nX113 is free\nX122 is free\ng(c(s(s(s(s(s(s(0)))))), T45)) !=? g(c(X113, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ','(p(T45, X122), g(c(s(s(s(s(s(s(s(0))))))), X122))), SCOPE: 16, CLAUSE: 5 | ','(p(T45, X122), g(c(s(s(s(s(s(s(s(0))))))), X122))), SCOPE: 16, CLAUSE: 6\n\nT45 is ground\nX113 is free\nX122 is free\ng(c(s(s(s(s(s(s(0)))))), T45)) !=? g(c(X113, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
75 [label="75: ','(p(T45, X122), g(c(s(s(s(s(s(s(s(0))))))), X122))), SCOPE: 16, CLAUSE: 6\n\nT45 is ground\nX113 is free\nX122 is free\ng(c(s(s(s(s(s(s(0)))))), T45)) !=? g(c(X113, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: g(c(s(s(s(s(s(s(s(0))))))), T48))\n\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
78 [label="78: g(c(s(s(s(s(s(s(s(0))))))), T48)), SCOPE: 17, CLAUSE: 3 | g(c(s(s(s(s(s(s(s(0))))))), T48)), SCOPE: 17, CLAUSE: 4\n\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: !_17 | g(c(s(s(s(s(s(s(s(0))))))), 0)), SCOPE: 17, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: g(c(s(s(s(s(s(s(s(0))))))), T48)), SCOPE: 17, CLAUSE: 4\n\nT48 is ground\nX130 is free\ng(c(s(s(s(s(s(s(s(0))))))), T48)) !=? g(c(X130, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
81 [label="81: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
83 [label="83: ','(p(T51, X139), g(c(s(s(s(s(s(s(s(s(0)))))))), X139)))\n\nT51 is ground\nX130 is free\nX139 is free\ng(c(s(s(s(s(s(s(s(0))))))), T51)) !=? g(c(X130, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
84 [label="84: ','(p(T51, X139), g(c(s(s(s(s(s(s(s(s(0)))))))), X139))), SCOPE: 18, CLAUSE: 5 | ','(p(T51, X139), g(c(s(s(s(s(s(s(s(s(0)))))))), X139))), SCOPE: 18, CLAUSE: 6\n\nT51 is ground\nX130 is free\nX139 is free\ng(c(s(s(s(s(s(s(s(0))))))), T51)) !=? g(c(X130, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(p(T51, X139), g(c(s(s(s(s(s(s(s(s(0)))))))), X139))), SCOPE: 18, CLAUSE: 6\n\nT51 is ground\nX130 is free\nX139 is free\ng(c(s(s(s(s(s(s(s(0))))))), T51)) !=? g(c(X130, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: g(c(s(s(s(s(s(s(s(s(0)))))))), T54))\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
87 [label="87: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: g(c(s(T55), T54))\n\nT54 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
89 [label="89: g(c(s(T55), T54)), SCOPE: 19, CLAUSE: 3 | g(c(s(T55), T54)), SCOPE: 19, CLAUSE: 4\n\nT54 is ground\nT55 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: !_19 | g(c(s(T59), 0)), SCOPE: 19, CLAUSE: 4\n\nT59 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: g(c(s(T55), T54)), SCOPE: 19, CLAUSE: 4\n\nT54 is ground\nT55 is ground\nX148 is free\ng(c(s(T55), T54)) !=? g(c(X148, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="92: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
93 [label="93: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
94 [label="94: ','(p(T66, X157), g(c(s(s(T65)), X157)))\n\nT65 is ground\nT66 is ground\nX148 is free\nX157 is free\ng(c(s(T65), T66)) !=? g(c(X148, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
95 [label="95: ','(p(T66, X157), g(c(s(s(T65)), X157))), SCOPE: 20, CLAUSE: 5 | ','(p(T66, X157), g(c(s(s(T65)), X157))), SCOPE: 20, CLAUSE: 6\n\nT65 is ground\nT66 is ground\nX148 is free\nX157 is free\ng(c(s(T65), T66)) !=? g(c(X148, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
96 [label="96: ','(p(T66, X157), g(c(s(s(T65)), X157))), SCOPE: 20, CLAUSE: 6\n\nT65 is ground\nT66 is ground\nX148 is free\nX157 is free\ng(c(s(T65), T66)) !=? g(c(X148, 0))", fontsize=16, style=filled, fillcolor=lightyellow];
97 [label="97: g(c(s(s(T65)), T70))\n\nT65 is ground\nT70 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
98 [label="98: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
99 [label="99: ','(','(p(T76, X170), f(c(X170, s(T77)))), g(c(T76, T77)))\n\nT76 is ground\nT77 is ground\nX7 is free\nX170 is free\nf(c(T76, T77)) !=? f(c(0, X7))", fontsize=16, style=filled, fillcolor=lightyellow];
100 [label="100: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
101 [label="101: ','(','(p(T76, X170), f(c(X170, s(T77)))), g(c(T76, T77))), SCOPE: 21, CLAUSE: 5 | ','(','(p(T76, X170), f(c(X170, s(T77)))), g(c(T76, T77))), SCOPE: 21, CLAUSE: 6\n\nT76 is ground\nT77 is ground\nX7 is free\nX170 is free\nf(c(T76, T77)) !=? f(c(0, X7))", fontsize=16, style=filled, fillcolor=lightyellow];
102 [label="102: ','(','(p(T76, X170), f(c(X170, s(T77)))), g(c(T76, T77))), SCOPE: 21, CLAUSE: 6\n\nT76 is ground\nT77 is ground\nX7 is free\nX170 is free\nf(c(T76, T77)) !=? f(c(0, X7))", fontsize=16, style=filled, fillcolor=lightyellow];
103 [label="103: ','(f(c(T81, s(T77))), g(c(s(T81), T77)))\n\nT77 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
104 [label="104: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
105 [label="105: f(c(T81, s(T77)))\n\nT77 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
106 [label="106: g(c(s(T81), T77))\n\nT77 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
107 [label="107: f(c(T81, s(T77))), SCOPE: 22, CLAUSE: 1 | f(c(T81, s(T77))), SCOPE: 22, CLAUSE: 2\n\nT77 is ground\nT81 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
108 [label="108: !_22 | f(c(0, s(T86))), SCOPE: 22, CLAUSE: 2\n\nT86 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
109 [label="109: f(c(T81, s(T77))), SCOPE: 22, CLAUSE: 2\n\nT77 is ground\nT81 is ground\nX179 is free\nf(c(T81, s(T77))) !=? f(c(0, X179))", fontsize=16, style=filled, fillcolor=lightyellow];
110 [label="110: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
111 [label="111: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
112 [label="112: ','(p(T92, X188), f(c(X188, s(s(T93)))))\n\nT92 is ground\nT93 is ground\nX179 is free\nX188 is free\nf(c(T92, s(T93))) !=? f(c(0, X179))", fontsize=16, style=filled, fillcolor=lightyellow];
113 [label="113: ','(p(T92, X188), f(c(X188, s(s(T93))))), SCOPE: 23, CLAUSE: 5 | ','(p(T92, X188), f(c(X188, s(s(T93))))), SCOPE: 23, CLAUSE: 6\n\nT92 is ground\nT93 is ground\nX179 is free\nX188 is free\nf(c(T92, s(T93))) !=? f(c(0, X179))", fontsize=16, style=filled, fillcolor=lightyellow];
114 [label="114: ','(p(T92, X188), f(c(X188, s(s(T93))))), SCOPE: 23, CLAUSE: 6\n\nT92 is ground\nT93 is ground\nX179 is free\nX188 is free\nf(c(T92, s(T93))) !=? f(c(0, X179))", fontsize=16, style=filled, fillcolor=lightyellow];
115 [label="115: f(c(T97, s(s(T93))))\n\nT93 is ground\nT97 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
116 [label="116: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
117 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 117 [arrowhead = none ];
117 -> 2;

