Left Termination proof of /tmp/tmppJURFa/list.pl

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

0 Prolog

↳1 PrologToPiTRSViaGraphTransformerProof (⇒, 65 ms)

↳2 PiTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 7 ms)

↳6 PiDP

↳7 UsableRulesProof (⇔, 0 ms)

↳8 PiDP

↳9 PiDPToQDPProof (⇔, 0 ms)

↳10 QDP

↳11 QDPSizeChangeProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

list([]).
list(.(X1, Ts)) :- list(Ts).

Query: list(g)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: list(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
5 [label="5: list(T1), SCOPE: 1, CLAUSE: 0 | list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true | list([]), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: list(T1), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nlist(T1) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: list([]), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: list(T5)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: list(T5), SCOPE: 2, CLAUSE: 0 | list(T5), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: list(T5), SCOPE: 2, CLAUSE: 0\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: list(T5), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: list(T11)\n\nT11 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 -> 5;

41 [label="EVAL with clause\nlist([]).\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 41 [arrowhead = none ];
41 -> 7;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 42 [arrowhead = none ];
42 -> 9;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 43 [arrowhead = none ];
43 -> 12;

44 [label="EVAL with clause\nlist(.(X6, X7)) :- list(X7).\nand substitutionX6 -> T4,\nX7 -> T5,\nT1 -> .(T4, T5)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 44 [arrowhead = none ];
44 -> 30;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 45 [arrowhead = none ];
45 -> 31;

46 [label="BACKTRACK\nfor clause: list(.(X1, Ts)) :- list(Ts)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 46 [arrowhead = none ];
46 -> 29;

47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
30 -> 47 [arrowhead = none ];
47 -> 32;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 48 [arrowhead = none ];
48 -> 33;

49 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 49 [arrowhead = none ];
49 -> 34;

50 [label="EVAL with clause\nlist([]).\nand substitutionT5 -> []", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 50 [arrowhead = none ];
50 -> 35;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 51 [arrowhead = none ];
51 -> 36;

52 [label="EVAL with clause\nlist(.(X12, X13)) :- list(X13).\nand substitutionX12 -> T10,\nX13 -> T11,\nT5 -> .(T10, T11)", fontsize=14, style = filled, fillcolor = lightblue];
34 -> 52 [arrowhead = none ];
52 -> 38;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 53 [arrowhead = none ];
53 -> 39;

54 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 54 [arrowhead = none ];
54 -> 37;

55 [label="INSTANCE with matching:\nT1 -> T11", fontsize=14, style = filled, fillcolor = lightgrey];
38 -> 55 [arrowhead = none , style=dashed];
55 -> 1[style=dashed];

}

(2) Obligation:

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

listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))

Pi is empty.

(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:

LISTA_IN_G(.(T4, .(T10, T11))) → U1_G(T4, T10, T11, listA_in_g(T11))
LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)

The TRS R consists of the following rules:

listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))

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

(4) Obligation:

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

LISTA_IN_G(.(T4, .(T10, T11))) → U1_G(T4, T10, T11, listA_in_g(T11))
LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)

The TRS R consists of the following rules:

listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))

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

(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:

LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)

The TRS R consists of the following rules:

listA_in_g([]) → listA_out_g([])
listA_in_g(.(T4, [])) → listA_out_g(.(T4, []))
listA_in_g(.(T4, .(T10, T11))) → U1_g(T4, T10, T11, listA_in_g(T11))
U1_g(T4, T10, T11, listA_out_g(T11)) → listA_out_g(.(T4, .(T10, T11)))

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

(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:

LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)

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

(9) PiDPToQDPProof (EQUIVALENT 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:

LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)

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:

LISTA_IN_G(.(T4, .(T10, T11))) → LISTA_IN_G(T11)
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 (EQUIVALENT transformation)

(10) Obligation:

(11) QDPSizeChangeProof (EQUIVALENT transformation)

(12) YES