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

(0) Obligation:

Clauses:

len([], 0).
len(Xs, s(N)) :- ','(no(empty(Xs)), ','(tail(Xs, Ys), len(Ys, N))).
tail([], []).
tail(.(X, Xs), Xs).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X1).
failure(b).

Query: len(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: len(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: len(T1, T2), SCOPE: 1, CLAUSE: 0 | len(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: true | len([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: len(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlen(T1, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: len([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(no(empty([])), ','(tail([], X6), len(X6, T5)))\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 5 | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(','(call(empty([])), ','(!_2, failure(a))), ','(tail([], X6), len(X6, T5))) | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X6), len(X6, T5)))) | ?_3 | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(empty([]), ','(','(!_2, failure(a)), ','(tail([], X6), len(X6, T5)))), SCOPE: 4, CLAUSE: 4 | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(','(!_2, failure(a)), ','(tail([], X6), len(X6, T5))) | ?_4 | ?_3 | ','(no(empty([])), ','(tail([], X6), len(X6, T5))), SCOPE: 2, CLAUSE: 6\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(failure(a), ','(tail([], X6), len(X6, T5)))\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: ','(failure(a), ','(tail([], X6), len(X6, T5))), SCOPE: 5, CLAUSE: 7\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: ','(no(empty(T8)), ','(tail(T8, X14), len(X14, T10)))\n\nT8 is ground\nX14 is free\nlen(T8, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ','(no(empty(T8)), ','(tail(T8, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 5 | ','(no(empty(T8)), ','(tail(T8, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT8 is ground\nX14 is free\nlen(T8, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: ','(','(call(empty(T13)), ','(!_6, failure(a))), ','(tail(T13, X14), len(X14, T10))) | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: ','(empty(T13), ','(','(!_6, failure(a)), ','(tail(T13, X14), len(X14, T10)))) | ?_7 | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: ','(empty(T13), ','(','(!_6, failure(a)), ','(tail(T13, X14), len(X14, T10)))), SCOPE: 8, CLAUSE: 4 | ?_8 | ?_7 | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
71 [label="71: ','(','(!_6, failure(a)), ','(tail([], X14), len(X14, T10))) | ?_8 | ?_7 | ','(no(empty([])), ','(tail([], X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
74 [label="74: ?_8 | ?_7 | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
76 [label="76: ','(failure(a), ','(tail([], X14), len(X14, T10)))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
77 [label="77: ','(failure(a), ','(tail([], X14), len(X14, T10))), SCOPE: 9, CLAUSE: 7\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
79 [label="79: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
80 [label="80: ?_7 | ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
82 [label="82: ','(no(empty(T13)), ','(tail(T13, X14), len(X14, T10))), SCOPE: 6, CLAUSE: 6\n\nT13 is ground\nX14 is free\nlen(T13, T2) !=? len([], 0)\nempty(T13) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
85 [label="85: ','(tail(T16, X14), len(X14, T10))\n\nT16 is ground\nX14 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
86 [label="86: ','(tail(T16, X14), len(X14, T10)), SCOPE: 10, CLAUSE: 2 | ','(tail(T16, X14), len(X14, T10)), SCOPE: 10, CLAUSE: 3\n\nT16 is ground\nX14 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
88 [label="88: ','(tail(T16, X14), len(X14, T10)), SCOPE: 10, CLAUSE: 3\n\nT16 is ground\nX14 is free\nlen(T16, T2) !=? len([], 0)\nempty(T16) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
90 [label="90: len(T22, T10)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
91 [label="91: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
92 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 92 [arrowhead = none ];
92 -> 6;

93 [label="EVAL with clause\nlen([], 0).\nand substitutionT1 -> [],\nT2 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 93 [arrowhead = none ];
93 -> 9;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 94 [arrowhead = none ];
94 -> 14;

95 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 95 [arrowhead = none ];
95 -> 16;

96 [label="EVAL with clause\nlen(X12, s(X13)) :- ','(no(empty(X12)), ','(tail(X12, X14), len(X14, X13))).\nand substitutionT1 -> T8,\nX12 -> T8,\nX13 -> T10,\nT2 -> s(T10),\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 96 [arrowhead = none ];
96 -> 54;

97 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 97 [arrowhead = none ];
97 -> 55;

98 [label="EVAL with clause\nlen(X4, s(X5)) :- ','(no(empty(X4)), ','(tail(X4, X6), len(X6, X5))).\nand substitutionX4 -> [],\nX5 -> T5,\nT2 -> s(T5),\nT4 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 98 [arrowhead = none ];
98 -> 19;

99 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 99 [arrowhead = none ];
99 -> 21;

100 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 100 [arrowhead = none ];
100 -> 25;

101 [label="ONLY EVAL with clause\nno(X9) :- ','(call(X9), ','(!_2, failure(a))).\nand substitutionX9 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 101 [arrowhead = none ];
101 -> 28;

102 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 102 [arrowhead = none ];
102 -> 30;

103 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 103 [arrowhead = none ];
103 -> 34;

104 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 104 [arrowhead = none ];
104 -> 38;

105 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 105 [arrowhead = none ];
105 -> 40;

106 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 106 [arrowhead = none ];
106 -> 44;

107 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 107 [arrowhead = none ];
107 -> 46;

108 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
54 -> 108 [arrowhead = none ];
108 -> 56;

109 [label="ONLY EVAL with clause\nno(X17) :- ','(call(X17), ','(!_6, failure(a))).\nand substitutionT8 -> T13,\nX17 -> empty(T13)", fontsize=14, style = filled, fillcolor = lightblue];
56 -> 109 [arrowhead = none ];
109 -> 62;

110 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
62 -> 110 [arrowhead = none ];
110 -> 65;

111 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
65 -> 111 [arrowhead = none ];
111 -> 68;

112 [label="EVAL with clause\nempty([]).\nand substitutionT13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
68 -> 112 [arrowhead = none ];
112 -> 71;

113 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
68 -> 113 [arrowhead = none ];
113 -> 74;

114 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
71 -> 114 [arrowhead = none ];
114 -> 76;

115 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
74 -> 115 [arrowhead = none ];
115 -> 80;

116 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
76 -> 116 [arrowhead = none ];
116 -> 77;

117 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
77 -> 117 [arrowhead = none ];
117 -> 79;

118 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
80 -> 118 [arrowhead = none ];
118 -> 82;

119 [label="ONLY EVAL with clause\nno(X20).\nand substitutionT13 -> T16,\nX20 -> empty(T16)", fontsize=14, style = filled, fillcolor = lightblue];
82 -> 119 [arrowhead = none ];
119 -> 85;

120 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
85 -> 120 [arrowhead = none ];
120 -> 86;

121 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (empty(T16), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
86 -> 121 [arrowhead = none ];
121 -> 88;

122 [label="EVAL with clause\ntail(.(X25, X26), X26).\nand substitutionX25 -> T21,\nX26 -> T22,\nT16 -> .(T21, T22),\nX14 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
88 -> 122 [arrowhead = none ];
122 -> 90;

123 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
88 -> 123 [arrowhead = none ];
123 -> 91;

124 [label="INSTANCE with matching:\nT1 -> T22\nT2 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
90 -> 124 [arrowhead = none , style=dashed];
124 -> 2[style=dashed];

}

(2) Obligation:

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

lenA_in_ga([], 0) → lenA_out_ga([], 0)
lenA_in_ga(.(T21, T22), s(T10)) → U1_ga(T21, T22, T10, lenA_in_ga(T22, T10))
U1_ga(T21, T22, T10, lenA_out_ga(T22, T10)) → lenA_out_ga(.(T21, T22), s(T10))

The argument filtering Pi contains the following mapping:
lenA_in_ga(x1, x2)  =  lenA_in_ga(x1)
[]  =  []
lenA_out_ga(x1, x2)  =  lenA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)

(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:

LENA_IN_GA(.(T21, T22), s(T10)) → U1_GA(T21, T22, T10, lenA_in_ga(T22, T10))
LENA_IN_GA(.(T21, T22), s(T10)) → LENA_IN_GA(T22, T10)

The TRS R consists of the following rules:

lenA_in_ga([], 0) → lenA_out_ga([], 0)
lenA_in_ga(.(T21, T22), s(T10)) → U1_ga(T21, T22, T10, lenA_in_ga(T22, T10))
U1_ga(T21, T22, T10, lenA_out_ga(T22, T10)) → lenA_out_ga(.(T21, T22), s(T10))

The argument filtering Pi contains the following mapping:
lenA_in_ga(x1, x2)  =  lenA_in_ga(x1)
[]  =  []
lenA_out_ga(x1, x2)  =  lenA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)
LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(4) Obligation:

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

LENA_IN_GA(.(T21, T22), s(T10)) → U1_GA(T21, T22, T10, lenA_in_ga(T22, T10))
LENA_IN_GA(.(T21, T22), s(T10)) → LENA_IN_GA(T22, T10)

The TRS R consists of the following rules:

lenA_in_ga([], 0) → lenA_out_ga([], 0)
lenA_in_ga(.(T21, T22), s(T10)) → U1_ga(T21, T22, T10, lenA_in_ga(T22, T10))
U1_ga(T21, T22, T10, lenA_out_ga(T22, T10)) → lenA_out_ga(.(T21, T22), s(T10))

The argument filtering Pi contains the following mapping:
lenA_in_ga(x1, x2)  =  lenA_in_ga(x1)
[]  =  []
lenA_out_ga(x1, x2)  =  lenA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)
LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x1, x2, x4)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

LENA_IN_GA(.(T21, T22), s(T10)) → LENA_IN_GA(T22, T10)

The TRS R consists of the following rules:

lenA_in_ga([], 0) → lenA_out_ga([], 0)
lenA_in_ga(.(T21, T22), s(T10)) → U1_ga(T21, T22, T10, lenA_in_ga(T22, T10))
U1_ga(T21, T22, T10, lenA_out_ga(T22, T10)) → lenA_out_ga(.(T21, T22), s(T10))

The argument filtering Pi contains the following mapping:
lenA_in_ga(x1, x2)  =  lenA_in_ga(x1)
[]  =  []
lenA_out_ga(x1, x2)  =  lenA_out_ga(x1, x2)
.(x1, x2)  =  .(x1, x2)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x1, x2, x4)
s(x1)  =  s(x1)
LENA_IN_GA(x1, x2)  =  LENA_IN_GA(x1)

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:

LENA_IN_GA(.(T21, T22), s(T10)) → LENA_IN_GA(T22, T10)

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

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:

LENA_IN_GA(.(T21, T22)) → LENA_IN_GA(T22)

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:

• LENA_IN_GA(.(T21, T22)) → LENA_IN_GA(T22)
  The graph contains the following edges 1 > 1

(12) YES