118 [label="ONLY EVAL with clause\nh(X4) :- ','(f(X4), g(X4)).\nand substitutionT1 -> T3,\nX4 -> T3", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 118 [arrowhead = none ];
118 -> 3;

119 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 119 [arrowhead = none ];
119 -> 4;

120 [label="EVAL with clause\nf(c(0, X7)) :- !_2.\nand substitutionX7 -> T6,\nT3 -> c(0, T6)", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 120 [arrowhead = none ];
120 -> 5;

121 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 121 [arrowhead = none ];
121 -> 6;

122 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 122 [arrowhead = none ];
122 -> 7;

123 [label="EVAL with clause\nf(c(X168, X169)) :- ','(p(X168, X170), f(c(X170, s(X169)))).\nand substitutionX168 -> T76,\nX169 -> T77,\nT3 -> c(T76, T77)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 123 [arrowhead = none ];
123 -> 99;

124 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 124 [arrowhead = none ];
124 -> 100;

125 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
7 -> 125 [arrowhead = none ];
125 -> 8;

126 [label="EVAL with clause\ng(c(X11, 0)) :- !_3.\nand substitutionX11 -> 0,\nT6 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 126 [arrowhead = none ];
126 -> 9;

127 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 127 [arrowhead = none ];
127 -> 10;

128 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 128 [arrowhead = none ];
128 -> 11;

129 [label="ONLY EVAL with clause\ng(c(X18, X19)) :- ','(p(X19, X20), g(c(s(X18), X20))).\nand substitutionX18 -> 0,\nT6 -> T9,\nX19 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 129 [arrowhead = none ];
129 -> 13;

130 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 130 [arrowhead = none ];
130 -> 12;

131 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 131 [arrowhead = none ];
131 -> 14;

132 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(0, T9)), g(c(X11, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 132 [arrowhead = none ];
132 -> 15;

133 [label="EVAL with clause\np(s(X24), X24).\nand substitutionX24 -> T12,\nT9 -> s(T12),\nX20 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 133 [arrowhead = none ];
133 -> 16;

134 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 134 [arrowhead = none ];
134 -> 17;

135 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 135 [arrowhead = none ];
135 -> 18;

136 [label="EVAL with clause\ng(c(X28, 0)) :- !_5.\nand substitutionX28 -> s(0),\nT12 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 136 [arrowhead = none ];
136 -> 19;

137 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 137 [arrowhead = none ];
137 -> 20;

138 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 138 [arrowhead = none ];
138 -> 21;

139 [label="ONLY EVAL with clause\ng(c(X35, X36)) :- ','(p(X36, X37), g(c(s(X35), X37))).\nand substitutionX35 -> s(0),\nT12 -> T15,\nX36 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 139 [arrowhead = none ];
139 -> 23;

140 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 140 [arrowhead = none ];
140 -> 22;

141 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 141 [arrowhead = none ];
141 -> 24;

142 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(s(0), T15)), g(c(X28, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 142 [arrowhead = none ];
142 -> 25;

143 [label="EVAL with clause\np(s(X41), X41).\nand substitutionX41 -> T18,\nT15 -> s(T18),\nX37 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 143 [arrowhead = none ];
143 -> 26;

144 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 144 [arrowhead = none ];
144 -> 27;

145 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 145 [arrowhead = none ];
145 -> 28;

146 [label="EVAL with clause\ng(c(X45, 0)) :- !_7.\nand substitutionX45 -> s(s(0)),\nT18 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 146 [arrowhead = none ];
146 -> 29;

