Left Termination of the query pattern countstack_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:

(0) Obligation:

Clauses:

countstack(empty, X) :- ','(!, eq(X, 0)).
countstack(S, X) :- ','(pop(S, nil), ','(!, ','(popped(S, Pd), countstack(Pd, X)))).
countstack(S, s(X)) :- ','(pop(S, P), ','(head(P, H), ','(tail(P, T), ','(popped(S, Pd), countstack(push(H, push(T, Pd)), X))))).
pop(empty, X1).
pop(push(P, X2), P).
popped(empty, empty).
popped(push(X3, Pd), Pd).
head(nil, X4).
head(cons(H, X5), H).
tail(nil, nil).
tail(cons(X6, T), T).
eq(X, X).

Queries:

countstack(g,a).

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: countstack(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: countstack(T1, T2), SCOPE: 1, CLAUSE: 0 | countstack(T1, T2), SCOPE: 1, CLAUSE: 1 | countstack(T1, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: ','(!_1, eq(T4, 0)) | countstack(empty, T2), SCOPE: 1, CLAUSE: 1 | countstack(empty, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: countstack(T1, T2), SCOPE: 1, CLAUSE: 1 | countstack(T1, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT1 is ground\nX8 is free; \ncountstack(T1, T2) !=? countstack(empty, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: eq(T4, 0)\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: eq(T4, 0), SCOPE: 2, CLAUSE: 11\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: \n\nX10 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ','(pop(T6, nil), ','(!_1, ','(popped(T6, X15), countstack(X15, T8)))) | countstack(T6, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT6 is ground\nX8 is free; \nX15 is free; \ncountstack(T6, T2) !=? countstack(empty, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ','(pop(T6, nil), ','(!_1, ','(popped(T6, X15), countstack(X15, T8)))), SCOPE: 3, CLAUSE: 3 | ','(pop(T6, nil), ','(!_1, ','(popped(T6, X15), countstack(X15, T8)))), SCOPE: 3, CLAUSE: 4 | countstack(T6, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT6 is ground\nX8 is free; \nX15 is free; \ncountstack(T6, T2) !=? countstack(empty, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(pop(T6, nil), ','(!_1, ','(popped(T6, X15), countstack(X15, T8)))), SCOPE: 3, CLAUSE: 4 | countstack(T6, T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT6 is ground\nX8 is free; \nX15 is free; \nX17 is free; \ncountstack(T6, T2) !=? countstack(empty, X8)\npop(T6, nil) !=? pop(empty, X17)", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(!_1, ','(popped(push(nil, T10), X15), countstack(X15, T8))) | countstack(push(nil, T10), T2), SCOPE: 1, CLAUSE: 2 | ?_1\n\nT10 is ground\nX8 is free; \nX15 is free; \nX17 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: countstack(T6, T2), SCOPE: 1, CLAUSE: 2\n\nT6 is ground\nX8 is free; \nX17 is free; \nX20 is free; \nX21 is free; \ncountstack(T6, T2) !=? countstack(empty, X8)\npop(T6, nil) !=? pop(empty, X17)\npop(T6, nil) !=? pop(push(X20, X21), X20)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(popped(push(nil, T10), X15), countstack(X15, T8))\n\nT10 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: ','(popped(push(nil, T10), X15), countstack(X15, T8)), SCOPE: 4, CLAUSE: 5 | ','(popped(push(nil, T10), X15), countstack(X15, T8)), SCOPE: 4, CLAUSE: 6\n\nT10 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(popped(push(nil, T10), X15), countstack(X15, T8)), SCOPE: 4, CLAUSE: 6\n\nT10 is ground\nX15 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: countstack(T11, T8)\n\nT11 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(pop(T12, X30), ','(head(X30, X31), ','(tail(X30, X32), ','(popped(T12, X33), countstack(push(X31, push(X32, X33)), T14)))))\n\nT12 is ground\nX8 is free; \nX17 is free; \nX20 is free; \nX21 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nX33 is free; \ncountstack(T12, T2) !=? countstack(empty, X8)\npop(T12, nil) !=? pop(empty, X17)\npop(T12, nil) !=? pop(push(X20, X21), X20)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\nX8 is free\nX17 is free; \nX20 is free; \nX21 is free; \nX28 is free; \nX29 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(pop(T12, X30), ','(head(X30, X31), ','(tail(X30, X32), ','(popped(T12, X33), countstack(push(X31, push(X32, X33)), T14))))), SCOPE: 5, CLAUSE: 3 | ','(pop(T12, X30), ','(head(X30, X31), ','(tail(X30, X32), ','(popped(T12, X33), countstack(push(X31, push(X32, X33)), T14))))), SCOPE: 5, CLAUSE: 4\n\nT12 is ground\nX8 is free; \nX17 is free; \nX20 is free; \nX21 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nX33 is free; \ncountstack(T12, T2) !=? countstack(empty, X8)\npop(T12, nil) !=? pop(empty, X17)\npop(T12, nil) !=? pop(push(X20, X21), X20)", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(pop(T12, X30), ','(head(X30, X31), ','(tail(X30, X32), ','(popped(T12, X33), countstack(push(X31, push(X32, X33)), T14))))), SCOPE: 5, CLAUSE: 4\n\nT12 is ground\nX8 is free; \nX17 is free; \nX20 is free; \nX21 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nX33 is free; \nX35 is free; \ncountstack(T12, T2) !=? countstack(empty, X8)\npop(T12, nil) !=? pop(empty, X17)\npop(T12, nil) !=? pop(push(X20, X21), X20)\npop(T12, X30) !=? pop(empty, X35)", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(head(T15, X31), ','(tail(T15, X32), ','(popped(push(T15, T16), X33), countstack(push(X31, push(X32, X33)), T14))))\n\nT15 is ground\nT16 is ground\nX8 is free; \nX17 is free; \nX20 is free; \nX21 is free; \nX30 is free; \nX31 is free; \nX32 is free; \nX33 is free; \nX35 is free; \npop(push(T15, T16), nil) !=? pop(push(X20, X21), X20)", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\nX8 is free\nX17 is free; \nX20 is free; \nX21 is free; \nX30 is free; \nX35 is free; \nX38 is free; \nX39 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(head(T15, X31), ','(tail(T15, X32), ','(popped(push(T15, T16), X33), countstack(push(X31, push(X32, X33)), T14)))), SCOPE: 6, CLAUSE: 7 | ','(head(T15, X31), ','(tail(T15, X32), ','(popped(push(T15, T16), X33), countstack(push(X31, push(X32, X33)), T14)))), SCOPE: 6, CLAUSE: 8\n\nT15 is ground\nT16 is ground\nX20 is free; \nX21 is free; \nX31 is free; \nX32 is free; \nX33 is free; \npop(push(T15, T16), nil) !=? pop(push(X20, X21), X20)", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(head(T15, X31), ','(tail(T15, X32), ','(popped(push(T15, T16), X33), countstack(push(X31, push(X32, X33)), T14)))), SCOPE: 6, CLAUSE: 8\n\nT15 is ground\nT16 is ground\nX20 is free; \nX21 is free; \nX31 is free; \nX32 is free; \nX33 is free; \nX41 is free; \npop(push(T15, T16), nil) !=? pop(push(X20, X21), X20)\nhead(T15, X31) !=? head(nil, X41)", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(tail(cons(T17, T18), X32), ','(popped(push(cons(T17, T18), T16), X33), countstack(push(T17, push(X32, X33)), T14)))\n\nT16 is ground\nT17 is ground\nT18 is ground\nX20 is free; \nX21 is free; \nX31 is free; \nX32 is free; \nX33 is free; \nX41 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\nX20 is free\nX21 is free; \nX31 is free; \nX41 is free; \nX44 is free; \nX45 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(tail(cons(T17, T18), X32), ','(popped(push(cons(T17, T18), T16), X33), countstack(push(T17, push(X32, X33)), T14))), SCOPE: 7, CLAUSE: 9 | ','(tail(cons(T17, T18), X32), ','(popped(push(cons(T17, T18), T16), X33), countstack(push(T17, push(X32, X33)), T14))), SCOPE: 7, CLAUSE: 10\n\nT16 is ground\nT17 is ground\nT18 is ground\nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(tail(cons(T17, T18), X32), ','(popped(push(cons(T17, T18), T16), X33), countstack(push(T17, push(X32, X33)), T14))), SCOPE: 7, CLAUSE: 10\n\nT16 is ground\nT17 is ground\nT18 is ground\nX32 is free; \nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(popped(push(cons(T19, T20), T16), X33), countstack(push(T19, push(T20, X33)), T14))\n\nT16 is ground\nT19 is ground\nT20 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: ','(popped(push(cons(T19, T20), T16), X33), countstack(push(T19, push(T20, X33)), T14)), SCOPE: 8, CLAUSE: 5 | ','(popped(push(cons(T19, T20), T16), X33), countstack(push(T19, push(T20, X33)), T14)), SCOPE: 8, CLAUSE: 6\n\nT16 is ground\nT19 is ground\nT20 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ','(popped(push(cons(T19, T20), T16), X33), countstack(push(T19, push(T20, X33)), T14)), SCOPE: 8, CLAUSE: 6\n\nT16 is ground\nT19 is ground\nT20 is ground\nX33 is free; ", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: countstack(push(T21, push(T22, T23)), T14)\n\nT21 is ground\nT22 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 35 [arrowhead = none ];
35 -> 2;

