Left Termination proof of /tmp/tmpaB2zMd/duplicate2.pl

Left Termination of the query pattern
duplicate(g,a)
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 (⇔, 0 ms)

↳6 PiDP

↳7 DependencyGraphProof (⇔, 0 ms)

↳8 PiDP

↳9 UsableRulesProof (⇔, 0 ms)

↳10 PiDP

↳11 PiDPToQDPProof (⇒, 18 ms)

↳12 QDP

↳13 QDPSizeChangeProof (⇔, 0 ms)

↳14 YES

(0) Obligation:

Clauses:

duplicate([], []).
duplicate(X, .(H, .(H, Z))) :- ','(no(empty(X)), ','(head(X, H), ','(tail(X, T), duplicate(T, Z)))).
head([], X1).
head(.(H, X2), H).
tail([], []).
tail(.(X3, T), T).
empty([]).
no(X) :- ','(X, ','(!, failure(a))).
no(X4).
failure(b).

Query: duplicate(g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: duplicate(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: duplicate(T1, T2), SCOPE: 1, CLAUSE: 0 | duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | duplicate([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: duplicate(T1, T2), SCOPE: 1, CLAUSE: 1\n\nT1 is ground\nduplicate(T1, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: duplicate([], T2), SCOPE: 1, CLAUSE: 1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8))))\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 7 | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ','(','(call(empty([])), ','(!_2, failure(a))), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))) | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8))))) | ?_3 | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(empty([]), ','(','(!_2, failure(a)), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8))))), SCOPE: 4, CLAUSE: 6 | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(','(!_2, failure(a)), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))) | ?_4 | ?_3 | ','(no(empty([])), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 2, CLAUSE: 8\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(failure(a), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8))))\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(failure(a), ','(head([], T7), ','(tail([], X11), duplicate(X11, T8)))), SCOPE: 5, CLAUSE: 9\n\nX11 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(no(empty(T12)), ','(head(T12, T15), ','(tail(T12, X21), duplicate(X21, T16))))\n\nT12 is ground\nX21 is free\nduplicate(T12, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(no(empty(T12)), ','(head(T12, T15), ','(tail(T12, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 7 | ','(no(empty(T12)), ','(head(T12, T15), ','(tail(T12, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT12 is ground\nX21 is free\nduplicate(T12, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(','(call(empty(T19)), ','(!_6, failure(a))), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))) | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(empty(T19), ','(','(!_6, failure(a)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16))))) | ?_7 | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(empty(T19), ','(','(!_6, failure(a)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16))))), SCOPE: 8, CLAUSE: 6 | ?_8 | ?_7 | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(','(!_6, failure(a)), ','(head([], T15), ','(tail([], X21), duplicate(X21, T16)))) | ?_8 | ?_7 | ','(no(empty([])), ','(head([], T15), ','(tail([], X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ?_8 | ?_7 | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(failure(a), ','(head([], T15), ','(tail([], X21), duplicate(X21, T16))))\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(failure(a), ','(head([], T15), ','(tail([], X21), duplicate(X21, T16)))), SCOPE: 9, CLAUSE: 9\n\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ?_7 | ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(no(empty(T19)), ','(head(T19, T15), ','(tail(T19, X21), duplicate(X21, T16)))), SCOPE: 6, CLAUSE: 8\n\nT19 is ground\nX21 is free\nduplicate(T19, T2) !=? duplicate([], [])\nempty(T19) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(head(T22, T15), ','(tail(T22, X21), duplicate(X21, T16)))\n\nT22 is ground\nX21 is free\nduplicate(T22, T2) !=? duplicate([], [])\nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(head(T22, T15), ','(tail(T22, X21), duplicate(X21, T16))), SCOPE: 10, CLAUSE: 2 | ','(head(T22, T15), ','(tail(T22, X21), duplicate(X21, T16))), SCOPE: 10, CLAUSE: 3\n\nT22 is ground\nX21 is free\nduplicate(T22, T2) !=? duplicate([], [])\nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(head(T22, T15), ','(tail(T22, X21), duplicate(X21, T16))), SCOPE: 10, CLAUSE: 3\n\nT22 is ground\nX21 is free\nduplicate(T22, T2) !=? duplicate([], [])\nempty(T22) !=? empty([])", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(tail(.(T28, T29), X21), duplicate(X21, T30))\n\nT28 is ground\nT29 is ground\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(tail(.(T28, T29), X21), duplicate(X21, T30)), SCOPE: 11, CLAUSE: 4 | ','(tail(.(T28, T29), X21), duplicate(X21, T30)), SCOPE: 11, CLAUSE: 5\n\nT28 is ground\nT29 is ground\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(tail(.(T28, T29), X21), duplicate(X21, T30)), SCOPE: 11, CLAUSE: 5\n\nT28 is ground\nT29 is ground\nX21 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: duplicate(T36, T30)\n\nT36 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 37 [arrowhead = none ];
37 -> 2;

38 [label="EVAL with clause\nduplicate([], []).\nand substitutionT1 -> [],\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 38 [arrowhead = none ];
38 -> 3;

39 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 39 [arrowhead = none ];
39 -> 4;

40 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 40 [arrowhead = none ];
40 -> 5;

41 [label="EVAL with clause\nduplicate(X18, .(X19, .(X19, X20))) :- ','(no(empty(X18)), ','(head(X18, X19), ','(tail(X18, X21), duplicate(X21, X20)))).\nand substitutionT1 -> T12,\nX18 -> T12,\nX19 -> T15,\nX20 -> T16,\nT2 -> .(T15, .(T15, T16)),\nT13 -> T15,\nT14 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 41 [arrowhead = none ];
41 -> 16;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 42 [arrowhead = none ];
42 -> 17;

43 [label="EVAL with clause\nduplicate(X8, .(X9, .(X9, X10))) :- ','(no(empty(X8)), ','(head(X8, X9), ','(tail(X8, X11), duplicate(X11, X10)))).\nand substitutionX8 -> [],\nX9 -> T7,\nX10 -> T8,\nT2 -> .(T7, .(T7, T8)),\nT5 -> T7,\nT6 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 43 [arrowhead = none ];
43 -> 6;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 44 [arrowhead = none ];
44 -> 7;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 45 [arrowhead = none ];
45 -> 8;

46 [label="ONLY EVAL with clause\nno(X14) :- ','(call(X14), ','(!_2, failure(a))).\nand substitutionX14 -> empty([])", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 46 [arrowhead = none ];
46 -> 9;

47 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 47 [arrowhead = none ];
47 -> 10;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 48 [arrowhead = none ];
48 -> 11;

49 [label="ONLY EVAL with clause\nempty([]).\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 49 [arrowhead = none ];
49 -> 12;

50 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 50 [arrowhead = none ];
50 -> 13;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 51 [arrowhead = none ];
51 -> 14;

52 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 52 [arrowhead = none ];
52 -> 15;

53 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
16 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="ONLY EVAL with clause\nno(X24) :- ','(call(X24), ','(!_6, failure(a))).\nand substitutionT12 -> T19,\nX24 -> empty(T19)", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 54 [arrowhead = none ];
54 -> 19;

55 [label="CALL", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 55 [arrowhead = none ];
55 -> 20;

56 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 56 [arrowhead = none ];
56 -> 21;

57 [label="EVAL with clause\nempty([]).\nand substitutionT19 -> []", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 57 [arrowhead = none ];
57 -> 22;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 58 [arrowhead = none ];
58 -> 23;

59 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 59 [arrowhead = none ];
59 -> 24;

60 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 60 [arrowhead = none ];
60 -> 27;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 61 [arrowhead = none ];
61 -> 25;

62 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 62 [arrowhead = none ];
62 -> 26;

63 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 63 [arrowhead = none ];
63 -> 28;

64 [label="ONLY EVAL with clause\nno(X27).\nand substitutionT19 -> T22,\nX27 -> empty(T22)", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 64 [arrowhead = none ];
64 -> 29;

65 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 65 [arrowhead = none ];
65 -> 30;

66 [label="BACKTRACK\nfor clause: head([], X1)\nwith clash: (empty(T22), empty([]))", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 66 [arrowhead = none ];
66 -> 31;

67 [label="EVAL with clause\nhead(.(X33, X34), X33).\nand substitutionX33 -> T28,\nX34 -> T29,\nT22 -> .(T28, T29),\nT15 -> T28,\nT16 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 67 [arrowhead = none ];
67 -> 32;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 69 [arrowhead = none ];
69 -> 34;

70 [label="BACKTRACK\nfor clause: tail([], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 70 [arrowhead = none ];
70 -> 35;

71 [label="ONLY EVAL with clause\ntail(.(X39, X40), X40).\nand substitutionT28 -> T35,\nX39 -> T35,\nT29 -> T36,\nX40 -> T36,\nX21 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 71 [arrowhead = none ];
71 -> 36;

72 [label="INSTANCE with matching:\nT1 -> T36\nT2 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
36 -> 72 [arrowhead = none , style=dashed];
72 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

duplicateA([], []).
duplicateA(.(T35, T36), .(T35, .(T35, T30))) :- duplicateA(T36, T30).