147 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 147 [arrowhead = none ];
147 -> 30;

148 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 148 [arrowhead = none ];
148 -> 31;

149 [label="ONLY EVAL with clause\ng(c(X52, X53)) :- ','(p(X53, X54), g(c(s(X52), X54))).\nand substitutionX52 -> s(s(0)),\nT18 -> T21,\nX53 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 149 [arrowhead = none ];
149 -> 33;

150 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 150 [arrowhead = none ];
150 -> 32;

151 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 151 [arrowhead = none ];
151 -> 34;

152 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(s(s(0)), T21)), g(c(X45, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 152 [arrowhead = none ];
152 -> 35;

153 [label="EVAL with clause\np(s(X58), X58).\nand substitutionX58 -> T24,\nT21 -> s(T24),\nX54 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 153 [arrowhead = none ];
153 -> 36;

154 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 154 [arrowhead = none ];
154 -> 37;

155 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 155 [arrowhead = none ];
155 -> 38;

156 [label="EVAL with clause\ng(c(X62, 0)) :- !_9.\nand substitutionX62 -> s(s(s(0))),\nT24 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 156 [arrowhead = none ];
156 -> 39;

157 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 157 [arrowhead = none ];
157 -> 40;

158 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 158 [arrowhead = none ];
158 -> 41;

159 [label="ONLY EVAL with clause\ng(c(X69, X70)) :- ','(p(X70, X71), g(c(s(X69), X71))).\nand substitutionX69 -> s(s(s(0))),\nT24 -> T27,\nX70 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 159 [arrowhead = none ];
159 -> 43;

160 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 160 [arrowhead = none ];
160 -> 42;

161 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
43 -> 161 [arrowhead = none ];
161 -> 44;

162 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(s(s(s(0))), T27)), g(c(X62, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 162 [arrowhead = none ];
162 -> 45;

163 [label="EVAL with clause\np(s(X75), X75).\nand substitutionX75 -> T30,\nT27 -> s(T30),\nX71 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
45 -> 163 [arrowhead = none ];
163 -> 46;

164 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 164 [arrowhead = none ];
164 -> 47;

165 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
46 -> 165 [arrowhead = none ];
165 -> 48;

166 [label="EVAL with clause\ng(c(X79, 0)) :- !_11.\nand substitutionX79 -> s(s(s(s(0)))),\nT30 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 166 [arrowhead = none ];
166 -> 49;

167 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 167 [arrowhead = none ];
167 -> 50;

168 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 168 [arrowhead = none ];
168 -> 51;

169 [label="ONLY EVAL with clause\ng(c(X86, X87)) :- ','(p(X87, X88), g(c(s(X86), X88))).\nand substitutionX86 -> s(s(s(s(0)))),\nT30 -> T33,\nX87 -> T33", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 169 [arrowhead = none ];
169 -> 53;

170 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 170 [arrowhead = none ];
170 -> 52;

171 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
53 -> 171 [arrowhead = none ];
171 -> 54;

172 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(s(s(s(s(0)))), T33)), g(c(X79, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 172 [arrowhead = none ];
172 -> 55;

173 [label="EVAL with clause\np(s(X92), X92).\nand substitutionX92 -> T36,\nT33 -> s(T36),\nX88 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
55 -> 173 [arrowhead = none ];
173 -> 56;

174 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 174 [arrowhead = none ];
174 -> 57;

175 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
56 -> 175 [arrowhead = none ];
175 -> 58;

176 [label="EVAL with clause\ng(c(X96, 0)) :- !_13.\nand substitutionX96 -> s(s(s(s(s(0))))),\nT36 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
58 -> 176 [arrowhead = none ];
176 -> 59;

177 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 177 [arrowhead = none ];
177 -> 60;

178 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 178 [arrowhead = none ];
178 -> 61;

179 [label="ONLY EVAL with clause\ng(c(X103, X104)) :- ','(p(X104, X105), g(c(s(X103), X105))).\nand substitutionX103 -> s(s(s(s(s(0))))),\nT36 -> T39,\nX104 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
60 -> 179 [arrowhead = none ];
179 -> 63;

180 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 180 [arrowhead = none ];
180 -> 62;

181 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
63 -> 181 [arrowhead = none ];
181 -> 64;

182 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(s(s(s(s(s(0))))), T39)), g(c(X96, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 182 [arrowhead = none ];
182 -> 65;

183 [label="EVAL with clause\np(s(X109), X109).\nand substitutionX109 -> T42,\nT39 -> s(T42),\nX105 -> T42", fontsize=14, style = filled, fillcolor = lightblue];
65 -> 183 [arrowhead = none ];
183 -> 66;

184 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 184 [arrowhead = none ];
184 -> 67;

185 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
66 -> 185 [arrowhead = none ];
185 -> 68;

186 [label="EVAL with clause\ng(c(X113, 0)) :- !_15.\nand substitutionX113 -> s(s(s(s(s(s(0)))))),\nT42 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 186 [arrowhead = none ];
186 -> 69;

187 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 187 [arrowhead = none ];
187 -> 70;

188 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
69 -> 188 [arrowhead = none ];
188 -> 71;

189 [label="ONLY EVAL with clause\ng(c(X120, X121)) :- ','(p(X121, X122), g(c(s(X120), X122))).\nand substitutionX120 -> s(s(s(s(s(s(0)))))),\nT42 -> T45,\nX121 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
70 -> 189 [arrowhead = none ];
189 -> 73;

190 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 190 [arrowhead = none ];
190 -> 72;

191 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
73 -> 191 [arrowhead = none ];
191 -> 74;

192 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(s(s(s(s(s(s(0)))))), T45)), g(c(X113, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 192 [arrowhead = none ];
192 -> 75;

