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

(0) Obligation:

Clauses:

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

Query: minimum(g,a)

(1) PrologToPiTRSViaGraphTransformerProof (SOUND transformation)

Transformed Prolog program to (Pi-)TRS.

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];
5 [label="5: minimum(T1, T2), SCOPE: 1, CLAUSE: 0 | minimum(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true | minimum(tree(T5, void, T6), T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: 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];
13 [label="13: minimum(tree(T5, void, T6), T2), SCOPE: 1, CLAUSE: 1\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: minimum(void, T13)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: minimum(void, T13), SCOPE: 2, CLAUSE: 0 | minimum(void, T13), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: minimum(void, T13), SCOPE: 2, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: 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];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: 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];
34 [label="34: 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];
36 [label="36: 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];
38 [label="38: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: minimum(T42, T45)\n\nT42 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 47 [arrowhead = none ];
47 -> 5;

48 [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];
5 -> 48 [arrowhead = none ];
48 -> 7;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 49 [arrowhead = none ];
49 -> 10;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 50 [arrowhead = none ];
50 -> 13;

51 [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];
10 -> 51 [arrowhead = none ];
51 -> 28;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 52 [arrowhead = none ];
52 -> 30;

53 [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];
13 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 54 [arrowhead = none ];
54 -> 20;

55 [label="BACKTRACK\nfor clause: minimum(tree(X, void, X1), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 55 [arrowhead = none ];
55 -> 22;

56 [label="BACKTRACK\nfor clause: minimum(tree(X2, Left, X3), X) :- minimum(Left, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 56 [arrowhead = none ];
56 -> 25;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 57 [arrowhead = none ];
57 -> 32;

58 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 58 [arrowhead = none ];
58 -> 34;

59 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
32 -> 59 [arrowhead = none ];
59 -> 36;

60 [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];
34 -> 60 [arrowhead = none ];
60 -> 38;

61 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 61 [arrowhead = none ];
61 -> 40;

62 [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];
36 -> 62 [arrowhead = none ];
62 -> 44;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 63 [arrowhead = none ];
63 -> 46;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 64 [arrowhead = none ];
64 -> 41;

65 [label="INSTANCE with matching:\nT1 -> T42\nT2 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 65 [arrowhead = none , style=dashed];
65 -> 1[style=dashed];

}

(2) Obligation:

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

minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimumA_out_ga(x1, x2)  =  minimumA_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)

(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:

MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → U1_GA(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUMA_IN_GA(T42, T45)

The TRS R consists of the following rules:

minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimumA_out_ga(x1, x2)  =  minimumA_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)
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

(4) Obligation:

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

MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → U1_GA(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUMA_IN_GA(T42, T45)

The TRS R consists of the following rules:

minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimumA_out_ga(x1, x2)  =  minimumA_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)
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

(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(T18, tree(T41, T42, T43), T20), T45) → MINIMUMA_IN_GA(T42, T45)

The TRS R consists of the following rules:

minimumA_in_ga(tree(T5, void, T6), T5) → minimumA_out_ga(tree(T5, void, T6), T5)
minimumA_in_ga(tree(T18, tree(T31, void, T32), T20), T31) → minimumA_out_ga(tree(T18, tree(T31, void, T32), T20), T31)
minimumA_in_ga(tree(T18, tree(T41, T42, T43), T20), T45) → U1_ga(T18, T41, T42, T43, T20, T45, minimumA_in_ga(T42, T45))
U1_ga(T18, T41, T42, T43, T20, T45, minimumA_out_ga(T42, T45)) → minimumA_out_ga(tree(T18, tree(T41, T42, T43), T20), T45)

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)
void  =  void
minimumA_out_ga(x1, x2)  =  minimumA_out_ga(x1, x2)
U1_ga(x1, x2, x3, x4, x5, x6, x7)  =  U1_ga(x1, x2, x3, x4, x5, x7)
MINIMUMA_IN_GA(x1, x2)  =  MINIMUMA_IN_GA(x1)

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:

MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20), T45) → MINIMUMA_IN_GA(T42, T45)

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

(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:

MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20)) → MINIMUMA_IN_GA(T42)

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:

• MINIMUMA_IN_GA(tree(T18, tree(T41, T42, T43), T20)) → MINIMUMA_IN_GA(T42)
  The graph contains the following edges 1 > 1

(12) YES