Left Termination proof of /tmp/tmp5gX9Si/append.pl

Left Termination of the query pattern
append3(a,a,g)
w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 54 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 7 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 24 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

append1([], X, X).
append1(.(X, Y), U, .(X, Z)) :- append1(Y, U, Z).
append2([], X, X).
append2(.(X, Y), U, .(X, Z)) :- append2(Y, U, Z).
append3([], X, X).
append3(.(X, Y), U, .(X, Z)) :- append3(Y, U, Z).

Query: append3(a,a,g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: append3(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: append3(T1, T2, T3), SCOPE: 1, CLAUSE: 4 | append3(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | append3(T1, T2, T5), SCOPE: 1, CLAUSE: 5\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: append3(T1, T2, T3), SCOPE: 1, CLAUSE: 5\n\nT3 is ground\nX2 is free\nappend3(T1, T2, T3) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: append3(T1, T2, T5), SCOPE: 1, CLAUSE: 5\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: append3(T14, T15, T13)\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: append3(T14, T15, T13), SCOPE: 2, CLAUSE: 4 | append3(T14, T15, T13), SCOPE: 2, CLAUSE: 5\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: append3(T14, T15, T13), SCOPE: 2, CLAUSE: 4\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: append3(T14, T15, T13), SCOPE: 2, CLAUSE: 5\n\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: append3(T33, T34, T32)\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: append3(T45, T46, T44)\n\nT41 is ground\nT44 is ground\nX2 is free\nappend3(T1, T2, .(T41, T44)) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: append3(T45, T46, T44), SCOPE: 3, CLAUSE: 4 | append3(T45, T46, T44), SCOPE: 3, CLAUSE: 5\n\nT41 is ground\nT44 is ground\nX2 is free\nappend3(T1, T2, .(T41, T44)) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: append3(T45, T46, T44), SCOPE: 3, CLAUSE: 4\n\nT41 is ground\nT44 is ground\nX2 is free\nappend3(T1, T2, .(T41, T44)) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: append3(T45, T46, T44), SCOPE: 3, CLAUSE: 5\n\nT41 is ground\nT44 is ground\nX2 is free\nappend3(T1, T2, .(T41, T44)) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: append3(T64, T65, T63)\n\nT41 is ground\nT60 is ground\nT63 is ground\nX2 is free\nappend3(T1, T2, .(T41, .(T60, T63))) !=? append3([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 51 [arrowhead = none ];
51 -> 6;

52 [label="EVAL with clause\nappend3([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 52 [arrowhead = none ];
52 -> 10;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 53 [arrowhead = none ];
53 -> 14;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 54 [arrowhead = none ];
54 -> 15;

55 [label="EVAL with clause\nappend3(.(X34, X35), X36, .(X34, X37)) :- append3(X35, X36, X37).\nand substitutionX34 -> T41,\nX35 -> T45,\nT1 -> .(T41, T45),\nT2 -> T46,\nX36 -> T46,\nX37 -> T44,\nT3 -> .(T41, T44),\nT42 -> T45,\nT43 -> T46", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 55 [arrowhead = none ];
55 -> 40;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 56 [arrowhead = none ];
56 -> 41;

57 [label="EVAL with clause\nappend3(.(X7, X8), X9, .(X7, X10)) :- append3(X8, X9, X10).\nand substitutionX7 -> T10,\nX8 -> T14,\nT1 -> .(T10, T14),\nT2 -> T15,\nX9 -> T15,\nX10 -> T13,\nT5 -> .(T10, T13),\nT11 -> T14,\nT12 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 57 [arrowhead = none ];
57 -> 24;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 58 [arrowhead = none ];
58 -> 29;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 59 [arrowhead = none ];
59 -> 30;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 60 [arrowhead = none ];
60 -> 31;

61 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
30 -> 61 [arrowhead = none ];
61 -> 32;

62 [label="EVAL with clause\nappend3([], X15, X15).\nand substitutionT14 -> [],\nT15 -> T20,\nX15 -> T20,\nT13 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 62 [arrowhead = none ];
62 -> 33;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 63 [arrowhead = none ];
63 -> 34;

64 [label="EVAL with clause\nappend3(.(X24, X25), X26, .(X24, X27)) :- append3(X25, X26, X27).\nand substitutionX24 -> T29,\nX25 -> T33,\nT14 -> .(T29, T33),\nT15 -> T34,\nX26 -> T34,\nX27 -> T32,\nT13 -> .(T29, T32),\nT30 -> T33,\nT31 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 64 [arrowhead = none ];
64 -> 38;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 65 [arrowhead = none ];
65 -> 39;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 66 [arrowhead = none ];
66 -> 35;

67 [label="INSTANCE with matching:\nT1 -> T33\nT2 -> T34\nT3 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 67 [arrowhead = none , style=dashed];
67 -> 2[style=dashed];

68 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
40 -> 68 [arrowhead = none ];
68 -> 42;

69 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 69 [arrowhead = none ];
69 -> 43;

70 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
42 -> 70 [arrowhead = none ];
70 -> 44;

71 [label="EVAL with clause\nappend3([], X42, X42).\nand substitutionT45 -> [],\nT46 -> T51,\nX42 -> T51,\nT44 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
43 -> 71 [arrowhead = none ];
71 -> 45;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 72 [arrowhead = none ];
72 -> 46;

73 [label="EVAL with clause\nappend3(.(X51, X52), X53, .(X51, X54)) :- append3(X52, X53, X54).\nand substitutionX51 -> T60,\nX52 -> T64,\nT45 -> .(T60, T64),\nT46 -> T65,\nX53 -> T65,\nX54 -> T63,\nT44 -> .(T60, T63),\nT61 -> T64,\nT62 -> T65", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 73 [arrowhead = none ];
73 -> 49;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 74 [arrowhead = none ];
74 -> 50;