193 [label="EVAL with clause\np(s(X126), X126).\nand substitutionX126 -> T48,\nT45 -> s(T48),\nX122 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
75 -> 193 [arrowhead = none ];
193 -> 76;

194 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
75 -> 194 [arrowhead = none ];
194 -> 77;

195 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 195 [arrowhead = none ];
195 -> 78;

196 [label="EVAL with clause\ng(c(X130, 0)) :- !_17.\nand substitutionX130 -> s(s(s(s(s(s(s(0))))))),\nT48 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
78 -> 196 [arrowhead = none ];
196 -> 79;

197 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
78 -> 197 [arrowhead = none ];
197 -> 80;

198 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
79 -> 198 [arrowhead = none ];
198 -> 81;

199 [label="ONLY EVAL with clause\ng(c(X137, X138)) :- ','(p(X138, X139), g(c(s(X137), X139))).\nand substitutionX137 -> s(s(s(s(s(s(s(0))))))),\nT48 -> T51,\nX138 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
80 -> 199 [arrowhead = none ];
199 -> 83;

200 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
81 -> 200 [arrowhead = none ];
200 -> 82;

201 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
83 -> 201 [arrowhead = none ];
201 -> 84;

202 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(s(s(s(s(s(s(s(0))))))), T51)), g(c(X130, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
84 -> 202 [arrowhead = none ];
202 -> 85;

203 [label="EVAL with clause\np(s(X143), X143).\nand substitutionX143 -> T54,\nT51 -> s(T54),\nX139 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
85 -> 203 [arrowhead = none ];
203 -> 86;

204 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
85 -> 204 [arrowhead = none ];
204 -> 87;

205 [label="GENERALIZATION\nT55 <-- s(s(s(s(s(s(s(0)))))))\n\nNew Knowledge:\nT55 is ground", fontsize=14, style = filled, fillcolor = violet];
86 -> 205 [arrowhead = none ];
205 -> 88;

206 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
88 -> 206 [arrowhead = none ];
206 -> 89;

207 [label="EVAL with clause\ng(c(X148, 0)) :- !_19.\nand substitutionT55 -> T59,\nX148 -> s(T59),\nT54 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
89 -> 207 [arrowhead = none ];
207 -> 90;

208 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
89 -> 208 [arrowhead = none ];
208 -> 91;

209 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
90 -> 209 [arrowhead = none ];
209 -> 92;

210 [label="ONLY EVAL with clause\ng(c(X155, X156)) :- ','(p(X156, X157), g(c(s(X155), X157))).\nand substitutionT55 -> T65,\nX155 -> s(T65),\nT54 -> T66,\nX156 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
91 -> 210 [arrowhead = none ];
210 -> 94;

211 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
92 -> 211 [arrowhead = none ];
211 -> 93;

212 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
94 -> 212 [arrowhead = none ];
212 -> 95;

213 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (g(c(s(T65), T66)), g(c(X148, 0)))", fontsize=14, style = filled, fillcolor = lightgreen];
95 -> 213 [arrowhead = none ];
213 -> 96;

214 [label="EVAL with clause\np(s(X161), X161).\nand substitutionX161 -> T70,\nT66 -> s(T70),\nX157 -> T70", fontsize=14, style = filled, fillcolor = lightblue];
96 -> 214 [arrowhead = none ];
214 -> 97;

215 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
96 -> 215 [arrowhead = none ];
215 -> 98;

216 [label="INSTANCE with matching:\nT55 -> s(T65)\nT54 -> T70", fontsize=14, style = filled, fillcolor = lightgrey];
97 -> 216 [arrowhead = none , style=dashed];
216 -> 88[style=dashed];

217 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
99 -> 217 [arrowhead = none ];
217 -> 101;

218 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (f(c(T76, T77)), f(c(0, X7)))", fontsize=14, style = filled, fillcolor = lightgreen];
101 -> 218 [arrowhead = none ];
218 -> 102;

219 [label="EVAL with clause\np(s(X174), X174).\nand substitutionX174 -> T81,\nT76 -> s(T81),\nX170 -> T81", fontsize=14, style = filled, fillcolor = lightblue];
102 -> 219 [arrowhead = none ];
219 -> 103;

220 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
102 -> 220 [arrowhead = none ];
220 -> 104;

221 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
103 -> 221 [arrowhead = none ];
221 -> 105;

222 [label="SPLIT 2\nnew knowledge:\nT81 is ground\nT77 is ground", fontsize=14, style = filled, fillcolor = violet];
103 -> 222 [arrowhead = none ];
222 -> 106;

223 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
105 -> 223 [arrowhead = none ];
223 -> 107;

224 [label="INSTANCE with matching:\nT55 -> T81\nT54 -> T77", fontsize=14, style = filled, fillcolor = lightgrey];
106 -> 224 [arrowhead = none , style=dashed];
224 -> 88[style=dashed];

225 [label="EVAL with clause\nf(c(0, X179)) :- !_22.\nand substitutionT81 -> 0,\nT77 -> T86,\nX179 -> s(T86)", fontsize=14, style = filled, fillcolor = lightblue];
107 -> 225 [arrowhead = none ];
225 -> 108;

226 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
107 -> 226 [arrowhead = none ];
226 -> 109;

227 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
108 -> 227 [arrowhead = none ];
227 -> 110;

228 [label="ONLY EVAL with clause\nf(c(X186, X187)) :- ','(p(X186, X188), f(c(X188, s(X187)))).\nand substitutionT81 -> T92,\nX186 -> T92,\nT77 -> T93,\nX187 -> s(T93)", fontsize=14, style = filled, fillcolor = lightblue];
109 -> 228 [arrowhead = none ];
228 -> 112;

