Left Termination proof of /tmp/tmp787m1j/flatten.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 83 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 48 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 PiDPToQDPProof (⇒, 0 ms)

↳8 QDP

↳9 UsableRulesReductionPairsProof (⇔, 97 ms)

↳10 QDP

↳11 MRRProof (⇔, 31 ms)

↳12 QDP

↳13 PisEmptyProof (⇔, 0 ms)

↳14 YES

(0) Obligation:

Clauses:

flatten(atom(X), .(X, [])).
flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y).
flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X).

Query: flatten(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: flatten(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: flatten(T1, T2), SCOPE: 1, CLAUSE: 0 | flatten(T1, T2), SCOPE: 1, CLAUSE: 1 | flatten(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | flatten(atom(T4), T2), SCOPE: 1, CLAUSE: 1 | flatten(atom(T4), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: flatten(T1, T2), SCOPE: 1, CLAUSE: 1 | flatten(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX2 is free\nflatten(T1, T2) !=? flatten(atom(X2), .(X2, []))", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: flatten(atom(T4), T2), SCOPE: 1, CLAUSE: 1 | flatten(atom(T4), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: flatten(atom(T4), T2), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: flatten(T9, T11) | flatten(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: flatten(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX2 is free\nX13 is free\nX14 is free\nX15 is free\nflatten(T1, T2) !=? flatten(atom(X2), .(X2, []))\nflatten(T1, T2) !=? flatten(cons(atom(X13), X14), .(X13, X15))", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: flatten(T9, T11), SCOPE: 2, CLAUSE: 0 | flatten(T9, T11), SCOPE: 2, CLAUSE: 1 | flatten(T9, T11), SCOPE: 2, CLAUSE: 2 | ?_2 | flatten(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: flatten(T9, T11), SCOPE: 2, CLAUSE: 0\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: flatten(T9, T11), SCOPE: 2, CLAUSE: 1 | flatten(T9, T11), SCOPE: 2, CLAUSE: 2 | ?_2 | flatten(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: flatten(T9, T11), SCOPE: 2, CLAUSE: 1\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: flatten(T9, T11), SCOPE: 2, CLAUSE: 2 | ?_2 | flatten(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: flatten(T30, T32)\n\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: flatten(T9, T11), SCOPE: 2, CLAUSE: 2\n\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | flatten(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: flatten(cons(T51, cons(T52, T53)), T55)\n\nT51 is ground\nT52 is ground\nT53 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: flatten(cons(atom(T8), T9), T2), SCOPE: 1, CLAUSE: 2\n\nT8 is ground\nT9 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: flatten(cons(T60, cons(T61, T62)), T64)\n\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: flatten(cons(T60, cons(T61, T62)), T64), SCOPE: 3, CLAUSE: 0 | flatten(cons(T60, cons(T61, T62)), T64), SCOPE: 3, CLAUSE: 1 | flatten(cons(T60, cons(T61, T62)), T64), SCOPE: 3, CLAUSE: 2\n\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: flatten(cons(T60, cons(T61, T62)), T64), SCOPE: 3, CLAUSE: 1 | flatten(cons(T60, cons(T61, T62)), T64), SCOPE: 3, CLAUSE: 2\n\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: flatten(cons(T60, cons(T61, T62)), T64), SCOPE: 3, CLAUSE: 1\n\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: flatten(cons(T60, cons(T61, T62)), T64), SCOPE: 3, CLAUSE: 2\n\nT60 is ground\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: flatten(cons(T82, T83), T85)\n\nT82 is ground\nT83 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: flatten(cons(T96, cons(T97, cons(T98, T99))), T101)\n\nT96 is ground\nT97 is ground\nT98 is ground\nT99 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 36 [arrowhead = none ];
36 -> 2;

37 [label="EVAL with clause\nflatten(atom(X2), .(X2, [])).\nand substitutionX2 -> T4,\nT1 -> atom(T4),\nT2 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 37 [arrowhead = none ];
37 -> 3;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 38 [arrowhead = none ];
38 -> 4;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 39 [arrowhead = none ];
39 -> 5;

40 [label="EVAL with clause\nflatten(cons(atom(X13), X14), .(X13, X15)) :- flatten(X14, X15).\nand substitutionX13 -> T8,\nX14 -> T9,\nT1 -> cons(atom(T8), T9),\nX15 -> T11,\nT2 -> .(T8, T11),\nT10 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 40 [arrowhead = none ];
40 -> 8;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 41 [arrowhead = none ];
41 -> 9;

42 [label="BACKTRACK\nfor clause: flatten(cons(atom(X), U), .(X, Y)) :- flatten(U, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="BACKTRACK\nfor clause: flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 43 [arrowhead = none ];
43 -> 7;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 44 [arrowhead = none ];
44 -> 10;

45 [label="EVAL with clause\nflatten(cons(cons(X68, X69), X70), X71) :- flatten(cons(X68, cons(X69, X70)), X71).\nand substitutionX68 -> T60,\nX69 -> T61,\nX70 -> T62,\nT1 -> cons(cons(T60, T61), T62),\nT2 -> T64,\nX71 -> T64,\nT63 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 45 [arrowhead = none ];
45 -> 26;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 46 [arrowhead = none ];
46 -> 27;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 47 [arrowhead = none ];
47 -> 11;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 48 [arrowhead = none ];
48 -> 12;

49 [label="EVAL with clause\nflatten(atom(X20), .(X20, [])).\nand substitutionX20 -> T16,\nT9 -> atom(T16),\nT11 -> .(T16, [])", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 49 [arrowhead = none ];
49 -> 13;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 50 [arrowhead = none ];
50 -> 14;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 51 [arrowhead = none ];
51 -> 16;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 53 [arrowhead = none ];
53 -> 15;

54 [label="EVAL with clause\nflatten(cons(atom(X33), X34), .(X33, X35)) :- flatten(X34, X35).\nand substitutionX33 -> T29,\nX34 -> T30,\nT9 -> cons(atom(T29), T30),\nX35 -> T32,\nT11 -> .(T29, T32),\nT31 -> T32", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 54 [arrowhead = none ];
54 -> 18;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 55 [arrowhead = none ];
55 -> 19;

56 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 56 [arrowhead = none ];
56 -> 20;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 57 [arrowhead = none ];
57 -> 21;

58 [label="INSTANCE with matching:\nT1 -> T30\nT2 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 58 [arrowhead = none , style=dashed];
58 -> 1[style=dashed];

59 [label="EVAL with clause\nflatten(cons(cons(X54, X55), X56), X57) :- flatten(cons(X54, cons(X55, X56)), X57).\nand substitutionX54 -> T51,\nX55 -> T52,\nX56 -> T53,\nT9 -> cons(cons(T51, T52), T53),\nT11 -> T55,\nX57 -> T55,\nT54 -> T55", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 59 [arrowhead = none ];
59 -> 22;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 60 [arrowhead = none ];
60 -> 23;

61 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 61 [arrowhead = none ];
61 -> 24;

62 [label="INSTANCE with matching:\nT1 -> cons(T51, cons(T52, T53))\nT2 -> T55", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 62 [arrowhead = none , style=dashed];
62 -> 1[style=dashed];

63 [label="BACKTRACK\nfor clause: flatten(cons(cons(U, V), W), X) :- flatten(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 63 [arrowhead = none ];
63 -> 25;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 64 [arrowhead = none ];
64 -> 28;

65 [label="BACKTRACK\nfor clause: flatten(atom(X), .(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 65 [arrowhead = none ];
65 -> 29;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 66 [arrowhead = none ];
66 -> 30;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 67 [arrowhead = none ];
67 -> 31;

68 [label="EVAL with clause\nflatten(cons(atom(X85), X86), .(X85, X87)) :- flatten(X86, X87).\nand substitutionX85 -> T81,\nT60 -> atom(T81),\nT61 -> T82,\nT62 -> T83,\nX86 -> cons(T82, T83),\nX87 -> T85,\nT64 -> .(T81, T85),\nT84 -> T85", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 68 [arrowhead = none ];
68 -> 32;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 69 [arrowhead = none ];
69 -> 33;

