Left Termination of the query pattern goal_in_1(g) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

goal(X) :- ','(s2l(X, Xs), list(Xs)).
list([]) :- !.
list(X) :- ','(tail(X, T), list(T)).
s2l(0, L) :- ','(!, eq(L, [])).
s2l(X, .(X1, Xs)) :- ','(p(X, P), s2l(P, Xs)).
tail([], []).
tail(.(X2, Xs), Xs).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

goal(g).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: goal(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: goal(T1), SCOPE: 1, CLAUSE: 0\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(s2l(T2, X5), list(X5))\n\nT2 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: s2l(T2, X5)\n\nT2 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: list(T3)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: s2l(T2, X5), SCOPE: 2, CLAUSE: 3 | s2l(T2, X5), SCOPE: 2, CLAUSE: 4 | ?_2\n\nT2 is ground\nX5 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ','(!_2, eq(X8, [])) | s2l(0, X5), SCOPE: 2, CLAUSE: 4 | ?_2\n\nX5 is free\nX8 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: s2l(T2, X5), SCOPE: 2, CLAUSE: 4\n\nT2 is ground\nX5 is free; \nX7 is free; \ns2l(T2, X5) !=? s2l(0, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: eq(X8, [])\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: eq(X8, []), SCOPE: 3, CLAUSE: 9\n\nX8 is free", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(p(T4, X18), s2l(X18, X20))\n\nT4 is ground\nX5 is free; \nX7 is free; \nX20 is free; \nX18 is free; \ns2l(T4, X5) !=? s2l(0, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: ','(p(T4, X18), s2l(X18, X20)), SCOPE: 4, CLAUSE: 7 | ','(p(T4, X18), s2l(X18, X20)), SCOPE: 4, CLAUSE: 8\n\nT4 is ground\nX5 is free; \nX7 is free; \nX20 is free; \nX18 is free; \ns2l(T4, X5) !=? s2l(0, X7)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(p(T4, X18), s2l(X18, X20)), SCOPE: 4, CLAUSE: 8\n\nT4 is ground\nX5 is free; \nX7 is free; \nX20 is free; \nX18 is free; \ns2l(T4, X5) !=? s2l(0, X7)\np(T4, X18) !=? p(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: s2l(T5, X20)\n\nT5 is ground\nX5 is free; \nX7 is free; \nX20 is free; \nX18 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: \n\nX5 is free\nX7 is free; \nX18 is free; \nX22 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: list(T3), SCOPE: 5, CLAUSE: 1 | list(T3), SCOPE: 5, CLAUSE: 2 | ?_5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: !_5 | list(T3), SCOPE: 5, CLAUSE: 2 | ?_5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: list(T3), SCOPE: 5, CLAUSE: 2\n\nlist(T3) !=? list([])", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(tail(T7, X25), list(X25))\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(tail(T7, X25), list(X25)), SCOPE: 6, CLAUSE: 5 | ','(tail(T7, X25), list(X25)), SCOPE: 6, CLAUSE: 6\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: list([]) | ','(tail(T7, X25), list(X25)), SCOPE: 6, CLAUSE: 6\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(tail(T7, X25), list(X25)), SCOPE: 6, CLAUSE: 6\n\nX25 is free\ntail(T7, X25) !=? tail([], [])", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: list([])\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ','(tail(T7, X25), list(X25)), SCOPE: 6, CLAUSE: 6\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: list([]), SCOPE: 7, CLAUSE: 1 | list([]), SCOPE: 7, CLAUSE: 2 | ?_7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: !_7 | list([]), SCOPE: 7, CLAUSE: 2 | ?_7\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: list(T10)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\nX25 is free\nX28 is free; \nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: list(T13)\n\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: \n\nX25 is free\nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 37 [arrowhead = none ];
37 -> 2;

38 [label="EVAL with clause\ngoal(X4) :- ','(s2l(X4, X5), list(X5)).\nand substitution\nT1 -> T2\nX4 -> T2", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 38 [arrowhead = none ];
38 -> 3;

39 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
3 -> 39 [arrowhead = none ];
39 -> 4;

40 [label="SPLIT 2\nreplacements:\nX5 -> T3", fontsize=14, style = filled, fillcolor = violet];
3 -> 40 [arrowhead = none ];
40 -> 5;

