Left Termination of the query pattern star_in_2(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).

Queries:

star(g,g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

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 | ?_1\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: !_1 | star(T3, []), SCOPE: 1, CLAUSE: 1 | star(T3, []), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT3 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 | ?_1\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, =(T4, [])) | star([], T4), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT4 is ground\nX2 is free; \nstar([], T4) !=? 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: =(T4, [])\n\nT4 is ground\nX2 is free; \nstar([], T4) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: =(T4, []), SCOPE: 2, CLAUSE: 5\n\nT4 is ground\nX2 is free; \nstar([], T4) !=? star(X2, [])", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\nX2 is free\nX6 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(app(T5, X11, T6), star(T5, X11))\n\nT5 is ground\nT6 is ground\nX2 is free; \nX4 is free; \nX11 is free; \nstar(T5, T6) !=? star(X2, [])\nstar(T5, T6) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: app(T5, X11, T6)\n\nT5 is ground\nT6 is ground\nX2 is free; \nX4 is free; \nX11 is free; \nstar(T5, T6) !=? star(X2, [])\nstar(T5, T6) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: star(T5, T7)\n\nT5 is ground\nT7 is ground\nX2 is free; \nX4 is free; \nstar(T5, T6) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: app(T5, X11, T6), SCOPE: 3, CLAUSE: 3 | app(T5, X11, T6), SCOPE: 3, CLAUSE: 4\n\nT5 is ground\nT6 is ground\nX2 is free; \nX4 is free; \nX11 is free; \nstar(T5, T6) !=? star(X2, [])\nstar(T5, T6) !=? star([], X4)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: app(T5, X11, T6), SCOPE: 3, CLAUSE: 4\n\nT5 is ground\nT6 is ground\nX2 is free; \nX4 is free; \nX11 is free; \nX13 is free; \nstar(T5, T6) !=? star(X2, [])\nstar(T5, T6) !=? star([], X4)\napp(T5, X11, T6) !=? app([], X13, X13)", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: app(T9, X22, T10)\n\nT9 is ground\nT10 is ground\nX2 is free; \nX4 is free; \nX11 is free; \nX13 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\nX2 is free\nX4 is free; \nX11 is free; \nX13 is free; \nX18 is free; \nX19 is free; \nX20 is free; \nX21 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: app(T9, X22, T10), SCOPE: 4, CLAUSE: 3 | app(T9, X22, T10), SCOPE: 4, CLAUSE: 4\n\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: app(T9, X22, T10), SCOPE: 4, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: app(T9, X22, T10), SCOPE: 4, CLAUSE: 4\n\nT9 is ground\nT10 is ground\nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\nX22 is free\nX25 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app(T13, X35, T14)\n\nT13 is ground\nT14 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\nX22 is free\nX31 is free; \nX32 is free; \nX33 is free; \nX34 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T13, X35, T14), SCOPE: 5, CLAUSE: 3 | app(T13, X35, T14), SCOPE: 5, CLAUSE: 4\n\nT13 is ground\nT14 is ground\nX35 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 28 [arrowhead = none ];
28 -> 2;

29 [label="EVAL with clause\nstar(X2, []) :- !.\nand substitution\nT1 -> T3\nX2 -> T3\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 29 [arrowhead = none ];
29 -> 3;

30 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 30 [arrowhead = none ];
30 -> 4;

31 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 31 [arrowhead = none ];
31 -> 5;

32 [label="EVAL with clause\nstar([], X4) :- ','(!, =(X4, [])).\nand substitution\nT1 -> []\nT2 -> T4\nX4 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 32 [arrowhead = none ];
32 -> 7;

33 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 33 [arrowhead = none ];
33 -> 8;

34 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 34 [arrowhead = none ];
34 -> 6;

35 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 35 [arrowhead = none ];
35 -> 9;

36 [label="EVAL with clause\nstar(X9, X10) :- ','(app(X9, X11, X10), star(X9, X11)).\nand substitution\nT1 -> T5\nX9 -> T5\nT2 -> T6\nX10 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 36 [arrowhead = none ];
36 -> 12;

37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 37 [arrowhead = none ];
37 -> 10;

