Left Termination proof of /tmp/tmpwaeJeV/map.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 0 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 15 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 PiDPToQDPProof (⇒, 0 ms)

↳8 QDP

↳9 QDPSizeChangeProof (⇔, 0 ms)

↳10 YES

(0) Obligation:

Clauses:

p(val_i, val_j).
map(.(X, Xs), .(Y, Ys)) :- ','(p(X, Y), map(Xs, Ys)).
map([], []).

Query: map(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: map(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: map(T1, T2), SCOPE: 1, CLAUSE: 1 | map(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(p(T7, T11), map(T8, T12)) | map(.(T7, T8), T2), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: map(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX5 is free\nX6 is free\nX7 is free\nX8 is free\nmap(T1, T2) !=? map(.(X5, X6), .(X7, X8))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: ','(p(T7, T11), map(T8, T12)), SCOPE: 2, CLAUSE: 0 | ?_2 | map(.(T7, T8), T2), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(p(T7, T11), map(T8, T12)), SCOPE: 2, CLAUSE: 0\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ?_2 | map(.(T7, T8), T2), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: map(T8, T13)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: map(.(T7, T8), T2), SCOPE: 1, CLAUSE: 2\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 15 [arrowhead = none ];
15 -> 2;

16 [label="EVAL with clause\nmap(.(X5, X6), .(X7, X8)) :- ','(p(X5, X7), map(X6, X8)).\nand substitutionX5 -> T7,\nX6 -> T8,\nT1 -> .(T7, T8),\nX7 -> T11,\nX8 -> T12,\nT2 -> .(T11, T12),\nT9 -> T11,\nT10 -> T12", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 16 [arrowhead = none ];
16 -> 3;

17 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 17 [arrowhead = none ];
17 -> 4;

18 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 18 [arrowhead = none ];
18 -> 5;

19 [label="EVAL with clause\nmap([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 19 [arrowhead = none ];
19 -> 12;

20 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 20 [arrowhead = none ];
20 -> 13;

21 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 21 [arrowhead = none ];
21 -> 6;

22 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
5 -> 22 [arrowhead = none ];
22 -> 7;

23 [label="EVAL with clause\np(val_i, val_j).\nand substitutionT7 -> val_i,\nT11 -> val_j,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 23 [arrowhead = none ];
23 -> 8;

24 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 24 [arrowhead = none ];
24 -> 9;

25 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 25 [arrowhead = none ];
25 -> 10;

26 [label="INSTANCE with matching:\nT1 -> T8\nT2 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
8 -> 26 [arrowhead = none , style=dashed];
26 -> 1[style=dashed];

27 [label="BACKTRACK\nfor clause: map([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 27 [arrowhead = none ];
27 -> 11;

28 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 28 [arrowhead = none ];
28 -> 14;

}

(2) Obligation:

Triples:

mapA(.(val_i, X1), .(val_j, X2)) :- mapA(X1, X2).

Clauses:

mapcA(.(val_i, X1), .(val_j, X2)) :- mapcA(X1, X2).
mapcA([], []).

Afs:

mapA(x1, x2) = mapA(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
mapA_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MAPA_IN_GA(.(val_i, X1), .(val_j, X2)) → U1_GA(X1, X2, mapA_in_ga(X1, X2))
MAPA_IN_GA(.(val_i, X1), .(val_j, X2)) → MAPA_IN_GA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
mapA_in_ga(x1, x2) = mapA_in_ga(x1)
.(x1, x2) = .(x1, x2)
val_i = val_i
val_j = val_j
MAPA_IN_GA(x1, x2) = MAPA_IN_GA(x1)
U1_GA(x1, x2, x3) = U1_GA(x1, x3)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

MAPA_IN_GA(.(val_i, X1), .(val_j, X2)) → U1_GA(X1, X2, mapA_in_ga(X1, X2))
MAPA_IN_GA(.(val_i, X1), .(val_j, X2)) → MAPA_IN_GA(X1, X2)

(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:

MAPA_IN_GA(.(val_i, X1), .(val_j, X2)) → MAPA_IN_GA(X1, X2)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
val_i = val_i
val_j = val_j
MAPA_IN_GA(x1, x2) = MAPA_IN_GA(x1)

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

(7) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(8) Obligation:

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

MAPA_IN_GA(.(val_i, X1)) → MAPA_IN_GA(X1)

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

(9) 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:

MAPA_IN_GA(.(val_i, X1)) → MAPA_IN_GA(X1)
The graph contains the following edges 1 > 1