Left Termination proof of /tmp/tmpY8riKR/select1.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 95 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 12 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 28 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

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

Query: select(g,g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: select(T1, T2, T3)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: select(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true | select(T6, .(T6, T7), T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: select(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nT2 is ground\nX3 is free\nX4 is free\nselect(T1, T2, T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: select(T6, .(T6, T7), T3), SCOPE: 1, CLAUSE: 1\n\nT6 is ground\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: select(T11, T12, T14)\n\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: select(T11, T12, T14), SCOPE: 2, CLAUSE: 0 | select(T11, T12, T14), SCOPE: 2, CLAUSE: 1\n\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: select(T11, T12, T14), SCOPE: 2, CLAUSE: 0\n\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: select(T11, T12, T14), SCOPE: 2, CLAUSE: 1\n\nT11 is ground\nT12 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: select(T33, T35, T37)\n\nT33 is ground\nT35 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: select(T46, T48, T50)\n\nT46 is ground\nT47 is ground\nT48 is ground\nX3 is free\nX4 is free\nselect(T46, .(T47, T48), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: select(T46, T48, T50), SCOPE: 3, CLAUSE: 0 | select(T46, T48, T50), SCOPE: 3, CLAUSE: 1\n\nT46 is ground\nT47 is ground\nT48 is ground\nX3 is free\nX4 is free\nselect(T46, .(T47, T48), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: select(T46, T48, T50), SCOPE: 3, CLAUSE: 0\n\nT46 is ground\nT47 is ground\nT48 is ground\nX3 is free\nX4 is free\nselect(T46, .(T47, T48), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: select(T46, T48, T50), SCOPE: 3, CLAUSE: 1\n\nT46 is ground\nT47 is ground\nT48 is ground\nX3 is free\nX4 is free\nselect(T46, .(T47, T48), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: select(T69, T71, T73)\n\nT47 is ground\nT69 is ground\nT70 is ground\nT71 is ground\nX3 is free\nX4 is free\nselect(T69, .(T47, .(T70, T71)), T3) !=? select(X3, .(X3, X4), X4)", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 60 [arrowhead = none ];
60 -> 6;

61 [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];
6 -> 61 [arrowhead = none ];
61 -> 7;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 62 [arrowhead = none ];
62 -> 10;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="EVAL with clause\nselect(X43, .(X44, X45), .(X44, X46)) :- select(X43, X45, X46).\nand substitutionT1 -> T46,\nX43 -> T46,\nX44 -> T47,\nX45 -> T48,\nT2 -> .(T47, T48),\nX46 -> T50,\nT3 -> .(T47, T50),\nT49 -> T50", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 64 [arrowhead = none ];
64 -> 40;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 65 [arrowhead = none ];
65 -> 41;

66 [label="EVAL with clause\nselect(X9, .(X10, X11), .(X10, X12)) :- select(X9, X11, X12).\nand substitutionT6 -> T11,\nX9 -> T11,\nX10 -> T11,\nT7 -> T12,\nX11 -> T12,\nX12 -> T14,\nT3 -> .(T11, T14),\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 66 [arrowhead = none ];
66 -> 18;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 67 [arrowhead = none ];
67 -> 20;

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 68 [arrowhead = none ];
68 -> 22;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 69 [arrowhead = none ];
69 -> 26;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
22 -> 70 [arrowhead = none ];
70 -> 27;

71 [label="EVAL with clause\nselect(X21, .(X21, X22), X22).\nand substitutionT11 -> T23,\nX21 -> T23,\nX22 -> T24,\nT12 -> .(T23, T24),\nT14 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 71 [arrowhead = none ];
71 -> 30;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 72 [arrowhead = none ];
72 -> 31;

73 [label="EVAL with clause\nselect(X31, .(X32, X33), .(X32, X34)) :- select(X31, X33, X34).\nand substitutionT11 -> T33,\nX31 -> T33,\nX32 -> T34,\nX33 -> T35,\nT12 -> .(T34, T35),\nX34 -> T37,\nT14 -> .(T34, T37),\nT36 -> T37", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 73 [arrowhead = none ];
73 -> 36;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 74 [arrowhead = none ];
74 -> 37;

75 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 75 [arrowhead = none ];
75 -> 32;

76 [label="INSTANCE with matching:\nT1 -> T33\nT2 -> T35\nT3 -> T37", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 76 [arrowhead = none , style=dashed];
76 -> 2[style=dashed];

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 77 [arrowhead = none ];
77 -> 45;

78 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 78 [arrowhead = none ];
78 -> 48;

79 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
45 -> 79 [arrowhead = none ];
79 -> 49;

80 [label="EVAL with clause\nselect(X55, .(X55, X56), X56).\nand substitutionT46 -> T59,\nX55 -> T59,\nX56 -> T60,\nT48 -> .(T59, T60),\nT50 -> T60", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 80 [arrowhead = none ];
80 -> 51;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 81 [arrowhead = none ];
81 -> 54;

82 [label="EVAL with clause\nselect(X65, .(X66, X67), .(X66, X68)) :- select(X65, X67, X68).\nand substitutionT46 -> T69,\nX65 -> T69,\nX66 -> T70,\nX67 -> T71,\nT48 -> .(T70, T71),\nX68 -> T73,\nT50 -> .(T70, T73),\nT72 -> T73", fontsize=14, style = filled, fillcolor = lightblue];
49 -> 82 [arrowhead = none ];
82 -> 58;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 83 [arrowhead = none ];
83 -> 59;

