Runtime Complexity of the query pattern
transpose(g,a)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^1)).

(0) Obligation:

Clauses:

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

Query: transpose(g,a)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

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

Query: transpose(g,a)

(3) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: transpose(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: transpose(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: transpose_aux(T8, [], T10)\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: transpose_aux(T8, [], T10), SCOPE: 2, CLAUSE: 0 | transpose_aux(T8, [], T10), SCOPE: 2, CLAUSE: 3\n\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | transpose_aux([], [], T10), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: transpose_aux(T8, [], T10), SCOPE: 2, CLAUSE: 3\n\nT8 is ground\nX14 is free\ntranspose_aux(T8, [], T10) !=? transpose_aux([], X14, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: transpose_aux([], [], T10), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(row2col(T19, .(T23, T24), X37, X38), transpose_aux(T20, X38, X37))\n\nT19 is ground\nT20 is ground\nX37 is free\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: row2col(T19, .(T23, T24), X37, X38)\n\nT19 is ground\nX37 is free\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: transpose_aux(T20, T26, T25)\n\nT20 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: row2col(T19, .(T23, T24), X37, X38), SCOPE: 3, CLAUSE: 1 | row2col(T19, .(T23, T24), X37, X38), SCOPE: 3, CLAUSE: 4\n\nT19 is ground\nX37 is free\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: row2col(T19, .(T23, T24), X37, X38), SCOPE: 3, CLAUSE: 4\n\nT19 is ground\nX37 is free\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: row2col(T36, T39, X61, X62)\n\nT36 is ground\nX61 is free\nX62 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: row2col(T36, T39, X61, X62), SCOPE: 4, CLAUSE: 1 | row2col(T36, T39, X61, X62), SCOPE: 4, CLAUSE: 4\n\nT36 is ground\nX61 is free\nX62 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true | row2col([], T39, X61, X62), SCOPE: 4, CLAUSE: 4\n\nX61 is free\nX62 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: row2col(T36, T39, X61, X62), SCOPE: 4, CLAUSE: 4\n\nT36 is ground\nX61 is free\nX62 is free\nrow2col(T36, T39, X61, X62) !=? row2col([], [], [], [])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: row2col([], T39, X61, X62), SCOPE: 4, CLAUSE: 4\n\nX61 is free\nX62 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: row2col(T49, T52, X91, X92)\n\nT49 is ground\nX91 is free\nX92 is free", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: transpose_aux(T20, T26, T25), SCOPE: 5, CLAUSE: 0 | transpose_aux(T20, T26, T25), SCOPE: 5, CLAUSE: 3\n\nT20 is ground\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true | transpose_aux([], T57, T25), SCOPE: 5, CLAUSE: 3\n\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: transpose_aux(T20, T26, T25), SCOPE: 5, CLAUSE: 3\n\nT20 is ground\nT26 is ground\nX97 is free\ntranspose_aux(T20, T26, T25) !=? transpose_aux([], X97, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: transpose_aux([], T57, T25), SCOPE: 5, CLAUSE: 3\n\nT57 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(row2col(T68, .(T73, T74), X120, X121), transpose_aux(T69, X121, X120))\n\nT68 is ground\nT69 is ground\nX120 is free\nX121 is free", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 61 [arrowhead = none ];
61 -> 4;

62 [label="ONLY EVAL with clause\ntranspose(X8, X9) :- transpose_aux(X8, [], X9).\nand substitutionT1 -> T8,\nX8 -> T8,\nT2 -> T10,\nX9 -> T10,\nT9 -> T10", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 62 [arrowhead = none ];
62 -> 10;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
10 -> 63 [arrowhead = none ];
63 -> 14;

64 [label="EVAL with clause\ntranspose_aux([], X14, X14).\nand substitutionT8 -> [],\nX14 -> [],\nT10 -> []", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 64 [arrowhead = none ];
64 -> 16;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 65 [arrowhead = none ];
65 -> 18;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 66 [arrowhead = none ];
66 -> 19;

67 [label="EVAL with clause\ntranspose_aux(.(X32, X33), X34, .(X35, X36)) :- ','(row2col(X32, .(X35, X36), X37, X38), transpose_aux(X33, X38, X37)).\nand substitutionX32 -> T19,\nX33 -> T20,\nT8 -> .(T19, T20),\nX34 -> [],\nX35 -> T23,\nX36 -> T24,\nT10 -> .(T23, T24),\nT21 -> T23,\nT22 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 67 [arrowhead = none ];
67 -> 23;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 68 [arrowhead = none ];
68 -> 24;

69 [label="BACKTRACK\nfor clause: transpose_aux(.(R, Rs), X1, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 69 [arrowhead = none ];
69 -> 20;

70 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 70 [arrowhead = none ];
70 -> 25;

71 [label="SPLIT 2\nnew knowledge:\nT19 is ground\nT26 is ground\nreplacements:X37 -> T25,\nX38 -> T26", fontsize=14, style = filled, fillcolor = violet];
23 -> 71 [arrowhead = none ];
71 -> 26;