229 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
110 -> 229 [arrowhead = none ];
229 -> 111;

230 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
112 -> 230 [arrowhead = none ];
230 -> 113;

231 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (f(c(T92, s(T93))), f(c(0, X179)))", fontsize=14, style = filled, fillcolor = lightgreen];
113 -> 231 [arrowhead = none ];
231 -> 114;

232 [label="EVAL with clause\np(s(X192), X192).\nand substitutionX192 -> T97,\nT92 -> s(T97),\nX188 -> T97", fontsize=14, style = filled, fillcolor = lightblue];
114 -> 232 [arrowhead = none ];
232 -> 115;

233 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
114 -> 233 [arrowhead = none ];
233 -> 116;

234 [label="INSTANCE with matching:\nT81 -> T97\nT77 -> s(T93)", fontsize=14, style = filled, fillcolor = lightgrey];
115 -> 234 [arrowhead = none , style=dashed];
234 -> 105[style=dashed];

}

(2) Obligation:

Clauses:

gA(T59, 0).
gA(T65, s(T70)) :- gA(s(T65), T70).
fB(0, T86).
fB(s(T97), T93) :- fB(T97, s(T93)).
hC(c(0, 0)).
hC(c(0, s(0))).
hC(c(0, s(s(0)))).
hC(c(0, s(s(s(0))))).
hC(c(0, s(s(s(s(0)))))).
hC(c(0, s(s(s(s(s(0))))))).
hC(c(0, s(s(s(s(s(s(0)))))))).
hC(c(0, s(s(s(s(s(s(s(0))))))))).
hC(c(0, s(s(s(s(s(s(s(s(T54)))))))))) :- gA(s(s(s(s(s(s(s(0))))))), T54).
hC(c(s(T81), T77)) :- fB(T81, T77).
hC(c(s(T81), T77)) :- ','(fB(T81, T77), gA(T81, T77)).

Query: hC(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:
hC_in: (b)
gA_in: (b,b)
fB_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

hC_in_g(c(0, 0)) → hC_out_g(c(0, 0))
hC_in_g(c(0, s(0))) → hC_out_g(c(0, s(0)))
hC_in_g(c(0, s(s(0)))) → hC_out_g(c(0, s(s(0))))
hC_in_g(c(0, s(s(s(0))))) → hC_out_g(c(0, s(s(s(0)))))
hC_in_g(c(0, s(s(s(s(0)))))) → hC_out_g(c(0, s(s(s(s(0))))))
hC_in_g(c(0, s(s(s(s(s(0))))))) → hC_out_g(c(0, s(s(s(s(s(0)))))))
hC_in_g(c(0, s(s(s(s(s(s(0)))))))) → hC_out_g(c(0, s(s(s(s(s(s(0))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(0))))))))) → hC_out_g(c(0, s(s(s(s(s(s(s(0)))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → U3_g(T54, gA_in_gg(s(s(s(s(s(s(s(0))))))), T54))
gA_in_gg(T59, 0) → gA_out_gg(T59, 0)
gA_in_gg(T65, s(T70)) → U1_gg(T65, T70, gA_in_gg(s(T65), T70))
U1_gg(T65, T70, gA_out_gg(s(T65), T70)) → gA_out_gg(T65, s(T70))
U3_g(T54, gA_out_gg(s(s(s(s(s(s(s(0))))))), T54)) → hC_out_g(c(0, s(s(s(s(s(s(s(s(T54))))))))))
hC_in_g(c(s(T81), T77)) → U4_g(T81, T77, fB_in_gg(T81, T77))
fB_in_gg(0, T86) → fB_out_gg(0, T86)
fB_in_gg(s(T97), T93) → U2_gg(T97, T93, fB_in_gg(T97, s(T93)))
U2_gg(T97, T93, fB_out_gg(T97, s(T93))) → fB_out_gg(s(T97), T93)
U4_g(T81, T77, fB_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))
U4_g(T81, T77, fB_out_gg(T81, T77)) → U5_g(T81, T77, gA_in_gg(T81, T77))
U5_g(T81, T77, gA_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))

The argument filtering Pi contains the following mapping:
hC_in_g(x1)  =  hC_in_g(x1)
c(x1, x2)  =  c(x1, x2)
0  =  0
hC_out_g(x1)  =  hC_out_g
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
gA_in_gg(x1, x2)  =  gA_in_gg(x1, x2)
gA_out_gg(x1, x2)  =  gA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U4_g(x1, x2, x3)  =  U4_g(x1, x2, x3)
fB_in_gg(x1, x2)  =  fB_in_gg(x1, x2)
fB_out_gg(x1, x2)  =  fB_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U5_g(x1, x2, x3)  =  U5_g(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

hC_in_g(c(0, 0)) → hC_out_g(c(0, 0))
hC_in_g(c(0, s(0))) → hC_out_g(c(0, s(0)))
hC_in_g(c(0, s(s(0)))) → hC_out_g(c(0, s(s(0))))
hC_in_g(c(0, s(s(s(0))))) → hC_out_g(c(0, s(s(s(0)))))
hC_in_g(c(0, s(s(s(s(0)))))) → hC_out_g(c(0, s(s(s(s(0))))))
hC_in_g(c(0, s(s(s(s(s(0))))))) → hC_out_g(c(0, s(s(s(s(s(0)))))))
hC_in_g(c(0, s(s(s(s(s(s(0)))))))) → hC_out_g(c(0, s(s(s(s(s(s(0))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(0))))))))) → hC_out_g(c(0, s(s(s(s(s(s(s(0)))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → U3_g(T54, gA_in_gg(s(s(s(s(s(s(s(0))))))), T54))
gA_in_gg(T59, 0) → gA_out_gg(T59, 0)
gA_in_gg(T65, s(T70)) → U1_gg(T65, T70, gA_in_gg(s(T65), T70))
U1_gg(T65, T70, gA_out_gg(s(T65), T70)) → gA_out_gg(T65, s(T70))
U3_g(T54, gA_out_gg(s(s(s(s(s(s(s(0))))))), T54)) → hC_out_g(c(0, s(s(s(s(s(s(s(s(T54))))))))))
hC_in_g(c(s(T81), T77)) → U4_g(T81, T77, fB_in_gg(T81, T77))
fB_in_gg(0, T86) → fB_out_gg(0, T86)
fB_in_gg(s(T97), T93) → U2_gg(T97, T93, fB_in_gg(T97, s(T93)))
U2_gg(T97, T93, fB_out_gg(T97, s(T93))) → fB_out_gg(s(T97), T93)
U4_g(T81, T77, fB_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))
U4_g(T81, T77, fB_out_gg(T81, T77)) → U5_g(T81, T77, gA_in_gg(T81, T77))
U5_g(T81, T77, gA_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))

