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

(0) Obligation:

Clauses:

star(U, []) :- !.
star([], W) :- ','(!, =(W, [])).
star(U, W) :- ','(app(U, V, W), star(U, V)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).
=(X, X).

Query: star(g,g)

(1) BuiltinConflictTransformerProof (EQUIVALENT transformation)

Renamed defined predicates conflicting with built-in predicates [PROLOG].

(2) Obligation:

Clauses:

star(U, []) :- !.
star([], W) :- ','(!, user_defined_=(W, [])).
star(U, W) :- ','(app(U, V, W), star(U, V)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).
user_defined_=(X, X).

Query: star(g,g)

(3) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: star(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: star(T1, T2), SCOPE: 1, CLAUSE: 0 | star(T1, T2), SCOPE: 1, CLAUSE: 1 | star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | star(T4, []), SCOPE: 1, CLAUSE: 1 | star(T4, []), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: star(T1, T2), SCOPE: 1, CLAUSE: 1 | star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX2 is free\nstar(T1, T2) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_1, user_defined_=(T6, [])) | star([], T6), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX2 is free\nstar([], T6) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX2 is free\nX4 is free\nstar(T1, T2) !=? star(X2, [])\nstar(T1, T2) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: user_defined_=(T6, [])\n\nT6 is ground\nX2 is free\nstar([], T6) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: user_defined_=(T6, []), SCOPE: 2, CLAUSE: 5\n\nT6 is ground\nX2 is free\nstar([], T6) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(app(T10, X10, T11), star(T10, X10))\n\nT10 is ground\nT11 is ground\nX2 is free\nX4 is free\nX10 is free\nstar(T10, T11) !=? star(X2, [])\nstar(T10, T11) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(app(T10, X10, T11), star(T10, X10)), SCOPE: 3, CLAUSE: 3 | ','(app(T10, X10, T11), star(T10, X10)), SCOPE: 3, CLAUSE: 4\n\nT10 is ground\nT11 is ground\nX2 is free\nX4 is free\nX10 is free\nstar(T10, T11) !=? star(X2, [])\nstar(T10, T11) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(app(T10, X10, T11), star(T10, X10)), SCOPE: 3, CLAUSE: 4\n\nT10 is ground\nT11 is ground\nX2 is free\nX4 is free\nX10 is free\nstar(T10, T11) !=? star(X2, [])\nstar(T10, T11) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(app(T20, X27, T21), star(.(T19, T20), X27))\n\nT19 is ground\nT20 is ground\nT21 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app(T20, X27, T21)\n\nT20 is ground\nT21 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: star(.(T19, T20), T24)\n\nT19 is ground\nT20 is ground\nT24 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(T20, X27, T21), SCOPE: 4, CLAUSE: 3 | app(T20, X27, T21), SCOPE: 4, CLAUSE: 4\n\nT20 is ground\nT21 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T20, X27, T21), SCOPE: 4, CLAUSE: 3\n\nT20 is ground\nT21 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(T20, X27, T21), SCOPE: 4, CLAUSE: 4\n\nT20 is ground\nT21 is ground\nX27 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app(T39, X60, T40)\n\nT39 is ground\nT40 is ground\nX60 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 27 [arrowhead = none ];
27 -> 2;

28 [label="EVAL with clause\nstar(X2, []) :- !_1.\nand substitutionT1 -> T4,\nX2 -> T4,\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 28 [arrowhead = none ];
28 -> 3;

29 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 29 [arrowhead = none ];
29 -> 4;

30 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 30 [arrowhead = none ];
30 -> 5;

31 [label="EVAL with clause\nstar([], X4) :- ','(!_1, user_defined_=(X4, [])).\nand substitutionT1 -> [],\nT2 -> T6,\nX4 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 31 [arrowhead = none ];
31 -> 7;

32 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 32 [arrowhead = none ];
32 -> 8;

33 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 33 [arrowhead = none ];
33 -> 6;

34 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 34 [arrowhead = none ];
34 -> 9;

35 [label="ONLY EVAL with clause\nstar(X8, X9) :- ','(app(X8, X10, X9), star(X8, X10)).\nand substitutionT1 -> T10,\nX8 -> T10,\nT2 -> T11,\nX9 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 35 [arrowhead = none ];
35 -> 12;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 36 [arrowhead = none ];
36 -> 10;

37 [label="BACKTRACK\nfor clause: user_defined_=(X, X)\nwith clash: (star([], T6), star(X2, []))", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 37 [arrowhead = none ];
37 -> 11;

38 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 38 [arrowhead = none ];
38 -> 13;

39 [label="BACKTRACK\nfor clause: app([], L, L)\nwith clash: (star(T10, T11), star([], X4))", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 39 [arrowhead = none ];
39 -> 14;

40 [label="EVAL with clause\napp(.(X23, X24), X25, .(X23, X26)) :- app(X24, X25, X26).\nand substitutionX23 -> T19,\nX24 -> T20,\nT10 -> .(T19, T20),\nX10 -> X27,\nX25 -> X27,\nX26 -> T21,\nT11 -> .(T19, T21)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 40 [arrowhead = none ];
40 -> 15;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 41 [arrowhead = none ];
41 -> 16;

42 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
15 -> 42 [arrowhead = none ];
42 -> 17;

43 [label="SPLIT 2\nnew knowledge:\nT20 is ground\nT24 is ground\nT21 is ground\nreplacements:X27 -> T24", fontsize=14, style = filled, fillcolor = violet];
15 -> 43 [arrowhead = none ];
43 -> 18;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 44 [arrowhead = none ];
44 -> 19;

45 [label="INSTANCE with matching:\nT1 -> .(T19, T20)\nT2 -> T24", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 45 [arrowhead = none , style=dashed];
45 -> 1[style=dashed];

46 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 46 [arrowhead = none ];
46 -> 20;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 47 [arrowhead = none ];
47 -> 21;

48 [label="EVAL with clause\napp([], X44, X44).\nand substitutionT20 -> [],\nX27 -> T31,\nX44 -> T31,\nT21 -> T31,\nX45 -> T31", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 48 [arrowhead = none ];
48 -> 22;

49 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 49 [arrowhead = none ];
49 -> 23;

50 [label="EVAL with clause\napp(.(X56, X57), X58, .(X56, X59)) :- app(X57, X58, X59).\nand substitutionX56 -> T38,\nX57 -> T39,\nT20 -> .(T38, T39),\nX27 -> X60,\nX58 -> X60,\nX59 -> T40,\nT21 -> .(T38, T40)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 50 [arrowhead = none ];
50 -> 25;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 51 [arrowhead = none ];
51 -> 26;

52 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 52 [arrowhead = none ];
52 -> 24;

53 [label="INSTANCE with matching:\nT20 -> T39\nX27 -> X60\nT21 -> T40", fontsize=14, style = filled, fillcolor = lightgrey];
25 -> 53 [arrowhead = none , style=dashed];
53 -> 17[style=dashed];

}

(4) Obligation:

Clauses:

appA([], T31, T31).
appA(.(T38, T39), X60, .(T38, T40)) :- appA(T39, X60, T40).
starB(T4, []).
starB(.(T19, T20), .(T19, T21)) :- appA(T20, X27, T21).
starB(.(T19, T20), .(T19, T21)) :- ','(appA(T20, T24, T21), starB(.(T19, T20), T24)).

Query: starB(g,g)

(5) PrologToPiTRSProof (SOUND transformation)

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

starB_in_gg(T4, []) → starB_out_gg(T4, [])
starB_in_gg(.(T19, T20), .(T19, T21)) → U2_gg(T19, T20, T21, appA_in_gag(T20, X27, T21))
appA_in_gag([], T31, T31) → appA_out_gag([], T31, T31)
appA_in_gag(.(T38, T39), X60, .(T38, T40)) → U1_gag(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
U1_gag(T38, T39, X60, T40, appA_out_gag(T39, X60, T40)) → appA_out_gag(.(T38, T39), X60, .(T38, T40))
U2_gg(T19, T20, T21, appA_out_gag(T20, X27, T21)) → starB_out_gg(.(T19, T20), .(T19, T21))
starB_in_gg(.(T19, T20), .(T19, T21)) → U3_gg(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_gg(T19, T20, T21, appA_out_gag(T20, T24, T21)) → U4_gg(T19, T20, T21, starB_in_gg(.(T19, T20), T24))
U4_gg(T19, T20, T21, starB_out_gg(.(T19, T20), T24)) → starB_out_gg(.(T19, T20), .(T19, T21))

The argument filtering Pi contains the following mapping:
starB_in_gg(x1, x2)  =  starB_in_gg(x1, x2)
[]  =  []
starB_out_gg(x1, x2)  =  starB_out_gg
.(x1, x2)  =  .(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
appA_in_gag(x1, x2, x3)  =  appA_in_gag(x1, x3)
appA_out_gag(x1, x2, x3)  =  appA_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x4)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(6) Obligation:

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

starB_in_gg(T4, []) → starB_out_gg(T4, [])
starB_in_gg(.(T19, T20), .(T19, T21)) → U2_gg(T19, T20, T21, appA_in_gag(T20, X27, T21))
appA_in_gag([], T31, T31) → appA_out_gag([], T31, T31)
appA_in_gag(.(T38, T39), X60, .(T38, T40)) → U1_gag(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
U1_gag(T38, T39, X60, T40, appA_out_gag(T39, X60, T40)) → appA_out_gag(.(T38, T39), X60, .(T38, T40))
U2_gg(T19, T20, T21, appA_out_gag(T20, X27, T21)) → starB_out_gg(.(T19, T20), .(T19, T21))
starB_in_gg(.(T19, T20), .(T19, T21)) → U3_gg(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_gg(T19, T20, T21, appA_out_gag(T20, T24, T21)) → U4_gg(T19, T20, T21, starB_in_gg(.(T19, T20), T24))
U4_gg(T19, T20, T21, starB_out_gg(.(T19, T20), T24)) → starB_out_gg(.(T19, T20), .(T19, T21))

The argument filtering Pi contains the following mapping:
starB_in_gg(x1, x2)  =  starB_in_gg(x1, x2)
[]  =  []
starB_out_gg(x1, x2)  =  starB_out_gg
.(x1, x2)  =  .(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
appA_in_gag(x1, x2, x3)  =  appA_in_gag(x1, x3)
appA_out_gag(x1, x2, x3)  =  appA_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x4)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)

(7) 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:

STARB_IN_GG(.(T19, T20), .(T19, T21)) → U2_GG(T19, T20, T21, appA_in_gag(T20, X27, T21))
STARB_IN_GG(.(T19, T20), .(T19, T21)) → APPA_IN_GAG(T20, X27, T21)
APPA_IN_GAG(.(T38, T39), X60, .(T38, T40)) → U1_GAG(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
APPA_IN_GAG(.(T38, T39), X60, .(T38, T40)) → APPA_IN_GAG(T39, X60, T40)
STARB_IN_GG(.(T19, T20), .(T19, T21)) → U3_GG(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_GG(T19, T20, T21, appA_out_gag(T20, T24, T21)) → U4_GG(T19, T20, T21, starB_in_gg(.(T19, T20), T24))
U3_GG(T19, T20, T21, appA_out_gag(T20, T24, T21)) → STARB_IN_GG(.(T19, T20), T24)

The TRS R consists of the following rules:

starB_in_gg(T4, []) → starB_out_gg(T4, [])
starB_in_gg(.(T19, T20), .(T19, T21)) → U2_gg(T19, T20, T21, appA_in_gag(T20, X27, T21))
appA_in_gag([], T31, T31) → appA_out_gag([], T31, T31)
appA_in_gag(.(T38, T39), X60, .(T38, T40)) → U1_gag(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
U1_gag(T38, T39, X60, T40, appA_out_gag(T39, X60, T40)) → appA_out_gag(.(T38, T39), X60, .(T38, T40))
U2_gg(T19, T20, T21, appA_out_gag(T20, X27, T21)) → starB_out_gg(.(T19, T20), .(T19, T21))
starB_in_gg(.(T19, T20), .(T19, T21)) → U3_gg(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_gg(T19, T20, T21, appA_out_gag(T20, T24, T21)) → U4_gg(T19, T20, T21, starB_in_gg(.(T19, T20), T24))
U4_gg(T19, T20, T21, starB_out_gg(.(T19, T20), T24)) → starB_out_gg(.(T19, T20), .(T19, T21))

The argument filtering Pi contains the following mapping:
starB_in_gg(x1, x2)  =  starB_in_gg(x1, x2)
[]  =  []
starB_out_gg(x1, x2)  =  starB_out_gg
.(x1, x2)  =  .(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
appA_in_gag(x1, x2, x3)  =  appA_in_gag(x1, x3)
appA_out_gag(x1, x2, x3)  =  appA_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x4)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
STARB_IN_GG(x1, x2)  =  STARB_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)
APPA_IN_GAG(x1, x2, x3)  =  APPA_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x5)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)

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

(8) Obligation:

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

STARB_IN_GG(.(T19, T20), .(T19, T21)) → U2_GG(T19, T20, T21, appA_in_gag(T20, X27, T21))
STARB_IN_GG(.(T19, T20), .(T19, T21)) → APPA_IN_GAG(T20, X27, T21)
APPA_IN_GAG(.(T38, T39), X60, .(T38, T40)) → U1_GAG(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
APPA_IN_GAG(.(T38, T39), X60, .(T38, T40)) → APPA_IN_GAG(T39, X60, T40)
STARB_IN_GG(.(T19, T20), .(T19, T21)) → U3_GG(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_GG(T19, T20, T21, appA_out_gag(T20, T24, T21)) → U4_GG(T19, T20, T21, starB_in_gg(.(T19, T20), T24))
U3_GG(T19, T20, T21, appA_out_gag(T20, T24, T21)) → STARB_IN_GG(.(T19, T20), T24)

The TRS R consists of the following rules:

starB_in_gg(T4, []) → starB_out_gg(T4, [])
starB_in_gg(.(T19, T20), .(T19, T21)) → U2_gg(T19, T20, T21, appA_in_gag(T20, X27, T21))
appA_in_gag([], T31, T31) → appA_out_gag([], T31, T31)
appA_in_gag(.(T38, T39), X60, .(T38, T40)) → U1_gag(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
U1_gag(T38, T39, X60, T40, appA_out_gag(T39, X60, T40)) → appA_out_gag(.(T38, T39), X60, .(T38, T40))
U2_gg(T19, T20, T21, appA_out_gag(T20, X27, T21)) → starB_out_gg(.(T19, T20), .(T19, T21))
starB_in_gg(.(T19, T20), .(T19, T21)) → U3_gg(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_gg(T19, T20, T21, appA_out_gag(T20, T24, T21)) → U4_gg(T19, T20, T21, starB_in_gg(.(T19, T20), T24))
U4_gg(T19, T20, T21, starB_out_gg(.(T19, T20), T24)) → starB_out_gg(.(T19, T20), .(T19, T21))

The argument filtering Pi contains the following mapping:
starB_in_gg(x1, x2)  =  starB_in_gg(x1, x2)
[]  =  []
starB_out_gg(x1, x2)  =  starB_out_gg
.(x1, x2)  =  .(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
appA_in_gag(x1, x2, x3)  =  appA_in_gag(x1, x3)
appA_out_gag(x1, x2, x3)  =  appA_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x4)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
STARB_IN_GG(x1, x2)  =  STARB_IN_GG(x1, x2)
U2_GG(x1, x2, x3, x4)  =  U2_GG(x4)
APPA_IN_GAG(x1, x2, x3)  =  APPA_IN_GAG(x1, x3)
U1_GAG(x1, x2, x3, x4, x5)  =  U1_GAG(x5)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x4)
U4_GG(x1, x2, x3, x4)  =  U4_GG(x4)

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

(9) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 4 less nodes.

(10) Complex Obligation (AND)

(11) Obligation:

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

APPA_IN_GAG(.(T38, T39), X60, .(T38, T40)) → APPA_IN_GAG(T39, X60, T40)

The TRS R consists of the following rules:

starB_in_gg(T4, []) → starB_out_gg(T4, [])
starB_in_gg(.(T19, T20), .(T19, T21)) → U2_gg(T19, T20, T21, appA_in_gag(T20, X27, T21))
appA_in_gag([], T31, T31) → appA_out_gag([], T31, T31)
appA_in_gag(.(T38, T39), X60, .(T38, T40)) → U1_gag(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
U1_gag(T38, T39, X60, T40, appA_out_gag(T39, X60, T40)) → appA_out_gag(.(T38, T39), X60, .(T38, T40))
U2_gg(T19, T20, T21, appA_out_gag(T20, X27, T21)) → starB_out_gg(.(T19, T20), .(T19, T21))
starB_in_gg(.(T19, T20), .(T19, T21)) → U3_gg(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_gg(T19, T20, T21, appA_out_gag(T20, T24, T21)) → U4_gg(T19, T20, T21, starB_in_gg(.(T19, T20), T24))
U4_gg(T19, T20, T21, starB_out_gg(.(T19, T20), T24)) → starB_out_gg(.(T19, T20), .(T19, T21))

The argument filtering Pi contains the following mapping:
starB_in_gg(x1, x2)  =  starB_in_gg(x1, x2)
[]  =  []
starB_out_gg(x1, x2)  =  starB_out_gg
.(x1, x2)  =  .(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
appA_in_gag(x1, x2, x3)  =  appA_in_gag(x1, x3)
appA_out_gag(x1, x2, x3)  =  appA_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x4)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
APPA_IN_GAG(x1, x2, x3)  =  APPA_IN_GAG(x1, x3)

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

(12) UsableRulesProof (EQUIVALENT transformation)

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

(13) Obligation:

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

APPA_IN_GAG(.(T38, T39), X60, .(T38, T40)) → APPA_IN_GAG(T39, X60, T40)

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

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

(14) PiDPToQDPProof (SOUND transformation)

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

(15) Obligation:

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

APPA_IN_GAG(.(T38, T39), .(T38, T40)) → APPA_IN_GAG(T39, T40)

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

(16) 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:

• APPA_IN_GAG(.(T38, T39), .(T38, T40)) → APPA_IN_GAG(T39, T40)
  The graph contains the following edges 1 > 1, 2 > 2

(17) YES

(18) Obligation:

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

STARB_IN_GG(.(T19, T20), .(T19, T21)) → U3_GG(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_GG(T19, T20, T21, appA_out_gag(T20, T24, T21)) → STARB_IN_GG(.(T19, T20), T24)

The TRS R consists of the following rules:

starB_in_gg(T4, []) → starB_out_gg(T4, [])
starB_in_gg(.(T19, T20), .(T19, T21)) → U2_gg(T19, T20, T21, appA_in_gag(T20, X27, T21))
appA_in_gag([], T31, T31) → appA_out_gag([], T31, T31)
appA_in_gag(.(T38, T39), X60, .(T38, T40)) → U1_gag(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
U1_gag(T38, T39, X60, T40, appA_out_gag(T39, X60, T40)) → appA_out_gag(.(T38, T39), X60, .(T38, T40))
U2_gg(T19, T20, T21, appA_out_gag(T20, X27, T21)) → starB_out_gg(.(T19, T20), .(T19, T21))
starB_in_gg(.(T19, T20), .(T19, T21)) → U3_gg(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_gg(T19, T20, T21, appA_out_gag(T20, T24, T21)) → U4_gg(T19, T20, T21, starB_in_gg(.(T19, T20), T24))
U4_gg(T19, T20, T21, starB_out_gg(.(T19, T20), T24)) → starB_out_gg(.(T19, T20), .(T19, T21))