70 [label="EVAL with clause\nflatten(cons(cons(X98, X99), X100), X101) :- flatten(cons(X98, cons(X99, X100)), X101).\nand substitutionX98 -> T96,\nX99 -> T97,\nT60 -> cons(T96, T97),\nT61 -> T98,\nT62 -> T99,\nX100 -> cons(T98, T99),\nT64 -> T101,\nX101 -> T101,\nT100 -> T101", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 70 [arrowhead = none ];
70 -> 34;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 71 [arrowhead = none ];
71 -> 35;

72 [label="INSTANCE with matching:\nT1 -> cons(T82, T83)\nT2 -> T85", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 72 [arrowhead = none , style=dashed];
72 -> 1[style=dashed];

73 [label="INSTANCE with matching:\nT1 -> cons(T96, cons(T97, cons(T98, T99)))\nT2 -> T101", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 73 [arrowhead = none , style=dashed];
73 -> 1[style=dashed];

}

(2) Obligation:

Triples:

flattenA(cons(atom(X1), cons(atom(X2), X3)), .(X1, .(X2, X4))) :- flattenA(X3, X4).
flattenA(cons(atom(X1), cons(cons(X2, X3), X4)), .(X1, X5)) :- flattenA(cons(X2, cons(X3, X4)), X5).
flattenA(cons(cons(atom(X1), X2), X3), .(X1, X4)) :- flattenA(cons(X2, X3), X4).
flattenA(cons(cons(cons(X1, X2), X3), X4), X5) :- flattenA(cons(X1, cons(X2, cons(X3, X4))), X5).

Clauses:

flattencA(atom(X1), .(X1, [])).
flattencA(cons(atom(X1), atom(X2)), .(X1, .(X2, []))).
flattencA(cons(atom(X1), cons(atom(X2), X3)), .(X1, .(X2, X4))) :- flattencA(X3, X4).
flattencA(cons(atom(X1), cons(cons(X2, X3), X4)), .(X1, X5)) :- flattencA(cons(X2, cons(X3, X4)), X5).
flattencA(cons(cons(atom(X1), X2), X3), .(X1, X4)) :- flattencA(cons(X2, X3), X4).
flattencA(cons(cons(cons(X1, X2), X3), X4), X5) :- flattencA(cons(X1, cons(X2, cons(X3, X4))), X5).

Afs:

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

FLATTENA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), .(X1, .(X2, X4))) → U1_GA(X1, X2, X3, X4, flattenA_in_ga(X3, X4))
FLATTENA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), .(X1, .(X2, X4))) → FLATTENA_IN_GA(X3, X4)
FLATTENA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), .(X1, X5)) → U2_GA(X1, X2, X3, X4, X5, flattenA_in_ga(cons(X2, cons(X3, X4)), X5))
FLATTENA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), .(X1, X5)) → FLATTENA_IN_GA(cons(X2, cons(X3, X4)), X5)
FLATTENA_IN_GA(cons(cons(atom(X1), X2), X3), .(X1, X4)) → U3_GA(X1, X2, X3, X4, flattenA_in_ga(cons(X2, X3), X4))
FLATTENA_IN_GA(cons(cons(atom(X1), X2), X3), .(X1, X4)) → FLATTENA_IN_GA(cons(X2, X3), X4)
FLATTENA_IN_GA(cons(cons(cons(X1, X2), X3), X4), X5) → U4_GA(X1, X2, X3, X4, X5, flattenA_in_ga(cons(X1, cons(X2, cons(X3, X4))), X5))
FLATTENA_IN_GA(cons(cons(cons(X1, X2), X3), X4), X5) → FLATTENA_IN_GA(cons(X1, cons(X2, cons(X3, X4))), X5)

R is empty.
The argument filtering Pi contains the following mapping:
flattenA_in_ga(x1, x2) = flattenA_in_ga(x1)
cons(x1, x2) = cons(x1, x2)
atom(x1) = atom(x1)
.(x1, x2) = .(x1, x2)
FLATTENA_IN_GA(x1, x2) = FLATTENA_IN_GA(x1)
U1_GA(x1, x2, x3, x4, x5) = U1_GA(x1, x2, x3, x5)
U2_GA(x1, x2, x3, x4, x5, x6) = U2_GA(x1, x2, x3, x4, x6)
U3_GA(x1, x2, x3, x4, x5) = U3_GA(x1, x2, x3, x5)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x1, x2, x3, x4, x6)

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:

FLATTENA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), .(X1, .(X2, X4))) → U1_GA(X1, X2, X3, X4, flattenA_in_ga(X3, X4))
FLATTENA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), .(X1, .(X2, X4))) → FLATTENA_IN_GA(X3, X4)
FLATTENA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), .(X1, X5)) → U2_GA(X1, X2, X3, X4, X5, flattenA_in_ga(cons(X2, cons(X3, X4)), X5))
FLATTENA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), .(X1, X5)) → FLATTENA_IN_GA(cons(X2, cons(X3, X4)), X5)
FLATTENA_IN_GA(cons(cons(atom(X1), X2), X3), .(X1, X4)) → U3_GA(X1, X2, X3, X4, flattenA_in_ga(cons(X2, X3), X4))
FLATTENA_IN_GA(cons(cons(atom(X1), X2), X3), .(X1, X4)) → FLATTENA_IN_GA(cons(X2, X3), X4)
FLATTENA_IN_GA(cons(cons(cons(X1, X2), X3), X4), X5) → U4_GA(X1, X2, X3, X4, X5, flattenA_in_ga(cons(X1, cons(X2, cons(X3, X4))), X5))
FLATTENA_IN_GA(cons(cons(cons(X1, X2), X3), X4), X5) → FLATTENA_IN_GA(cons(X1, cons(X2, cons(X3, X4))), X5)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(6) Obligation:

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

FLATTENA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4)), .(X1, X5)) → FLATTENA_IN_GA(cons(X2, cons(X3, X4)), X5)
FLATTENA_IN_GA(cons(atom(X1), cons(atom(X2), X3)), .(X1, .(X2, X4))) → FLATTENA_IN_GA(X3, X4)
FLATTENA_IN_GA(cons(cons(atom(X1), X2), X3), .(X1, X4)) → FLATTENA_IN_GA(cons(X2, X3), X4)
FLATTENA_IN_GA(cons(cons(cons(X1, X2), X3), X4), X5) → FLATTENA_IN_GA(cons(X1, cons(X2, cons(X3, X4))), X5)

R is empty.
The argument filtering Pi contains the following mapping:
cons(x1, x2) = cons(x1, x2)
atom(x1) = atom(x1)
.(x1, x2) = .(x1, x2)
FLATTENA_IN_GA(x1, x2) = FLATTENA_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:

FLATTENA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → FLATTENA_IN_GA(cons(X2, cons(X3, X4)))
FLATTENA_IN_GA(cons(atom(X1), cons(atom(X2), X3))) → FLATTENA_IN_GA(X3)
FLATTENA_IN_GA(cons(cons(atom(X1), X2), X3)) → FLATTENA_IN_GA(cons(X2, X3))
FLATTENA_IN_GA(cons(cons(cons(X1, X2), X3), X4)) → FLATTENA_IN_GA(cons(X1, cons(X2, cons(X3, X4))))

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

(9) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

The following dependency pairs can be deleted:

FLATTENA_IN_GA(cons(atom(X1), cons(cons(X2, X3), X4))) → FLATTENA_IN_GA(cons(X2, cons(X3, X4)))
FLATTENA_IN_GA(cons(atom(X1), cons(atom(X2), X3))) → FLATTENA_IN_GA(X3)
FLATTENA_IN_GA(cons(cons(atom(X1), X2), X3)) → FLATTENA_IN_GA(cons(X2, X3))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(FLATTENA_IN_GA(x₁)) = 2·x₁
POL(atom(x₁)) = x₁
POL(cons(x₁, x₂)) = x₁ + x₂

(10) Obligation:

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

FLATTENA_IN_GA(cons(cons(cons(X1, X2), X3), X4)) → FLATTENA_IN_GA(cons(X1, cons(X2, cons(X3, X4))))

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

(11) MRRProof (EQUIVALENT transformation)

By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

FLATTENA_IN_GA(cons(cons(cons(X1, X2), X3), X4)) → FLATTENA_IN_GA(cons(X1, cons(X2, cons(X3, X4))))

Used ordering: Polynomial interpretation [POLO]:

POL(FLATTENA_IN_GA(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = 2 + 2·x₁ + x₂

(12) Obligation:

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

(13) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.