Left Termination proof of /tmp/tmpu7DqbH/append-bff.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 48 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 3 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇒, 0 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

app([], X, X).
app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs).

Query: app(g,a,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="A: app(T1, T2, T3)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
6 [label="6: app(T1, T2, T3), SCOPE: 1, CLAUSE: 0 | app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: true | app([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: app(T1, T2, T3), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX2 is free\napp(T1, T2, T3) !=? app([], X2, X2)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app([], T2, T3), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: app(T11, T14, T15)\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T11, T14, T15), SCOPE: 2, CLAUSE: 0 | app(T11, T14, T15), SCOPE: 2, CLAUSE: 1\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: app(T11, T14, T15), SCOPE: 2, CLAUSE: 0\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(T11, T14, T15), SCOPE: 2, CLAUSE: 1\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: app(T30, T33, T34)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="EVAL with clause\napp([], X2, X2).\nand substitutionT1 -> [],\nT2 -> T5,\nX2 -> T5,\nT3 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 43 [arrowhead = none ];
43 -> 10;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 44 [arrowhead = none ];
44 -> 11;

45 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 45 [arrowhead = none ];
45 -> 17;

46 [label="EVAL with clause\napp(.(X11, X12), X13, .(X11, X14)) :- app(X12, X13, X14).\nand substitutionX11 -> T10,\nX12 -> T11,\nT1 -> .(T10, T11),\nT2 -> T14,\nX13 -> T14,\nX14 -> T15,\nT3 -> .(T10, T15),\nT12 -> T14,\nT13 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 46 [arrowhead = none ];
46 -> 30;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 47 [arrowhead = none ];
47 -> 31;

48 [label="BACKTRACK\nfor clause: app(.(X, Xs), Ys, .(X, Zs)) :- app(Xs, Ys, Zs)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 48 [arrowhead = none ];
48 -> 19;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 49 [arrowhead = none ];
49 -> 32;

50 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 50 [arrowhead = none ];
50 -> 33;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 51 [arrowhead = none ];
51 -> 34;

52 [label="EVAL with clause\napp([], X19, X19).\nand substitutionT11 -> [],\nT14 -> T20,\nX19 -> T20,\nT15 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 52 [arrowhead = none ];
52 -> 37;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 53 [arrowhead = none ];
53 -> 38;

54 [label="EVAL with clause\napp(.(X28, X29), X30, .(X28, X31)) :- app(X29, X30, X31).\nand substitutionX28 -> T29,\nX29 -> T30,\nT11 -> .(T29, T30),\nT14 -> T33,\nX30 -> T33,\nX31 -> T34,\nT15 -> .(T29, T34),\nT31 -> T33,\nT32 -> T34", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 54 [arrowhead = none ];
54 -> 40;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 55 [arrowhead = none ];
55 -> 41;

56 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 56 [arrowhead = none ];
56 -> 39;

57 [label="INSTANCE with matching:\nT1 -> T30\nT2 -> T33\nT3 -> T34", fontsize=14, style = filled, fillcolor = lightgrey];
40 -> 57 [arrowhead = none , style=dashed];
57 -> 2[style=dashed];

}

(2) Obligation:

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

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T10, []), T20, .(T10, T20)) → appA_out_gaa(.(T10, []), T20, .(T10, T20))
appA_in_gaa(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → U1_gaa(T10, T29, T30, T33, T34, appA_in_gaa(T30, T33, T34))
U1_gaa(T10, T29, T30, T33, T34, appA_out_gaa(T30, T33, T34)) → appA_out_gaa(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34)))

(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:

APPA_IN_GAA(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → U1_GAA(T10, T29, T30, T33, T34, appA_in_gaa(T30, T33, T34))
APPA_IN_GAA(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → APPA_IN_GAA(T30, T33, T34)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T10, []), T20, .(T10, T20)) → appA_out_gaa(.(T10, []), T20, .(T10, T20))
appA_in_gaa(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → U1_gaa(T10, T29, T30, T33, T34, appA_in_gaa(T30, T33, T34))
U1_gaa(T10, T29, T30, T33, T34, appA_out_gaa(T30, T33, T34)) → appA_out_gaa(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34)))

The argument filtering Pi contains the following mapping:
appA_in_gaa(x1, x2, x3) = appA_in_gaa(x1)
[] = []
appA_out_gaa(x1, x2, x3) = appA_out_gaa(x1)
.(x1, x2) = .(x1, x2)
U1_gaa(x1, x2, x3, x4, x5, x6) = U1_gaa(x1, x2, x3, x6)
APPA_IN_GAA(x1, x2, x3) = APPA_IN_GAA(x1)
U1_GAA(x1, x2, x3, x4, x5, x6) = U1_GAA(x1, x2, x3, x6)

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

(4) Obligation:

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

APPA_IN_GAA(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → U1_GAA(T10, T29, T30, T33, T34, appA_in_gaa(T30, T33, T34))
APPA_IN_GAA(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → APPA_IN_GAA(T30, T33, T34)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T10, []), T20, .(T10, T20)) → appA_out_gaa(.(T10, []), T20, .(T10, T20))
appA_in_gaa(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → U1_gaa(T10, T29, T30, T33, T34, appA_in_gaa(T30, T33, T34))
U1_gaa(T10, T29, T30, T33, T34, appA_out_gaa(T30, T33, T34)) → appA_out_gaa(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34)))

(5) DependencyGraphProof (EQUIVALENT transformation)

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

(6) Obligation:

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

APPA_IN_GAA(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → APPA_IN_GAA(T30, T33, T34)

The TRS R consists of the following rules:

appA_in_gaa([], T5, T5) → appA_out_gaa([], T5, T5)
appA_in_gaa(.(T10, []), T20, .(T10, T20)) → appA_out_gaa(.(T10, []), T20, .(T10, T20))
appA_in_gaa(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → U1_gaa(T10, T29, T30, T33, T34, appA_in_gaa(T30, T33, T34))
U1_gaa(T10, T29, T30, T33, T34, appA_out_gaa(T30, T33, T34)) → appA_out_gaa(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34)))

(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:

APPA_IN_GAA(.(T10, .(T29, T30)), T33, .(T10, .(T29, T34))) → APPA_IN_GAA(T30, T33, T34)

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

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:

APPA_IN_GAA(.(T10, .(T29, T30))) → APPA_IN_GAA(T30)

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:

APPA_IN_GAA(.(T10, .(T29, T30))) → APPA_IN_GAA(T30)
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