72 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
25 -> 72 [arrowhead = none ];
72 -> 27;

73 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
26 -> 73 [arrowhead = none ];
73 -> 48;

74 [label="BACKTRACK\nfor clause: row2col([], [], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 74 [arrowhead = none ];
74 -> 28;

75 [label="EVAL with clause\nrow2col(.(X55, X56), .(.(X55, X57), X58), .(X57, X59), .([], X60)) :- row2col(X56, X58, X59, X60).\nand substitutionX55 -> T35,\nX56 -> T36,\nT19 -> .(T35, T36),\nX57 -> T37,\nT23 -> .(T35, T37),\nT24 -> T39,\nX58 -> T39,\nX59 -> X61,\nX37 -> .(T37, X61),\nX60 -> X62,\nX38 -> .([], X62),\nT38 -> T39", fontsize=14, style = filled, fillcolor = lightblue];
28 -> 75 [arrowhead = none ];
75 -> 32;

76 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 76 [arrowhead = none ];
76 -> 34;

77 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 77 [arrowhead = none ];
77 -> 37;

78 [label="EVAL with clause\nrow2col([], [], [], []).\nand substitutionT36 -> [],\nT39 -> [],\nX61 -> [],\nX62 -> []", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 78 [arrowhead = none ];
78 -> 39;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 79 [arrowhead = none ];
79 -> 40;

80 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 80 [arrowhead = none ];
80 -> 41;

81 [label="EVAL with clause\nrow2col(.(X85, X86), .(.(X85, X87), X88), .(X87, X89), .([], X90)) :- row2col(X86, X88, X89, X90).\nand substitutionX85 -> T48,\nX86 -> T49,\nT36 -> .(T48, T49),\nX87 -> T50,\nX88 -> T52,\nT39 -> .(.(T48, T50), T52),\nX89 -> X91,\nX61 -> .(T50, X91),\nX90 -> X92,\nX62 -> .([], X92),\nT51 -> T52", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 81 [arrowhead = none ];
81 -> 45;

82 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 82 [arrowhead = none ];
82 -> 46;

83 [label="BACKTRACK\nfor clause: row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) :- row2col(Xs, Cols, Cols1, As)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 83 [arrowhead = none ];
83 -> 42;

84 [label="INSTANCE with matching:\nT36 -> T49\nT39 -> T52\nX61 -> X91\nX62 -> X92", fontsize=14, style = filled, fillcolor = lightgrey];
45 -> 84 [arrowhead = none , style=dashed];
84 -> 32[style=dashed];

85 [label="EVAL with clause\ntranspose_aux([], X97, X97).\nand substitutionT20 -> [],\nT26 -> T57,\nX97 -> T57,\nT25 -> T57", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 85 [arrowhead = none ];
85 -> 52;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 86 [arrowhead = none ];
86 -> 54;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 87 [arrowhead = none ];
87 -> 55;

88 [label="EVAL with clause\ntranspose_aux(.(X115, X116), X117, .(X118, X119)) :- ','(row2col(X115, .(X118, X119), X120, X121), transpose_aux(X116, X121, X120)).\nand substitutionX115 -> T68,\nX116 -> T69,\nT20 -> .(T68, T69),\nT26 -> T70,\nX117 -> T70,\nX118 -> T73,\nX119 -> T74,\nT25 -> .(T73, T74),\nT71 -> T73,\nT72 -> T74", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 88 [arrowhead = none ];
88 -> 59;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 89 [arrowhead = none ];
89 -> 60;

90 [label="BACKTRACK\nfor clause: transpose_aux(.(R, Rs), X1, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 90 [arrowhead = none ];
90 -> 56;

91 [label="INSTANCE with matching:\nT19 -> T68\nT23 -> T73\nT24 -> T74\nX37 -> X120\nX38 -> X121\nT20 -> T69", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 91 [arrowhead = none , style=dashed];
91 -> 23[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f23_in(z0, z1), .(z0, z1))
U1(f23_out1(z0), .(z1, z2)) → f1_out1
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))
f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
f26_in([], z0) → f26_out1
f26_in(.(z0, z1), z2) → U4(f23_in(z0, z1), .(z0, z1), z2)
U4(f23_out1(z0), .(z1, z2), z3) → f26_out1
f23_in(z0, z1) → U5(f25_in(z0), z0, z1)
U5(f25_out1(z0), z1, z2) → U6(f26_in(z2, z0), z1, z2, z0)
U6(f26_out1, z0, z1, z2) → f23_out1(z2)

Tuples:

F1_IN(.(z0, z1)) → c1(U1'(f23_in(z0, z1), .(z0, z1)), F23_IN(z0, z1))
F32_IN(.(z0, z1)) → c4(U2'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F25_IN(.(z0, z1)) → c6(U3'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(U4'(f23_in(z0, z1), .(z0, z1), z2), F23_IN(z0, z1))
F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
U5'(f25_out1(z0), z1, z2) → c12(U6'(f26_in(z2, z0), z1, z2, z0), F26_IN(z2, z0))

