Left Termination proof of /tmp/tmp1cmeth/countstack.pl

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

0 Prolog

↳1 PrologToDTProblemTransformerProof (⇒, 89 ms)

↳2 TRIPLES

↳3 TriplesToPiDPProof (⇒, 72 ms)

↳4 PiDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 PiDP

↳7 PiDPToQDPProof (⇒, 0 ms)

↳8 QDP

↳9 UsableRulesReductionPairsProof (⇔, 129 ms)

↳10 QDP

↳11 PisEmptyProof (⇔, 0 ms)

↳12 YES

(0) Obligation:

Clauses:

countstack(empty, 0).
countstack(push(nil, T), X) :- countstack(T, X).
countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X).

Query: countstack(g,a)

(1) PrologToDTProblemTransformerProof (SOUND transformation)

Built DT problem from termination graph DT10.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="A: 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\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: true | countstack(empty, T2), SCOPE: 1, CLAUSE: 1 | countstack(empty, T2), SCOPE: 1, CLAUSE: 2\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\n\nT1 is ground\ncountstack(T1, T2) !=? countstack(empty, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: countstack(empty, T2), SCOPE: 1, CLAUSE: 1 | countstack(empty, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: countstack(empty, T2), SCOPE: 1, CLAUSE: 2\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: countstack(T5, T7) | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: countstack(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nX9 is free\nX10 is free\ncountstack(T1, T2) !=? countstack(empty, 0)\ncountstack(T1, T2) !=? countstack(push(nil, X9), X10)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: countstack(T5, T7), SCOPE: 2, CLAUSE: 0 | countstack(T5, T7), SCOPE: 2, CLAUSE: 1 | countstack(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: countstack(T5, T7), SCOPE: 2, CLAUSE: 0\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: countstack(T5, T7), SCOPE: 2, CLAUSE: 1 | countstack(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: countstack(T5, T7), SCOPE: 2, CLAUSE: 1\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: countstack(T5, T7), SCOPE: 2, CLAUSE: 2 | ?_2 | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: countstack(T16, T18)\n\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: countstack(T5, T7), SCOPE: 2, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ?_2 | countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: countstack(push(T35, push(T36, T37)), T39)\n\nT35 is ground\nT36 is ground\nT37 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: countstack(push(nil, T5), T2), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: countstack(push(T44, push(T45, T46)), T48)\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 0 | countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 1 | countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 2\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 1 | countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 2\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 1\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: countstack(push(T44, push(T45, T46)), T48), SCOPE: 3, CLAUSE: 2\n\nT44 is ground\nT45 is ground\nT46 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: countstack(push(T61, T62), T64)\n\nT61 is ground\nT62 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: countstack(push(T75, push(T76, push(T77, T78))), T80)\n\nT75 is ground\nT76 is ground\nT77 is ground\nT78 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 36 [arrowhead = none ];
36 -> 2;

37 [label="EVAL with clause\ncountstack(empty, 0).\nand substitutionT1 -> empty,\nT2 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 37 [arrowhead = none ];
37 -> 3;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 38 [arrowhead = none ];
38 -> 4;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 39 [arrowhead = none ];
39 -> 5;

40 [label="EVAL with clause\ncountstack(push(nil, X9), X10) :- countstack(X9, X10).\nand substitutionX9 -> T5,\nT1 -> push(nil, T5),\nT2 -> T7,\nX10 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 40 [arrowhead = none ];
40 -> 8;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 41 [arrowhead = none ];
41 -> 9;

42 [label="BACKTRACK\nfor clause: countstack(push(nil, T), X) :- countstack(T, X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 42 [arrowhead = none ];
42 -> 6;

43 [label="BACKTRACK\nfor clause: countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 43 [arrowhead = none ];
43 -> 7;

44 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
8 -> 44 [arrowhead = none ];
44 -> 10;