36 [label="EVAL with clause\ncountstack(empty, X8) :- ','(!, eq(X8, 0)).\nand substitution\nT1 -> empty\nT2 -> T4\nX8 -> T4\nT3 -> T4", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 36 [arrowhead = none ];
36 -> 3;

37 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 37 [arrowhead = none ];
37 -> 4;

38 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 38 [arrowhead = none ];
38 -> 5;

39 [label="EVAL with clause\ncountstack(X13, X14) :- ','(pop(X13, nil), ','(!, ','(popped(X13, X15), countstack(X15, X14)))).\nand substitution\nT1 -> T6\nX13 -> T6\nT2 -> T8\nX14 -> T8\nT7 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 39 [arrowhead = none ];
39 -> 10;

40 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 40 [arrowhead = none ];
40 -> 6;

41 [label="EVAL with clause\neq(X10, X10).\nand substitution\nT4 -> 0\nX10 -> 0\nT5 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 41 [arrowhead = none ];
41 -> 7;

42 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 42 [arrowhead = none ];
42 -> 8;

43 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 43 [arrowhead = none ];
43 -> 9;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 44 [arrowhead = none ];
44 -> 11;

45 [label="BACKTRACK\nfor clause: pop(empty, X1)\nwith clash: (countstack(T6, T2), countstack(empty, X8))", fontsize=14, style = filled, fillcolor = white];
11 -> 45 [arrowhead = none ];
45 -> 12;

46 [label="EVAL with clause\npop(push(X20, X21), X20).\nand substitution\nX20 -> nil\nX21 -> T10\nT6 -> push(nil, T10)\nT9 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 46 [arrowhead = none ];
46 -> 13;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 47 [arrowhead = none ];
47 -> 14;

48 [label="CUT", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 48 [arrowhead = none ];
48 -> 15;

49 [label="EVAL with clause\ncountstack(X28, s(X29)) :- ','(pop(X28, X30), ','(head(X30, X31), ','(tail(X30, X32), ','(popped(X28, X33), countstack(push(X31, push(X32, X33)), X29))))).\nand substitution\nT6 -> T12\nX28 -> T12\nX29 -> T14\nT2 -> s(T14)\nT13 -> T14", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 49 [arrowhead = none ];
49 -> 19;

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

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 51 [arrowhead = none ];
51 -> 16;

52 [label="BACKTRACK\nfor clause: popped(empty, empty)because of non-unification", fontsize=14, style = filled, fillcolor = white];
16 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="EVAL with clause\npopped(push(X24, X25), X25).\nand substitution\nX24 -> nil\nT10 -> T11\nX25 -> T11\nX15 -> T11", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 53 [arrowhead = none ];
53 -> 18;

54 [label="INSTANCE with matching:\nT1 -> T11\nT2 -> T8", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 54 [arrowhead = none , style=dashed];
54 -> 1[style=dashed];

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 55 [arrowhead = none ];
55 -> 21;

