Left Termination proof of /tmp/tmpQJEtBD/minimum-bf.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 88 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 0 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 PiDPToQDPProof (⇒, 0 ms)

↳8 QDP

↳9 QDPSizeChangeProof (⇔, 0 ms)

↳10 YES

(0) Obligation:

Clauses:

minimum(tree(X, void, X1), X).
minimum(tree(X2, Left, X3), X) :- minimum(Left, X).

Query: minimum(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: minimum(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: minimum(T1, T2), SCOPE: 1, CLAUSE: 0 | minimum(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | minimum(tree(T5, void, T6), T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: minimum(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nX6 is free\nX7 is free\nminimum(T1, T2) !=? minimum(tree(X6, void, X7), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: minimum(tree(T5, void, T6), T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: minimum(void, T13)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: minimum(void, T13), SCOPE: 2, CLAUSE: 0 | minimum(void, T13), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: minimum(void, T13), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: minimum(T19, T22)\n\nT18 is ground\nT19 is ground\nT20 is ground\nX6 is free\nX7 is free\nminimum(tree(T18, T19, T20), T2) !=? minimum(tree(X6, void, X7), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: minimum(T19, T22), SCOPE: 3, CLAUSE: 0 | minimum(T19, T22), SCOPE: 3, CLAUSE: 1\n\nT18 is ground\nT19 is ground\nT20 is ground\nX6 is free\nX7 is free\nminimum(tree(T18, T19, T20), T2) !=? minimum(tree(X6, void, X7), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: minimum(T19, T22), SCOPE: 3, CLAUSE: 0\n\nT18 is ground\nT19 is ground\nT20 is ground\nX6 is free\nX7 is free\nminimum(tree(T18, T19, T20), T2) !=? minimum(tree(X6, void, X7), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: minimum(T19, T22), SCOPE: 3, CLAUSE: 1\n\nT18 is ground\nT19 is ground\nT20 is ground\nX6 is free\nX7 is free\nminimum(tree(T18, T19, T20), T2) !=? minimum(tree(X6, void, X7), X6)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: minimum(T42, T45)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 20 [arrowhead = none ];
20 -> 2;

21 [label="EVAL with clause\nminimum(tree(X6, void, X7), X6).\nand substitutionX6 -> T5,\nX7 -> T6,\nT1 -> tree(T5, void, T6),\nT2 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 21 [arrowhead = none ];
21 -> 3;

22 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 22 [arrowhead = none ];
22 -> 4;

23 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 23 [arrowhead = none ];
23 -> 5;

24 [label="EVAL with clause\nminimum(tree(X26, X27, X28), X29) :- minimum(X27, X29).\nand substitutionX26 -> T18,\nX27 -> T19,\nX28 -> T20,\nT1 -> tree(T18, T19, T20),\nT2 -> T22,\nX29 -> T22,\nT21 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 24 [arrowhead = none ];
24 -> 10;

25 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 25 [arrowhead = none ];
25 -> 11;

26 [label="ONLY EVAL with clause\nminimum(tree(X12, X13, X14), X15) :- minimum(X13, X15).\nand substitutionT5 -> T10,\nX12 -> T10,\nX13 -> void,\nT6 -> T11,\nX14 -> T11,\nT2 -> T13,\nX15 -> T13,\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 26 [arrowhead = none ];
26 -> 6;

27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 27 [arrowhead = none ];
27 -> 7;

28 [label="BACKTRACK\nfor clause: minimum(tree(X, void, X1), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 28 [arrowhead = none ];
28 -> 8;

29 [label="BACKTRACK\nfor clause: minimum(tree(X2, Left, X3), X) :- minimum(Left, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 29 [arrowhead = none ];
29 -> 9;

30 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 30 [arrowhead = none ];
30 -> 12;

31 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 31 [arrowhead = none ];
31 -> 13;

32 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 32 [arrowhead = none ];
32 -> 14;

33 [label="EVAL with clause\nminimum(tree(X38, void, X39), X38).\nand substitutionX38 -> T31,\nX39 -> T32,\nT19 -> tree(T31, void, T32),\nT22 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 33 [arrowhead = none ];
33 -> 15;

34 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 34 [arrowhead = none ];
34 -> 16;

35 [label="EVAL with clause\nminimum(tree(X48, X49, X50), X51) :- minimum(X49, X51).\nand substitutionX48 -> T41,\nX49 -> T42,\nX50 -> T43,\nT19 -> tree(T41, T42, T43),\nT22 -> T45,\nX51 -> T45,\nT44 -> T45", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 35 [arrowhead = none ];
35 -> 18;

36 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 36 [arrowhead = none ];
36 -> 19;

37 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 37 [arrowhead = none ];
37 -> 17;

38 [label="INSTANCE with matching:\nT1 -> T42\nT2 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 38 [arrowhead = none , style=dashed];
38 -> 1[style=dashed];

}

(2) Obligation:

Triples:

minimumA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumA(X3, X6).

Clauses:

minimumcA(tree(X1, void, X2), X1).
minimumcA(tree(X1, tree(X2, void, X3), X4), X2).
minimumcA(tree(X1, tree(X2, X3, X4), X5), X6) :- minimumcA(X3, X6).

Afs:

minimumA(x1, x2) = minimumA(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:
minimumA_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

MINIMUMA_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → U1_GA(X1, X2, X3, X4, X5, X6, minimumA_in_ga(X3, X6))
MINIMUMA_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → MINIMUMA_IN_GA(X3, X6)

R is empty.
The argument filtering Pi contains the following mapping:
minimumA_in_ga(x1, x2) = minimumA_in_ga(x1)
tree(x1, x2, x3) = tree(x1, x2, x3)
MINIMUMA_IN_GA(x1, x2) = MINIMUMA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5, x6, x7) = U1_GA(x1, x2, x3, x4, x5, x7)

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:

MINIMUMA_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → U1_GA(X1, X2, X3, X4, X5, X6, minimumA_in_ga(X3, X6))
MINIMUMA_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → MINIMUMA_IN_GA(X3, X6)

(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:

MINIMUMA_IN_GA(tree(X1, tree(X2, X3, X4), X5), X6) → MINIMUMA_IN_GA(X3, X6)

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

MINIMUMA_IN_GA(tree(X1, tree(X2, X3, X4), X5)) → MINIMUMA_IN_GA(X3)

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:

MINIMUMA_IN_GA(tree(X1, tree(X2, X3, X4), X5)) → MINIMUMA_IN_GA(X3)
The graph contains the following edges 1 > 1