The argument filtering Pi contains the following mapping:
hC_in_g(x1)  =  hC_in_g(x1)
c(x1, x2)  =  c(x1, x2)
0  =  0
hC_out_g(x1)  =  hC_out_g
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
gA_in_gg(x1, x2)  =  gA_in_gg(x1, x2)
gA_out_gg(x1, x2)  =  gA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U4_g(x1, x2, x3)  =  U4_g(x1, x2, x3)
fB_in_gg(x1, x2)  =  fB_in_gg(x1, x2)
fB_out_gg(x1, x2)  =  fB_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U5_g(x1, x2, x3)  =  U5_g(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:

HC_IN_G(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → U3_G(T54, gA_in_gg(s(s(s(s(s(s(s(0))))))), T54))
HC_IN_G(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → GA_IN_GG(s(s(s(s(s(s(s(0))))))), T54)
GA_IN_GG(T65, s(T70)) → U1_GG(T65, T70, gA_in_gg(s(T65), T70))
GA_IN_GG(T65, s(T70)) → GA_IN_GG(s(T65), T70)
HC_IN_G(c(s(T81), T77)) → U4_G(T81, T77, fB_in_gg(T81, T77))
HC_IN_G(c(s(T81), T77)) → FB_IN_GG(T81, T77)
FB_IN_GG(s(T97), T93) → U2_GG(T97, T93, fB_in_gg(T97, s(T93)))
FB_IN_GG(s(T97), T93) → FB_IN_GG(T97, s(T93))
U4_G(T81, T77, fB_out_gg(T81, T77)) → U5_G(T81, T77, gA_in_gg(T81, T77))
U4_G(T81, T77, fB_out_gg(T81, T77)) → GA_IN_GG(T81, T77)

The TRS R consists of the following rules:

hC_in_g(c(0, 0)) → hC_out_g(c(0, 0))
hC_in_g(c(0, s(0))) → hC_out_g(c(0, s(0)))
hC_in_g(c(0, s(s(0)))) → hC_out_g(c(0, s(s(0))))
hC_in_g(c(0, s(s(s(0))))) → hC_out_g(c(0, s(s(s(0)))))
hC_in_g(c(0, s(s(s(s(0)))))) → hC_out_g(c(0, s(s(s(s(0))))))
hC_in_g(c(0, s(s(s(s(s(0))))))) → hC_out_g(c(0, s(s(s(s(s(0)))))))
hC_in_g(c(0, s(s(s(s(s(s(0)))))))) → hC_out_g(c(0, s(s(s(s(s(s(0))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(0))))))))) → hC_out_g(c(0, s(s(s(s(s(s(s(0)))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → U3_g(T54, gA_in_gg(s(s(s(s(s(s(s(0))))))), T54))
gA_in_gg(T59, 0) → gA_out_gg(T59, 0)
gA_in_gg(T65, s(T70)) → U1_gg(T65, T70, gA_in_gg(s(T65), T70))
U1_gg(T65, T70, gA_out_gg(s(T65), T70)) → gA_out_gg(T65, s(T70))
U3_g(T54, gA_out_gg(s(s(s(s(s(s(s(0))))))), T54)) → hC_out_g(c(0, s(s(s(s(s(s(s(s(T54))))))))))
hC_in_g(c(s(T81), T77)) → U4_g(T81, T77, fB_in_gg(T81, T77))
fB_in_gg(0, T86) → fB_out_gg(0, T86)
fB_in_gg(s(T97), T93) → U2_gg(T97, T93, fB_in_gg(T97, s(T93)))
U2_gg(T97, T93, fB_out_gg(T97, s(T93))) → fB_out_gg(s(T97), T93)
U4_g(T81, T77, fB_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))
U4_g(T81, T77, fB_out_gg(T81, T77)) → U5_g(T81, T77, gA_in_gg(T81, T77))
U5_g(T81, T77, gA_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))

The argument filtering Pi contains the following mapping:
hC_in_g(x1)  =  hC_in_g(x1)
c(x1, x2)  =  c(x1, x2)
0  =  0
hC_out_g(x1)  =  hC_out_g
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
gA_in_gg(x1, x2)  =  gA_in_gg(x1, x2)
gA_out_gg(x1, x2)  =  gA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U4_g(x1, x2, x3)  =  U4_g(x1, x2, x3)
fB_in_gg(x1, x2)  =  fB_in_gg(x1, x2)
fB_out_gg(x1, x2)  =  fB_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U5_g(x1, x2, x3)  =  U5_g(x3)
HC_IN_G(x1)  =  HC_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
GA_IN_GG(x1, x2)  =  GA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U4_G(x1, x2, x3)  =  U4_G(x1, x2, x3)
FB_IN_GG(x1, x2)  =  FB_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U5_G(x1, x2, x3)  =  U5_G(x3)