56 [label="BACKTRACK\nfor clause: pop(empty, X1)\nwith clash: (countstack(T12, T2), countstack(empty, X8))", fontsize=14, style = filled, fillcolor = white];
21 -> 56 [arrowhead = none ];
56 -> 22;

57 [label="EVAL with clause\npop(push(X38, X39), X38).\nand substitution\nX38 -> T15\nX39 -> T16\nT12 -> push(T15, T16)\nX30 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 57 [arrowhead = none ];
57 -> 23;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 58 [arrowhead = none ];
58 -> 24;

59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 59 [arrowhead = none ];
59 -> 25;

60 [label="BACKTRACK\nfor clause: head(nil, X4)\nwith clash: (pop(push(T15, T16), nil), pop(push(X20, X21), X20))", fontsize=14, style = filled, fillcolor = white];
25 -> 60 [arrowhead = none ];
60 -> 26;

61 [label="EVAL with clause\nhead(cons(X44, X45), X44).\nand substitution\nX44 -> T17\nX45 -> T18\nT15 -> cons(T17, T18)\nX31 -> T17", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 61 [arrowhead = none ];
61 -> 27;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 62 [arrowhead = none ];
62 -> 28;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
27 -> 63 [arrowhead = none ];
63 -> 29;

64 [label="BACKTRACK\nfor clause: tail(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = white];
29 -> 64 [arrowhead = none ];
64 -> 30;

65 [label="EVAL with clause\ntail(cons(X48, X49), X49).\nand substitution\nT17 -> T19\nX48 -> T19\nT18 -> T20\nX49 -> T20\nX32 -> T20", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 65 [arrowhead = none ];
65 -> 31;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
31 -> 66 [arrowhead = none ];
66 -> 32;

67 [label="BACKTRACK\nfor clause: popped(empty, empty)because of non-unification", fontsize=14, style = filled, fillcolor = white];
32 -> 67 [arrowhead = none ];
67 -> 33;

68 [label="EVAL with clause\npopped(push(X52, X53), X53).\nand substitution\nT19 -> T21\nT20 -> T22\nX52 -> cons(T21, T22)\nT16 -> T23\nX53 -> T23\nX33 -> T23", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 68 [arrowhead = none ];
68 -> 34;

69 [label="INSTANCE with matching:\nT1 -> push(T21, push(T22, T23))\nT2 -> T14", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 69 [arrowhead = none , style=dashed];
69 -> 1[style=dashed];

}

(2) Obligation:

Clauses:

countstack1(empty, 0).
countstack1(push(nil, T11), T8) :- countstack1(T11, T8).
countstack1(push(cons(T21, T22), T23), s(T14)) :- countstack1(push(T21, push(T22, T23)), T14).

Queries:

countstack1(g,a).

(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:
countstack1_in: (b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

countstack1_in_ga(empty, 0) → countstack1_out_ga(empty, 0)
countstack1_in_ga(push(nil, T11), T8) → U1_ga(T11, T8, countstack1_in_ga(T11, T8))
countstack1_in_ga(push(cons(T21, T22), T23), s(T14)) → U2_ga(T21, T22, T23, T14, countstack1_in_ga(push(T21, push(T22, T23)), T14))
U2_ga(T21, T22, T23, T14, countstack1_out_ga(push(T21, push(T22, T23)), T14)) → countstack1_out_ga(push(cons(T21, T22), T23), s(T14))
U1_ga(T11, T8, countstack1_out_ga(T11, T8)) → countstack1_out_ga(push(nil, T11), T8)

The argument filtering Pi contains the following mapping:
countstack1_in_ga(x1, x2)  =  countstack1_in_ga(x1)
empty  =  empty
countstack1_out_ga(x1, x2)  =  countstack1_out_ga(x1, x2)
push(x1, x2)  =  push(x1, x2)
nil  =  nil
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
cons(x1, x2)  =  cons(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

countstack1_in_ga(empty, 0) → countstack1_out_ga(empty, 0)
countstack1_in_ga(push(nil, T11), T8) → U1_ga(T11, T8, countstack1_in_ga(T11, T8))
countstack1_in_ga(push(cons(T21, T22), T23), s(T14)) → U2_ga(T21, T22, T23, T14, countstack1_in_ga(push(T21, push(T22, T23)), T14))
U2_ga(T21, T22, T23, T14, countstack1_out_ga(push(T21, push(T22, T23)), T14)) → countstack1_out_ga(push(cons(T21, T22), T23), s(T14))
U1_ga(T11, T8, countstack1_out_ga(T11, T8)) → countstack1_out_ga(push(nil, T11), T8)