S tuples:

F1_IN(.(z0, z1)) → c1(U1'(f23_in(z0, z1), .(z0, z1)), F23_IN(z0, z1))
F32_IN(.(z0, z1)) → c4(U2'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F25_IN(.(z0, z1)) → c6(U3'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(U4'(f23_in(z0, z1), .(z0, z1), z2), F23_IN(z0, z1))
F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
U5'(f25_out1(z0), z1, z2) → c12(U6'(f26_in(z2, z0), z1, z2, z0), F26_IN(z2, z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f23_in, U5, U6

Defined Pair Symbols:

F1_IN, F32_IN, F25_IN, F26_IN, F23_IN, U5'

Compound Symbols:

c1, c4, c6, c9, c11, c12

(5) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f23_in(z0, z1), .(z0, z1))
U1(f23_out1(z0), .(z1, z2)) → f1_out1
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))
f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
f26_in([], z0) → f26_out1
f26_in(.(z0, z1), z2) → U4(f23_in(z0, z1), .(z0, z1), z2)
U4(f23_out1(z0), .(z1, z2), z3) → f26_out1
f23_in(z0, z1) → U5(f25_in(z0), z0, z1)
U5(f25_out1(z0), z1, z2) → U6(f26_in(z2, z0), z1, z2, z0)
U6(f26_out1, z0, z1, z2) → f23_out1(z2)

Tuples:

F32_IN(.(z0, z1)) → c4(U2'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(U4'(f23_in(z0, z1), .(z0, z1), z2), F23_IN(z0, z1))
F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
U5'(f25_out1(z0), z1, z2) → c12(U6'(f26_in(z2, z0), z1, z2, z0), F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c(U1'(f23_in(z0, z1), .(z0, z1)))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(U3'(f32_in(z1), .(z0, z1)))
F25_IN(.(z0, z1)) → c(F32_IN(z1))

S tuples:

F32_IN(.(z0, z1)) → c4(U2'(f32_in(z1), .(z0, z1)), F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(U4'(f23_in(z0, z1), .(z0, z1), z2), F23_IN(z0, z1))
F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
U5'(f25_out1(z0), z1, z2) → c12(U6'(f26_in(z2, z0), z1, z2, z0), F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c(U1'(f23_in(z0, z1), .(z0, z1)))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(U3'(f32_in(z1), .(z0, z1)))
F25_IN(.(z0, z1)) → c(F32_IN(z1))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f23_in, U5, U6

Defined Pair Symbols:

F32_IN, F26_IN, F23_IN, U5', F1_IN, F25_IN

Compound Symbols:

c4, c9, c11, c12, c

(7) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f23_in(z0, z1), .(z0, z1))
U1(f23_out1(z0), .(z1, z2)) → f1_out1
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))
f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
f26_in([], z0) → f26_out1
f26_in(.(z0, z1), z2) → U4(f23_in(z0, z1), .(z0, z1), z2)
U4(f23_out1(z0), .(z1, z2), z3) → f26_out1
f23_in(z0, z1) → U5(f25_in(z0), z0, z1)
U5(f25_out1(z0), z1, z2) → U6(f26_in(z2, z0), z1, z2, z0)
U6(f26_out1, z0, z1, z2) → f23_out1(z2)

Tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c

S tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f23_in, U5, U6

Defined Pair Symbols:

F23_IN, F1_IN, F25_IN, F32_IN, F26_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(9) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F1_IN(.(z0, z1)) → c

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f23_in(z0, z1), .(z0, z1))
U1(f23_out1(z0), .(z1, z2)) → f1_out1
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))
f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
f26_in([], z0) → f26_out1
f26_in(.(z0, z1), z2) → U4(f23_in(z0, z1), .(z0, z1), z2)
U4(f23_out1(z0), .(z1, z2), z3) → f26_out1
f23_in(z0, z1) → U5(f25_in(z0), z0, z1)
U5(f25_out1(z0), z1, z2) → U6(f26_in(z2, z0), z1, z2, z0)
U6(f26_out1, z0, z1, z2) → f23_out1(z2)

Tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c

S tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F25_IN(.(z0, z1)) → c

K tuples:

F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F1_IN(.(z0, z1)) → c

Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f23_in, U5, U6

Defined Pair Symbols:

F23_IN, F1_IN, F25_IN, F32_IN, F26_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))

We considered the (Usable) Rules:

f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))

And the Tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [1] + x₂   
POL(F1_IN(x₁)) = [2]x₁   
POL(F23_IN(x₁, x₂)) = [2]x₂   
POL(F25_IN(x₁)) = 0   
POL(F26_IN(x₁, x₂)) = [2]x₁   
POL(F32_IN(x₁)) = 0   
POL(U2(x₁, x₂)) = 0   
POL(U3(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = [2]x₃   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f25_in(x₁)) = [1]   
POL(f25_out1(x₁)) = 0   
POL(f32_in(x₁)) = 0   
POL(f32_out1(x₁)) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f23_in(z0, z1), .(z0, z1))
U1(f23_out1(z0), .(z1, z2)) → f1_out1
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))
f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
f26_in([], z0) → f26_out1
f26_in(.(z0, z1), z2) → U4(f23_in(z0, z1), .(z0, z1), z2)
U4(f23_out1(z0), .(z1, z2), z3) → f26_out1
f23_in(z0, z1) → U5(f25_in(z0), z0, z1)
U5(f25_out1(z0), z1, z2) → U6(f26_in(z2, z0), z1, z2, z0)
U6(f26_out1, z0, z1, z2) → f23_out1(z2)

Tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c

S tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F25_IN(.(z0, z1)) → c

K tuples:

F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F1_IN(.(z0, z1)) → c
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))

Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f23_in, U5, U6

Defined Pair Symbols:

F23_IN, F1_IN, F25_IN, F32_IN, F26_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(13) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F25_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c(F32_IN(z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F25_IN(.(z0, z1)) → c
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f23_in(z0, z1), .(z0, z1))
U1(f23_out1(z0), .(z1, z2)) → f1_out1
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))
f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
f26_in([], z0) → f26_out1
f26_in(.(z0, z1), z2) → U4(f23_in(z0, z1), .(z0, z1), z2)
U4(f23_out1(z0), .(z1, z2), z3) → f26_out1
f23_in(z0, z1) → U5(f25_in(z0), z0, z1)
U5(f25_out1(z0), z1, z2) → U6(f26_in(z2, z0), z1, z2, z0)
U6(f26_out1, z0, z1, z2) → f23_out1(z2)

Tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c

S tuples:

F32_IN(.(z0, z1)) → c4(F32_IN(z1))

K tuples:

F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F1_IN(.(z0, z1)) → c
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F25_IN(.(z0, z1)) → c

Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f23_in, U5, U6

Defined Pair Symbols:

F23_IN, F1_IN, F25_IN, F32_IN, F26_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(15) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F32_IN(.(z0, z1)) → c4(F32_IN(z1))

We considered the (Usable) Rules:

f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))

And the Tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(F1_IN(x₁)) = [3] + x₁   
POL(F23_IN(x₁, x₂)) = [1] + x₁ + x₂   
POL(F25_IN(x₁)) = [1] + x₁   
POL(F26_IN(x₁, x₂)) = x₁   
POL(F32_IN(x₁)) = [2] + x₁   
POL(U2(x₁, x₂)) = 0   
POL(U3(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = x₃   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f25_in(x₁)) = 0   
POL(f25_out1(x₁)) = 0   
POL(f32_in(x₁)) = 0   
POL(f32_out1(x₁)) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, z1)) → U1(f23_in(z0, z1), .(z0, z1))
U1(f23_out1(z0), .(z1, z2)) → f1_out1
f32_in([]) → f32_out1([])
f32_in(.(z0, z1)) → U2(f32_in(z1), .(z0, z1))
U2(f32_out1(z0), .(z1, z2)) → f32_out1(.([], z0))
f25_in(.(z0, z1)) → U3(f32_in(z1), .(z0, z1))
U3(f32_out1(z0), .(z1, z2)) → f25_out1(.([], z0))
f26_in([], z0) → f26_out1
f26_in(.(z0, z1), z2) → U4(f23_in(z0, z1), .(z0, z1), z2)
U4(f23_out1(z0), .(z1, z2), z3) → f26_out1
f23_in(z0, z1) → U5(f25_in(z0), z0, z1)
U5(f25_out1(z0), z1, z2) → U6(f26_in(z2, z0), z1, z2, z0)
U6(f26_out1, z0, z1, z2) → f23_out1(z2)

Tuples:

F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
F32_IN(.(z0, z1)) → c4(F32_IN(z1))
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F1_IN(.(z0, z1)) → c
F25_IN(.(z0, z1)) → c

S tuples:none
K tuples:

F1_IN(.(z0, z1)) → c(F23_IN(z0, z1))
F1_IN(.(z0, z1)) → c
F26_IN(.(z0, z1), z2) → c9(F23_IN(z0, z1))
F23_IN(z0, z1) → c11(U5'(f25_in(z0), z0, z1), F25_IN(z0))
F25_IN(.(z0, z1)) → c(F32_IN(z1))
U5'(f25_out1(z0), z1, z2) → c12(F26_IN(z2, z0))
F25_IN(.(z0, z1)) → c
F32_IN(.(z0, z1)) → c4(F32_IN(z1))

Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f23_in, U5, U6

Defined Pair Symbols:

F23_IN, F1_IN, F25_IN, F32_IN, F26_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(17) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(18) BOUNDS(O(1), O(1))