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

(6) Obligation:

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

HC_IN_G(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → U3_G(T54, gA_in_gg(s(s(s(s(s(s(s(0))))))), T54))
HC_IN_G(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → GA_IN_GG(s(s(s(s(s(s(s(0))))))), T54)
GA_IN_GG(T65, s(T70)) → U1_GG(T65, T70, gA_in_gg(s(T65), T70))
GA_IN_GG(T65, s(T70)) → GA_IN_GG(s(T65), T70)
HC_IN_G(c(s(T81), T77)) → U4_G(T81, T77, fB_in_gg(T81, T77))
HC_IN_G(c(s(T81), T77)) → FB_IN_GG(T81, T77)
FB_IN_GG(s(T97), T93) → U2_GG(T97, T93, fB_in_gg(T97, s(T93)))
FB_IN_GG(s(T97), T93) → FB_IN_GG(T97, s(T93))
U4_G(T81, T77, fB_out_gg(T81, T77)) → U5_G(T81, T77, gA_in_gg(T81, T77))
U4_G(T81, T77, fB_out_gg(T81, T77)) → GA_IN_GG(T81, T77)

The TRS R consists of the following rules:

hC_in_g(c(0, 0)) → hC_out_g(c(0, 0))
hC_in_g(c(0, s(0))) → hC_out_g(c(0, s(0)))
hC_in_g(c(0, s(s(0)))) → hC_out_g(c(0, s(s(0))))
hC_in_g(c(0, s(s(s(0))))) → hC_out_g(c(0, s(s(s(0)))))
hC_in_g(c(0, s(s(s(s(0)))))) → hC_out_g(c(0, s(s(s(s(0))))))
hC_in_g(c(0, s(s(s(s(s(0))))))) → hC_out_g(c(0, s(s(s(s(s(0)))))))
hC_in_g(c(0, s(s(s(s(s(s(0)))))))) → hC_out_g(c(0, s(s(s(s(s(s(0))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(0))))))))) → hC_out_g(c(0, s(s(s(s(s(s(s(0)))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → U3_g(T54, gA_in_gg(s(s(s(s(s(s(s(0))))))), T54))
gA_in_gg(T59, 0) → gA_out_gg(T59, 0)
gA_in_gg(T65, s(T70)) → U1_gg(T65, T70, gA_in_gg(s(T65), T70))
U1_gg(T65, T70, gA_out_gg(s(T65), T70)) → gA_out_gg(T65, s(T70))
U3_g(T54, gA_out_gg(s(s(s(s(s(s(s(0))))))), T54)) → hC_out_g(c(0, s(s(s(s(s(s(s(s(T54))))))))))
hC_in_g(c(s(T81), T77)) → U4_g(T81, T77, fB_in_gg(T81, T77))
fB_in_gg(0, T86) → fB_out_gg(0, T86)
fB_in_gg(s(T97), T93) → U2_gg(T97, T93, fB_in_gg(T97, s(T93)))
U2_gg(T97, T93, fB_out_gg(T97, s(T93))) → fB_out_gg(s(T97), T93)
U4_g(T81, T77, fB_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))
U4_g(T81, T77, fB_out_gg(T81, T77)) → U5_g(T81, T77, gA_in_gg(T81, T77))
U5_g(T81, T77, gA_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))

The argument filtering Pi contains the following mapping:
hC_in_g(x1)  =  hC_in_g(x1)
c(x1, x2)  =  c(x1, x2)
0  =  0
hC_out_g(x1)  =  hC_out_g
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
gA_in_gg(x1, x2)  =  gA_in_gg(x1, x2)
gA_out_gg(x1, x2)  =  gA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U4_g(x1, x2, x3)  =  U4_g(x1, x2, x3)
fB_in_gg(x1, x2)  =  fB_in_gg(x1, x2)
fB_out_gg(x1, x2)  =  fB_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U5_g(x1, x2, x3)  =  U5_g(x3)
HC_IN_G(x1)  =  HC_IN_G(x1)
U3_G(x1, x2)  =  U3_G(x2)
GA_IN_GG(x1, x2)  =  GA_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
U4_G(x1, x2, x3)  =  U4_G(x1, x2, x3)
FB_IN_GG(x1, x2)  =  FB_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x3)
U5_G(x1, x2, x3)  =  U5_G(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 8 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

FB_IN_GG(s(T97), T93) → FB_IN_GG(T97, s(T93))

The TRS R consists of the following rules:

hC_in_g(c(0, 0)) → hC_out_g(c(0, 0))
hC_in_g(c(0, s(0))) → hC_out_g(c(0, s(0)))
hC_in_g(c(0, s(s(0)))) → hC_out_g(c(0, s(s(0))))
hC_in_g(c(0, s(s(s(0))))) → hC_out_g(c(0, s(s(s(0)))))
hC_in_g(c(0, s(s(s(s(0)))))) → hC_out_g(c(0, s(s(s(s(0))))))
hC_in_g(c(0, s(s(s(s(s(0))))))) → hC_out_g(c(0, s(s(s(s(s(0)))))))
hC_in_g(c(0, s(s(s(s(s(s(0)))))))) → hC_out_g(c(0, s(s(s(s(s(s(0))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(0))))))))) → hC_out_g(c(0, s(s(s(s(s(s(s(0)))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → U3_g(T54, gA_in_gg(s(s(s(s(s(s(s(0))))))), T54))
gA_in_gg(T59, 0) → gA_out_gg(T59, 0)
gA_in_gg(T65, s(T70)) → U1_gg(T65, T70, gA_in_gg(s(T65), T70))
U1_gg(T65, T70, gA_out_gg(s(T65), T70)) → gA_out_gg(T65, s(T70))
U3_g(T54, gA_out_gg(s(s(s(s(s(s(s(0))))))), T54)) → hC_out_g(c(0, s(s(s(s(s(s(s(s(T54))))))))))
hC_in_g(c(s(T81), T77)) → U4_g(T81, T77, fB_in_gg(T81, T77))
fB_in_gg(0, T86) → fB_out_gg(0, T86)
fB_in_gg(s(T97), T93) → U2_gg(T97, T93, fB_in_gg(T97, s(T93)))
U2_gg(T97, T93, fB_out_gg(T97, s(T93))) → fB_out_gg(s(T97), T93)
U4_g(T81, T77, fB_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))
U4_g(T81, T77, fB_out_gg(T81, T77)) → U5_g(T81, T77, gA_in_gg(T81, T77))
U5_g(T81, T77, gA_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))