The argument filtering Pi contains the following mapping:
countstack1_in_ga(x1, x2)  =  countstack1_in_ga(x1)
empty  =  empty
countstack1_out_ga(x1, x2)  =  countstack1_out_ga(x1, x2)
push(x1, x2)  =  push(x1, x2)
nil  =  nil
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
cons(x1, x2)  =  cons(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)

(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:

COUNTSTACK1_IN_GA(push(nil, T11), T8) → U1_GA(T11, T8, countstack1_in_ga(T11, T8))
COUNTSTACK1_IN_GA(push(nil, T11), T8) → COUNTSTACK1_IN_GA(T11, T8)
COUNTSTACK1_IN_GA(push(cons(T21, T22), T23), s(T14)) → U2_GA(T21, T22, T23, T14, countstack1_in_ga(push(T21, push(T22, T23)), T14))
COUNTSTACK1_IN_GA(push(cons(T21, T22), T23), s(T14)) → COUNTSTACK1_IN_GA(push(T21, push(T22, T23)), T14)

The TRS R consists of the following rules:

countstack1_in_ga(empty, 0) → countstack1_out_ga(empty, 0)
countstack1_in_ga(push(nil, T11), T8) → U1_ga(T11, T8, countstack1_in_ga(T11, T8))
countstack1_in_ga(push(cons(T21, T22), T23), s(T14)) → U2_ga(T21, T22, T23, T14, countstack1_in_ga(push(T21, push(T22, T23)), T14))
U2_ga(T21, T22, T23, T14, countstack1_out_ga(push(T21, push(T22, T23)), T14)) → countstack1_out_ga(push(cons(T21, T22), T23), s(T14))
U1_ga(T11, T8, countstack1_out_ga(T11, T8)) → countstack1_out_ga(push(nil, T11), T8)

The argument filtering Pi contains the following mapping:
countstack1_in_ga(x1, x2)  =  countstack1_in_ga(x1)
empty  =  empty
countstack1_out_ga(x1, x2)  =  countstack1_out_ga(x1, x2)
push(x1, x2)  =  push(x1, x2)
nil  =  nil
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
cons(x1, x2)  =  cons(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
COUNTSTACK1_IN_GA(x1, x2)  =  COUNTSTACK1_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)

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

(6) Obligation:

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

COUNTSTACK1_IN_GA(push(nil, T11), T8) → U1_GA(T11, T8, countstack1_in_ga(T11, T8))
COUNTSTACK1_IN_GA(push(nil, T11), T8) → COUNTSTACK1_IN_GA(T11, T8)
COUNTSTACK1_IN_GA(push(cons(T21, T22), T23), s(T14)) → U2_GA(T21, T22, T23, T14, countstack1_in_ga(push(T21, push(T22, T23)), T14))
COUNTSTACK1_IN_GA(push(cons(T21, T22), T23), s(T14)) → COUNTSTACK1_IN_GA(push(T21, push(T22, T23)), T14)

The TRS R consists of the following rules:

countstack1_in_ga(empty, 0) → countstack1_out_ga(empty, 0)
countstack1_in_ga(push(nil, T11), T8) → U1_ga(T11, T8, countstack1_in_ga(T11, T8))
countstack1_in_ga(push(cons(T21, T22), T23), s(T14)) → U2_ga(T21, T22, T23, T14, countstack1_in_ga(push(T21, push(T22, T23)), T14))
U2_ga(T21, T22, T23, T14, countstack1_out_ga(push(T21, push(T22, T23)), T14)) → countstack1_out_ga(push(cons(T21, T22), T23), s(T14))
U1_ga(T11, T8, countstack1_out_ga(T11, T8)) → countstack1_out_ga(push(nil, T11), T8)