45 [label="EVAL with clause\ncountstack(push(cons(X49, X50), X51), s(X52)) :- countstack(push(X49, push(X50, X51)), X52).\nand substitutionX49 -> T44,\nX50 -> T45,\nX51 -> T46,\nT1 -> push(cons(T44, T45), T46),\nX52 -> T48,\nT2 -> s(T48),\nT47 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 45 [arrowhead = none ];
45 -> 26;

46 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 46 [arrowhead = none ];
46 -> 27;

47 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 47 [arrowhead = none ];
47 -> 11;

48 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
10 -> 48 [arrowhead = none ];
48 -> 12;

49 [label="EVAL with clause\ncountstack(empty, 0).\nand substitutionT5 -> empty,\nT7 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 49 [arrowhead = none ];
49 -> 13;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 50 [arrowhead = none ];
50 -> 14;

51 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 51 [arrowhead = none ];
51 -> 16;

52 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
12 -> 52 [arrowhead = none ];
52 -> 17;

53 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 53 [arrowhead = none ];
53 -> 15;

54 [label="EVAL with clause\ncountstack(push(nil, X19), X20) :- countstack(X19, X20).\nand substitutionX19 -> T16,\nT5 -> push(nil, T16),\nT7 -> T18,\nX20 -> T18,\nT17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
16 -> 54 [arrowhead = none ];
54 -> 18;

55 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 55 [arrowhead = none ];
55 -> 19;

56 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 56 [arrowhead = none ];
56 -> 20;

57 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
17 -> 57 [arrowhead = none ];
57 -> 21;

58 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> T18", fontsize=14, style = filled, fillcolor = lightgrey];
18 -> 58 [arrowhead = none , style=dashed];
58 -> 1[style=dashed];

59 [label="EVAL with clause\ncountstack(push(cons(X37, X38), X39), s(X40)) :- countstack(push(X37, push(X38, X39)), X40).\nand substitutionX37 -> T35,\nX38 -> T36,\nX39 -> T37,\nT5 -> push(cons(T35, T36), T37),\nX40 -> T39,\nT7 -> s(T39),\nT38 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 59 [arrowhead = none ];
59 -> 22;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 60 [arrowhead = none ];
60 -> 23;

61 [label="FAILURE", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 61 [arrowhead = none ];
61 -> 24;

62 [label="INSTANCE with matching:\nT1 -> push(T35, push(T36, T37))\nT2 -> T39", fontsize=14, style = filled, fillcolor = lightgrey];
22 -> 62 [arrowhead = none , style=dashed];
62 -> 1[style=dashed];

63 [label="BACKTRACK\nfor clause: countstack(push(cons(U, V), T), s(X)) :- countstack(push(U, push(V, T)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 63 [arrowhead = none ];
63 -> 25;

64 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 64 [arrowhead = none ];
64 -> 28;

65 [label="BACKTRACK\nfor clause: countstack(empty, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 65 [arrowhead = none ];
65 -> 29;

66 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 66 [arrowhead = none ];
66 -> 30;

67 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
29 -> 67 [arrowhead = none ];
67 -> 31;

68 [label="EVAL with clause\ncountstack(push(nil, X61), X62) :- countstack(X61, X62).\nand substitutionT44 -> nil,\nT45 -> T61,\nT46 -> T62,\nX61 -> push(T61, T62),\nT48 -> T64,\nX62 -> T64,\nT63 -> T64", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 68 [arrowhead = none ];
68 -> 32;

69 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 69 [arrowhead = none ];
69 -> 33;

70 [label="EVAL with clause\ncountstack(push(cons(X71, X72), X73), s(X74)) :- countstack(push(X71, push(X72, X73)), X74).\nand substitutionX71 -> T75,\nX72 -> T76,\nT44 -> cons(T75, T76),\nT45 -> T77,\nT46 -> T78,\nX73 -> push(T77, T78),\nX74 -> T80,\nT48 -> s(T80),\nT79 -> T80", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 70 [arrowhead = none ];
70 -> 34;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 71 [arrowhead = none ];
71 -> 35;

72 [label="INSTANCE with matching:\nT1 -> push(T61, T62)\nT2 -> T64", fontsize=14, style = filled, fillcolor = lightgrey];
32 -> 72 [arrowhead = none , style=dashed];
72 -> 1[style=dashed];

73 [label="INSTANCE with matching:\nT1 -> push(T75, push(T76, push(T77, T78)))\nT2 -> T80", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 73 [arrowhead = none , style=dashed];
73 -> 1[style=dashed];

}

(2) Obligation:

Triples:

countstackA(push(nil, push(nil, X1)), X2) :- countstackA(X1, X2).
countstackA(push(nil, push(cons(X1, X2), X3)), s(X4)) :- countstackA(push(X1, push(X2, X3)), X4).
countstackA(push(cons(nil, X1), X2), s(X3)) :- countstackA(push(X1, X2), X3).
countstackA(push(cons(cons(X1, X2), X3), X4), s(s(X5))) :- countstackA(push(X1, push(X2, push(X3, X4))), X5).

Clauses:

countstackcA(empty, 0).
countstackcA(push(nil, empty), 0).
countstackcA(push(nil, push(nil, X1)), X2) :- countstackcA(X1, X2).
countstackcA(push(nil, push(cons(X1, X2), X3)), s(X4)) :- countstackcA(push(X1, push(X2, X3)), X4).
countstackcA(push(cons(nil, X1), X2), s(X3)) :- countstackcA(push(X1, X2), X3).
countstackcA(push(cons(cons(X1, X2), X3), X4), s(s(X5))) :- countstackcA(push(X1, push(X2, push(X3, X4))), X5).

Afs:

countstackA(x1, x2) = countstackA(x1)

(3) TriplesToPiDPProof (SOUND transformation)

We use the technique of [DT09]. With regard to the inferred argument filtering the predicates were used in the following modes:
countstackA_in: (b,f)
Transforming TRIPLES into the following Term Rewriting System:
Pi DP problem:
The TRS P consists of the following rules:

COUNTSTACKA_IN_GA(push(nil, push(nil, X1)), X2) → U1_GA(X1, X2, countstackA_in_ga(X1, X2))
COUNTSTACKA_IN_GA(push(nil, push(nil, X1)), X2) → COUNTSTACKA_IN_GA(X1, X2)
COUNTSTACKA_IN_GA(push(nil, push(cons(X1, X2), X3)), s(X4)) → U2_GA(X1, X2, X3, X4, countstackA_in_ga(push(X1, push(X2, X3)), X4))
COUNTSTACKA_IN_GA(push(nil, push(cons(X1, X2), X3)), s(X4)) → COUNTSTACKA_IN_GA(push(X1, push(X2, X3)), X4)
COUNTSTACKA_IN_GA(push(cons(nil, X1), X2), s(X3)) → U3_GA(X1, X2, X3, countstackA_in_ga(push(X1, X2), X3))
COUNTSTACKA_IN_GA(push(cons(nil, X1), X2), s(X3)) → COUNTSTACKA_IN_GA(push(X1, X2), X3)
COUNTSTACKA_IN_GA(push(cons(cons(X1, X2), X3), X4), s(s(X5))) → U4_GA(X1, X2, X3, X4, X5, countstackA_in_ga(push(X1, push(X2, push(X3, X4))), X5))
COUNTSTACKA_IN_GA(push(cons(cons(X1, X2), X3), X4), s(s(X5))) → COUNTSTACKA_IN_GA(push(X1, push(X2, push(X3, X4))), X5)

R is empty.
The argument filtering Pi contains the following mapping:
countstackA_in_ga(x1, x2) = countstackA_in_ga(x1)
push(x1, x2) = push(x1, x2)
nil = nil
cons(x1, x2) = cons(x1, x2)
s(x1) = s(x1)
COUNTSTACKA_IN_GA(x1, x2) = COUNTSTACKA_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)
U3_GA(x1, x2, x3, x4) = U3_GA(x1, x2, x4)
U4_GA(x1, x2, x3, x4, x5, x6) = U4_GA(x1, x2, x3, x4, x6)

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

Infinitary Constructor Rewriting Termination of PiDP implies Termination of TRIPLES

(4) Obligation:

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

COUNTSTACKA_IN_GA(push(nil, push(nil, X1)), X2) → U1_GA(X1, X2, countstackA_in_ga(X1, X2))
COUNTSTACKA_IN_GA(push(nil, push(nil, X1)), X2) → COUNTSTACKA_IN_GA(X1, X2)
COUNTSTACKA_IN_GA(push(nil, push(cons(X1, X2), X3)), s(X4)) → U2_GA(X1, X2, X3, X4, countstackA_in_ga(push(X1, push(X2, X3)), X4))
COUNTSTACKA_IN_GA(push(nil, push(cons(X1, X2), X3)), s(X4)) → COUNTSTACKA_IN_GA(push(X1, push(X2, X3)), X4)
COUNTSTACKA_IN_GA(push(cons(nil, X1), X2), s(X3)) → U3_GA(X1, X2, X3, countstackA_in_ga(push(X1, X2), X3))
COUNTSTACKA_IN_GA(push(cons(nil, X1), X2), s(X3)) → COUNTSTACKA_IN_GA(push(X1, X2), X3)
COUNTSTACKA_IN_GA(push(cons(cons(X1, X2), X3), X4), s(s(X5))) → U4_GA(X1, X2, X3, X4, X5, countstackA_in_ga(push(X1, push(X2, push(X3, X4))), X5))
COUNTSTACKA_IN_GA(push(cons(cons(X1, X2), X3), X4), s(s(X5))) → COUNTSTACKA_IN_GA(push(X1, push(X2, push(X3, X4))), X5)

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(6) Obligation:

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

COUNTSTACKA_IN_GA(push(nil, push(cons(X1, X2), X3)), s(X4)) → COUNTSTACKA_IN_GA(push(X1, push(X2, X3)), X4)
COUNTSTACKA_IN_GA(push(nil, push(nil, X1)), X2) → COUNTSTACKA_IN_GA(X1, X2)
COUNTSTACKA_IN_GA(push(cons(nil, X1), X2), s(X3)) → COUNTSTACKA_IN_GA(push(X1, X2), X3)
COUNTSTACKA_IN_GA(push(cons(cons(X1, X2), X3), X4), s(s(X5))) → COUNTSTACKA_IN_GA(push(X1, push(X2, push(X3, X4))), X5)

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)
COUNTSTACKA_IN_GA(x1, x2) = COUNTSTACKA_IN_GA(x1)

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

(7) PiDPToQDPProof (SOUND transformation)

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

(8) Obligation:

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

COUNTSTACKA_IN_GA(push(nil, push(cons(X1, X2), X3))) → COUNTSTACKA_IN_GA(push(X1, push(X2, X3)))
COUNTSTACKA_IN_GA(push(nil, push(nil, X1))) → COUNTSTACKA_IN_GA(X1)
COUNTSTACKA_IN_GA(push(cons(nil, X1), X2)) → COUNTSTACKA_IN_GA(push(X1, X2))
COUNTSTACKA_IN_GA(push(cons(cons(X1, X2), X3), X4)) → COUNTSTACKA_IN_GA(push(X1, push(X2, push(X3, X4))))

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

(9) 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:

COUNTSTACKA_IN_GA(push(nil, push(cons(X1, X2), X3))) → COUNTSTACKA_IN_GA(push(X1, push(X2, X3)))
COUNTSTACKA_IN_GA(push(nil, push(nil, X1))) → COUNTSTACKA_IN_GA(X1)
COUNTSTACKA_IN_GA(push(cons(nil, X1), X2)) → COUNTSTACKA_IN_GA(push(X1, X2))
COUNTSTACKA_IN_GA(push(cons(cons(X1, X2), X3), X4)) → COUNTSTACKA_IN_GA(push(X1, push(X2, push(X3, X4))))

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

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

(10) Obligation:

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

(11) PisEmptyProof (EQUIVALENT transformation)

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