The argument filtering Pi contains the following mapping:
starB_in_gg(x1, x2)  =  starB_in_gg(x1, x2)
[]  =  []
starB_out_gg(x1, x2)  =  starB_out_gg
.(x1, x2)  =  .(x1, x2)
U2_gg(x1, x2, x3, x4)  =  U2_gg(x4)
appA_in_gag(x1, x2, x3)  =  appA_in_gag(x1, x3)
appA_out_gag(x1, x2, x3)  =  appA_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
U3_gg(x1, x2, x3, x4)  =  U3_gg(x1, x2, x4)
U4_gg(x1, x2, x3, x4)  =  U4_gg(x4)
STARB_IN_GG(x1, x2)  =  STARB_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x4)

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

(19) UsableRulesProof (EQUIVALENT transformation)

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

(20) Obligation:

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

STARB_IN_GG(.(T19, T20), .(T19, T21)) → U3_GG(T19, T20, T21, appA_in_gag(T20, T24, T21))
U3_GG(T19, T20, T21, appA_out_gag(T20, T24, T21)) → STARB_IN_GG(.(T19, T20), T24)

The TRS R consists of the following rules:

appA_in_gag([], T31, T31) → appA_out_gag([], T31, T31)
appA_in_gag(.(T38, T39), X60, .(T38, T40)) → U1_gag(T38, T39, X60, T40, appA_in_gag(T39, X60, T40))
U1_gag(T38, T39, X60, T40, appA_out_gag(T39, X60, T40)) → appA_out_gag(.(T38, T39), X60, .(T38, T40))

The argument filtering Pi contains the following mapping:
[]  =  []
.(x1, x2)  =  .(x1, x2)
appA_in_gag(x1, x2, x3)  =  appA_in_gag(x1, x3)
appA_out_gag(x1, x2, x3)  =  appA_out_gag(x2)
U1_gag(x1, x2, x3, x4, x5)  =  U1_gag(x5)
STARB_IN_GG(x1, x2)  =  STARB_IN_GG(x1, x2)
U3_GG(x1, x2, x3, x4)  =  U3_GG(x1, x2, x4)

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

(21) PiDPToQDPProof (SOUND transformation)

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

(22) Obligation:

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

STARB_IN_GG(.(T19, T20), .(T19, T21)) → U3_GG(T19, T20, appA_in_gag(T20, T21))
U3_GG(T19, T20, appA_out_gag(T24)) → STARB_IN_GG(.(T19, T20), T24)

The TRS R consists of the following rules:

appA_in_gag([], T31) → appA_out_gag(T31)
appA_in_gag(.(T38, T39), .(T38, T40)) → U1_gag(appA_in_gag(T39, T40))
U1_gag(appA_out_gag(X60)) → appA_out_gag(X60)

The set Q consists of the following terms:

appA_in_gag(x0, x1)
U1_gag(x0)

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

STARB_IN_GG(.(T19, T20), .(T19, T21)) → U3_GG(T19, T20, appA_in_gag(T20, T21))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(.(x₁, x₂)) = 1 + x₂   
POL(STARB_IN_GG(x₁, x₂)) = x₂   
POL(U1_gag(x₁)) = x₁   
POL(U3_GG(x₁, x₂, x₃)) = x₃   
POL([]) = 0   
POL(appA_in_gag(x₁, x₂)) = x₂   
POL(appA_out_gag(x₁)) = x₁   

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

appA_in_gag([], T31) → appA_out_gag(T31)
appA_in_gag(.(T38, T39), .(T38, T40)) → U1_gag(appA_in_gag(T39, T40))
U1_gag(appA_out_gag(X60)) → appA_out_gag(X60)

(24) Obligation:

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

U3_GG(T19, T20, appA_out_gag(T24)) → STARB_IN_GG(.(T19, T20), T24)

The TRS R consists of the following rules:

appA_in_gag([], T31) → appA_out_gag(T31)
appA_in_gag(.(T38, T39), .(T38, T40)) → U1_gag(appA_in_gag(T39, T40))
U1_gag(appA_out_gag(X60)) → appA_out_gag(X60)

The set Q consists of the following terms:

appA_in_gag(x0, x1)
U1_gag(x0)

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

(25) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(26) TRUE