Left Termination proof of /tmp/tmpT6CH0N/len2.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 0 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 3 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 0 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 YES

(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) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: len(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: len(T1, T2), SCOPE: 1, CLAUSE: 0 | len(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | len([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: len(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlen(T1, T2) !=? len([], 0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: len([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(no(empty([])), ','(tail([], X6), len(X6, T5)))\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(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];
9 [label="9: ','(','(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];
10 [label="10: ','(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];
11 [label="11: ','(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];
12 [label="12: ','(','(!_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];
13 [label="13: ','(failure(a), ','(tail([], X6), len(X6, T5)))\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(failure(a), ','(tail([], X6), len(X6, T5))), SCOPE: 5, CLAUSE: 7\n\nX6 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(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];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(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];
19 [label="19: ','(','(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];
20 [label="20: ','(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];
21 [label="21: ','(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];
22 [label="22: ','(','(!_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];
23 [label="23: ?_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];
24 [label="24: ','(failure(a), ','(tail([], X14), len(X14, T10)))\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(failure(a), ','(tail([], X14), len(X14, T10))), SCOPE: 9, CLAUSE: 7\n\nX14 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ?_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];
28 [label="28: ','(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];
29 [label="29: ','(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];
30 [label="30: ','(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];
31 [label="31: ','(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];
32 [label="32: len(T22, T10)\n\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 34 [arrowhead = none ];
34 -> 2;

35 [label="EVAL with clause\nlen([], 0).\nand substitutionT1 -> [],\nT2 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 35 [arrowhead = none ];
35 -> 3;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 36 [arrowhead = none ];
36 -> 4;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 37 [arrowhead = none ];
37 -> 5;

38 [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];
4 -> 38 [arrowhead = none ];
38 -> 16;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 39 [arrowhead = none ];
39 -> 17;

40 [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];
5 -> 40 [arrowhead = none ];
40 -> 6;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 41 [arrowhead = none ];
41 -> 7;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 42 [arrowhead = none ];
42 -> 8;

43 [label="ONLY EVAL with clause\nno(X9) :- ','(call(X9), ','(!_2, failure(a))).\nand substitutionX9 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 43 [arrowhead = none ];
43 -> 9;

44 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 44 [arrowhead = none ];
44 -> 10;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 45 [arrowhead = none ];
45 -> 11;

46 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 46 [arrowhead = none ];
46 -> 12;

47 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 47 [arrowhead = none ];
47 -> 13;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 48 [arrowhead = none ];
48 -> 14;

49 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 49 [arrowhead = none ];
49 -> 15;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 50 [arrowhead = none ];
50 -> 18;

51 [label="ONLY EVAL with clause\nno(X17) :- ','(call(X17), ','(!_6, failure(a))).\nand substitutionT8 -> T13,\nX17 -> empty(T13)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 51 [arrowhead = none ];
51 -> 19;

52 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 52 [arrowhead = none ];
52 -> 20;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 53 [arrowhead = none ];
53 -> 21;

54 [label="EVAL with clause\nempty([]).\nand substitutionT13 -> []", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 54 [arrowhead = none ];
54 -> 22;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 55 [arrowhead = none ];
55 -> 23;

56 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 56 [arrowhead = none ];
56 -> 24;

57 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 57 [arrowhead = none ];
57 -> 27;

58 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 58 [arrowhead = none ];
58 -> 25;

59 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 59 [arrowhead = none ];
59 -> 26;

60 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 60 [arrowhead = none ];
60 -> 28;

61 [label="ONLY EVAL with clause\nno(X20).\nand substitutionT13 -> T16,\nX20 -> empty(T16)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 61 [arrowhead = none ];
61 -> 29;

62 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 62 [arrowhead = none ];
62 -> 30;

63 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (empty(T16), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 63 [arrowhead = none ];
63 -> 31;

64 [label="EVAL with clause\ntail(.(X25, X26), X26).\nand substitutionX25 -> T21,\nX26 -> T22,\nT16 -> .(T21, T22),\nX14 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 64 [arrowhead = none ];
64 -> 32;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 65 [arrowhead = none ];
65 -> 33;

66 [label="INSTANCE with matching:\nT1 -> T22\nT2 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 66 [arrowhead = none , style=dashed];
66 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

lenA([], 0).
lenA(.(T21, T22), s(T10)) :- lenA(T22, T10).

Query: lenA(g,a)

(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:
lenA_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
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))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) 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))

(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:

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(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x4)
s(x1) = s(x1)
LENA_IN_GA(x1, x2) = LENA_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x4)

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

(6) 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))

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(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)

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))

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) 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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) 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.

(13) 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

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES