Left Termination proof of /tmp/tmph2ocKu/delmin-ffb.pl

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

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇒, 89 ms)

↳2 Prolog

↳3 PrologToPiTRSProof (⇒, 0 ms)

↳4 PiTRS

↳5 DependencyPairsProof (⇔, 17 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 0 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 YES

(0) Obligation:

Clauses:

delete(X, tree(X, void, Right), Right).
delete(X, tree(X, Left, void), Left).
delete(X, tree(X, Left, Right), tree(Y, Left, Right1)) :- delmin(Right, Y, Right1).
delete(X, tree(Y, Left, Right), tree(Y, Left1, Right)) :- ','(less(X, Y), delete(X, Left, Left1)).
delete(X, tree(Y, Left, Right), tree(Y, Left, Right1)) :- ','(less(Y, X), delete(X, Right, Right1)).
delmin(tree(Y, void, Right), Y, Right).
delmin(tree(X, Left, X1), Y, tree(X, Left1, X2)) :- delmin(Left, Y, Left1).
less(0, s(X3)).
less(s(X), s(Y)) :- less(X, Y).

Query: delmin(a,a,g)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: delmin(T1, T2, T3)\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: delmin(T1, T2, T3), SCOPE: 1, CLAUSE: 5 | delmin(T1, T2, T3), SCOPE: 1, CLAUSE: 6\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | delmin(T1, T2, T7), SCOPE: 1, CLAUSE: 6\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: delmin(T1, T2, T3), SCOPE: 1, CLAUSE: 6\n\nT3 is ground\nX6 is free\nX7 is free\ndelmin(T1, T2, T3) !=? delmin(tree(X6, void, X7), X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: delmin(T1, T2, T7), SCOPE: 1, CLAUSE: 6\n\nT7 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: delmin(T20, T21, T18)\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: delmin(T20, T21, T18), SCOPE: 2, CLAUSE: 5 | delmin(T20, T21, T18), SCOPE: 2, CLAUSE: 6\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: delmin(T20, T21, T18), SCOPE: 2, CLAUSE: 5\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: delmin(T20, T21, T18), SCOPE: 2, CLAUSE: 6\n\nT18 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: delmin(T50, T51, T48)\n\nT48 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: delmin(T68, T69, T66)\n\nT62 is ground\nT66 is ground\nT67 is ground\nX6 is free\nX7 is free\ndelmin(T1, T2, tree(T62, T66, T67)) !=? delmin(tree(X6, void, X7), X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: delmin(T68, T69, T66), SCOPE: 3, CLAUSE: 5 | delmin(T68, T69, T66), SCOPE: 3, CLAUSE: 6\n\nT62 is ground\nT66 is ground\nT67 is ground\nX6 is free\nX7 is free\ndelmin(T1, T2, tree(T62, T66, T67)) !=? delmin(tree(X6, void, X7), X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: delmin(T68, T69, T66), SCOPE: 3, CLAUSE: 5\n\nT62 is ground\nT66 is ground\nT67 is ground\nX6 is free\nX7 is free\ndelmin(T1, T2, tree(T62, T66, T67)) !=? delmin(tree(X6, void, X7), X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: delmin(T68, T69, T66), SCOPE: 3, CLAUSE: 6\n\nT62 is ground\nT66 is ground\nT67 is ground\nX6 is free\nX7 is free\ndelmin(T1, T2, tree(T62, T66, T67)) !=? delmin(tree(X6, void, X7), X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: delmin(T98, T99, T96)\n\nT62 is ground\nT67 is ground\nT92 is ground\nT96 is ground\nT97 is ground\nX6 is free\nX7 is free\ndelmin(T1, T2, tree(T62, tree(T92, T96, T97), T67)) !=? delmin(tree(X6, void, X7), X6, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 26 [arrowhead = none ];
26 -> 2;

27 [label="EVAL with clause\ndelmin(tree(X6, void, X7), X6, X7).\nand substitutionX6 -> T6,\nX7 -> T7,\nT1 -> tree(T6, void, T7),\nT2 -> T6,\nT3 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 27 [arrowhead = none ];
27 -> 3;

28 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 28 [arrowhead = none ];
28 -> 4;

29 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 29 [arrowhead = none ];
29 -> 5;

30 [label="EVAL with clause\ndelmin(tree(X58, X59, X60), X61, tree(X58, X62, X63)) :- delmin(X59, X61, X62).\nand substitutionX58 -> T62,\nX59 -> T68,\nX60 -> T64,\nT1 -> tree(T62, T68, T64),\nT2 -> T69,\nX61 -> T69,\nX62 -> T66,\nX63 -> T67,\nT3 -> tree(T62, T66, T67),\nT63 -> T68,\nT65 -> T69", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 30 [arrowhead = none ];
30 -> 16;