Query: duplicateA(g,a)

(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:
duplicateA_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T35, T36), .(T35, .(T35, T30))) → U1_ga(T35, T36, T30, duplicateA_in_ga(T36, T30))
U1_ga(T35, T36, T30, duplicateA_out_ga(T36, T30)) → duplicateA_out_ga(.(T35, T36), .(T35, .(T35, T30)))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T35, T36), .(T35, .(T35, T30))) → U1_ga(T35, T36, T30, duplicateA_in_ga(T36, T30))
U1_ga(T35, T36, T30, duplicateA_out_ga(T36, T30)) → duplicateA_out_ga(.(T35, T36), .(T35, .(T35, T30)))

(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:

DUPLICATEA_IN_GA(.(T35, T36), .(T35, .(T35, T30))) → U1_GA(T35, T36, T30, duplicateA_in_ga(T36, T30))
DUPLICATEA_IN_GA(.(T35, T36), .(T35, .(T35, T30))) → DUPLICATEA_IN_GA(T36, T30)

The TRS R consists of the following rules:

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T35, T36), .(T35, .(T35, T30))) → U1_ga(T35, T36, T30, duplicateA_in_ga(T36, T30))
U1_ga(T35, T36, T30, duplicateA_out_ga(T36, T30)) → duplicateA_out_ga(.(T35, T36), .(T35, .(T35, T30)))

The argument filtering Pi contains the following mapping:
duplicateA_in_ga(x1, x2) = duplicateA_in_ga(x1)
[] = []
duplicateA_out_ga(x1, x2) = duplicateA_out_ga(x2)
.(x1, x2) = .(x1, x2)
U1_ga(x1, x2, x3, x4) = U1_ga(x1, x4)
DUPLICATEA_IN_GA(x1, x2) = DUPLICATEA_IN_GA(x1)
U1_GA(x1, x2, x3, x4) = U1_GA(x1, x4)

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

(6) Obligation:

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

DUPLICATEA_IN_GA(.(T35, T36), .(T35, .(T35, T30))) → U1_GA(T35, T36, T30, duplicateA_in_ga(T36, T30))
DUPLICATEA_IN_GA(.(T35, T36), .(T35, .(T35, T30))) → DUPLICATEA_IN_GA(T36, T30)

The TRS R consists of the following rules:

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T35, T36), .(T35, .(T35, T30))) → U1_ga(T35, T36, T30, duplicateA_in_ga(T36, T30))
U1_ga(T35, T36, T30, duplicateA_out_ga(T36, T30)) → duplicateA_out_ga(.(T35, T36), .(T35, .(T35, T30)))

(7) DependencyGraphProof (EQUIVALENT transformation)

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

(8) Obligation:

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

DUPLICATEA_IN_GA(.(T35, T36), .(T35, .(T35, T30))) → DUPLICATEA_IN_GA(T36, T30)

The TRS R consists of the following rules:

duplicateA_in_ga([], []) → duplicateA_out_ga([], [])
duplicateA_in_ga(.(T35, T36), .(T35, .(T35, T30))) → U1_ga(T35, T36, T30, duplicateA_in_ga(T36, T30))
U1_ga(T35, T36, T30, duplicateA_out_ga(T36, T30)) → duplicateA_out_ga(.(T35, T36), .(T35, .(T35, T30)))

(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:

DUPLICATEA_IN_GA(.(T35, T36), .(T35, .(T35, T30))) → DUPLICATEA_IN_GA(T36, T30)

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

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:

DUPLICATEA_IN_GA(.(T35, T36)) → DUPLICATEA_IN_GA(T36)

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:

DUPLICATEA_IN_GA(.(T35, T36)) → DUPLICATEA_IN_GA(T36)
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