(19) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
2 [label="2: transpose(T1, T2)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: transpose(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: transpose_aux(T5, [], T7)\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 0 | transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 3\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true | transpose_aux([], [], T7), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: transpose_aux(T5, [], T7), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nX8 is free\ntranspose_aux(T5, [], T7) !=? transpose_aux([], X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: transpose_aux([], [], T7), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ','(row2col(T16, .(T20, T21), X31, X32), transpose_aux(T17, X32, X31))\n\nT16 is ground\nT17 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: row2col(T16, .(T20, T21), X31, X32)\n\nT16 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: transpose_aux(T17, T23, T22)\n\nT17 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: row2col(T16, .(T20, T21), X31, X32), SCOPE: 3, CLAUSE: 1 | row2col(T16, .(T20, T21), X31, X32), SCOPE: 3, CLAUSE: 4\n\nT16 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: row2col(T16, .(T20, T21), X31, X32), SCOPE: 3, CLAUSE: 4\n\nT16 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: row2col(T33, T36, X55, X56)\n\nT33 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: row2col(T33, T36, X55, X56), SCOPE: 4, CLAUSE: 1 | row2col(T33, T36, X55, X56), SCOPE: 4, CLAUSE: 4\n\nT33 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: true | row2col([], T36, X55, X56), SCOPE: 4, CLAUSE: 4\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: row2col(T33, T36, X55, X56), SCOPE: 4, CLAUSE: 4\n\nT33 is ground\nX55 is free\nX56 is free\nrow2col(T33, T36, X55, X56) !=? row2col([], [], [], [])", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: row2col([], T36, X55, X56), SCOPE: 4, CLAUSE: 4\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: row2col(T46, T49, X85, X86)\n\nT46 is ground\nX85 is free\nX86 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: transpose_aux(T17, T23, T22), SCOPE: 5, CLAUSE: 0 | transpose_aux(T17, T23, T22), SCOPE: 5, CLAUSE: 3\n\nT17 is ground\nT23 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true | transpose_aux([], T54, T22), SCOPE: 5, CLAUSE: 3\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: transpose_aux(T17, T23, T22), SCOPE: 5, CLAUSE: 3\n\nT17 is ground\nT23 is ground\nX91 is free\ntranspose_aux(T17, T23, T22) !=? transpose_aux([], X91, X91)", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: transpose_aux([], T54, T22), SCOPE: 5, CLAUSE: 3\n\nT54 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(row2col(T65, .(T70, T71), X114, X115), transpose_aux(T66, X115, X114))\n\nT65 is ground\nT66 is ground\nX114 is free\nX115 is free", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 59 [arrowhead = none ];
59 -> 3;

60 [label="ONLY EVAL with clause\ntranspose(X4, X5) :- transpose_aux(X4, [], X5).\nand substitutionT1 -> T5,\nX4 -> T5,\nT2 -> T7,\nX5 -> T7,\nT6 -> T7", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 60 [arrowhead = none ];
60 -> 5;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 61 [arrowhead = none ];
61 -> 6;

62 [label="EVAL with clause\ntranspose_aux([], X8, X8).\nand substitutionT5 -> [],\nX8 -> [],\nT7 -> []", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 62 [arrowhead = none ];
62 -> 7;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 63 [arrowhead = none ];
63 -> 8;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 64 [arrowhead = none ];
64 -> 9;

65 [label="EVAL with clause\ntranspose_aux(.(X26, X27), X28, .(X29, X30)) :- ','(row2col(X26, .(X29, X30), X31, X32), transpose_aux(X27, X32, X31)).\nand substitutionX26 -> T16,\nX27 -> T17,\nT5 -> .(T16, T17),\nX28 -> [],\nX29 -> T20,\nX30 -> T21,\nT7 -> .(T20, T21),\nT18 -> T20,\nT19 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 65 [arrowhead = none ];
65 -> 12;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 66 [arrowhead = none ];
66 -> 13;

67 [label="BACKTRACK\nfor clause: transpose_aux(.(R, Rs), X1, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 67 [arrowhead = none ];
67 -> 11;

68 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
12 -> 68 [arrowhead = none ];
68 -> 15;

69 [label="SPLIT 2\nnew knowledge:\nT16 is ground\nT23 is ground\nreplacements:X31 -> T22,\nX32 -> T23", fontsize=14, style = filled, fillcolor = violet];
12 -> 69 [arrowhead = none ];
69 -> 17;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 70 [arrowhead = none ];
70 -> 21;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
17 -> 71 [arrowhead = none ];
71 -> 47;

72 [label="BACKTRACK\nfor clause: row2col([], [], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 72 [arrowhead = none ];
72 -> 22;

73 [label="EVAL with clause\nrow2col(.(X49, X50), .(.(X49, X51), X52), .(X51, X53), .([], X54)) :- row2col(X50, X52, X53, X54).\nand substitutionX49 -> T32,\nX50 -> T33,\nT16 -> .(T32, T33),\nX51 -> T34,\nT20 -> .(T32, T34),\nT21 -> T36,\nX52 -> T36,\nX53 -> X55,\nX31 -> .(T34, X55),\nX54 -> X56,\nX32 -> .([], X56),\nT35 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
22 -> 73 [arrowhead = none ];
73 -> 29;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
22 -> 74 [arrowhead = none ];
74 -> 30;

75 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
29 -> 75 [arrowhead = none ];
75 -> 31;

76 [label="EVAL with clause\nrow2col([], [], [], []).\nand substitutionT33 -> [],\nT36 -> [],\nX55 -> [],\nX56 -> []", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 76 [arrowhead = none ];
76 -> 33;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 77 [arrowhead = none ];
77 -> 35;

78 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 78 [arrowhead = none ];
78 -> 36;

79 [label="EVAL with clause\nrow2col(.(X79, X80), .(.(X79, X81), X82), .(X81, X83), .([], X84)) :- row2col(X80, X82, X83, X84).\nand substitutionX79 -> T45,\nX80 -> T46,\nT33 -> .(T45, T46),\nX81 -> T47,\nX82 -> T49,\nT36 -> .(.(T45, T47), T49),\nX83 -> X85,\nX55 -> .(T47, X85),\nX84 -> X86,\nX56 -> .([], X86),\nT48 -> T49", fontsize=14, style = filled, fillcolor = lightblue];
35 -> 79 [arrowhead = none ];
79 -> 43;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 80 [arrowhead = none ];
80 -> 44;

81 [label="BACKTRACK\nfor clause: row2col(.(X, Xs), .(.(X, Ys), Cols), .(Ys, Cols1), .([], As)) :- row2col(Xs, Cols, Cols1, As)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 81 [arrowhead = none ];
81 -> 38;

82 [label="INSTANCE with matching:\nT33 -> T46\nT36 -> T49\nX55 -> X85\nX56 -> X86", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 82 [arrowhead = none , style=dashed];
82 -> 29[style=dashed];

83 [label="EVAL with clause\ntranspose_aux([], X91, X91).\nand substitutionT17 -> [],\nT23 -> T54,\nX91 -> T54,\nT22 -> T54", fontsize=14, style = filled, fillcolor = lightblue];
47 -> 83 [arrowhead = none ];
83 -> 49;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 84 [arrowhead = none ];
84 -> 50;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 85 [arrowhead = none ];
85 -> 51;

86 [label="EVAL with clause\ntranspose_aux(.(X109, X110), X111, .(X112, X113)) :- ','(row2col(X109, .(X112, X113), X114, X115), transpose_aux(X110, X115, X114)).\nand substitutionX109 -> T65,\nX110 -> T66,\nT17 -> .(T65, T66),\nT23 -> T67,\nX111 -> T67,\nX112 -> T70,\nX113 -> T71,\nT22 -> .(T70, T71),\nT68 -> T70,\nT69 -> T71", fontsize=14, style = filled, fillcolor = lightblue];
50 -> 86 [arrowhead = none ];
86 -> 57;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
50 -> 87 [arrowhead = none ];
87 -> 58;

88 [label="BACKTRACK\nfor clause: transpose_aux(.(R, Rs), X1, .(C, Cs)) :- ','(row2col(R, .(C, Cs), Cols1, Accm), transpose_aux(Rs, Accm, Cols1))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 88 [arrowhead = none ];
88 -> 53;

89 [label="INSTANCE with matching:\nT16 -> T65\nT20 -> T70\nT21 -> T71\nX31 -> X114\nX32 -> X115\nT17 -> T66", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 89 [arrowhead = none , style=dashed];
89 -> 12[style=dashed];

}

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f12_in(z0, z1), .(z0, z1))
U1(f12_out1(z0), .(z1, z2)) → f2_out1
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))
f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
f17_in([], z0) → f17_out1
f17_in(.(z0, z1), z2) → U4(f12_in(z0, z1), .(z0, z1), z2)
U4(f12_out1(z0), .(z1, z2), z3) → f17_out1
f12_in(z0, z1) → U5(f15_in(z0), z0, z1)
U5(f15_out1(z0), z1, z2) → U6(f17_in(z2, z0), z1, z2, z0)
U6(f17_out1, z0, z1, z2) → f12_out1(z2)

Tuples:

F2_IN(.(z0, z1)) → c1(U1'(f12_in(z0, z1), .(z0, z1)), F12_IN(z0, z1))
F29_IN(.(z0, z1)) → c4(U2'(f29_in(z1), .(z0, z1)), F29_IN(z1))
F15_IN(.(z0, z1)) → c6(U3'(f29_in(z1), .(z0, z1)), F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(U4'(f12_in(z0, z1), .(z0, z1), z2), F12_IN(z0, z1))
F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
U5'(f15_out1(z0), z1, z2) → c12(U6'(f17_in(z2, z0), z1, z2, z0), F17_IN(z2, z0))

S tuples:

F2_IN(.(z0, z1)) → c1(U1'(f12_in(z0, z1), .(z0, z1)), F12_IN(z0, z1))
F29_IN(.(z0, z1)) → c4(U2'(f29_in(z1), .(z0, z1)), F29_IN(z1))
F15_IN(.(z0, z1)) → c6(U3'(f29_in(z1), .(z0, z1)), F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(U4'(f12_in(z0, z1), .(z0, z1), z2), F12_IN(z0, z1))
F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
U5'(f15_out1(z0), z1, z2) → c12(U6'(f17_in(z2, z0), z1, z2, z0), F17_IN(z2, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f29_in, U2, f15_in, U3, f17_in, U4, f12_in, U5, U6

Defined Pair Symbols:

F2_IN, F29_IN, F15_IN, F17_IN, F12_IN, U5'

Compound Symbols:

c1, c4, c6, c9, c11, c12

(21) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID) transformation)

Split RHS of tuples not part of any SCC

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f12_in(z0, z1), .(z0, z1))
U1(f12_out1(z0), .(z1, z2)) → f2_out1
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))
f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
f17_in([], z0) → f17_out1
f17_in(.(z0, z1), z2) → U4(f12_in(z0, z1), .(z0, z1), z2)
U4(f12_out1(z0), .(z1, z2), z3) → f17_out1
f12_in(z0, z1) → U5(f15_in(z0), z0, z1)
U5(f15_out1(z0), z1, z2) → U6(f17_in(z2, z0), z1, z2, z0)
U6(f17_out1, z0, z1, z2) → f12_out1(z2)

Tuples:

F29_IN(.(z0, z1)) → c4(U2'(f29_in(z1), .(z0, z1)), F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(U4'(f12_in(z0, z1), .(z0, z1), z2), F12_IN(z0, z1))
F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
U5'(f15_out1(z0), z1, z2) → c12(U6'(f17_in(z2, z0), z1, z2, z0), F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c(U1'(f12_in(z0, z1), .(z0, z1)))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(U3'(f29_in(z1), .(z0, z1)))
F15_IN(.(z0, z1)) → c(F29_IN(z1))

S tuples:

F29_IN(.(z0, z1)) → c4(U2'(f29_in(z1), .(z0, z1)), F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(U4'(f12_in(z0, z1), .(z0, z1), z2), F12_IN(z0, z1))
F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
U5'(f15_out1(z0), z1, z2) → c12(U6'(f17_in(z2, z0), z1, z2, z0), F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c(U1'(f12_in(z0, z1), .(z0, z1)))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(U3'(f29_in(z1), .(z0, z1)))
F15_IN(.(z0, z1)) → c(F29_IN(z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f29_in, U2, f15_in, U3, f17_in, U4, f12_in, U5, U6

Defined Pair Symbols:

F29_IN, F17_IN, F12_IN, U5', F2_IN, F15_IN

Compound Symbols:

c4, c9, c11, c12, c

(23) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)

Removed 5 trailing tuple parts

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f12_in(z0, z1), .(z0, z1))
U1(f12_out1(z0), .(z1, z2)) → f2_out1
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))
f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
f17_in([], z0) → f17_out1
f17_in(.(z0, z1), z2) → U4(f12_in(z0, z1), .(z0, z1), z2)
U4(f12_out1(z0), .(z1, z2), z3) → f17_out1
f12_in(z0, z1) → U5(f15_in(z0), z0, z1)
U5(f15_out1(z0), z1, z2) → U6(f17_in(z2, z0), z1, z2, z0)
U6(f17_out1, z0, z1, z2) → f12_out1(z2)

Tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c

S tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, f29_in, U2, f15_in, U3, f17_in, U4, f12_in, U5, U6

Defined Pair Symbols:

F12_IN, F2_IN, F15_IN, F29_IN, F17_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(25) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F2_IN(.(z0, z1)) → c

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f12_in(z0, z1), .(z0, z1))
U1(f12_out1(z0), .(z1, z2)) → f2_out1
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))
f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
f17_in([], z0) → f17_out1
f17_in(.(z0, z1), z2) → U4(f12_in(z0, z1), .(z0, z1), z2)
U4(f12_out1(z0), .(z1, z2), z3) → f17_out1
f12_in(z0, z1) → U5(f15_in(z0), z0, z1)
U5(f15_out1(z0), z1, z2) → U6(f17_in(z2, z0), z1, z2, z0)
U6(f17_out1, z0, z1, z2) → f12_out1(z2)

Tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c

S tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F15_IN(.(z0, z1)) → c

K tuples:

F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F2_IN(.(z0, z1)) → c

Defined Rule Symbols:

f2_in, U1, f29_in, U2, f15_in, U3, f17_in, U4, f12_in, U5, U6

Defined Pair Symbols:

F12_IN, F2_IN, F15_IN, F29_IN, F17_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(27) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))

We considered the (Usable) Rules:

f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))

And the Tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [1] + x₂   
POL(F12_IN(x₁, x₂)) = [2]x₂   
POL(F15_IN(x₁)) = 0   
POL(F17_IN(x₁, x₂)) = [2]x₁   
POL(F29_IN(x₁)) = 0   
POL(F2_IN(x₁)) = [2]x₁   
POL(U2(x₁, x₂)) = 0   
POL(U3(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = [2]x₃   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f15_in(x₁)) = [1]   
POL(f15_out1(x₁)) = 0   
POL(f29_in(x₁)) = 0   
POL(f29_out1(x₁)) = 0   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f12_in(z0, z1), .(z0, z1))
U1(f12_out1(z0), .(z1, z2)) → f2_out1
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))
f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
f17_in([], z0) → f17_out1
f17_in(.(z0, z1), z2) → U4(f12_in(z0, z1), .(z0, z1), z2)
U4(f12_out1(z0), .(z1, z2), z3) → f17_out1
f12_in(z0, z1) → U5(f15_in(z0), z0, z1)
U5(f15_out1(z0), z1, z2) → U6(f17_in(z2, z0), z1, z2, z0)
U6(f17_out1, z0, z1, z2) → f12_out1(z2)

Tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c

S tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F15_IN(.(z0, z1)) → c

K tuples:

F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F2_IN(.(z0, z1)) → c
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f29_in, U2, f15_in, U3, f17_in, U4, f12_in, U5, U6

Defined Pair Symbols:

F12_IN, F2_IN, F15_IN, F29_IN, F17_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(29) CdtKnowledgeProof (EQUIVALENT transformation)

The following tuples could be moved from S to K by knowledge propagation:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F15_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c(F29_IN(z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F15_IN(.(z0, z1)) → c
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f12_in(z0, z1), .(z0, z1))
U1(f12_out1(z0), .(z1, z2)) → f2_out1
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))
f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
f17_in([], z0) → f17_out1
f17_in(.(z0, z1), z2) → U4(f12_in(z0, z1), .(z0, z1), z2)
U4(f12_out1(z0), .(z1, z2), z3) → f17_out1
f12_in(z0, z1) → U5(f15_in(z0), z0, z1)
U5(f15_out1(z0), z1, z2) → U6(f17_in(z2, z0), z1, z2, z0)
U6(f17_out1, z0, z1, z2) → f12_out1(z2)

Tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c

S tuples:

F29_IN(.(z0, z1)) → c4(F29_IN(z1))

K tuples:

F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F2_IN(.(z0, z1)) → c
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F15_IN(.(z0, z1)) → c

Defined Rule Symbols:

f2_in, U1, f29_in, U2, f15_in, U3, f17_in, U4, f12_in, U5, U6

Defined Pair Symbols:

F12_IN, F2_IN, F15_IN, F29_IN, F17_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c

(31) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)

Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.

F29_IN(.(z0, z1)) → c4(F29_IN(z1))

We considered the (Usable) Rules:

f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))

And the Tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c

The order we found is given by the following interpretation:
Polynomial interpretation :

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(F12_IN(x₁, x₂)) = [1] + x₁ + x₂   
POL(F15_IN(x₁)) = [1] + x₁   
POL(F17_IN(x₁, x₂)) = x₁   
POL(F29_IN(x₁)) = [2] + x₁   
POL(F2_IN(x₁)) = [3] + x₁   
POL(U2(x₁, x₂)) = 0   
POL(U3(x₁, x₂)) = 0   
POL(U5'(x₁, x₂, x₃)) = x₃   
POL([]) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(f15_in(x₁)) = 0   
POL(f15_out1(x₁)) = 0   
POL(f29_in(x₁)) = 0   
POL(f29_out1(x₁)) = 0   

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, z1)) → U1(f12_in(z0, z1), .(z0, z1))
U1(f12_out1(z0), .(z1, z2)) → f2_out1
f29_in([]) → f29_out1([])
f29_in(.(z0, z1)) → U2(f29_in(z1), .(z0, z1))
U2(f29_out1(z0), .(z1, z2)) → f29_out1(.([], z0))
f15_in(.(z0, z1)) → U3(f29_in(z1), .(z0, z1))
U3(f29_out1(z0), .(z1, z2)) → f15_out1(.([], z0))
f17_in([], z0) → f17_out1
f17_in(.(z0, z1), z2) → U4(f12_in(z0, z1), .(z0, z1), z2)
U4(f12_out1(z0), .(z1, z2), z3) → f17_out1
f12_in(z0, z1) → U5(f15_in(z0), z0, z1)
U5(f15_out1(z0), z1, z2) → U6(f17_in(z2, z0), z1, z2, z0)
U6(f17_out1, z0, z1, z2) → f12_out1(z2)

Tuples:

F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
F29_IN(.(z0, z1)) → c4(F29_IN(z1))
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F2_IN(.(z0, z1)) → c
F15_IN(.(z0, z1)) → c

S tuples:none
K tuples:

F2_IN(.(z0, z1)) → c(F12_IN(z0, z1))
F2_IN(.(z0, z1)) → c
F17_IN(.(z0, z1), z2) → c9(F12_IN(z0, z1))
F12_IN(z0, z1) → c11(U5'(f15_in(z0), z0, z1), F15_IN(z0))
F15_IN(.(z0, z1)) → c(F29_IN(z1))
U5'(f15_out1(z0), z1, z2) → c12(F17_IN(z2, z0))
F15_IN(.(z0, z1)) → c
F29_IN(.(z0, z1)) → c4(F29_IN(z1))

Defined Rule Symbols:

f2_in, U1, f29_in, U2, f15_in, U3, f17_in, U4, f12_in, U5, U6

Defined Pair Symbols:

F12_IN, F2_IN, F15_IN, F29_IN, F17_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c