84 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 84 [arrowhead = none ];
84 -> 55;

85 [label="INSTANCE with matching:\nT1 -> T69\nT2 -> T71\nT3 -> T73", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 85 [arrowhead = none , style=dashed];
85 -> 2[style=dashed];

}

(2) Obligation:

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

selectA_in_gga(T6, .(T6, T7), T7) → selectA_out_gga(T6, .(T6, T7), T7)
selectA_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → selectA_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
selectA_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, selectA_in_gga(T33, T35, T37))
selectA_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → selectA_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
selectA_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, selectA_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, selectA_out_gga(T69, T71, T73)) → selectA_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, selectA_out_gga(T33, T35, T37)) → selectA_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

(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:

SELECTA_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_GGA(T33, T34, T35, T37, selectA_in_gga(T33, T35, T37))
SELECTA_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → SELECTA_IN_GGA(T33, T35, T37)
SELECTA_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_GGA(T69, T47, T70, T71, T73, selectA_in_gga(T69, T71, T73))
SELECTA_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → SELECTA_IN_GGA(T69, T71, T73)

The TRS R consists of the following rules:

selectA_in_gga(T6, .(T6, T7), T7) → selectA_out_gga(T6, .(T6, T7), T7)
selectA_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → selectA_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
selectA_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, selectA_in_gga(T33, T35, T37))
selectA_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → selectA_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
selectA_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, selectA_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, selectA_out_gga(T69, T71, T73)) → selectA_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, selectA_out_gga(T33, T35, T37)) → selectA_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

The argument filtering Pi contains the following mapping:
selectA_in_gga(x1, x2, x3) = selectA_in_gga(x1, x2)
.(x1, x2) = .(x1, x2)
selectA_out_gga(x1, x2, x3) = selectA_out_gga(x1, x2, x3)
U1_gga(x1, x2, x3, x4, x5) = U1_gga(x1, x2, x3, x5)
U2_gga(x1, x2, x3, x4, x5, x6) = U2_gga(x1, x2, x3, x4, x6)
SELECTA_IN_GGA(x1, x2, x3) = SELECTA_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3, x4, x5) = U1_GGA(x1, x2, x3, x5)
U2_GGA(x1, x2, x3, x4, x5, x6) = U2_GGA(x1, x2, x3, x4, x6)

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

(4) Obligation:

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

SELECTA_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_GGA(T33, T34, T35, T37, selectA_in_gga(T33, T35, T37))
SELECTA_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → SELECTA_IN_GGA(T33, T35, T37)
SELECTA_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_GGA(T69, T47, T70, T71, T73, selectA_in_gga(T69, T71, T73))
SELECTA_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → SELECTA_IN_GGA(T69, T71, T73)

The TRS R consists of the following rules:

selectA_in_gga(T6, .(T6, T7), T7) → selectA_out_gga(T6, .(T6, T7), T7)
selectA_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → selectA_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
selectA_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, selectA_in_gga(T33, T35, T37))
selectA_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → selectA_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
selectA_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, selectA_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, selectA_out_gga(T69, T71, T73)) → selectA_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, selectA_out_gga(T33, T35, T37)) → selectA_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

SELECTA_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → SELECTA_IN_GGA(T69, T71, T73)
SELECTA_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → SELECTA_IN_GGA(T33, T35, T37)

The TRS R consists of the following rules:

selectA_in_gga(T6, .(T6, T7), T7) → selectA_out_gga(T6, .(T6, T7), T7)
selectA_in_gga(T23, .(T23, .(T23, T24)), .(T23, T24)) → selectA_out_gga(T23, .(T23, .(T23, T24)), .(T23, T24))
selectA_in_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → U1_gga(T33, T34, T35, T37, selectA_in_gga(T33, T35, T37))
selectA_in_gga(T59, .(T47, .(T59, T60)), .(T47, T60)) → selectA_out_gga(T59, .(T47, .(T59, T60)), .(T47, T60))
selectA_in_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → U2_gga(T69, T47, T70, T71, T73, selectA_in_gga(T69, T71, T73))
U2_gga(T69, T47, T70, T71, T73, selectA_out_gga(T69, T71, T73)) → selectA_out_gga(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73)))
U1_gga(T33, T34, T35, T37, selectA_out_gga(T33, T35, T37)) → selectA_out_gga(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37)))

(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:

SELECTA_IN_GGA(T69, .(T47, .(T70, T71)), .(T47, .(T70, T73))) → SELECTA_IN_GGA(T69, T71, T73)
SELECTA_IN_GGA(T33, .(T33, .(T34, T35)), .(T33, .(T34, T37))) → SELECTA_IN_GGA(T33, T35, T37)

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

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:

SELECTA_IN_GGA(T69, .(T47, .(T70, T71))) → SELECTA_IN_GGA(T69, T71)
SELECTA_IN_GGA(T33, .(T33, .(T34, T35))) → SELECTA_IN_GGA(T33, T35)

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:

SELECTA_IN_GGA(T69, .(T47, .(T70, T71))) → SELECTA_IN_GGA(T69, T71)
The graph contains the following edges 1 >= 1, 2 > 2
SELECTA_IN_GGA(T33, .(T33, .(T34, T35))) → SELECTA_IN_GGA(T33, T35)
The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2

(0) Obligation:

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) PiDPToQDPProof (SOUND transformation)

(10) Obligation:

(11) QDPSizeChangeProof (EQUIVALENT transformation)

(12) YES