38 [label="BACKTRACK\nfor clause: =(X, X)\nwith clash: (star([], T4), star(X2, []))", fontsize=14, style = filled, fillcolor = white];
10 -> 38 [arrowhead = none ];
38 -> 11;

39 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 39 [arrowhead = none ];
39 -> 13;

40 [label="SPLIT 2\nnew knowledge:\nT7 is ground\nreplacements:\nX11 -> T7", fontsize=14, style = filled, fillcolor = violet];
12 -> 40 [arrowhead = none ];
40 -> 14;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 41 [arrowhead = none ];
41 -> 15;

42 [label="INSTANCE with matching:\nT1 -> T5\nT2 -> T7", fontsize=14, style = filled, fillcolor = lightgrey];
14 -> 42 [arrowhead = none , style=dashed];
42 -> 1[style=dashed];

43 [label="BACKTRACK\nfor clause: app([], L, L)\nwith clash: (star(T5, T6), star([], X4))", fontsize=14, style = filled, fillcolor = white];
15 -> 43 [arrowhead = none ];
43 -> 16;

44 [label="EVAL with clause\napp(.(X18, X19), X20, .(X18, X21)) :- app(X19, X20, X21).\nand substitution\nX18 -> T8\nX19 -> T9\nT5 -> .(T8, T9)\nX11 -> X22\nX20 -> X22\nX21 -> T10\nT6 -> .(T8, T10)", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 44 [arrowhead = none ];
44 -> 17;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 45 [arrowhead = none ];
45 -> 18;

46 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 46 [arrowhead = none ];
46 -> 19;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 47 [arrowhead = none ];
47 -> 20;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 48 [arrowhead = none ];
48 -> 21;

49 [label="EVAL with clause\napp([], X25, X25).\nand substitution\nT9 -> []\nX22 -> T11\nX25 -> T11\nT10 -> T11\nX26 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 49 [arrowhead = none ];
49 -> 22;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 50 [arrowhead = none ];
50 -> 23;

51 [label="EVAL with clause\napp(.(X31, X32), X33, .(X31, X34)) :- app(X32, X33, X34).\nand substitution\nX31 -> T12\nX32 -> T13\nT9 -> .(T12, T13)\nX22 -> X35\nX33 -> X35\nX34 -> T14\nT10 -> .(T12, T14)", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 51 [arrowhead = none ];
51 -> 25;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
21 -> 52 [arrowhead = none ];
52 -> 26;

53 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 53 [arrowhead = none ];
53 -> 24;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 54 [arrowhead = none ];
54 -> 27;

55 [label="INSTANCE with matching:\nT9 -> T13\nX22 -> X35\nT10 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
27 -> 55 [arrowhead = none , style=dashed];
55 -> 19[style=dashed];

}

(2) Obligation:

Clauses:

star1(T3, []).
star1(T5, T6) :- app13(T5, X11, T6).
star1(T5, T6) :- ','(app13(T5, T7, T6), star1(T5, T7)).
app19([], T11, T11).
app19(.(T12, T13), X35, .(T12, T14)) :- app19(T13, X35, T14).
app13(.(T8, []), T11, .(T8, T11)).
app13(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) :- app19(T13, X35, T14).

Queries:

star1(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:
star1_in: (b,b)
app13_in: (b,f,b)
app19_in: (b,f,b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)

(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:

STAR1_IN_GG(T5, T6) → U1_GG(T5, T6, app13_in_gag(T5, X11, T6))
STAR1_IN_GG(T5, T6) → APP13_IN_GAG(T5, X11, T6)
APP13_IN_GAG(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_GAG(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
APP13_IN_GAG(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → APP19_IN_GAG(T13, X35, T14)
APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → U4_GAG(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → APP19_IN_GAG(T13, X35, T14)
STAR1_IN_GG(T5, T6) → U2_GG(T5, T6, app13_in_gag(T5, T7, T6))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → U3_GG(T5, T6, star1_in_gg(T5, T7))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
APP13_IN_GAG(x1, x2, x3)  =  APP13_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
APP19_IN_GAG(x1, x2, x3)  =  APP19_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4, x5)  =  U4_GAG(x5)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x3)
U3_GG(x1, x2, x3)  =  U3_GG(x3)

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

(6) Obligation:

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

STAR1_IN_GG(T5, T6) → U1_GG(T5, T6, app13_in_gag(T5, X11, T6))
STAR1_IN_GG(T5, T6) → APP13_IN_GAG(T5, X11, T6)
APP13_IN_GAG(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_GAG(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
APP13_IN_GAG(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → APP19_IN_GAG(T13, X35, T14)
APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → U4_GAG(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → APP19_IN_GAG(T13, X35, T14)
STAR1_IN_GG(T5, T6) → U2_GG(T5, T6, app13_in_gag(T5, T7, T6))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → U3_GG(T5, T6, star1_in_gg(T5, T7))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U1_GG(x1, x2, x3)  =  U1_GG(x3)
APP13_IN_GAG(x1, x2, x3)  =  APP13_IN_GAG(x1, x3)
U5_GAG(x1, x2, x3, x4, x5, x6)  =  U5_GAG(x6)
APP19_IN_GAG(x1, x2, x3)  =  APP19_IN_GAG(x1, x3)
U4_GAG(x1, x2, x3, x4, x5)  =  U4_GAG(x5)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x3)
U3_GG(x1, x2, x3)  =  U3_GG(x3)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 6 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → APP19_IN_GAG(T13, X35, T14)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
APP19_IN_GAG(x1, x2, x3)  =  APP19_IN_GAG(x1, x3)

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

(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:

APP19_IN_GAG(.(T12, T13), X35, .(T12, T14)) → APP19_IN_GAG(T13, X35, T14)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
APP19_IN_GAG(x1, x2, x3)  =  APP19_IN_GAG(x1, x3)

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

(12) PiDPToQDPProof (SOUND 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:

APP19_IN_GAG(.(T12, T13), .(T12, T14)) → APP19_IN_GAG(T13, T14)

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:

• APP19_IN_GAG(.(T12, T13), .(T12, T14)) → APP19_IN_GAG(T13, T14)
  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:

STAR1_IN_GG(T5, T6) → U2_GG(T5, T6, app13_in_gag(T5, T7, T6))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

star1_in_gg(T3, []) → star1_out_gg(T3, [])
star1_in_gg(T5, T6) → U1_gg(T5, T6, app13_in_gag(T5, X11, T6))
app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
U1_gg(T5, T6, app13_out_gag(T5, X11, T6)) → star1_out_gg(T5, T6)
star1_in_gg(T5, T6) → U2_gg(T5, T6, app13_in_gag(T5, T7, T6))
U2_gg(T5, T6, app13_out_gag(T5, T7, T6)) → U3_gg(T5, T6, star1_in_gg(T5, T7))
U3_gg(T5, T6, star1_out_gg(T5, T7)) → star1_out_gg(T5, T6)

The argument filtering Pi contains the following mapping:
star1_in_gg(x1, x2)  =  star1_in_gg(x1, x2)
[]  =  []
star1_out_gg(x1, x2)  =  star1_out_gg
U1_gg(x1, x2, x3)  =  U1_gg(x3)
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
U2_gg(x1, x2, x3)  =  U2_gg(x1, x3)
U3_gg(x1, x2, x3)  =  U3_gg(x3)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x3)

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

(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:

STAR1_IN_GG(T5, T6) → U2_GG(T5, T6, app13_in_gag(T5, T7, T6))
U2_GG(T5, T6, app13_out_gag(T5, T7, T6)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

app13_in_gag(.(T8, []), T11, .(T8, T11)) → app13_out_gag(.(T8, []), T11, .(T8, T11))
app13_in_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14))) → U5_gag(T8, T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U5_gag(T8, T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app13_out_gag(.(T8, .(T12, T13)), X35, .(T8, .(T12, T14)))
app19_in_gag([], T11, T11) → app19_out_gag([], T11, T11)
app19_in_gag(.(T12, T13), X35, .(T12, T14)) → U4_gag(T12, T13, X35, T14, app19_in_gag(T13, X35, T14))
U4_gag(T12, T13, X35, T14, app19_out_gag(T13, X35, T14)) → app19_out_gag(.(T12, T13), X35, .(T12, T14))

The argument filtering Pi contains the following mapping:
[]  =  []
app13_in_gag(x1, x2, x3)  =  app13_in_gag(x1, x3)
.(x1, x2)  =  .(x1, x2)
app13_out_gag(x1, x2, x3)  =  app13_out_gag(x2)
U5_gag(x1, x2, x3, x4, x5, x6)  =  U5_gag(x6)
app19_in_gag(x1, x2, x3)  =  app19_in_gag(x1, x3)
app19_out_gag(x1, x2, x3)  =  app19_out_gag(x2)
U4_gag(x1, x2, x3, x4, x5)  =  U4_gag(x5)
STAR1_IN_GG(x1, x2)  =  STAR1_IN_GG(x1, x2)
U2_GG(x1, x2, x3)  =  U2_GG(x1, x3)

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

(19) PiDPToQDPProof (SOUND 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:

STAR1_IN_GG(T5, T6) → U2_GG(T5, app13_in_gag(T5, T6))
U2_GG(T5, app13_out_gag(T7)) → STAR1_IN_GG(T5, T7)

The TRS R consists of the following rules:

app13_in_gag(.(T8, []), .(T8, T11)) → app13_out_gag(T11)
app13_in_gag(.(T8, .(T12, T13)), .(T8, .(T12, T14))) → U5_gag(app19_in_gag(T13, T14))
U5_gag(app19_out_gag(X35)) → app13_out_gag(X35)
app19_in_gag([], T11) → app19_out_gag(T11)
app19_in_gag(.(T12, T13), .(T12, T14)) → U4_gag(app19_in_gag(T13, T14))
U4_gag(app19_out_gag(X35)) → app19_out_gag(X35)

The set Q consists of the following terms:

app13_in_gag(x0, x1)
U5_gag(x0)
app19_in_gag(x0, x1)
U4_gag(x0)

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

(21) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

U2_GG(T5, app13_out_gag(T7)) → STAR1_IN_GG(T5, T7)

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

POL(.(x₁, x₂)) = 1 + x₂   
POL(STAR1_IN_GG(x₁, x₂)) = x₂   
POL(U2_GG(x₁, x₂)) = x₂   
POL(U4_gag(x₁)) = 1 + x₁   
POL(U5_gag(x₁)) = 1 + x₁   
POL([]) = 0   
POL(app13_in_gag(x₁, x₂)) = x₂   
POL(app13_out_gag(x₁)) = 1 + x₁   
POL(app19_in_gag(x₁, x₂)) = 1 + x₂   
POL(app19_out_gag(x₁)) = 1 + x₁   

The following usable rules [FROCOS05] were oriented:

app13_in_gag(.(T8, []), .(T8, T11)) → app13_out_gag(T11)
app13_in_gag(.(T8, .(T12, T13)), .(T8, .(T12, T14))) → U5_gag(app19_in_gag(T13, T14))
app19_in_gag([], T11) → app19_out_gag(T11)
app19_in_gag(.(T12, T13), .(T12, T14)) → U4_gag(app19_in_gag(T13, T14))
U5_gag(app19_out_gag(X35)) → app13_out_gag(X35)
U4_gag(app19_out_gag(X35)) → app19_out_gag(X35)

(22) Obligation:

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

STAR1_IN_GG(T5, T6) → U2_GG(T5, app13_in_gag(T5, T6))

The TRS R consists of the following rules:

app13_in_gag(.(T8, []), .(T8, T11)) → app13_out_gag(T11)
app13_in_gag(.(T8, .(T12, T13)), .(T8, .(T12, T14))) → U5_gag(app19_in_gag(T13, T14))
U5_gag(app19_out_gag(X35)) → app13_out_gag(X35)
app19_in_gag([], T11) → app19_out_gag(T11)
app19_in_gag(.(T12, T13), .(T12, T14)) → U4_gag(app19_in_gag(T13, T14))
U4_gag(app19_out_gag(X35)) → app19_out_gag(X35)

The set Q consists of the following terms:

app13_in_gag(x0, x1)
U5_gag(x0)
app19_in_gag(x0, x1)
U4_gag(x0)

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

(23) DependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE