Left Termination proof of /tmp/tmpDOCtiy/overlap1.pl

Left Termination of the query pattern overlap_in_2(g, g) w.r.t. the given Prolog program could successfully be proven:

0 Prolog

↳1 PrologToPrologProblemTransformerProof (⇐)

↳2 Prolog

↳3 PrologToPiTRSProof (⇐)

↳4 PiTRS

↳5 DependencyPairsProof (⇔)

↳6 PiDP

↳7 DependencyGraphProof (⇔)

↳8 AND

    ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

            ↳11 PiDP

                ↳12 PiDPToQDPProof (⇔)

                    ↳13 QDP

                        ↳14 QDPSizeChangeProof (⇔)

                            ↳15 TRUE

    ↳16 PiDP

        ↳17 UsableRulesProof (⇔)

            ↳18 PiDP

                ↳19 PiDPToQDPProof (⇔)

                    ↳20 QDP

                        ↳21 QDPSizeChangeProof (⇔)

                            ↳22 TRUE

    ↳23 PiDP

        ↳24 UsableRulesProof (⇔)

            ↳25 PiDP

                ↳26 PiDPToQDPProof (⇐)

                    ↳27 QDP

                        ↳28 QDPSizeChangeProof (⇔)

                            ↳29 TRUE

(0) Obligation:

Clauses:

overlap(Xs, Ys) :- ','(member(X, Xs), member(X, Ys)).
member(X1, []) :- ','(!, failure(a)).
member(X, Y) :- head(Y, X).
member(X, Y) :- ','(tail(Y, T), member(X, T)).
head([], X2).
head(.(H, X3), H).
tail([], []).
tail(.(X4, T), T).
failure(b).

Queries:

overlap(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: overlap(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: overlap(T1, T2), SCOPE: 1, CLAUSE: 0\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(member(X9, T3), member(X9, T4))\n\nT3 is ground\nT4 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: member(X9, T3)\n\nT3 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: member(T5, T4)\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: member(X9, T3), SCOPE: 2, CLAUSE: 1 | member(X9, T3), SCOPE: 2, CLAUSE: 2 | member(X9, T3), SCOPE: 2, CLAUSE: 3 | ?_2\n\nT3 is ground\nX9 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, failure(a)) | member(X9, []), SCOPE: 2, CLAUSE: 2 | member(X9, []), SCOPE: 2, CLAUSE: 3 | ?_2\n\nX9 is free", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: member(X9, T3), SCOPE: 2, CLAUSE: 2 | member(X9, T3), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX9 is free; \nX11 is free; \nmember(X9, T3) !=? member(X11, [])", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: failure(a), SCOPE: 3, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: member(X9, T3), SCOPE: 2, CLAUSE: 2\n\nT3 is ground\nX9 is free; \nX11 is free; \nmember(X9, T3) !=? member(X11, [])", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: member(X9, T3), SCOPE: 2, CLAUSE: 3\n\nT3 is ground\nX9 is free; \nX11 is free; \nmember(X9, T3) !=? member(X11, [])", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: head(T6, X19)\n\nT6 is ground\nX9 is free; \nX11 is free; \nX19 is free; \nmember(X9, T6) !=? member(X11, [])", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: head(T6, X19), SCOPE: 4, CLAUSE: 4 | head(T6, X19), SCOPE: 4, CLAUSE: 5\n\nT6 is ground\nX9 is free; \nX11 is free; \nX19 is free; \nmember(X9, T6) !=? member(X11, [])", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: head(T6, X19), SCOPE: 4, CLAUSE: 5\n\nT6 is ground\nX9 is free; \nX11 is free; \nX19 is free; \nX21 is free; \nmember(X9, T6) !=? member(X11, [])\nhead(T6, X19) !=? head([], X21)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: true\n\nX9 is free\nX11 is free; \nX19 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\nX9 is free\nX11 is free; \nX19 is free; \nX21 is free; \nX24 is free; \nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(tail(T9, X30), member(X31, X30))\n\nT9 is ground\nX9 is free; \nX11 is free; \nX31 is free; \nX30 is free; \nmember(X9, T9) !=? member(X11, [])", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(tail(T9, X30), member(X31, X30)), SCOPE: 5, CLAUSE: 6 | ','(tail(T9, X30), member(X31, X30)), SCOPE: 5, CLAUSE: 7\n\nT9 is ground\nX9 is free; \nX11 is free; \nX31 is free; \nX30 is free; \nmember(X9, T9) !=? member(X11, [])", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(tail(T9, X30), member(X31, X30)), SCOPE: 5, CLAUSE: 7\n\nT9 is ground\nX9 is free; \nX11 is free; \nX31 is free; \nX30 is free; \nmember(X9, T9) !=? member(X11, [])\ntail(T9, X30) !=? tail([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: member(X31, T11)\n\nT11 is ground\nX9 is free; \nX11 is free; \nX31 is free; \nX30 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\nX9 is free\nX11 is free; \nX30 is free; \nX34 is free; \nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: member(T5, T4), SCOPE: 6, CLAUSE: 1 | member(T5, T4), SCOPE: 6, CLAUSE: 2 | member(T5, T4), SCOPE: 6, CLAUSE: 3 | ?_6\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(!_6, failure(a)) | member(T5, []), SCOPE: 6, CLAUSE: 2 | member(T5, []), SCOPE: 6, CLAUSE: 3 | ?_6\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: member(T5, T4), SCOPE: 6, CLAUSE: 2 | member(T5, T4), SCOPE: 6, CLAUSE: 3\n\nT4 is ground\nX37 is free; \nmember(T5, T4) !=? member(X37, [])", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: failure(a)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: failure(a), SCOPE: 7, CLAUSE: 8\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: member(T5, T4), SCOPE: 6, CLAUSE: 2\n\nT4 is ground\nX37 is free; \nmember(T5, T4) !=? member(X37, [])", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: member(T5, T4), SCOPE: 6, CLAUSE: 3\n\nT4 is ground\nX37 is free; \nmember(T5, T4) !=? member(X37, [])", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: head(T14, T15)\n\nT14 is ground\nX37 is free; \nmember(T5, T14) !=? member(X37, [])", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: head(T14, T15), SCOPE: 8, CLAUSE: 4 | head(T14, T15), SCOPE: 8, CLAUSE: 5\n\nT14 is ground\nX37 is free; \nmember(T5, T14) !=? member(X37, [])", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: head(T14, T15), SCOPE: 8, CLAUSE: 5\n\nT14 is ground\nX37 is free; \nX45 is free; \nmember(T5, T14) !=? member(X37, [])\nhead(T14, T15) !=? head([], X45)", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: true\n\nX37 is free\nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\nX37 is free\nX45 is free; \nX48 is free; \nX49 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ','(tail(T19, X54), member(T20, X54))\n\nT19 is ground\nX37 is free; \nX54 is free; \nmember(T5, T19) !=? member(X37, [])", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: ','(tail(T19, X54), member(T20, X54)), SCOPE: 9, CLAUSE: 6 | ','(tail(T19, X54), member(T20, X54)), SCOPE: 9, CLAUSE: 7\n\nT19 is ground\nX37 is free; \nX54 is free; \nmember(T5, T19) !=? member(X37, [])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(tail(T19, X54), member(T20, X54)), SCOPE: 9, CLAUSE: 7\n\nT19 is ground\nX37 is free; \nX54 is free; \nmember(T5, T19) !=? member(X37, [])\ntail(T19, X54) !=? tail([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: member(T20, T22)\n\nT22 is ground\nX37 is free; \nX54 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: \n\nX37 is free\nX54 is free; \nX57 is free; \nX58 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 44 [arrowhead = none ];
44 -> 2;

45 [label="EVAL with clause\noverlap(X7, X8) :- ','(member(X9, X7), member(X9, X8)).\nand substitution\nT1 -> T3\nX7 -> T3\nT2 -> T4\nX8 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 45 [arrowhead = none ];
45 -> 3;

46 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 46 [arrowhead = none ];
46 -> 4;

47 [label="SPLIT 2\nreplacements:\nX9 -> T5", fontsize=14, style = filled, fillcolor = violet];
3 -> 47 [arrowhead = none ];
47 -> 5;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 48 [arrowhead = none ];
48 -> 6;

49 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 49 [arrowhead = none ];
49 -> 25;

50 [label="EVAL with clause\nmember(X11, []) :- ','(!, failure(a)).\nand substitution\nX9 -> X12\nX11 -> X12\nT3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 50 [arrowhead = none ];
50 -> 7;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 51 [arrowhead = none ];
51 -> 8;

52 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 52 [arrowhead = none ];
52 -> 9;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 53 [arrowhead = none ];
53 -> 12;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
8 -> 54 [arrowhead = none ];
54 -> 13;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 55 [arrowhead = none ];
55 -> 10;

56 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
10 -> 56 [arrowhead = none ];
56 -> 11;

57 [label="EVAL with clause\nmember(X17, X18) :- head(X18, X17).\nand substitution\nX9 -> X19\nX17 -> X19\nT3 -> T6\nX18 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 57 [arrowhead = none ];
57 -> 14;

58 [label="EVAL with clause\nmember(X28, X29) :- ','(tail(X29, X30), member(X28, X30)).\nand substitution\nX9 -> X31\nX28 -> X31\nT3 -> T9\nX29 -> T9", fontsize=14, style = filled, fillcolor = lightblue];
13 -> 58 [arrowhead = none ];
58 -> 20;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
14 -> 59 [arrowhead = none ];
59 -> 15;

60 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (member(X9, T6), member(X11, []))", fontsize=14, style = filled, fillcolor = white];
15 -> 60 [arrowhead = none ];
60 -> 16;

61 [label="EVAL with clause\nhead(.(X24, X25), X24).\nand substitution\nX24 -> T7\nX25 -> T8\nT6 -> .(T7, T8)\nX19 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 61 [arrowhead = none ];
61 -> 17;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 62 [arrowhead = none ];
62 -> 18;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 63 [arrowhead = none ];
63 -> 19;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 64 [arrowhead = none ];
64 -> 21;

65 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (member(X9, T9), member(X11, []))", fontsize=14, style = filled, fillcolor = white];
21 -> 65 [arrowhead = none ];
65 -> 22;

66 [label="EVAL with clause\ntail(.(X34, X35), X35).\nand substitution\nX34 -> T10\nX35 -> T11\nT9 -> .(T10, T11)\nX30 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 66 [arrowhead = none ];
66 -> 23;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 67 [arrowhead = none ];
67 -> 24;

68 [label="INSTANCE with matching:\nX9 -> X31\nT3 -> T11", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 68 [arrowhead = none , style=dashed];
68 -> 4[style=dashed];

69 [label="EVAL with clause\nmember(X37, []) :- ','(!, failure(a)).\nand substitution\nT5 -> T12\nX37 -> T12\nT4 -> []", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 69 [arrowhead = none ];
69 -> 26;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 70 [arrowhead = none ];
70 -> 27;

71 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 71 [arrowhead = none ];
71 -> 28;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 72 [arrowhead = none ];
72 -> 31;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
27 -> 73 [arrowhead = none ];
73 -> 32;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 74 [arrowhead = none ];
74 -> 29;

75 [label="BACKTRACK\nfor clause: failure(b)because of non-unification", fontsize=14, style = filled, fillcolor = white];
29 -> 75 [arrowhead = none ];
75 -> 30;

76 [label="EVAL with clause\nmember(X42, X43) :- head(X43, X42).\nand substitution\nT5 -> T15\nX42 -> T15\nT4 -> T14\nX43 -> T14\nT13 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 76 [arrowhead = none ];
76 -> 33;

77 [label="EVAL with clause\nmember(X52, X53) :- ','(tail(X53, X54), member(X52, X54)).\nand substitution\nT5 -> T20\nX52 -> T20\nT4 -> T19\nX53 -> T19\nT18 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
32 -> 77 [arrowhead = none ];
77 -> 39;

78 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
33 -> 78 [arrowhead = none ];
78 -> 34;

79 [label="BACKTRACK\nfor clause: head([], X2)\nwith clash: (member(T5, T14), member(X37, []))", fontsize=14, style = filled, fillcolor = white];
34 -> 79 [arrowhead = none ];
79 -> 35;

80 [label="EVAL with clause\nhead(.(X48, X49), X48).\nand substitution\nX48 -> T16\nX49 -> T17\nT14 -> .(T16, T17)\nT15 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 80 [arrowhead = none ];
80 -> 36;

81 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 81 [arrowhead = none ];
81 -> 37;

82 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 82 [arrowhead = none ];
82 -> 38;

83 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
39 -> 83 [arrowhead = none ];
83 -> 40;

84 [label="BACKTRACK\nfor clause: tail([], [])\nwith clash: (member(T5, T19), member(X37, []))", fontsize=14, style = filled, fillcolor = white];
40 -> 84 [arrowhead = none ];
84 -> 41;

85 [label="EVAL with clause\ntail(.(X57, X58), X58).\nand substitution\nX57 -> T21\nX58 -> T22\nT19 -> .(T21, T22)\nX54 -> T22", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 85 [arrowhead = none ];
85 -> 42;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
41 -> 86 [arrowhead = none ];
86 -> 43;

87 [label="INSTANCE with matching:\nT5 -> T20\nT4 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
42 -> 87 [arrowhead = none , style=dashed];
87 -> 5[style=dashed];

}

(2) Obligation:

Clauses:

member4(T7, .(T7, T8)).
member4(X31, .(T10, T11)) :- member4(X31, T11).
member5(T16, .(T16, T17)).
member5(T20, .(T21, T22)) :- member5(T20, T22).
overlap1(T3, T4) :- member4(X9, T3).
overlap1(T3, .(T16, T17)) :- member4(T16, T3).
overlap1(T3, .(T21, T22)) :- ','(member4(T20, T3), member5(T20, T22)).

Queries:

overlap1(g,g).

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
overlap1_in: (b,b)
member4_in: (f,b) (b,b)
member5_in: (b,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

(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:

OVERLAP1_IN_GG(T3, T4) → U3_GG(T3, T4, member4_in_ag(X9, T3))
OVERLAP1_IN_GG(T3, T4) → MEMBER4_IN_AG(X9, T3)
MEMBER4_IN_AG(X31, .(T10, T11)) → U1_AG(X31, T10, T11, member4_in_ag(X31, T11))
MEMBER4_IN_AG(X31, .(T10, T11)) → MEMBER4_IN_AG(X31, T11)
OVERLAP1_IN_GG(T3, .(T16, T17)) → U4_GG(T3, T16, T17, member4_in_gg(T16, T3))
OVERLAP1_IN_GG(T3, .(T16, T17)) → MEMBER4_IN_GG(T16, T3)
MEMBER4_IN_GG(X31, .(T10, T11)) → U1_GG(X31, T10, T11, member4_in_gg(X31, T11))
MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)
OVERLAP1_IN_GG(T3, .(T21, T22)) → U5_GG(T3, T21, T22, member4_in_ag(T20, T3))
OVERLAP1_IN_GG(T3, .(T21, T22)) → MEMBER4_IN_AG(T20, T3)
U5_GG(T3, T21, T22, member4_out_ag(T20, T3)) → U6_GG(T3, T21, T22, member5_in_gg(T20, T22))
U5_GG(T3, T21, T22, member4_out_ag(T20, T3)) → MEMBER5_IN_GG(T20, T22)
MEMBER5_IN_GG(T20, .(T21, T22)) → U2_GG(T20, T21, T22, member5_in_gg(T20, T22))
MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

The argument filtering Pi contains the following mapping:
overlap1_in_gg(x1, x2) = overlap1_in_gg(x1, x2)
U3_gg(x1, x2, x3) = U3_gg(x3)
member4_in_ag(x1, x2) = member4_in_ag(x2)
.(x1, x2) = .(x1, x2)
member4_out_ag(x1, x2) = member4_out_ag(x1)
U1_ag(x1, x2, x3, x4) = U1_ag(x4)
overlap1_out_gg(x1, x2) = overlap1_out_gg
U4_gg(x1, x2, x3, x4) = U4_gg(x4)
member4_in_gg(x1, x2) = member4_in_gg(x1, x2)
member4_out_gg(x1, x2) = member4_out_gg
U1_gg(x1, x2, x3, x4) = U1_gg(x4)
U5_gg(x1, x2, x3, x4) = U5_gg(x3, x4)
U6_gg(x1, x2, x3, x4) = U6_gg(x4)
member5_in_gg(x1, x2) = member5_in_gg(x1, x2)
member5_out_gg(x1, x2) = member5_out_gg
U2_gg(x1, x2, x3, x4) = U2_gg(x4)
OVERLAP1_IN_GG(x1, x2) = OVERLAP1_IN_GG(x1, x2)
U3_GG(x1, x2, x3) = U3_GG(x3)
MEMBER4_IN_AG(x1, x2) = MEMBER4_IN_AG(x2)
U1_AG(x1, x2, x3, x4) = U1_AG(x4)
U4_GG(x1, x2, x3, x4) = U4_GG(x4)
MEMBER4_IN_GG(x1, x2) = MEMBER4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4) = U1_GG(x4)
U5_GG(x1, x2, x3, x4) = U5_GG(x3, x4)
U6_GG(x1, x2, x3, x4) = U6_GG(x4)
MEMBER5_IN_GG(x1, x2) = MEMBER5_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4) = U2_GG(x4)

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

(6) Obligation:

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

OVERLAP1_IN_GG(T3, T4) → U3_GG(T3, T4, member4_in_ag(X9, T3))
OVERLAP1_IN_GG(T3, T4) → MEMBER4_IN_AG(X9, T3)
MEMBER4_IN_AG(X31, .(T10, T11)) → U1_AG(X31, T10, T11, member4_in_ag(X31, T11))
MEMBER4_IN_AG(X31, .(T10, T11)) → MEMBER4_IN_AG(X31, T11)
OVERLAP1_IN_GG(T3, .(T16, T17)) → U4_GG(T3, T16, T17, member4_in_gg(T16, T3))
OVERLAP1_IN_GG(T3, .(T16, T17)) → MEMBER4_IN_GG(T16, T3)
MEMBER4_IN_GG(X31, .(T10, T11)) → U1_GG(X31, T10, T11, member4_in_gg(X31, T11))
MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)
OVERLAP1_IN_GG(T3, .(T21, T22)) → U5_GG(T3, T21, T22, member4_in_ag(T20, T3))
OVERLAP1_IN_GG(T3, .(T21, T22)) → MEMBER4_IN_AG(T20, T3)
U5_GG(T3, T21, T22, member4_out_ag(T20, T3)) → U6_GG(T3, T21, T22, member5_in_gg(T20, T22))
U5_GG(T3, T21, T22, member4_out_ag(T20, T3)) → MEMBER5_IN_GG(T20, T22)
MEMBER5_IN_GG(T20, .(T21, T22)) → U2_GG(T20, T21, T22, member5_in_gg(T20, T22))
MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 11 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

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

(12) PiDPToQDPProof (EQUIVALENT transformation)

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

(13) Obligation:

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

MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)

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

(14) 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:

MEMBER5_IN_GG(T20, .(T21, T22)) → MEMBER5_IN_GG(T20, T22)
The graph contains the following edges 1 >= 1, 2 > 2

(15) TRUE

(16) Obligation:

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

MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

(17) UsableRulesProof (EQUIVALENT transformation)

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

(18) Obligation:

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

MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)

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

(19) PiDPToQDPProof (EQUIVALENT transformation)

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

(20) Obligation:

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

MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)

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

(21) 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:

MEMBER4_IN_GG(X31, .(T10, T11)) → MEMBER4_IN_GG(X31, T11)
The graph contains the following edges 1 >= 1, 2 > 2

(22) TRUE

(23) Obligation:

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

MEMBER4_IN_AG(X31, .(T10, T11)) → MEMBER4_IN_AG(X31, T11)

The TRS R consists of the following rules:

overlap1_in_gg(T3, T4) → U3_gg(T3, T4, member4_in_ag(X9, T3))
member4_in_ag(T7, .(T7, T8)) → member4_out_ag(T7, .(T7, T8))
member4_in_ag(X31, .(T10, T11)) → U1_ag(X31, T10, T11, member4_in_ag(X31, T11))
U1_ag(X31, T10, T11, member4_out_ag(X31, T11)) → member4_out_ag(X31, .(T10, T11))
U3_gg(T3, T4, member4_out_ag(X9, T3)) → overlap1_out_gg(T3, T4)
overlap1_in_gg(T3, .(T16, T17)) → U4_gg(T3, T16, T17, member4_in_gg(T16, T3))
member4_in_gg(T7, .(T7, T8)) → member4_out_gg(T7, .(T7, T8))
member4_in_gg(X31, .(T10, T11)) → U1_gg(X31, T10, T11, member4_in_gg(X31, T11))
U1_gg(X31, T10, T11, member4_out_gg(X31, T11)) → member4_out_gg(X31, .(T10, T11))
U4_gg(T3, T16, T17, member4_out_gg(T16, T3)) → overlap1_out_gg(T3, .(T16, T17))
overlap1_in_gg(T3, .(T21, T22)) → U5_gg(T3, T21, T22, member4_in_ag(T20, T3))
U5_gg(T3, T21, T22, member4_out_ag(T20, T3)) → U6_gg(T3, T21, T22, member5_in_gg(T20, T22))
member5_in_gg(T16, .(T16, T17)) → member5_out_gg(T16, .(T16, T17))
member5_in_gg(T20, .(T21, T22)) → U2_gg(T20, T21, T22, member5_in_gg(T20, T22))
U2_gg(T20, T21, T22, member5_out_gg(T20, T22)) → member5_out_gg(T20, .(T21, T22))
U6_gg(T3, T21, T22, member5_out_gg(T20, T22)) → overlap1_out_gg(T3, .(T21, T22))

(24) UsableRulesProof (EQUIVALENT transformation)

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

(25) Obligation:

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

MEMBER4_IN_AG(X31, .(T10, T11)) → MEMBER4_IN_AG(X31, T11)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2) = .(x1, x2)
MEMBER4_IN_AG(x1, x2) = MEMBER4_IN_AG(x2)

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

(26) PiDPToQDPProof (SOUND transformation)

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

(27) Obligation:

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

MEMBER4_IN_AG(.(T10, T11)) → MEMBER4_IN_AG(T11)

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

(28) 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:

MEMBER4_IN_AG(.(T10, T11)) → MEMBER4_IN_AG(T11)
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) Complex Obligation (AND)

(9) Obligation:

(10) UsableRulesProof (EQUIVALENT transformation)

(11) Obligation:

(12) PiDPToQDPProof (EQUIVALENT transformation)

(13) Obligation:

(14) QDPSizeChangeProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) UsableRulesProof (EQUIVALENT transformation)

(18) Obligation:

(19) PiDPToQDPProof (EQUIVALENT transformation)

(20) Obligation:

(21) QDPSizeChangeProof (EQUIVALENT transformation)

(22) TRUE

(23) Obligation:

(24) UsableRulesProof (EQUIVALENT transformation)

(25) Obligation:

(26) PiDPToQDPProof (SOUND transformation)

(27) Obligation:

(28) QDPSizeChangeProof (EQUIVALENT transformation)

(29) TRUE