Left Termination proof of /tmp/tmpIrTf9b/fold3.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 83 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 0 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 PiDPToQDPProof (⇒, 11 ms)

↳8 QDP

↳9 QDPSizeChangeProof (⇔, 0 ms)

↳10 YES

(0) Obligation:

Clauses:

fold(X, [], Z) :- ','(!, eq(X, Z)).
fold(X, Y, Z) :- ','(head(Y, H), ','(tail(Y, T), ','(myop(X, H, V), fold(V, T, Z)))).
myop(a, b, a).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
eq(X, X).

Query: fold(g,g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="B: fold(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T6, T8)) | fold(T6, [], T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: fold(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX6 is free\nX7 is free\nfold(T1, T2, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T6, T8)\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T6, T8), SCOPE: 2, CLAUSE: 7\n\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18))))\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 3 | ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(head(T16, X17), ','(tail(T16, X18), ','(myop(T15, X17, X19), fold(X19, X18, T18)))), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT16 is ground\nX6 is free\nX7 is free\nX17 is free\nX18 is free\nX19 is free\nfold(T15, T16, T3) !=? fold(X6, [], X7)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18)))\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 5 | ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(tail(.(T23, T24), X18), ','(myop(T15, T23, X19), fold(X19, X18, T18))), SCOPE: 4, CLAUSE: 6\n\nT15 is ground\nT23 is ground\nT24 is ground\nX18 is free\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(myop(T15, T31, X19), fold(X19, T32, T18))\n\nT15 is ground\nT31 is ground\nT32 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(myop(T15, T31, X19), fold(X19, T32, T18)), SCOPE: 5, CLAUSE: 2\n\nT15 is ground\nT31 is ground\nT32 is ground\nX19 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="A: fold(a, T32, T18)\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: fold(a, T32, T18), SCOPE: 6, CLAUSE: 0 | fold(a, T32, T18), SCOPE: 6, CLAUSE: 1\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(!_6, eq(a, T38)) | fold(a, [], T18), SCOPE: 6, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: fold(a, T32, T18), SCOPE: 6, CLAUSE: 1\n\nT32 is ground\nX49 is free\nX50 is free\nfold(a, T32, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: eq(a, T38)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: eq(a, T38), SCOPE: 7, CLAUSE: 7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47))))\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 3 | ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(head(T45, X69), ','(tail(T45, X70), ','(myop(a, X69, X71), fold(X71, X70, T47)))), SCOPE: 8, CLAUSE: 4\n\nT45 is ground\nX49 is free\nX50 is free\nX69 is free\nX70 is free\nX71 is free\nfold(a, T45, T18) !=? fold(X49, [], X50)", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47)))\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 5 | ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 6\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(tail(.(T53, T54), X70), ','(myop(a, T53, X71), fold(X71, X70, T47))), SCOPE: 9, CLAUSE: 6\n\nT53 is ground\nT54 is ground\nX70 is free\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ','(myop(a, T61, X71), fold(X71, T62, T47))\n\nT61 is ground\nT62 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: ','(myop(a, T61, X71), fold(X71, T62, T47)), SCOPE: 10, CLAUSE: 2\n\nT61 is ground\nT62 is ground\nX71 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: fold(a, T62, T47)\n\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 40 [arrowhead = none ];
40 -> 2;

41 [label="EVAL with clause\nfold(X6, [], X7) :- ','(!_1, eq(X6, X7)).\nand substitutionT1 -> T6,\nX6 -> T6,\nT2 -> [],\nT3 -> T8,\nX7 -> T8,\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 41 [arrowhead = none ];
41 -> 3;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 42 [arrowhead = none ];
42 -> 4;

43 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 43 [arrowhead = none ];
43 -> 5;

44 [label="ONLY EVAL with clause\nfold(X14, X15, X16) :- ','(head(X15, X17), ','(tail(X15, X18), ','(myop(X14, X17, X19), fold(X19, X18, X16)))).\nand substitutionT1 -> T15,\nX14 -> T15,\nT2 -> T16,\nX15 -> T16,\nT3 -> T18,\nX16 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 44 [arrowhead = none ];
44 -> 10;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 45 [arrowhead = none ];
45 -> 6;

46 [label="EVAL with clause\neq(X10, X10).\nand substitutionT6 -> T11,\nX10 -> T11,\nT8 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 46 [arrowhead = none ];
46 -> 7;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 47 [arrowhead = none ];
47 -> 8;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 48 [arrowhead = none ];
48 -> 9;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 49 [arrowhead = none ];
49 -> 11;

50 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(T15, T16, T3), fold(X6, [], X7))", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 50 [arrowhead = none ];
50 -> 12;

51 [label="EVAL with clause\nhead(.(X26, X27), X26).\nand substitutionX26 -> T23,\nX27 -> T24,\nT16 -> .(T23, T24),\nX17 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 51 [arrowhead = none ];
51 -> 13;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 52 [arrowhead = none ];
52 -> 14;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 53 [arrowhead = none ];
53 -> 15;

54 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 54 [arrowhead = none ];
54 -> 16;

55 [label="ONLY EVAL with clause\ntail(.(X38, X39), X39).\nand substitutionT23 -> T31,\nX38 -> T31,\nT24 -> T32,\nX39 -> T32,\nX18 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 55 [arrowhead = none ];
55 -> 17;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="EVAL with clause\nmyop(a, b, a).\nand substitutionT15 -> a,\nT31 -> b,\nX19 -> a", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 57 [arrowhead = none ];
57 -> 19;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 58 [arrowhead = none ];
58 -> 20;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 59 [arrowhead = none ];
59 -> 21;

60 [label="EVAL with clause\nfold(X49, [], X50) :- ','(!_6, eq(X49, X50)).\nand substitutionX49 -> a,\nT32 -> [],\nT18 -> T38,\nX50 -> T38,\nT37 -> T38", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 60 [arrowhead = none ];
60 -> 22;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 61 [arrowhead = none ];
61 -> 23;

62 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 62 [arrowhead = none ];
62 -> 24;

63 [label="ONLY EVAL with clause\nfold(X66, X67, X68) :- ','(head(X67, X69), ','(tail(X67, X70), ','(myop(X66, X69, X71), fold(X71, X70, X68)))).\nand substitutionX66 -> a,\nT32 -> T45,\nX67 -> T45,\nT18 -> T47,\nX68 -> T47,\nT46 -> T47", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 63 [arrowhead = none ];
63 -> 29;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 64 [arrowhead = none ];
64 -> 25;

65 [label="EVAL with clause\neq(X53, X53).\nand substitutionX53 -> a,\nT38 -> a", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 65 [arrowhead = none ];
65 -> 26;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 66 [arrowhead = none ];
66 -> 27;

67 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 67 [arrowhead = none ];
67 -> 28;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 68 [arrowhead = none ];
68 -> 30;

69 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (fold(a, T45, T18), fold(X49, [], X50))", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 69 [arrowhead = none ];
69 -> 31;

70 [label="EVAL with clause\nhead(.(X81, X82), X81).\nand substitutionX81 -> T53,\nX82 -> T54,\nT45 -> .(T53, T54),\nX69 -> T53", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 70 [arrowhead = none ];
70 -> 32;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 71 [arrowhead = none ];
71 -> 33;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 72 [arrowhead = none ];
72 -> 34;

73 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 73 [arrowhead = none ];
73 -> 35;

74 [label="ONLY EVAL with clause\ntail(.(X93, X94), X94).\nand substitutionT53 -> T61,\nX93 -> T61,\nT54 -> T62,\nX94 -> T62,\nX70 -> T62", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 74 [arrowhead = none ];
74 -> 36;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
36 -> 75 [arrowhead = none ];
75 -> 37;

76 [label="EVAL with clause\nmyop(a, b, a).\nand substitutionT61 -> b,\nX71 -> a", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 76 [arrowhead = none ];
76 -> 38;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 77 [arrowhead = none ];
77 -> 39;

78 [label="INSTANCE with matching:\nT32 -> T62\nT18 -> T47", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 78 [arrowhead = none , style=dashed];
78 -> 19[style=dashed];

}

(2) Obligation:

Triples:

foldA(.(b, X1), X2) :- foldA(X1, X2).
foldB(a, .(b, X1), X2) :- foldA(X1, X2).

Clauses:

foldcA([], a).
foldcA(.(b, X1), X2) :- foldcA(X1, X2).

Afs:

foldB(x1, x2, x3) = foldB(x1, x2)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
foldB_in: (b,b,f)
foldA_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

FOLDB_IN_GGA(a, .(b, X1), X2) → U2_GGA(X1, X2, foldA_in_ga(X1, X2))
FOLDB_IN_GGA(a, .(b, X1), X2) → FOLDA_IN_GA(X1, X2)
FOLDA_IN_GA(.(b, X1), X2) → U1_GA(X1, X2, foldA_in_ga(X1, X2))
FOLDA_IN_GA(.(b, X1), X2) → FOLDA_IN_GA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
a = a
.(x1, x2) = .(x1, x2)
b = b
foldA_in_ga(x1, x2) = foldA_in_ga(x1)
FOLDB_IN_GGA(x1, x2, x3) = FOLDB_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3) = U2_GGA(x1, x3)
FOLDA_IN_GA(x1, x2) = FOLDA_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

FOLDB_IN_GGA(a, .(b, X1), X2) → U2_GGA(X1, X2, foldA_in_ga(X1, X2))
FOLDB_IN_GGA(a, .(b, X1), X2) → FOLDA_IN_GA(X1, X2)
FOLDA_IN_GA(.(b, X1), X2) → U1_GA(X1, X2, foldA_in_ga(X1, X2))
FOLDA_IN_GA(.(b, X1), X2) → FOLDA_IN_GA(X1, X2)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 3 less nodes.

(6) Obligation:

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

FOLDA_IN_GA(.(b, X1), X2) → FOLDA_IN_GA(X1, X2)

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

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

FOLDA_IN_GA(.(b, X1)) → FOLDA_IN_GA(X1)

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

(9) 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:

FOLDA_IN_GA(.(b, X1)) → FOLDA_IN_GA(X1)
The graph contains the following edges 1 > 1