75 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
45 -> 75 [arrowhead = none ];
75 -> 47;

76 [label="INSTANCE with matching:\nT1 -> T64\nT2 -> T65\nT3 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
49 -> 76 [arrowhead = none , style=dashed];
76 -> 2[style=dashed];

}

(2) Obligation:

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

append3A_in_aag([], T5, T5) → append3A_out_aag([], T5, T5)
append3A_in_aag(.(T10, []), T20, .(T10, T20)) → append3A_out_aag(.(T10, []), T20, .(T10, T20))
append3A_in_aag(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → U1_aag(T10, T29, T33, T34, T32, append3A_in_aag(T33, T34, T32))
append3A_in_aag(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) → U2_aag(T41, T60, T64, T65, T63, append3A_in_aag(T64, T65, T63))
U2_aag(T41, T60, T64, T65, T63, append3A_out_aag(T64, T65, T63)) → append3A_out_aag(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63)))
U1_aag(T10, T29, T33, T34, T32, append3A_out_aag(T33, T34, T32)) → append3A_out_aag(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32)))

(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:

APPEND3A_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → U1_AAG(T10, T29, T33, T34, T32, append3A_in_aag(T33, T34, T32))
APPEND3A_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → APPEND3A_IN_AAG(T33, T34, T32)
APPEND3A_IN_AAG(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) → U2_AAG(T41, T60, T64, T65, T63, append3A_in_aag(T64, T65, T63))

The TRS R consists of the following rules:

append3A_in_aag([], T5, T5) → append3A_out_aag([], T5, T5)
append3A_in_aag(.(T10, []), T20, .(T10, T20)) → append3A_out_aag(.(T10, []), T20, .(T10, T20))
append3A_in_aag(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → U1_aag(T10, T29, T33, T34, T32, append3A_in_aag(T33, T34, T32))
append3A_in_aag(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) → U2_aag(T41, T60, T64, T65, T63, append3A_in_aag(T64, T65, T63))
U2_aag(T41, T60, T64, T65, T63, append3A_out_aag(T64, T65, T63)) → append3A_out_aag(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63)))
U1_aag(T10, T29, T33, T34, T32, append3A_out_aag(T33, T34, T32)) → append3A_out_aag(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32)))

The argument filtering Pi contains the following mapping:
append3A_in_aag(x1, x2, x3) = append3A_in_aag(x3)
append3A_out_aag(x1, x2, x3) = append3A_out_aag(x1, x2, x3)
.(x1, x2) = .(x1, x2)
U1_aag(x1, x2, x3, x4, x5, x6) = U1_aag(x1, x2, x5, x6)
U2_aag(x1, x2, x3, x4, x5, x6) = U2_aag(x1, x2, x5, x6)
APPEND3A_IN_AAG(x1, x2, x3) = APPEND3A_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5, x6) = U1_AAG(x1, x2, x5, x6)
U2_AAG(x1, x2, x3, x4, x5, x6) = U2_AAG(x1, x2, x5, x6)

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

(4) Obligation:

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

APPEND3A_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → U1_AAG(T10, T29, T33, T34, T32, append3A_in_aag(T33, T34, T32))
APPEND3A_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → APPEND3A_IN_AAG(T33, T34, T32)
APPEND3A_IN_AAG(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) → U2_AAG(T41, T60, T64, T65, T63, append3A_in_aag(T64, T65, T63))

The TRS R consists of the following rules:

append3A_in_aag([], T5, T5) → append3A_out_aag([], T5, T5)
append3A_in_aag(.(T10, []), T20, .(T10, T20)) → append3A_out_aag(.(T10, []), T20, .(T10, T20))
append3A_in_aag(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → U1_aag(T10, T29, T33, T34, T32, append3A_in_aag(T33, T34, T32))
append3A_in_aag(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) → U2_aag(T41, T60, T64, T65, T63, append3A_in_aag(T64, T65, T63))
U2_aag(T41, T60, T64, T65, T63, append3A_out_aag(T64, T65, T63)) → append3A_out_aag(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63)))
U1_aag(T10, T29, T33, T34, T32, append3A_out_aag(T33, T34, T32)) → append3A_out_aag(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32)))

(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:

APPEND3A_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → APPEND3A_IN_AAG(T33, T34, T32)

The TRS R consists of the following rules:

append3A_in_aag([], T5, T5) → append3A_out_aag([], T5, T5)
append3A_in_aag(.(T10, []), T20, .(T10, T20)) → append3A_out_aag(.(T10, []), T20, .(T10, T20))
append3A_in_aag(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → U1_aag(T10, T29, T33, T34, T32, append3A_in_aag(T33, T34, T32))
append3A_in_aag(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63))) → U2_aag(T41, T60, T64, T65, T63, append3A_in_aag(T64, T65, T63))
U2_aag(T41, T60, T64, T65, T63, append3A_out_aag(T64, T65, T63)) → append3A_out_aag(.(T41, .(T60, T64)), T65, .(T41, .(T60, T63)))
U1_aag(T10, T29, T33, T34, T32, append3A_out_aag(T33, T34, T32)) → append3A_out_aag(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32)))

(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:

APPEND3A_IN_AAG(.(T10, .(T29, T33)), T34, .(T10, .(T29, T32))) → APPEND3A_IN_AAG(T33, T34, T32)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
APPEND3A_IN_AAG(x1, x2, x3) = APPEND3A_IN_AAG(x3)

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:

APPEND3A_IN_AAG(.(T10, .(T29, T32))) → APPEND3A_IN_AAG(T32)

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:

APPEND3A_IN_AAG(.(T10, .(T29, T32))) → APPEND3A_IN_AAG(T32)
The graph contains the following edges 1 > 1

(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