The argument filtering Pi contains the following mapping:
countstack1_in_ga(x1, x2)  =  countstack1_in_ga(x1)
empty  =  empty
countstack1_out_ga(x1, x2)  =  countstack1_out_ga(x1, x2)
push(x1, x2)  =  push(x1, x2)
nil  =  nil
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
cons(x1, x2)  =  cons(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
COUNTSTACK1_IN_GA(x1, x2)  =  COUNTSTACK1_IN_GA(x1)
U1_GA(x1, x2, x3)  =  U1_GA(x1, x3)
U2_GA(x1, x2, x3, x4, x5)  =  U2_GA(x1, x2, x3, x5)

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

COUNTSTACK1_IN_GA(push(cons(T21, T22), T23), s(T14)) → COUNTSTACK1_IN_GA(push(T21, push(T22, T23)), T14)
COUNTSTACK1_IN_GA(push(nil, T11), T8) → COUNTSTACK1_IN_GA(T11, T8)

The TRS R consists of the following rules:

countstack1_in_ga(empty, 0) → countstack1_out_ga(empty, 0)
countstack1_in_ga(push(nil, T11), T8) → U1_ga(T11, T8, countstack1_in_ga(T11, T8))
countstack1_in_ga(push(cons(T21, T22), T23), s(T14)) → U2_ga(T21, T22, T23, T14, countstack1_in_ga(push(T21, push(T22, T23)), T14))
U2_ga(T21, T22, T23, T14, countstack1_out_ga(push(T21, push(T22, T23)), T14)) → countstack1_out_ga(push(cons(T21, T22), T23), s(T14))
U1_ga(T11, T8, countstack1_out_ga(T11, T8)) → countstack1_out_ga(push(nil, T11), T8)

The argument filtering Pi contains the following mapping:
countstack1_in_ga(x1, x2)  =  countstack1_in_ga(x1)
empty  =  empty
countstack1_out_ga(x1, x2)  =  countstack1_out_ga(x1, x2)
push(x1, x2)  =  push(x1, x2)
nil  =  nil
U1_ga(x1, x2, x3)  =  U1_ga(x1, x3)
cons(x1, x2)  =  cons(x1, x2)
U2_ga(x1, x2, x3, x4, x5)  =  U2_ga(x1, x2, x3, x5)
s(x1)  =  s(x1)
COUNTSTACK1_IN_GA(x1, x2)  =  COUNTSTACK1_IN_GA(x1)

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

(9) UsableRulesProof (EQUIVALENT transformation)

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

(10) Obligation:

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

COUNTSTACK1_IN_GA(push(cons(T21, T22), T23), s(T14)) → COUNTSTACK1_IN_GA(push(T21, push(T22, T23)), T14)
COUNTSTACK1_IN_GA(push(nil, T11), T8) → COUNTSTACK1_IN_GA(T11, T8)

R is empty.
The argument filtering Pi contains the following mapping:
push(x1, x2)  =  push(x1, x2)
nil  =  nil
cons(x1, x2)  =  cons(x1, x2)
s(x1)  =  s(x1)
COUNTSTACK1_IN_GA(x1, x2)  =  COUNTSTACK1_IN_GA(x1)

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

(11) PiDPToQDPProof (SOUND transformation)

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

(12) Obligation:

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

COUNTSTACK1_IN_GA(push(cons(T21, T22), T23)) → COUNTSTACK1_IN_GA(push(T21, push(T22, T23)))
COUNTSTACK1_IN_GA(push(nil, T11)) → COUNTSTACK1_IN_GA(T11)

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

(13) UsableRulesReductionPairsProof (EQUIVALENT transformation)

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

The following dependency pairs can be deleted:

COUNTSTACK1_IN_GA(push(cons(T21, T22), T23)) → COUNTSTACK1_IN_GA(push(T21, push(T22, T23)))
COUNTSTACK1_IN_GA(push(nil, T11)) → COUNTSTACK1_IN_GA(T11)

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(COUNTSTACK1_IN_GA(x₁)) = 2·x₁   
POL(cons(x₁, x₂)) = 2 + 2·x₁ + 2·x₂   
POL(nil) = 0   
POL(push(x₁, x₂)) = 1 + 2·x₁ + x₂   

(14) Obligation:

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

(15) PisEmptyProof (EQUIVALENT transformation)

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

(16) TRUE