Left Termination proof of /tmp/tmpyXTk5B/select.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 89 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 0 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 3 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 YES

(0) Obligation:

Clauses:

select(X, .(X, Xs), Xs).
select(X, .(Y, Xs), .(Y, Zs)) :- select(X, Xs, Zs).

Query: select(a,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: select(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | select(T1, .(T6, T7), T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT2 is ground\nX3 is free\nX4 is free\nselect(T1, T2, T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: select(T1, .(T6, T7), T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: select(T16, T14, T17)\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: select(T16, T14, T17), SCOPE: 2, CLAUSE: 0 | select(T16, T14, T17), SCOPE: 2, CLAUSE: 1\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: select(T16, T14, T17), SCOPE: 2, CLAUSE: 0\n\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: select(T16, T14, T17), SCOPE: 2, CLAUSE: 1\n\nT14 is ground", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: select(T40, T38, T41)\n\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: select(T54, T52, T55)\n\nT51 is ground\nT52 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, T52), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: select(T54, T52, T55), SCOPE: 3, CLAUSE: 0 | select(T54, T52, T55), SCOPE: 3, CLAUSE: 1\n\nT51 is ground\nT52 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, T52), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: select(T54, T52, T55), SCOPE: 3, CLAUSE: 0\n\nT51 is ground\nT52 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, T52), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: select(T54, T52, T55), SCOPE: 3, CLAUSE: 1\n\nT51 is ground\nT52 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, T52), T3) !=? select(X3, .(X3, X4), X4)", 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: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: select(T78, T76, T79)\n\nT51 is ground\nT75 is ground\nT76 is ground\nX3 is free\nX4 is free\nselect(T1, .(T51, .(T75, T76)), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 26 [arrowhead = none ];
26 -> 2;

27 [label="EVAL with clause\nselect(X3, .(X3, X4), X4).\nand substitutionT1 -> T6,\nX3 -> T6,\nX4 -> T7,\nT2 -> .(T6, T7),\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 27 [arrowhead = none ];
27 -> 3;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 28 [arrowhead = none ];
28 -> 4;

29 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 29 [arrowhead = none ];
29 -> 5;

30 [label="EVAL with clause\nselect(X43, .(X44, X45), .(X44, X46)) :- select(X43, X45, X46).\nand substitutionT1 -> T54,\nX43 -> T54,\nX44 -> T51,\nX45 -> T52,\nT2 -> .(T51, T52),\nX46 -> T55,\nT3 -> .(T51, T55),\nT50 -> T54,\nT53 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 30 [arrowhead = none ];
30 -> 16;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 31 [arrowhead = none ];
31 -> 17;

32 [label="EVAL with clause\nselect(X9, .(X10, X11), .(X10, X12)) :- select(X9, X11, X12).\nand substitutionT1 -> T16,\nX9 -> T16,\nT6 -> T13,\nX10 -> T13,\nT7 -> T14,\nX11 -> T14,\nX12 -> T17,\nT3 -> .(T13, T17),\nT12 -> T16,\nT15 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 33 [arrowhead = none ];
33 -> 7;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 34 [arrowhead = none ];
34 -> 8;

35 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 35 [arrowhead = none ];
35 -> 9;

36 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="EVAL with clause\nselect(X21, .(X21, X22), X22).\nand substitutionT16 -> T26,\nX21 -> T26,\nX22 -> T27,\nT14 -> .(T26, T27),\nT17 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="EVAL with clause\nselect(X31, .(X32, X33), .(X32, X34)) :- select(X31, X33, X34).\nand substitutionT16 -> T40,\nX31 -> T40,\nX32 -> T37,\nX33 -> T38,\nT14 -> .(T37, T38),\nX34 -> T41,\nT17 -> .(T37, T41),\nT36 -> T40,\nT39 -> T41", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 39 [arrowhead = none ];
39 -> 14;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 40 [arrowhead = none ];
40 -> 15;

41 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 41 [arrowhead = none ];
41 -> 13;

42 [label="INSTANCE with matching:\nT1 -> T40\nT2 -> T38\nT3 -> T41", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 42 [arrowhead = none , style=dashed];
42 -> 1[style=dashed];

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 43 [arrowhead = none ];
43 -> 18;

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 44 [arrowhead = none ];
44 -> 19;

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 45 [arrowhead = none ];
45 -> 20;

46 [label="EVAL with clause\nselect(X55, .(X55, X56), X56).\nand substitutionT54 -> T64,\nX55 -> T64,\nX56 -> T65,\nT52 -> .(T64, T65),\nT55 -> T65", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 46 [arrowhead = none ];
46 -> 21;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 47 [arrowhead = none ];
47 -> 22;

48 [label="EVAL with clause\nselect(X65, .(X66, X67), .(X66, X68)) :- select(X65, X67, X68).\nand substitutionT54 -> T78,\nX65 -> T78,\nX66 -> T75,\nX67 -> T76,\nT52 -> .(T75, T76),\nX68 -> T79,\nT55 -> .(T75, T79),\nT74 -> T78,\nT77 -> T79", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 48 [arrowhead = none ];
48 -> 24;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 49 [arrowhead = none ];
49 -> 25;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 50 [arrowhead = none ];
50 -> 23;

51 [label="INSTANCE with matching:\nT1 -> T78\nT2 -> T76\nT3 -> T79", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 51 [arrowhead = none , style=dashed];
51 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

selectA(T6, .(T6, T7), T7).
selectA(T26, .(T13, .(T26, T27)), .(T13, T27)).
selectA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) :- selectA(T40, T38, T41).
selectA(T64, .(T51, .(T64, T65)), .(T51, T65)).
selectA(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) :- selectA(T78, T76, T79).

Query: selectA(a,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:
selectA_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

(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:

SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_AGA(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → SELECTA_IN_AGA(T40, T38, T41)
SELECTA_IN_AGA(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_AGA(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))

The TRS R consists of the following rules:

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

The argument filtering Pi contains the following mapping:
selectA_in_aga(x1, x2, x3) = selectA_in_aga(x2)
.(x1, x2) = .(x1, x2)
selectA_out_aga(x1, x2, x3) = selectA_out_aga(x1, x3)
U1_aga(x1, x2, x3, x4, x5, x6) = U1_aga(x2, x3, x6)
U2_aga(x1, x2, x3, x4, x5, x6) = U2_aga(x2, x3, x6)
SELECTA_IN_AGA(x1, x2, x3) = SELECTA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4, x5, x6) = U1_AGA(x2, x3, x6)
U2_AGA(x1, x2, x3, x4, x5, x6) = U2_AGA(x2, x3, x6)

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

(6) Obligation:

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

SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_AGA(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → SELECTA_IN_AGA(T40, T38, T41)
SELECTA_IN_AGA(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_AGA(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))

The TRS R consists of the following rules:

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → SELECTA_IN_AGA(T40, T38, T41)

The TRS R consists of the following rules:

selectA_in_aga(T6, .(T6, T7), T7) → selectA_out_aga(T6, .(T6, T7), T7)
selectA_in_aga(T26, .(T13, .(T26, T27)), .(T13, T27)) → selectA_out_aga(T26, .(T13, .(T26, T27)), .(T13, T27))
selectA_in_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → U1_aga(T40, T13, T37, T38, T41, selectA_in_aga(T40, T38, T41))
selectA_in_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79))) → U2_aga(T78, T51, T75, T76, T79, selectA_in_aga(T78, T76, T79))
U2_aga(T78, T51, T75, T76, T79, selectA_out_aga(T78, T76, T79)) → selectA_out_aga(T78, .(T51, .(T75, T76)), .(T51, .(T75, T79)))
U1_aga(T40, T13, T37, T38, T41, selectA_out_aga(T40, T38, T41)) → selectA_out_aga(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41)))

(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:

SELECTA_IN_AGA(T40, .(T13, .(T37, T38)), .(T13, .(T37, T41))) → SELECTA_IN_AGA(T40, T38, T41)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
SELECTA_IN_AGA(x1, x2, x3) = SELECTA_IN_AGA(x2)

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:

SELECTA_IN_AGA(.(T13, .(T37, T38))) → SELECTA_IN_AGA(T38)

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:

SELECTA_IN_AGA(.(T13, .(T37, T38))) → SELECTA_IN_AGA(T38)
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