31 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 31 [arrowhead = none ];
31 -> 17;

32 [label="EVAL with clause\ndelmin(tree(X14, X15, X16), X17, tree(X14, X18, X19)) :- delmin(X15, X17, X18).\nand substitutionX14 -> T14,\nX15 -> T20,\nX16 -> T16,\nT1 -> tree(T14, T20, T16),\nT2 -> T21,\nX17 -> T21,\nX18 -> T18,\nX19 -> T19,\nT7 -> tree(T14, T18, T19),\nT15 -> T20,\nT17 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 32 [arrowhead = none ];
32 -> 6;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 33 [arrowhead = none ];
33 -> 7;

34 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 34 [arrowhead = none ];
34 -> 8;

35 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 35 [arrowhead = none ];
35 -> 9;

36 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="EVAL with clause\ndelmin(tree(X28, void, X29), X28, X29).\nand substitutionX28 -> T30,\nX29 -> T31,\nT20 -> tree(T30, void, T31),\nT21 -> T30,\nT18 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 38 [arrowhead = none ];
38 -> 12;

39 [label="EVAL with clause\ndelmin(tree(X42, X43, X44), X45, tree(X42, X46, X47)) :- delmin(X43, X45, X46).\nand substitutionX42 -> T44,\nX43 -> T50,\nX44 -> T46,\nT20 -> tree(T44, T50, T46),\nT21 -> T51,\nX45 -> T51,\nX46 -> T48,\nX47 -> T49,\nT18 -> tree(T44, T48, T49),\nT45 -> T50,\nT47 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 39 [arrowhead = none ];
39 -> 14;

40 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 40 [arrowhead = none ];
40 -> 15;

41 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 41 [arrowhead = none ];
41 -> 13;

42 [label="INSTANCE with matching:\nT1 -> T50\nT2 -> T51\nT3 -> T48", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 42 [arrowhead = none , style=dashed];
42 -> 1[style=dashed];

43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 43 [arrowhead = none ];
43 -> 18;

44 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 44 [arrowhead = none ];
44 -> 19;

45 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 45 [arrowhead = none ];
45 -> 20;

46 [label="EVAL with clause\ndelmin(tree(X72, void, X73), X72, X73).\nand substitutionX72 -> T78,\nX73 -> T79,\nT68 -> tree(T78, void, T79),\nT69 -> T78,\nT66 -> T79", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 46 [arrowhead = none ];
46 -> 21;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 47 [arrowhead = none ];
47 -> 22;

48 [label="EVAL with clause\ndelmin(tree(X86, X87, X88), X89, tree(X86, X90, X91)) :- delmin(X87, X89, X90).\nand substitutionX86 -> T92,\nX87 -> T98,\nX88 -> T94,\nT68 -> tree(T92, T98, T94),\nT69 -> T99,\nX89 -> T99,\nX90 -> T96,\nX91 -> T97,\nT66 -> tree(T92, T96, T97),\nT93 -> T98,\nT95 -> T99", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 48 [arrowhead = none ];
48 -> 24;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 49 [arrowhead = none ];
49 -> 25;

50 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 50 [arrowhead = none ];
50 -> 23;

51 [label="INSTANCE with matching:\nT1 -> T98\nT2 -> T99\nT3 -> T96", fontsize=14, style = filled, fillcolor = lightgrey];
24 -> 51 [arrowhead = none , style=dashed];
51 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

delminA(tree(T6, void, T7), T6, T7).
delminA(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19)).
delminA(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) :- delminA(T50, T51, T48).
delminA(tree(T62, tree(T78, void, T79), T64), T78, tree(T62, T79, T67)).
delminA(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67)) :- delminA(T98, T99, T96).

