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

(0) Obligation:

Clauses:

transpose_aux(.(R, Rs), X1, .(C, Cs)) :- row2col(R, .(C, Cs), Cols1, [], Accm).
row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), A, B) :- row2col(Xs, Cols, Cols1, .([], A), B).

Query: transpose_aux(a,g,a)

(1) PrologDeterminacyProcessorProof (EQUIVALENT transformation)

The root node satisfies the determinacy criterion.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: transpose_aux(T1, T2, T3)\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
2 [label="2: transpose_aux(T1, T2, T3), SCOPE: 1, CLAUSE: 0\n\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
3 [label="3: row2col(T14, .(T15, T16), X12, [], X13)\n\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
4 [label="4: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: row2col(T14, .(T15, T16), X12, [], X13), SCOPE: 2, CLAUSE: 1\n\nX12 is free\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: row2col(T29, T30, X39, .([], []), X40)\n\nX39 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: row2col(T29, T30, X39, .([], []), X40), SCOPE: 3, CLAUSE: 1\n\nX39 is free\nX40 is free", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: row2col(T47, T48, X75, .([], .([], [])), X76)\n\nX75 is free\nX76 is free", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: row2col(T47, T48, X75, .([], .([], [])), X76), SCOPE: 4, CLAUSE: 1\n\nX75 is free\nX76 is free", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: row2col(T65, T66, X111, .([], .([], .([], []))), X112)\n\nX111 is free\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: row2col(T65, T66, X111, .([], .([], .([], []))), X112), SCOPE: 5, CLAUSE: 1\n\nX111 is free\nX112 is free", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: row2col(T83, T84, X147, .([], .([], .([], .([], [])))), X148)\n\nX147 is free\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: row2col(T83, T84, X147, .([], .([], .([], .([], [])))), X148), SCOPE: 6, CLAUSE: 1\n\nX147 is free\nX148 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: row2col(T101, T102, X183, .([], .([], .([], .([], .([], []))))), X184)\n\nX183 is free\nX184 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: row2col(T101, T102, X183, .([], .([], .([], .([], .([], []))))), X184), SCOPE: 7, CLAUSE: 1\n\nX183 is free\nX184 is free", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: row2col(T119, T120, X219, .([], .([], .([], .([], .([], .([], [])))))), X220)\n\nX219 is free\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: row2col(T119, T120, X219, .([], .([], .([], .([], .([], .([], [])))))), X220), SCOPE: 8, CLAUSE: 1\n\nX219 is free\nX220 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: row2col(T137, T138, X255, .([], .([], .([], .([], .([], .([], .([], []))))))), X256)\n\nX255 is free\nX256 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: row2col(T137, T138, X255, .([], .([], .([], .([], .([], .([], .([], []))))))), X256), SCOPE: 9, CLAUSE: 1\n\nX255 is free\nX256 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: row2col(T155, T156, X291, .([], .([], .([], .([], .([], .([], .([], .([], [])))))))), X292)\n\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: row2col(T155, T156, X291, .([], T161), X292)\n\nT161 is ground\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: row2col(T155, T156, X291, .([], T161), X292), SCOPE: 10, CLAUSE: 1\n\nT161 is ground\nX291 is free\nX292 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: row2col(T182, T183, X336, .([], .([], T181)), X337)\n\nT181 is ground\nX336 is free\nX337 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 33 [arrowhead = none ];
33 -> 2;

34 [label="EVAL with clause\ntranspose_aux(.(X7, X8), X9, .(X10, X11)) :- row2col(X7, .(X10, X11), X12, [], X13).\nand substitutionX7 -> T14,\nX8 -> T10,\nT1 -> .(T14, T10),\nT2 -> T11,\nX9 -> T11,\nX10 -> T15,\nX11 -> T16,\nT3 -> .(T15, T16),\nT9 -> T14,\nT12 -> T15,\nT13 -> T16", fontsize=14, style = filled, fillcolor = lightblue];
2 -> 34 [arrowhead = none ];
34 -> 3;

35 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
2 -> 35 [arrowhead = none ];
35 -> 4;

36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
3 -> 36 [arrowhead = none ];
36 -> 5;

37 [label="EVAL with clause\nrow2col(.(X32, X33), .(.(X32, X34), X35), .(X34, X36), X37, X38) :- row2col(X33, X35, X36, .([], X37), X38).\nand substitutionX32 -> T25,\nX33 -> T29,\nT14 -> .(T25, T29),\nX34 -> T27,\nT15 -> .(T25, T27),\nT16 -> T30,\nX35 -> T30,\nX36 -> X39,\nX12 -> .(T27, X39),\nX37 -> [],\nX13 -> X40,\nX38 -> X40,\nT26 -> T29,\nT28 -> T30", fontsize=14, style = filled, fillcolor = lightblue];
5 -> 37 [arrowhead = none ];
37 -> 6;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 38 [arrowhead = none ];
38 -> 7;

39 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 39 [arrowhead = none ];
39 -> 8;

40 [label="EVAL with clause\nrow2col(.(X68, X69), .(.(X68, X70), X71), .(X70, X72), X73, X74) :- row2col(X69, X71, X72, .([], X73), X74).\nand substitutionX68 -> T43,\nX69 -> T47,\nT29 -> .(T43, T47),\nX70 -> T45,\nX71 -> T48,\nT30 -> .(.(T43, T45), T48),\nX72 -> X75,\nX39 -> .(T45, X75),\nX73 -> .([], []),\nX40 -> X76,\nX74 -> X76,\nT44 -> T47,\nT46 -> T48", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 40 [arrowhead = none ];
40 -> 9;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 41 [arrowhead = none ];
41 -> 10;

42 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
9 -> 42 [arrowhead = none ];
42 -> 11;

43 [label="EVAL with clause\nrow2col(.(X104, X105), .(.(X104, X106), X107), .(X106, X108), X109, X110) :- row2col(X105, X107, X108, .([], X109), X110).\nand substitutionX104 -> T61,\nX105 -> T65,\nT47 -> .(T61, T65),\nX106 -> T63,\nX107 -> T66,\nT48 -> .(.(T61, T63), T66),\nX108 -> X111,\nX75 -> .(T63, X111),\nX109 -> .([], .([], [])),\nX76 -> X112,\nX110 -> X112,\nT62 -> T65,\nT64 -> T66", fontsize=14, style = filled, fillcolor = lightblue];
11 -> 43 [arrowhead = none ];
43 -> 12;

44 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 44 [arrowhead = none ];
44 -> 13;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
12 -> 45 [arrowhead = none ];
45 -> 14;

46 [label="EVAL with clause\nrow2col(.(X140, X141), .(.(X140, X142), X143), .(X142, X144), X145, X146) :- row2col(X141, X143, X144, .([], X145), X146).\nand substitutionX140 -> T79,\nX141 -> T83,\nT65 -> .(T79, T83),\nX142 -> T81,\nX143 -> T84,\nT66 -> .(.(T79, T81), T84),\nX144 -> X147,\nX111 -> .(T81, X147),\nX145 -> .([], .([], .([], []))),\nX112 -> X148,\nX146 -> X148,\nT80 -> T83,\nT82 -> T84", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 46 [arrowhead = none ];
46 -> 15;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 47 [arrowhead = none ];
47 -> 16;

48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 48 [arrowhead = none ];
48 -> 17;

49 [label="EVAL with clause\nrow2col(.(X176, X177), .(.(X176, X178), X179), .(X178, X180), X181, X182) :- row2col(X177, X179, X180, .([], X181), X182).\nand substitutionX176 -> T97,\nX177 -> T101,\nT83 -> .(T97, T101),\nX178 -> T99,\nX179 -> T102,\nT84 -> .(.(T97, T99), T102),\nX180 -> X183,\nX147 -> .(T99, X183),\nX181 -> .([], .([], .([], .([], [])))),\nX148 -> X184,\nX182 -> X184,\nT98 -> T101,\nT100 -> T102", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 49 [arrowhead = none ];
49 -> 18;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 50 [arrowhead = none ];
50 -> 19;

51 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
18 -> 51 [arrowhead = none ];
51 -> 20;

52 [label="EVAL with clause\nrow2col(.(X212, X213), .(.(X212, X214), X215), .(X214, X216), X217, X218) :- row2col(X213, X215, X216, .([], X217), X218).\nand substitutionX212 -> T115,\nX213 -> T119,\nT101 -> .(T115, T119),\nX214 -> T117,\nX215 -> T120,\nT102 -> .(.(T115, T117), T120),\nX216 -> X219,\nX183 -> .(T117, X219),\nX217 -> .([], .([], .([], .([], .([], []))))),\nX184 -> X220,\nX218 -> X220,\nT116 -> T119,\nT118 -> T120", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 52 [arrowhead = none ];
52 -> 21;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 53 [arrowhead = none ];
53 -> 22;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 54 [arrowhead = none ];
54 -> 23;

55 [label="EVAL with clause\nrow2col(.(X248, X249), .(.(X248, X250), X251), .(X250, X252), X253, X254) :- row2col(X249, X251, X252, .([], X253), X254).\nand substitutionX248 -> T133,\nX249 -> T137,\nT119 -> .(T133, T137),\nX250 -> T135,\nX251 -> T138,\nT120 -> .(.(T133, T135), T138),\nX252 -> X255,\nX219 -> .(T135, X255),\nX253 -> .([], .([], .([], .([], .([], .([], [])))))),\nX220 -> X256,\nX254 -> X256,\nT134 -> T137,\nT136 -> T138", fontsize=14, style = filled, fillcolor = lightblue];
23 -> 55 [arrowhead = none ];
55 -> 24;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 56 [arrowhead = none ];
56 -> 25;

57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
24 -> 57 [arrowhead = none ];
57 -> 26;

58 [label="EVAL with clause\nrow2col(.(X284, X285), .(.(X284, X286), X287), .(X286, X288), X289, X290) :- row2col(X285, X287, X288, .([], X289), X290).\nand substitutionX284 -> T151,\nX285 -> T155,\nT137 -> .(T151, T155),\nX286 -> T153,\nX287 -> T156,\nT138 -> .(.(T151, T153), T156),\nX288 -> X291,\nX255 -> .(T153, X291),\nX289 -> .([], .([], .([], .([], .([], .([], .([], []))))))),\nX256 -> X292,\nX290 -> X292,\nT152 -> T155,\nT154 -> T156", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 58 [arrowhead = none ];
58 -> 27;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 59 [arrowhead = none ];
59 -> 28;

60 [label="GENERALIZATION\nT161 <-- .([], .([], .([], .([], .([], .([], .([], [])))))))\n\nNew Knowledge:\nT161 is ground", fontsize=14, style = filled, fillcolor = violet];
27 -> 60 [arrowhead = none ];
60 -> 29;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 61 [arrowhead = none ];
61 -> 30;

62 [label="EVAL with clause\nrow2col(.(X329, X330), .(.(X329, X331), X332), .(X331, X333), X334, X335) :- row2col(X330, X332, X333, .([], X334), X335).\nand substitutionX329 -> T177,\nX330 -> T182,\nT155 -> .(T177, T182),\nX331 -> T179,\nX332 -> T183,\nT156 -> .(.(T177, T179), T183),\nX333 -> X336,\nX291 -> .(T179, X336),\nT161 -> T181,\nX334 -> .([], T181),\nX292 -> X337,\nX335 -> X337,\nT178 -> T182,\nT180 -> T183", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 62 [arrowhead = none ];
62 -> 31;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 63 [arrowhead = none ];
63 -> 32;

64 [label="INSTANCE with matching:\nT155 -> T182\nT156 -> T183\nX291 -> X336\nT161 -> .([], T181)\nX292 -> X337", fontsize=14, style = filled, fillcolor = lightgrey];
31 -> 64 [arrowhead = none , style=dashed];
64 -> 29[style=dashed];

}

(2) YES