41 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
4 -> 41 [arrowhead = none ];
41 -> 6;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 42 [arrowhead = none ];
42 -> 18;

43 [label="EVAL with clause\ns2l(0, X7) :- ','(!, eq(X7, [])).\nand substitution\nT2 -> 0\nX5 -> X8\nX7 -> X8", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 43 [arrowhead = none ];
43 -> 7;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 44 [arrowhead = none ];
44 -> 8;

45 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 45 [arrowhead = none ];
45 -> 9;

46 [label="EVAL with clause\ns2l(X15, .(X16, X17)) :- ','(p(X15, X18), s2l(X18, X17)).\nand substitution\nT2 -> T4\nX15 -> T4\nX16 -> X19\nX17 -> X20\nX5 -> .(X19, X20)", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 46 [arrowhead = none ];
46 -> 13;

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

48 [label="EVAL with clause\neq(X10, X10).\nand substitution\nX8 -> []\nX10 -> []\nX11 -> []", fontsize=14, style = filled, fillcolor = lightblue];
10 -> 48 [arrowhead = none ];
48 -> 11;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 49 [arrowhead = none ];
49 -> 12;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
13 -> 50 [arrowhead = none ];
50 -> 14;

51 [label="BACKTRACK\nfor clause: p(0, 0)\nwith clash: (s2l(T4, X5), s2l(0, X7))", fontsize=14, style = filled, fillcolor = white];
14 -> 51 [arrowhead = none ];
51 -> 15;

52 [label="EVAL with clause\np(s(X22), X22).\nand substitution\nX22 -> T5\nT4 -> s(T5)\nX18 -> T5", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 52 [arrowhead = none ];
52 -> 16;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
15 -> 53 [arrowhead = none ];
53 -> 17;

54 [label="INSTANCE with matching:\nT2 -> T5\nX5 -> X20", fontsize=14, style = filled, fillcolor = lightgrey];
16 -> 54 [arrowhead = none , style=dashed];
54 -> 4[style=dashed];

55 [label="EVAL with clause\nlist([]) :- !.\nand substitution\nT3 -> []", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 55 [arrowhead = none ];
55 -> 19;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 56 [arrowhead = none ];
56 -> 20;

57 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 57 [arrowhead = none ];
57 -> 21;

58 [label="EVAL with clause\nlist(X24) :- ','(tail(X24, X25), list(X25)).\nand substitution\nT3 -> T7\nX24 -> T7\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 58 [arrowhead = none ];
58 -> 23;

59 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 59 [arrowhead = none ];
59 -> 22;

60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 60 [arrowhead = none ];
60 -> 24;

61 [label="EVAL with clause\ntail([], []).\nand substitution\nT7 -> []\nX25 -> []", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 61 [arrowhead = none ];
61 -> 25;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 62 [arrowhead = none ];
62 -> 26;

63 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 63 [arrowhead = none ];
63 -> 27;

64 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
25 -> 64 [arrowhead = none ];
64 -> 28;

65 [label="EVAL with clause\ntail(.(X32, X33), X33).\nand substitution\nX32 -> T11\nX33 -> T13\nT7 -> .(T11, T13)\nX25 -> T13\nT12 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 65 [arrowhead = none ];
65 -> 35;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 66 [arrowhead = none ];
66 -> 36;

67 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 67 [arrowhead = none ];
67 -> 29;

68 [label="EVAL with clause\ntail(.(X28, X29), X29).\nand substitution\nX28 -> T8\nX29 -> T10\nT7 -> .(T8, T10)\nX25 -> T10\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 69 [arrowhead = none ];
69 -> 34;

70 [label="EVAL with clause\nlist([]) :- !.\nand substitution", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 70 [arrowhead = none ];
70 -> 30;

71 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 71 [arrowhead = none ];
71 -> 31;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 72 [arrowhead = none ];
72 -> 32;

73 [label="INSTANCE with matching:\nT3 -> T10", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 73 [arrowhead = none , style=dashed];
73 -> 5[style=dashed];

74 [label="INSTANCE with matching:\nT3 -> T13", fontsize=14, style = filled, fillcolor = lightgrey];
35 -> 74 [arrowhead = none , style=dashed];
74 -> 5[style=dashed];

}

(2) Obligation:

Clauses:

s2l4(0, []).
s2l4(s(T5), .(X19, X20)) :- s2l4(T5, X20).
list5([]).
list5([]).
list5(.(T8, T10)) :- list5(T10).
list5(.(T11, T13)) :- list5(T13).
goal1(T2) :- s2l4(T2, X5).
goal1(T2) :- s2l4(T2, []).
goal1(T2) :- s2l4(T2, []).
goal1(T2) :- ','(s2l4(T2, .(T8, T10)), list5(T10)).
goal1(T2) :- ','(s2l4(T2, .(T11, T13)), list5(T13)).

Queries:

goal1(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:
goal1_in: (b)
s2l4_in: (b,f) (b,b)
list5_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X5))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X19, X20)) → U1_ga(T5, X19, X20, s2l4_in_ga(T5, X20))
U1_ga(T5, X19, X20, s2l4_out_ga(T5, X20)) → s2l4_out_ga(s(T5), .(X19, X20))
U4_g(T2, s2l4_out_ga(T2, X5)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X19, X20)) → U1_gg(T5, X19, X20, s2l4_in_gg(T5, X20))
U1_gg(T5, X19, X20, s2l4_out_gg(T5, X20)) → s2l4_out_gg(s(T5), .(X19, X20))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T8, T10)))
U6_g(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_g(T2, list5_in_g(T10))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T8, T10)) → U2_g(T8, T10, list5_in_g(T10))
list5_in_g(.(T11, T13)) → U3_g(T11, T13, list5_in_g(T13))
U3_g(T11, T13, list5_out_g(T13)) → list5_out_g(.(T11, T13))
U2_g(T8, T10, list5_out_g(T10)) → list5_out_g(.(T8, T10))
U7_g(T2, list5_out_g(T10)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_g(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_g(T2, list5_in_g(T13))
U9_g(T2, list5_out_g(T13)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X5))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X19, X20)) → U1_ga(T5, X19, X20, s2l4_in_ga(T5, X20))
U1_ga(T5, X19, X20, s2l4_out_ga(T5, X20)) → s2l4_out_ga(s(T5), .(X19, X20))
U4_g(T2, s2l4_out_ga(T2, X5)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X19, X20)) → U1_gg(T5, X19, X20, s2l4_in_gg(T5, X20))
U1_gg(T5, X19, X20, s2l4_out_gg(T5, X20)) → s2l4_out_gg(s(T5), .(X19, X20))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T8, T10)))
U6_g(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_g(T2, list5_in_g(T10))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T8, T10)) → U2_g(T8, T10, list5_in_g(T10))
list5_in_g(.(T11, T13)) → U3_g(T11, T13, list5_in_g(T13))
U3_g(T11, T13, list5_out_g(T13)) → list5_out_g(.(T11, T13))
U2_g(T8, T10, list5_out_g(T10)) → list5_out_g(.(T8, T10))
U7_g(T2, list5_out_g(T10)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_g(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_g(T2, list5_in_g(T13))
U9_g(T2, list5_out_g(T13)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)

(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:

GOAL1_IN_G(T2) → U4_G(T2, s2l4_in_ga(T2, X5))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, X5)
S2L4_IN_GA(s(T5), .(X19, X20)) → U1_GA(T5, X19, X20, s2l4_in_ga(T5, X20))
S2L4_IN_GA(s(T5), .(X19, X20)) → S2L4_IN_GA(T5, X20)
GOAL1_IN_G(T2) → U5_G(T2, s2l4_in_gg(T2, []))
GOAL1_IN_G(T2) → S2L4_IN_GG(T2, [])
S2L4_IN_GG(s(T5), .(X19, X20)) → U1_GG(T5, X19, X20, s2l4_in_gg(T5, X20))
S2L4_IN_GG(s(T5), .(X19, X20)) → S2L4_IN_GG(T5, X20)
GOAL1_IN_G(T2) → U6_G(T2, s2l4_in_ga(T2, .(T8, T10)))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, .(T8, T10))
U6_G(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_G(T2, list5_in_g(T10))
U6_G(T2, s2l4_out_ga(T2, .(T8, T10))) → LIST5_IN_G(T10)
LIST5_IN_G(.(T8, T10)) → U2_G(T8, T10, list5_in_g(T10))
LIST5_IN_G(.(T8, T10)) → LIST5_IN_G(T10)
LIST5_IN_G(.(T11, T13)) → U3_G(T11, T13, list5_in_g(T13))
GOAL1_IN_G(T2) → U8_G(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_G(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_G(T2, list5_in_g(T13))
U8_G(T2, s2l4_out_ga(T2, .(T11, T13))) → LIST5_IN_G(T13)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X5))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X19, X20)) → U1_ga(T5, X19, X20, s2l4_in_ga(T5, X20))
U1_ga(T5, X19, X20, s2l4_out_ga(T5, X20)) → s2l4_out_ga(s(T5), .(X19, X20))
U4_g(T2, s2l4_out_ga(T2, X5)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X19, X20)) → U1_gg(T5, X19, X20, s2l4_in_gg(T5, X20))
U1_gg(T5, X19, X20, s2l4_out_gg(T5, X20)) → s2l4_out_gg(s(T5), .(X19, X20))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T8, T10)))
U6_g(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_g(T2, list5_in_g(T10))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T8, T10)) → U2_g(T8, T10, list5_in_g(T10))
list5_in_g(.(T11, T13)) → U3_g(T11, T13, list5_in_g(T13))
U3_g(T11, T13, list5_out_g(T13)) → list5_out_g(.(T11, T13))
U2_g(T8, T10, list5_out_g(T10)) → list5_out_g(.(T8, T10))
U7_g(T2, list5_out_g(T10)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_g(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_g(T2, list5_in_g(T13))
U9_g(T2, list5_out_g(T13)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U5_G(x1, x2)  =  U5_G(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)
U3_G(x1, x2, x3)  =  U3_G(x3)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)

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

(6) Obligation:

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

GOAL1_IN_G(T2) → U4_G(T2, s2l4_in_ga(T2, X5))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, X5)
S2L4_IN_GA(s(T5), .(X19, X20)) → U1_GA(T5, X19, X20, s2l4_in_ga(T5, X20))
S2L4_IN_GA(s(T5), .(X19, X20)) → S2L4_IN_GA(T5, X20)
GOAL1_IN_G(T2) → U5_G(T2, s2l4_in_gg(T2, []))
GOAL1_IN_G(T2) → S2L4_IN_GG(T2, [])
S2L4_IN_GG(s(T5), .(X19, X20)) → U1_GG(T5, X19, X20, s2l4_in_gg(T5, X20))
S2L4_IN_GG(s(T5), .(X19, X20)) → S2L4_IN_GG(T5, X20)
GOAL1_IN_G(T2) → U6_G(T2, s2l4_in_ga(T2, .(T8, T10)))
GOAL1_IN_G(T2) → S2L4_IN_GA(T2, .(T8, T10))
U6_G(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_G(T2, list5_in_g(T10))
U6_G(T2, s2l4_out_ga(T2, .(T8, T10))) → LIST5_IN_G(T10)
LIST5_IN_G(.(T8, T10)) → U2_G(T8, T10, list5_in_g(T10))
LIST5_IN_G(.(T8, T10)) → LIST5_IN_G(T10)
LIST5_IN_G(.(T11, T13)) → U3_G(T11, T13, list5_in_g(T13))
GOAL1_IN_G(T2) → U8_G(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_G(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_G(T2, list5_in_g(T13))
U8_G(T2, s2l4_out_ga(T2, .(T11, T13))) → LIST5_IN_G(T13)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X5))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X19, X20)) → U1_ga(T5, X19, X20, s2l4_in_ga(T5, X20))
U1_ga(T5, X19, X20, s2l4_out_ga(T5, X20)) → s2l4_out_ga(s(T5), .(X19, X20))
U4_g(T2, s2l4_out_ga(T2, X5)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X19, X20)) → U1_gg(T5, X19, X20, s2l4_in_gg(T5, X20))
U1_gg(T5, X19, X20, s2l4_out_gg(T5, X20)) → s2l4_out_gg(s(T5), .(X19, X20))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T8, T10)))
U6_g(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_g(T2, list5_in_g(T10))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T8, T10)) → U2_g(T8, T10, list5_in_g(T10))
list5_in_g(.(T11, T13)) → U3_g(T11, T13, list5_in_g(T13))
U3_g(T11, T13, list5_out_g(T13)) → list5_out_g(.(T11, T13))
U2_g(T8, T10, list5_out_g(T10)) → list5_out_g(.(T8, T10))
U7_g(T2, list5_out_g(T10)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_g(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_g(T2, list5_in_g(T13))
U9_g(T2, list5_out_g(T13)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
GOAL1_IN_G(x1)  =  GOAL1_IN_G(x1)
U4_G(x1, x2)  =  U4_G(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)
U1_GA(x1, x2, x3, x4)  =  U1_GA(x4)
U5_G(x1, x2)  =  U5_G(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)
U1_GG(x1, x2, x3, x4)  =  U1_GG(x4)
U6_G(x1, x2)  =  U6_G(x2)
U7_G(x1, x2)  =  U7_G(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x3)
U3_G(x1, x2, x3)  =  U3_G(x3)
U8_G(x1, x2)  =  U8_G(x2)
U9_G(x1, x2)  =  U9_G(x2)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 15 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

LIST5_IN_G(.(T8, T10)) → LIST5_IN_G(T10)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X5))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X19, X20)) → U1_ga(T5, X19, X20, s2l4_in_ga(T5, X20))
U1_ga(T5, X19, X20, s2l4_out_ga(T5, X20)) → s2l4_out_ga(s(T5), .(X19, X20))
U4_g(T2, s2l4_out_ga(T2, X5)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X19, X20)) → U1_gg(T5, X19, X20, s2l4_in_gg(T5, X20))
U1_gg(T5, X19, X20, s2l4_out_gg(T5, X20)) → s2l4_out_gg(s(T5), .(X19, X20))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T8, T10)))
U6_g(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_g(T2, list5_in_g(T10))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T8, T10)) → U2_g(T8, T10, list5_in_g(T10))
list5_in_g(.(T11, T13)) → U3_g(T11, T13, list5_in_g(T13))
U3_g(T11, T13, list5_out_g(T13)) → list5_out_g(.(T11, T13))
U2_g(T8, T10, list5_out_g(T10)) → list5_out_g(.(T8, T10))
U7_g(T2, list5_out_g(T10)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_g(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_g(T2, list5_in_g(T13))
U9_g(T2, list5_out_g(T13)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)

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:

LIST5_IN_G(.(T8, T10)) → LIST5_IN_G(T10)

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
LIST5_IN_G(x1)  =  LIST5_IN_G(x1)

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:

LIST5_IN_G(.(T10)) → LIST5_IN_G(T10)

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:

• LIST5_IN_G(.(T10)) → LIST5_IN_G(T10)
  The graph contains the following edges 1 > 1

(15) TRUE

(16) Obligation:

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

S2L4_IN_GG(s(T5), .(X19, X20)) → S2L4_IN_GG(T5, X20)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X5))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X19, X20)) → U1_ga(T5, X19, X20, s2l4_in_ga(T5, X20))
U1_ga(T5, X19, X20, s2l4_out_ga(T5, X20)) → s2l4_out_ga(s(T5), .(X19, X20))
U4_g(T2, s2l4_out_ga(T2, X5)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X19, X20)) → U1_gg(T5, X19, X20, s2l4_in_gg(T5, X20))
U1_gg(T5, X19, X20, s2l4_out_gg(T5, X20)) → s2l4_out_gg(s(T5), .(X19, X20))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T8, T10)))
U6_g(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_g(T2, list5_in_g(T10))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T8, T10)) → U2_g(T8, T10, list5_in_g(T10))
list5_in_g(.(T11, T13)) → U3_g(T11, T13, list5_in_g(T13))
U3_g(T11, T13, list5_out_g(T13)) → list5_out_g(.(T11, T13))
U2_g(T8, T10, list5_out_g(T10)) → list5_out_g(.(T8, T10))
U7_g(T2, list5_out_g(T10)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_g(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_g(T2, list5_in_g(T13))
U9_g(T2, list5_out_g(T13)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)

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:

S2L4_IN_GG(s(T5), .(X19, X20)) → S2L4_IN_GG(T5, X20)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L4_IN_GG(x1, x2)  =  S2L4_IN_GG(x1, x2)

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:

S2L4_IN_GG(s(T5), .(X20)) → S2L4_IN_GG(T5, X20)

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:

• S2L4_IN_GG(s(T5), .(X20)) → S2L4_IN_GG(T5, X20)
  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:

S2L4_IN_GA(s(T5), .(X19, X20)) → S2L4_IN_GA(T5, X20)

The TRS R consists of the following rules:

goal1_in_g(T2) → U4_g(T2, s2l4_in_ga(T2, X5))
s2l4_in_ga(0, []) → s2l4_out_ga(0, [])
s2l4_in_ga(s(T5), .(X19, X20)) → U1_ga(T5, X19, X20, s2l4_in_ga(T5, X20))
U1_ga(T5, X19, X20, s2l4_out_ga(T5, X20)) → s2l4_out_ga(s(T5), .(X19, X20))
U4_g(T2, s2l4_out_ga(T2, X5)) → goal1_out_g(T2)
goal1_in_g(T2) → U5_g(T2, s2l4_in_gg(T2, []))
s2l4_in_gg(0, []) → s2l4_out_gg(0, [])
s2l4_in_gg(s(T5), .(X19, X20)) → U1_gg(T5, X19, X20, s2l4_in_gg(T5, X20))
U1_gg(T5, X19, X20, s2l4_out_gg(T5, X20)) → s2l4_out_gg(s(T5), .(X19, X20))
U5_g(T2, s2l4_out_gg(T2, [])) → goal1_out_g(T2)
goal1_in_g(T2) → U6_g(T2, s2l4_in_ga(T2, .(T8, T10)))
U6_g(T2, s2l4_out_ga(T2, .(T8, T10))) → U7_g(T2, list5_in_g(T10))
list5_in_g([]) → list5_out_g([])
list5_in_g(.(T8, T10)) → U2_g(T8, T10, list5_in_g(T10))
list5_in_g(.(T11, T13)) → U3_g(T11, T13, list5_in_g(T13))
U3_g(T11, T13, list5_out_g(T13)) → list5_out_g(.(T11, T13))
U2_g(T8, T10, list5_out_g(T10)) → list5_out_g(.(T8, T10))
U7_g(T2, list5_out_g(T10)) → goal1_out_g(T2)
goal1_in_g(T2) → U8_g(T2, s2l4_in_ga(T2, .(T11, T13)))
U8_g(T2, s2l4_out_ga(T2, .(T11, T13))) → U9_g(T2, list5_in_g(T13))
U9_g(T2, list5_out_g(T13)) → goal1_out_g(T2)

The argument filtering Pi contains the following mapping:
goal1_in_g(x1)  =  goal1_in_g(x1)
U4_g(x1, x2)  =  U4_g(x2)
s2l4_in_ga(x1, x2)  =  s2l4_in_ga(x1)
0  =  0
s2l4_out_ga(x1, x2)  =  s2l4_out_ga(x2)
s(x1)  =  s(x1)
U1_ga(x1, x2, x3, x4)  =  U1_ga(x4)
.(x1, x2)  =  .(x2)
goal1_out_g(x1)  =  goal1_out_g
U5_g(x1, x2)  =  U5_g(x2)
s2l4_in_gg(x1, x2)  =  s2l4_in_gg(x1, x2)
[]  =  []
s2l4_out_gg(x1, x2)  =  s2l4_out_gg
U1_gg(x1, x2, x3, x4)  =  U1_gg(x4)
U6_g(x1, x2)  =  U6_g(x2)
U7_g(x1, x2)  =  U7_g(x2)
list5_in_g(x1)  =  list5_in_g(x1)
list5_out_g(x1)  =  list5_out_g
U2_g(x1, x2, x3)  =  U2_g(x3)
U3_g(x1, x2, x3)  =  U3_g(x3)
U8_g(x1, x2)  =  U8_g(x2)
U9_g(x1, x2)  =  U9_g(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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

(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:

S2L4_IN_GA(s(T5), .(X19, X20)) → S2L4_IN_GA(T5, X20)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
.(x1, x2)  =  .(x2)
S2L4_IN_GA(x1, x2)  =  S2L4_IN_GA(x1)

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:

S2L4_IN_GA(s(T5)) → S2L4_IN_GA(T5)

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:

• S2L4_IN_GA(s(T5)) → S2L4_IN_GA(T5)
  The graph contains the following edges 1 > 1

(29) TRUE