Runtime Complexity of the query pattern
transpose(g,g)
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,g)

(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,g)

(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\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: transpose(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: transpose_aux(T7, [], T8)\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: transpose_aux(T7, [], T8), SCOPE: 2, CLAUSE: 0 | transpose_aux(T7, [], T8), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: true | transpose_aux([], [], []), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: transpose_aux(T7, [], T8), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nX14 is free\ntranspose_aux(T7, [], T8) !=? transpose_aux([], X14, X14)", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: transpose_aux([], [], []), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(row2col(T17, .(T19, T20), X37, X38), transpose_aux(T18, X38, X37))\n\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground\nX37 is free\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: row2col(T17, .(T19, T20), X37, X38)\n\nT17 is ground\nT19 is ground\nT20 is ground\nX37 is free\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: transpose_aux(T18, T22, T21)\n\nT18 is ground\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: row2col(T17, .(T19, T20), X37, X38), SCOPE: 3, CLAUSE: 1 | row2col(T17, .(T19, T20), X37, X38), SCOPE: 3, CLAUSE: 4\n\nT17 is ground\nT19 is ground\nT20 is ground\nX37 is free\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: row2col(T17, .(T19, T20), X37, X38), SCOPE: 3, CLAUSE: 4\n\nT17 is ground\nT19 is ground\nT20 is ground\nX37 is free\nX38 is free", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: row2col(T32, T34, X61, X62)\n\nT32 is ground\nT34 is ground\nX61 is free\nX62 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: row2col(T32, T34, X61, X62), SCOPE: 4, CLAUSE: 1 | row2col(T32, T34, X61, X62), SCOPE: 4, CLAUSE: 4\n\nT32 is ground\nT34 is ground\nX61 is free\nX62 is free", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: true | row2col([], [], X61, X62), SCOPE: 4, CLAUSE: 4\n\nX61 is free\nX62 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: row2col(T32, T34, X61, X62), SCOPE: 4, CLAUSE: 4\n\nT32 is ground\nT34 is ground\nX61 is free\nX62 is free\nrow2col(T32, T34, X61, X62) !=? row2col([], [], [], [])", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: row2col([], [], 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];
44 [label="44: row2col(T44, T46, X91, X92)\n\nT44 is ground\nT46 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(T18, T22, T21), SCOPE: 5, CLAUSE: 0 | transpose_aux(T18, T22, T21), SCOPE: 5, CLAUSE: 3\n\nT18 is ground\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: true | transpose_aux([], T51, T51), SCOPE: 5, CLAUSE: 3\n\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: transpose_aux(T18, T22, T21), SCOPE: 5, CLAUSE: 3\n\nT18 is ground\nT21 is ground\nT22 is ground\nX97 is free\ntranspose_aux(T18, T22, T21) !=? transpose_aux([], X97, X97)", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: transpose_aux([], T51, T51), SCOPE: 5, CLAUSE: 3\n\nT51 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: ','(row2col(T62, .(T65, T66), X120, X121), transpose_aux(T63, X121, X120))\n\nT62 is ground\nT63 is ground\nT65 is ground\nT66 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 -> 3;

62 [label="ONLY EVAL with clause\ntranspose(X8, X9) :- transpose_aux(X8, [], X9).\nand substitutionT1 -> T7,\nX8 -> T7,\nT2 -> T8,\nX9 -> T8", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 62 [arrowhead = none ];
62 -> 6;

63 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
6 -> 63 [arrowhead = none ];
63 -> 12;

64 [label="EVAL with clause\ntranspose_aux([], X14, X14).\nand substitutionT7 -> [],\nX14 -> [],\nT8 -> []", fontsize=14, style = filled, fillcolor = lightblue];
12 -> 64 [arrowhead = none ];
64 -> 13;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 65 [arrowhead = none ];
65 -> 14;

66 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
13 -> 66 [arrowhead = none ];
66 -> 15;

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 -> T17,\nX33 -> T18,\nT7 -> .(T17, T18),\nX34 -> [],\nX35 -> T19,\nX36 -> T20,\nT8 -> .(T19, T20)", fontsize=14, style = filled, fillcolor = lightblue];
14 -> 67 [arrowhead = none ];
67 -> 21;

68 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
14 -> 68 [arrowhead = none ];
68 -> 22;

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];
15 -> 69 [arrowhead = none ];
69 -> 16;

70 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
21 -> 70 [arrowhead = none ];
70 -> 25;

71 [label="SPLIT 2\nnew knowledge:\nT17 is ground\nT19 is ground\nT20 is ground\nT21 is ground\nT22 is ground\nreplacements:X37 -> T21,\nX38 -> T22", fontsize=14, style = filled, fillcolor = violet];
21 -> 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 -> T31,\nX56 -> T32,\nT17 -> .(T31, T32),\nX57 -> T33,\nT19 -> .(T31, T33),\nT20 -> T34,\nX58 -> T34,\nX59 -> X61,\nX37 -> .(T33, X61),\nX60 -> X62,\nX38 -> .([], X62)", 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 -> 33;

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

78 [label="EVAL with clause\nrow2col([], [], [], []).\nand substitutionT32 -> [],\nT34 -> [],\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 -> T43,\nX86 -> T44,\nT32 -> .(T43, T44),\nX87 -> T45,\nX88 -> T46,\nT34 -> .(.(T43, T45), T46),\nX89 -> X91,\nX61 -> .(T45, X91),\nX90 -> X92,\nX62 -> .([], X92)", fontsize=14, style = filled, fillcolor = lightblue];
40 -> 81 [arrowhead = none ];
81 -> 44;

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:\nT32 -> T44\nT34 -> T46\nX61 -> X91\nX62 -> X92", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 84 [arrowhead = none , style=dashed];
84 -> 32[style=dashed];

85 [label="EVAL with clause\ntranspose_aux([], X97, X97).\nand substitutionT18 -> [],\nT22 -> T51,\nX97 -> T51,\nT21 -> T51", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 85 [arrowhead = none ];
85 -> 51;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 86 [arrowhead = none ];
86 -> 53;

87 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 87 [arrowhead = none ];
87 -> 54;

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 -> T62,\nX116 -> T63,\nT18 -> .(T62, T63),\nT22 -> T64,\nX117 -> T64,\nX118 -> T65,\nX119 -> T66,\nT21 -> .(T65, T66)", fontsize=14, style = filled, fillcolor = lightblue];
53 -> 88 [arrowhead = none ];
88 -> 59;

89 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
53 -> 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];
54 -> 90 [arrowhead = none ];
90 -> 56;

91 [label="INSTANCE with matching:\nT17 -> T62\nT19 -> T65\nT20 -> T66\nX37 -> X120\nX38 -> X121\nT18 -> T63", fontsize=14, style = filled, fillcolor = lightgrey];
59 -> 91 [arrowhead = none , style=dashed];
91 -> 21[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1
f1_in(.(z0, z1), .(z2, z3)) → U1(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f21_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))
f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
f26_in([], z0, z0) → f26_out1
f26_in(.(z0, z1), z2, .(z3, z4)) → U4(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f21_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f26_out1
f21_in(z0, z1, z2, z3) → U5(f25_in(z0, z1, z2), z0, z1, z2, z3)
U5(f25_out1(z0, z1), z2, z3, z4, z5) → U6(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f26_out1, z0, z1, z2, z3, z4, z5) → f21_out1(z4, z5)

Tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c1(U1'(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3)), F21_IN(z0, z2, z3, z1))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(U2'(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3)), F32_IN(z1, z3))
F25_IN(.(z0, z1), .(z0, z2), z3) → c6(U3'(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3), F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(U4'(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4)), F21_IN(z0, z3, z4, z1))
F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(U6'(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1), F26_IN(z5, z1, z0))

S tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c1(U1'(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3)), F21_IN(z0, z2, z3, z1))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(U2'(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3)), F32_IN(z1, z3))
F25_IN(.(z0, z1), .(z0, z2), z3) → c6(U3'(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3), F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(U4'(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4)), F21_IN(z0, z3, z4, z1))
F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(U6'(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1), F26_IN(z5, z1, z0))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f21_in, U5, U6

Defined Pair Symbols:

F1_IN, F32_IN, F25_IN, F26_IN, F21_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), .(z2, z3)) → U1(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f21_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))
f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
f26_in([], z0, z0) → f26_out1
f26_in(.(z0, z1), z2, .(z3, z4)) → U4(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f21_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f26_out1
f21_in(z0, z1, z2, z3) → U5(f25_in(z0, z1, z2), z0, z1, z2, z3)
U5(f25_out1(z0, z1), z2, z3, z4, z5) → U6(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f26_out1, z0, z1, z2, z3, z4, z5) → f21_out1(z4, z5)

Tuples:

F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(U2'(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3)), F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(U4'(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4)), F21_IN(z0, z3, z4, z1))
F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(U6'(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1), F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c(U1'(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3)))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(U3'(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))

S tuples:

F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(U2'(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3)), F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(U4'(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4)), F21_IN(z0, z3, z4, z1))
F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(U6'(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1), F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c(U1'(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3)))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(U3'(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f21_in, U5, U6

Defined Pair Symbols:

F32_IN, F26_IN, F21_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), .(z2, z3)) → U1(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f21_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))
f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
f26_in([], z0, z0) → f26_out1
f26_in(.(z0, z1), z2, .(z3, z4)) → U4(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f21_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f26_out1
f21_in(z0, z1, z2, z3) → U5(f25_in(z0, z1, z2), z0, z1, z2, z3)
U5(f25_out1(z0, z1), z2, z3, z4, z5) → U6(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f26_out1, z0, z1, z2, z3, z4, z5) → f21_out1(z4, z5)

Tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c
F25_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c
F25_IN(.(z0, z1), .(z0, z2), z3) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f21_in, U5, U6

Defined Pair Symbols:

F21_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), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F1_IN(.(z0, z1), .(z2, z3)) → c

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1
f1_in(.(z0, z1), .(z2, z3)) → U1(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f21_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))
f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
f26_in([], z0, z0) → f26_out1
f26_in(.(z0, z1), z2, .(z3, z4)) → U4(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f21_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f26_out1
f21_in(z0, z1, z2, z3) → U5(f25_in(z0, z1, z2), z0, z1, z2, z3)
U5(f25_out1(z0, z1), z2, z3, z4, z5) → U6(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f26_out1, z0, z1, z2, z3, z4, z5) → f21_out1(z4, z5)

Tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c
F25_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F25_IN(.(z0, z1), .(z0, z2), z3) → c

K tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F1_IN(.(z0, z1), .(z2, z3)) → c

Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f21_in, U5, U6

Defined Pair Symbols:

F21_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.

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c

We considered the (Usable) Rules:

f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))

And the Tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c
F25_IN(.(z0, z1), .(z0, z2), z3) → c

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂   
POL(F1_IN(x₁, x₂)) = [2]x₁ + [2]x₂   
POL(F21_IN(x₁, x₂, x₃, x₄)) = [2] + [2]x₁ + [2]x₄   
POL(F25_IN(x₁, x₂, x₃)) = [2]x₁   
POL(F26_IN(x₁, x₂, x₃)) = [1] + [2]x₁   
POL(F32_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3(x₁, x₂, x₃, x₄)) = 0   
POL(U5'(x₁, x₂, x₃, x₄, x₅)) = [1] + [2]x₅   
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, x₂, x₃)) = 0   
POL(f25_out1(x₁, x₂)) = 0   
POL(f32_in(x₁, x₂)) = 0   
POL(f32_out1(x₁, x₂)) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1
f1_in(.(z0, z1), .(z2, z3)) → U1(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f21_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))
f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
f26_in([], z0, z0) → f26_out1
f26_in(.(z0, z1), z2, .(z3, z4)) → U4(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f21_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f26_out1
f21_in(z0, z1, z2, z3) → U5(f25_in(z0, z1, z2), z0, z1, z2, z3)
U5(f25_out1(z0, z1), z2, z3, z4, z5) → U6(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f26_out1, z0, z1, z2, z3, z4, z5) → f21_out1(z4, z5)

Tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c
F25_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:

F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))

K tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F1_IN(.(z0, z1), .(z2, z3)) → c
F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c

Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f21_in, U5, U6

Defined Pair Symbols:

F21_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:

U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1
f1_in(.(z0, z1), .(z2, z3)) → U1(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f21_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))
f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
f26_in([], z0, z0) → f26_out1
f26_in(.(z0, z1), z2, .(z3, z4)) → U4(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f21_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f26_out1
f21_in(z0, z1, z2, z3) → U5(f25_in(z0, z1, z2), z0, z1, z2, z3)
U5(f25_out1(z0, z1), z2, z3, z4, z5) → U6(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f26_out1, z0, z1, z2, z3, z4, z5) → f21_out1(z4, z5)

Tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c
F25_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:

F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))

K tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F1_IN(.(z0, z1), .(z2, z3)) → c
F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))

Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f21_in, U5, U6

Defined Pair Symbols:

F21_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), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))

We considered the (Usable) Rules:

f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))

And the Tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c
F25_IN(.(z0, z1), .(z0, z2), z3) → c

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

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(F1_IN(x₁, x₂)) = x₁   
POL(F21_IN(x₁, x₂, x₃, x₄)) = [1] + x₁ + x₄   
POL(F25_IN(x₁, x₂, x₃)) = x₁   
POL(F26_IN(x₁, x₂, x₃)) = x₁   
POL(F32_IN(x₁, x₂)) = [1] + x₁   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3(x₁, x₂, x₃, x₄)) = 0   
POL(U5'(x₁, x₂, x₃, x₄, x₅)) = x₅   
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, x₂, x₃)) = 0   
POL(f25_out1(x₁, x₂)) = 0   
POL(f32_in(x₁, x₂)) = 0   
POL(f32_out1(x₁, x₂)) = 0   

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([], []) → f1_out1
f1_in(.(z0, z1), .(z2, z3)) → U1(f21_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f21_out1(z0, z1), .(z2, z3), .(z4, z5)) → f1_out1
f32_in([], []) → f32_out1([], [])
f32_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f32_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f32_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f32_out1(.(z4, z0), .([], z1))
f25_in(.(z0, z1), .(z0, z2), z3) → U3(f32_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f32_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f25_out1(.(z4, z0), .([], z1))
f26_in([], z0, z0) → f26_out1
f26_in(.(z0, z1), z2, .(z3, z4)) → U4(f21_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f21_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f26_out1
f21_in(z0, z1, z2, z3) → U5(f25_in(z0, z1, z2), z0, z1, z2, z3)
U5(f25_out1(z0, z1), z2, z3, z4, z5) → U6(f26_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f26_out1, z0, z1, z2, z3, z4, z5) → f21_out1(z4, z5)

Tuples:

F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F1_IN(.(z0, z1), .(z2, z3)) → c
F25_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:none
K tuples:

F1_IN(.(z0, z1), .(z2, z3)) → c(F21_IN(z0, z2, z3, z1))
F1_IN(.(z0, z1), .(z2, z3)) → c
F21_IN(z0, z1, z2, z3) → c11(U5'(f25_in(z0, z1, z2), z0, z1, z2, z3), F25_IN(z0, z1, z2))
F25_IN(.(z0, z1), .(z0, z2), z3) → c(F32_IN(z1, z3))
F26_IN(.(z0, z1), z2, .(z3, z4)) → c9(F21_IN(z0, z3, z4, z1))
F25_IN(.(z0, z1), .(z0, z2), z3) → c
U5'(f25_out1(z0, z1), z2, z3, z4, z5) → c12(F26_IN(z5, z1, z0))
F32_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F32_IN(z1, z3))

Defined Rule Symbols:

f1_in, U1, f32_in, U2, f25_in, U3, f26_in, U4, f21_in, U5, U6

Defined Pair Symbols:

F21_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\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: transpose(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: transpose_aux(T5, [], T6)\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: transpose_aux(T5, [], T6), SCOPE: 2, CLAUSE: 0 | transpose_aux(T5, [], T6), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: true | transpose_aux([], [], []), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: transpose_aux(T5, [], T6), SCOPE: 2, CLAUSE: 3\n\nT5 is ground\nT6 is ground\nX8 is free\ntranspose_aux(T5, [], T6) !=? transpose_aux([], X8, X8)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: transpose_aux([], [], []), SCOPE: 2, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(row2col(T15, .(T17, T18), X31, X32), transpose_aux(T16, X32, X31))\n\nT15 is ground\nT16 is ground\nT17 is ground\nT18 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: row2col(T15, .(T17, T18), X31, X32)\n\nT15 is ground\nT17 is ground\nT18 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: transpose_aux(T16, T20, T19)\n\nT16 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: row2col(T15, .(T17, T18), X31, X32), SCOPE: 3, CLAUSE: 1 | row2col(T15, .(T17, T18), X31, X32), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT17 is ground\nT18 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: row2col(T15, .(T17, T18), X31, X32), SCOPE: 3, CLAUSE: 4\n\nT15 is ground\nT17 is ground\nT18 is ground\nX31 is free\nX32 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: row2col(T30, T32, X55, X56)\n\nT30 is ground\nT32 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(T30, T32, X55, X56), SCOPE: 4, CLAUSE: 1 | row2col(T30, T32, X55, X56), SCOPE: 4, CLAUSE: 4\n\nT30 is ground\nT32 is ground\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: true | row2col([], [], X55, X56), SCOPE: 4, CLAUSE: 4\n\nX55 is free\nX56 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: row2col(T30, T32, X55, X56), SCOPE: 4, CLAUSE: 4\n\nT30 is ground\nT32 is ground\nX55 is free\nX56 is free\nrow2col(T30, T32, X55, X56) !=? row2col([], [], [], [])", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: row2col([], [], 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(T42, T44, X85, X86)\n\nT42 is ground\nT44 is ground\nX85 is free\nX86 is free", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: transpose_aux(T16, T20, T19), SCOPE: 5, CLAUSE: 0 | transpose_aux(T16, T20, T19), SCOPE: 5, CLAUSE: 3\n\nT16 is ground\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: true | transpose_aux([], T49, T49), SCOPE: 5, CLAUSE: 3\n\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: transpose_aux(T16, T20, T19), SCOPE: 5, CLAUSE: 3\n\nT16 is ground\nT19 is ground\nT20 is ground\nX91 is free\ntranspose_aux(T16, T20, T19) !=? transpose_aux([], X91, X91)", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: transpose_aux([], T49, T49), SCOPE: 5, CLAUSE: 3\n\nT49 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: ','(row2col(T60, .(T63, T64), X114, X115), transpose_aux(T61, X115, X114))\n\nT60 is ground\nT61 is ground\nT63 is ground\nT64 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 -> 4;

60 [label="ONLY EVAL with clause\ntranspose(X4, X5) :- transpose_aux(X4, [], X5).\nand substitutionT1 -> T5,\nX4 -> T5,\nT2 -> T6,\nX5 -> T6", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 60 [arrowhead = none ];
60 -> 5;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
5 -> 61 [arrowhead = none ];
61 -> 7;

62 [label="EVAL with clause\ntranspose_aux([], X8, X8).\nand substitutionT5 -> [],\nX8 -> [],\nT6 -> []", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 62 [arrowhead = none ];
62 -> 8;

63 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 63 [arrowhead = none ];
63 -> 9;

64 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 64 [arrowhead = none ];
64 -> 10;

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 -> T15,\nX27 -> T16,\nT5 -> .(T15, T16),\nX28 -> [],\nX29 -> T17,\nX30 -> T18,\nT6 -> .(T17, T18)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 65 [arrowhead = none ];
65 -> 17;

66 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 66 [arrowhead = none ];
66 -> 18;

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];
10 -> 67 [arrowhead = none ];
67 -> 11;

68 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
17 -> 68 [arrowhead = none ];
68 -> 19;

69 [label="SPLIT 2\nnew knowledge:\nT15 is ground\nT17 is ground\nT18 is ground\nT19 is ground\nT20 is ground\nreplacements:X31 -> T19,\nX32 -> T20", fontsize=14, style = filled, fillcolor = violet];
17 -> 69 [arrowhead = none ];
69 -> 20;

70 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
19 -> 70 [arrowhead = none ];
70 -> 23;

71 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
20 -> 71 [arrowhead = none ];
71 -> 47;

72 [label="BACKTRACK\nfor clause: row2col([], [], [], [])because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
23 -> 72 [arrowhead = none ];
72 -> 24;

73 [label="EVAL with clause\nrow2col(.(X49, X50), .(.(X49, X51), X52), .(X51, X53), .([], X54)) :- row2col(X50, X52, X53, X54).\nand substitutionX49 -> T29,\nX50 -> T30,\nT15 -> .(T29, T30),\nX51 -> T31,\nT17 -> .(T29, T31),\nT18 -> T32,\nX52 -> T32,\nX53 -> X55,\nX31 -> .(T31, X55),\nX54 -> X56,\nX32 -> .([], X56)", fontsize=14, style = filled, fillcolor = lightblue];
24 -> 73 [arrowhead = none ];
73 -> 29;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 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 substitutionT30 -> [],\nT32 -> [],\nX55 -> [],\nX56 -> []", fontsize=14, style = filled, fillcolor = lightblue];
31 -> 76 [arrowhead = none ];
76 -> 34;

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];
34 -> 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 -> T41,\nX80 -> T42,\nT30 -> .(T41, T42),\nX81 -> T43,\nX82 -> T44,\nT32 -> .(.(T41, T43), T44),\nX83 -> X85,\nX55 -> .(T43, X85),\nX84 -> X86,\nX56 -> .([], X86)", 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 -> 45;

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:\nT30 -> T42\nT32 -> T44\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 substitutionT16 -> [],\nT20 -> T49,\nX91 -> T49,\nT19 -> T49", 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 -> 52;

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 -> T60,\nX110 -> T61,\nT16 -> .(T60, T61),\nT20 -> T62,\nX111 -> T62,\nX112 -> T63,\nX113 -> T64,\nT19 -> .(T63, T64)", 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];
52 -> 88 [arrowhead = none ];
88 -> 55;

89 [label="INSTANCE with matching:\nT15 -> T60\nT17 -> T63\nT18 -> T64\nX31 -> X114\nX32 -> X115\nT16 -> T61", fontsize=14, style = filled, fillcolor = lightgrey];
57 -> 89 [arrowhead = none , style=dashed];
89 -> 17[style=dashed];

}

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1
f2_in(.(z0, z1), .(z2, z3)) → U1(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f17_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))
f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
f20_in([], z0, z0) → f20_out1
f20_in(.(z0, z1), z2, .(z3, z4)) → U4(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f17_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f20_out1
f17_in(z0, z1, z2, z3) → U5(f19_in(z0, z1, z2), z0, z1, z2, z3)
U5(f19_out1(z0, z1), z2, z3, z4, z5) → U6(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f20_out1, z0, z1, z2, z3, z4, z5) → f17_out1(z4, z5)

Tuples:

F2_IN(.(z0, z1), .(z2, z3)) → c1(U1'(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3)), F17_IN(z0, z2, z3, z1))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(U2'(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3)), F29_IN(z1, z3))
F19_IN(.(z0, z1), .(z0, z2), z3) → c6(U3'(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3), F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(U4'(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4)), F17_IN(z0, z3, z4, z1))
F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(U6'(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1), F20_IN(z5, z1, z0))

S tuples:

F2_IN(.(z0, z1), .(z2, z3)) → c1(U1'(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3)), F17_IN(z0, z2, z3, z1))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(U2'(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3)), F29_IN(z1, z3))
F19_IN(.(z0, z1), .(z0, z2), z3) → c6(U3'(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3), F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(U4'(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4)), F17_IN(z0, z3, z4, z1))
F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(U6'(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1), F20_IN(z5, z1, z0))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f29_in, U2, f19_in, U3, f20_in, U4, f17_in, U5, U6

Defined Pair Symbols:

F2_IN, F29_IN, F19_IN, F20_IN, F17_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), .(z2, z3)) → U1(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f17_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))
f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
f20_in([], z0, z0) → f20_out1
f20_in(.(z0, z1), z2, .(z3, z4)) → U4(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f17_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f20_out1
f17_in(z0, z1, z2, z3) → U5(f19_in(z0, z1, z2), z0, z1, z2, z3)
U5(f19_out1(z0, z1), z2, z3, z4, z5) → U6(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f20_out1, z0, z1, z2, z3, z4, z5) → f17_out1(z4, z5)

Tuples:

F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(U2'(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3)), F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(U4'(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4)), F17_IN(z0, z3, z4, z1))
F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(U6'(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1), F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c(U1'(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3)))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(U3'(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))

S tuples:

F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(U2'(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3)), F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(U4'(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4)), F17_IN(z0, z3, z4, z1))
F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(U6'(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1), F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c(U1'(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3)))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(U3'(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f29_in, U2, f19_in, U3, f20_in, U4, f17_in, U5, U6

Defined Pair Symbols:

F29_IN, F20_IN, F17_IN, U5', F2_IN, F19_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), .(z2, z3)) → U1(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f17_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))
f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
f20_in([], z0, z0) → f20_out1
f20_in(.(z0, z1), z2, .(z3, z4)) → U4(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f17_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f20_out1
f17_in(z0, z1, z2, z3) → U5(f19_in(z0, z1, z2), z0, z1, z2, z3)
U5(f19_out1(z0, z1), z2, z3, z4, z5) → U6(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f20_out1, z0, z1, z2, z3, z4, z5) → f17_out1(z4, z5)

Tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c
F19_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c
F19_IN(.(z0, z1), .(z0, z2), z3) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, f29_in, U2, f19_in, U3, f20_in, U4, f17_in, U5, U6

Defined Pair Symbols:

F17_IN, F2_IN, F19_IN, F29_IN, F20_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), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1
f2_in(.(z0, z1), .(z2, z3)) → U1(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f17_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))
f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
f20_in([], z0, z0) → f20_out1
f20_in(.(z0, z1), z2, .(z3, z4)) → U4(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f17_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f20_out1
f17_in(z0, z1, z2, z3) → U5(f19_in(z0, z1, z2), z0, z1, z2, z3)
U5(f19_out1(z0, z1), z2, z3, z4, z5) → U6(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f20_out1, z0, z1, z2, z3, z4, z5) → f17_out1(z4, z5)

Tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c
F19_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F19_IN(.(z0, z1), .(z0, z2), z3) → c

K tuples:

F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c

Defined Rule Symbols:

f2_in, U1, f29_in, U2, f19_in, U3, f20_in, U4, f17_in, U5, U6

Defined Pair Symbols:

F17_IN, F2_IN, F19_IN, F29_IN, F20_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, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c

We considered the (Usable) Rules:

f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))

And the Tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c
F19_IN(.(z0, z1), .(z0, z2), z3) → c

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂   
POL(F17_IN(x₁, x₂, x₃, x₄)) = [2] + [2]x₁ + [2]x₄   
POL(F19_IN(x₁, x₂, x₃)) = [2]x₁   
POL(F20_IN(x₁, x₂, x₃)) = [1] + [2]x₁   
POL(F29_IN(x₁, x₂)) = 0   
POL(F2_IN(x₁, x₂)) = [2]x₁ + [2]x₂   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3(x₁, x₂, x₃, x₄)) = 0   
POL(U5'(x₁, x₂, x₃, x₄, x₅)) = [1] + [2]x₅   
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(f19_in(x₁, x₂, x₃)) = 0   
POL(f19_out1(x₁, x₂)) = 0   
POL(f29_in(x₁, x₂)) = 0   
POL(f29_out1(x₁, x₂)) = 0   

(28) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1
f2_in(.(z0, z1), .(z2, z3)) → U1(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f17_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))
f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
f20_in([], z0, z0) → f20_out1
f20_in(.(z0, z1), z2, .(z3, z4)) → U4(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f17_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f20_out1
f17_in(z0, z1, z2, z3) → U5(f19_in(z0, z1, z2), z0, z1, z2, z3)
U5(f19_out1(z0, z1), z2, z3, z4, z5) → U6(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f20_out1, z0, z1, z2, z3, z4, z5) → f17_out1(z4, z5)

Tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c
F19_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:

F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))

K tuples:

F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c
F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c

Defined Rule Symbols:

f2_in, U1, f29_in, U2, f19_in, U3, f20_in, U4, f17_in, U5, U6

Defined Pair Symbols:

F17_IN, F2_IN, F19_IN, F29_IN, F20_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:

U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))

(30) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1
f2_in(.(z0, z1), .(z2, z3)) → U1(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f17_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))
f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
f20_in([], z0, z0) → f20_out1
f20_in(.(z0, z1), z2, .(z3, z4)) → U4(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f17_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f20_out1
f17_in(z0, z1, z2, z3) → U5(f19_in(z0, z1, z2), z0, z1, z2, z3)
U5(f19_out1(z0, z1), z2, z3, z4, z5) → U6(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f20_out1, z0, z1, z2, z3, z4, z5) → f17_out1(z4, z5)

Tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c
F19_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:

F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))

K tuples:

F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c
F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))

Defined Rule Symbols:

f2_in, U1, f29_in, U2, f19_in, U3, f20_in, U4, f17_in, U5, U6

Defined Pair Symbols:

F17_IN, F2_IN, F19_IN, F29_IN, F20_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), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))

We considered the (Usable) Rules:

f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))

And the Tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c
F19_IN(.(z0, z1), .(z0, z2), z3) → c

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

POL(.(x₁, x₂)) = [1] + x₁ + x₂   
POL(F17_IN(x₁, x₂, x₃, x₄)) = [1] + x₁ + x₄   
POL(F19_IN(x₁, x₂, x₃)) = x₁   
POL(F20_IN(x₁, x₂, x₃)) = x₁   
POL(F29_IN(x₁, x₂)) = [1] + x₁   
POL(F2_IN(x₁, x₂)) = x₁   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3(x₁, x₂, x₃, x₄)) = 0   
POL(U5'(x₁, x₂, x₃, x₄, x₅)) = x₅   
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(f19_in(x₁, x₂, x₃)) = 0   
POL(f19_out1(x₁, x₂)) = 0   
POL(f29_in(x₁, x₂)) = 0   
POL(f29_out1(x₁, x₂)) = 0   

(32) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([], []) → f2_out1
f2_in(.(z0, z1), .(z2, z3)) → U1(f17_in(z0, z2, z3, z1), .(z0, z1), .(z2, z3))
U1(f17_out1(z0, z1), .(z2, z3), .(z4, z5)) → f2_out1
f29_in([], []) → f29_out1([], [])
f29_in(.(z0, z1), .(.(z0, z2), z3)) → U2(f29_in(z1, z3), .(z0, z1), .(.(z0, z2), z3))
U2(f29_out1(z0, z1), .(z2, z3), .(.(z2, z4), z5)) → f29_out1(.(z4, z0), .([], z1))
f19_in(.(z0, z1), .(z0, z2), z3) → U3(f29_in(z1, z3), .(z0, z1), .(z0, z2), z3)
U3(f29_out1(z0, z1), .(z2, z3), .(z2, z4), z5) → f19_out1(.(z4, z0), .([], z1))
f20_in([], z0, z0) → f20_out1
f20_in(.(z0, z1), z2, .(z3, z4)) → U4(f17_in(z0, z3, z4, z1), .(z0, z1), z2, .(z3, z4))
U4(f17_out1(z0, z1), .(z2, z3), z4, .(z5, z6)) → f20_out1
f17_in(z0, z1, z2, z3) → U5(f19_in(z0, z1, z2), z0, z1, z2, z3)
U5(f19_out1(z0, z1), z2, z3, z4, z5) → U6(f20_in(z5, z1, z0), z2, z3, z4, z5, z0, z1)
U6(f20_out1, z0, z1, z2, z3, z4, z5) → f17_out1(z4, z5)

Tuples:

F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F2_IN(.(z0, z1), .(z2, z3)) → c
F19_IN(.(z0, z1), .(z0, z2), z3) → c

S tuples:none
K tuples:

F2_IN(.(z0, z1), .(z2, z3)) → c(F17_IN(z0, z2, z3, z1))
F2_IN(.(z0, z1), .(z2, z3)) → c
F17_IN(z0, z1, z2, z3) → c11(U5'(f19_in(z0, z1, z2), z0, z1, z2, z3), F19_IN(z0, z1, z2))
F19_IN(.(z0, z1), .(z0, z2), z3) → c(F29_IN(z1, z3))
F20_IN(.(z0, z1), z2, .(z3, z4)) → c9(F17_IN(z0, z3, z4, z1))
F19_IN(.(z0, z1), .(z0, z2), z3) → c
U5'(f19_out1(z0, z1), z2, z3, z4, z5) → c12(F20_IN(z5, z1, z0))
F29_IN(.(z0, z1), .(.(z0, z2), z3)) → c4(F29_IN(z1, z3))

Defined Rule Symbols:

f2_in, U1, f29_in, U2, f19_in, U3, f20_in, U4, f17_in, U5, U6

Defined Pair Symbols:

F17_IN, F2_IN, F19_IN, F29_IN, F20_IN, U5'

Compound Symbols:

c11, c, c4, c9, c12, c