The argument filtering Pi contains the following mapping:
hC_in_g(x1)  =  hC_in_g(x1)
c(x1, x2)  =  c(x1, x2)
0  =  0
hC_out_g(x1)  =  hC_out_g
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
gA_in_gg(x1, x2)  =  gA_in_gg(x1, x2)
gA_out_gg(x1, x2)  =  gA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U4_g(x1, x2, x3)  =  U4_g(x1, x2, x3)
fB_in_gg(x1, x2)  =  fB_in_gg(x1, x2)
fB_out_gg(x1, x2)  =  fB_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U5_g(x1, x2, x3)  =  U5_g(x3)
FB_IN_GG(x1, x2)  =  FB_IN_GG(x1, x2)

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:

FB_IN_GG(s(T97), T93) → FB_IN_GG(T97, s(T93))

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:

FB_IN_GG(s(T97), T93) → FB_IN_GG(T97, s(T93))

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:

• FB_IN_GG(s(T97), T93) → FB_IN_GG(T97, s(T93))
  The graph contains the following edges 1 > 1

(15) YES

(16) Obligation:

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

GA_IN_GG(T65, s(T70)) → GA_IN_GG(s(T65), T70)

The TRS R consists of the following rules:

hC_in_g(c(0, 0)) → hC_out_g(c(0, 0))
hC_in_g(c(0, s(0))) → hC_out_g(c(0, s(0)))
hC_in_g(c(0, s(s(0)))) → hC_out_g(c(0, s(s(0))))
hC_in_g(c(0, s(s(s(0))))) → hC_out_g(c(0, s(s(s(0)))))
hC_in_g(c(0, s(s(s(s(0)))))) → hC_out_g(c(0, s(s(s(s(0))))))
hC_in_g(c(0, s(s(s(s(s(0))))))) → hC_out_g(c(0, s(s(s(s(s(0)))))))
hC_in_g(c(0, s(s(s(s(s(s(0)))))))) → hC_out_g(c(0, s(s(s(s(s(s(0))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(0))))))))) → hC_out_g(c(0, s(s(s(s(s(s(s(0)))))))))
hC_in_g(c(0, s(s(s(s(s(s(s(s(T54)))))))))) → U3_g(T54, gA_in_gg(s(s(s(s(s(s(s(0))))))), T54))
gA_in_gg(T59, 0) → gA_out_gg(T59, 0)
gA_in_gg(T65, s(T70)) → U1_gg(T65, T70, gA_in_gg(s(T65), T70))
U1_gg(T65, T70, gA_out_gg(s(T65), T70)) → gA_out_gg(T65, s(T70))
U3_g(T54, gA_out_gg(s(s(s(s(s(s(s(0))))))), T54)) → hC_out_g(c(0, s(s(s(s(s(s(s(s(T54))))))))))
hC_in_g(c(s(T81), T77)) → U4_g(T81, T77, fB_in_gg(T81, T77))
fB_in_gg(0, T86) → fB_out_gg(0, T86)
fB_in_gg(s(T97), T93) → U2_gg(T97, T93, fB_in_gg(T97, s(T93)))
U2_gg(T97, T93, fB_out_gg(T97, s(T93))) → fB_out_gg(s(T97), T93)
U4_g(T81, T77, fB_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))
U4_g(T81, T77, fB_out_gg(T81, T77)) → U5_g(T81, T77, gA_in_gg(T81, T77))
U5_g(T81, T77, gA_out_gg(T81, T77)) → hC_out_g(c(s(T81), T77))

The argument filtering Pi contains the following mapping:
hC_in_g(x1)  =  hC_in_g(x1)
c(x1, x2)  =  c(x1, x2)
0  =  0
hC_out_g(x1)  =  hC_out_g
s(x1)  =  s(x1)
U3_g(x1, x2)  =  U3_g(x2)
gA_in_gg(x1, x2)  =  gA_in_gg(x1, x2)
gA_out_gg(x1, x2)  =  gA_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
U4_g(x1, x2, x3)  =  U4_g(x1, x2, x3)
fB_in_gg(x1, x2)  =  fB_in_gg(x1, x2)
fB_out_gg(x1, x2)  =  fB_out_gg
U2_gg(x1, x2, x3)  =  U2_gg(x3)
U5_g(x1, x2, x3)  =  U5_g(x3)
GA_IN_GG(x1, x2)  =  GA_IN_GG(x1, x2)

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:

GA_IN_GG(T65, s(T70)) → GA_IN_GG(s(T65), T70)

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

(19) PiDPToQDPProof (EQUIVALENT 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:

GA_IN_GG(T65, s(T70)) → GA_IN_GG(s(T65), T70)

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:

• GA_IN_GG(T65, s(T70)) → GA_IN_GG(s(T65), T70)
  The graph contains the following edges 2 > 2

(22) YES