Query: delminA(a,a,g)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
delminA_in: (f,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

delminA_in_aag(tree(T6, void, T7), T6, T7) → delminA_out_aag(tree(T6, void, T7), T6, T7)
delminA_in_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19)) → delminA_out_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19))
delminA_in_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_in_aag(T50, T51, T48))
delminA_in_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67)) → U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_in_aag(T98, T99, T96))
U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_out_aag(T98, T99, T96)) → delminA_out_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67))
U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_out_aag(T50, T51, T48)) → delminA_out_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

delminA_in_aag(tree(T6, void, T7), T6, T7) → delminA_out_aag(tree(T6, void, T7), T6, T7)
delminA_in_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19)) → delminA_out_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19))
delminA_in_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_in_aag(T50, T51, T48))
delminA_in_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67)) → U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_in_aag(T98, T99, T96))
U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_out_aag(T98, T99, T96)) → delminA_out_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67))
U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_out_aag(T50, T51, T48)) → delminA_out_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19))

(5) 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:

DELMINA_IN_AAG(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → U1_AAG(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_in_aag(T50, T51, T48))
DELMINA_IN_AAG(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → DELMINA_IN_AAG(T50, T51, T48)
DELMINA_IN_AAG(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67)) → U2_AAG(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_in_aag(T98, T99, T96))

The TRS R consists of the following rules:

delminA_in_aag(tree(T6, void, T7), T6, T7) → delminA_out_aag(tree(T6, void, T7), T6, T7)
delminA_in_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19)) → delminA_out_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19))
delminA_in_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_in_aag(T50, T51, T48))
delminA_in_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67)) → U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_in_aag(T98, T99, T96))
U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_out_aag(T98, T99, T96)) → delminA_out_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67))
U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_out_aag(T50, T51, T48)) → delminA_out_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19))

The argument filtering Pi contains the following mapping:
delminA_in_aag(x1, x2, x3) = delminA_in_aag(x3)
delminA_out_aag(x1, x2, x3) = delminA_out_aag
tree(x1, x2, x3) = tree(x2, x3)
U1_aag(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U1_aag(x10)
U2_aag(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U2_aag(x10)
DELMINA_IN_AAG(x1, x2, x3) = DELMINA_IN_AAG(x3)
U1_AAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U1_AAG(x10)
U2_AAG(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) = U2_AAG(x10)

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

(6) Obligation:

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

DELMINA_IN_AAG(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → U1_AAG(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_in_aag(T50, T51, T48))
DELMINA_IN_AAG(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → DELMINA_IN_AAG(T50, T51, T48)
DELMINA_IN_AAG(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67)) → U2_AAG(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_in_aag(T98, T99, T96))

The TRS R consists of the following rules:

delminA_in_aag(tree(T6, void, T7), T6, T7) → delminA_out_aag(tree(T6, void, T7), T6, T7)
delminA_in_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19)) → delminA_out_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19))
delminA_in_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_in_aag(T50, T51, T48))
delminA_in_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67)) → U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_in_aag(T98, T99, T96))
U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_out_aag(T98, T99, T96)) → delminA_out_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67))
U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_out_aag(T50, T51, T48)) → delminA_out_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

DELMINA_IN_AAG(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → DELMINA_IN_AAG(T50, T51, T48)

The TRS R consists of the following rules:

delminA_in_aag(tree(T6, void, T7), T6, T7) → delminA_out_aag(tree(T6, void, T7), T6, T7)
delminA_in_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19)) → delminA_out_aag(tree(T14, tree(T30, void, T31), T16), T30, tree(T14, T31, T19))
delminA_in_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_in_aag(T50, T51, T48))
delminA_in_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67)) → U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_in_aag(T98, T99, T96))
U2_aag(T62, T92, T98, T94, T64, T99, T96, T97, T67, delminA_out_aag(T98, T99, T96)) → delminA_out_aag(tree(T62, tree(T92, T98, T94), T64), T99, tree(T62, tree(T92, T96, T97), T67))
U1_aag(T14, T44, T50, T46, T16, T51, T48, T49, T19, delminA_out_aag(T50, T51, T48)) → delminA_out_aag(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19))

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

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

DELMINA_IN_AAG(tree(T14, tree(T44, T50, T46), T16), T51, tree(T14, tree(T44, T48, T49), T19)) → DELMINA_IN_AAG(T50, T51, T48)

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

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

DELMINA_IN_AAG(tree(tree(T48, T49), T19)) → DELMINA_IN_AAG(T48)

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

(13) 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:

DELMINA_IN_AAG(tree(tree(T48, T49), T19)) → DELMINA_IN_AAG(T48)
The graph contains the following edges 1 > 1

(0) Obligation:

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

(2) Obligation:

(3) PrologToPiTRSProof (SOUND transformation)

(4) Obligation:

(5) DependencyPairsProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) UsableRulesProof (EQUIVALENT transformation)

(10) Obligation:

(11) PiDPToQDPProof (SOUND transformation)

(12) Obligation:

(13) QDPSizeChangeProof (EQUIVALENT transformation)

(14) YES