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

(0) Obligation:

Clauses:

ordered([]).
ordered(.(X1, [])).
ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))).
less(0, s(X2)).
less(s(X), s(Y)) :- less(X, Y).

Query: ordered(g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

ordered([]).
ordered(.(X1, [])).
less(0, s(X2)).
ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs))).
less(s(X), s(Y)) :- less(X, Y).

Query: ordered(g)

(3) 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: ordered(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: ordered(T1), SCOPE: 1, CLAUSE: 0 | ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nordered(T1) !=? ordered([])", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ordered([]), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | ordered(.(T4, [])), SCOPE: 1, CLAUSE: 3\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ordered(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nX9 is free\nordered(T1) !=? ordered([])\nordered(T1) !=? ordered(.(X9, []))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ordered(.(T4, [])), SCOPE: 1, CLAUSE: 3\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: ','(less(T12, s(T13)), ordered(.(T13, T14)))\n\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: less(T12, s(T13))\n\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: ordered(.(T13, T14))\n\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: less(T12, s(T13)), SCOPE: 2, CLAUSE: 2 | less(T12, s(T13)), SCOPE: 2, CLAUSE: 4\n\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: true | less(0, s(T21)), SCOPE: 2, CLAUSE: 4\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: less(T12, s(T13)), SCOPE: 2, CLAUSE: 4\n\nT12 is ground\nT13 is ground\nX28 is free\nless(T12, s(T13)) !=? less(0, s(X28))", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: less(0, s(T21)), SCOPE: 2, CLAUSE: 4\n\nT21 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: less(T26, T27)\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: less(T26, T27), SCOPE: 3, CLAUSE: 2 | less(T26, T27), SCOPE: 3, CLAUSE: 4\n\nT26 is ground\nT27 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: true | less(0, s(T32)), SCOPE: 3, CLAUSE: 4\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: less(T26, T27), SCOPE: 3, CLAUSE: 4\n\nT26 is ground\nT27 is ground\nX41 is free\nless(T26, T27) !=? less(0, s(X41))", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: less(0, s(T32)), SCOPE: 3, CLAUSE: 4\n\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: less(T37, T38)\n\nT37 is ground\nT38 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 60 [arrowhead = none ];
60 -> 3;

61 [label="EVAL with clause\nordered([]).\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 61 [arrowhead = none ];
61 -> 5;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 62 [arrowhead = none ];
62 -> 8;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 63 [arrowhead = none ];
63 -> 9;

64 [label="EVAL with clause\nordered(.(X9, [])).\nand substitutionX9 -> T4,\nT1 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 64 [arrowhead = none ];
64 -> 16;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 65 [arrowhead = none ];
65 -> 18;

66 [label="BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 66 [arrowhead = none ];
66 -> 12;

67 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 67 [arrowhead = none ];
67 -> 14;

68 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 68 [arrowhead = none ];
68 -> 20;

69 [label="EVAL with clause\nordered(.(X19, .(X20, X21))) :- ','(less(X19, s(X20)), ordered(.(X20, X21))).\nand substitutionX19 -> T12,\nX20 -> T13,\nX21 -> T14,\nT1 -> .(T12, .(T13, T14))", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 69 [arrowhead = none ];
69 -> 25;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 70 [arrowhead = none ];
70 -> 27;

71 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 71 [arrowhead = none ];
71 -> 22;

72 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
25 -> 72 [arrowhead = none ];
72 -> 32;

73 [label="SPLIT 2\nnew knowledge:\nT12 is ground\nT13 is ground", fontsize=14, style = filled, fillcolor = violet];
25 -> 73 [arrowhead = none ];
73 -> 33;

74 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 74 [arrowhead = none ];
74 -> 36;

75 [label="INSTANCE with matching:\nT1 -> .(T13, T14)", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 75 [arrowhead = none , style=dashed];
75 -> 2[style=dashed];

76 [label="EVAL with clause\nless(0, s(X28)).\nand substitutionT12 -> 0,\nT13 -> T21,\nX28 -> T21", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 76 [arrowhead = none ];
76 -> 38;

77 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 77 [arrowhead = none ];
77 -> 39;

78 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 78 [arrowhead = none ];
78 -> 40;

79 [label="EVAL with clause\nless(s(X35), s(X36)) :- less(X35, X36).\nand substitutionX35 -> T26,\nT12 -> s(T26),\nT13 -> T27,\nX36 -> T27", fontsize=14, style = filled, fillcolor = lightblue];
39 -> 79 [arrowhead = none ];
79 -> 45;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 80 [arrowhead = none ];
80 -> 47;

81 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 81 [arrowhead = none ];
81 -> 42;

82 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
45 -> 82 [arrowhead = none ];
82 -> 51;

83 [label="EVAL with clause\nless(0, s(X41)).\nand substitutionT26 -> 0,\nX41 -> T32,\nT27 -> s(T32)", fontsize=14, style = filled, fillcolor = lightblue];
51 -> 83 [arrowhead = none ];
83 -> 52;

84 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
51 -> 84 [arrowhead = none ];
84 -> 54;

85 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
52 -> 85 [arrowhead = none ];
85 -> 55;

86 [label="EVAL with clause\nless(s(X48), s(X49)) :- less(X48, X49).\nand substitutionX48 -> T37,\nT26 -> s(T37),\nX49 -> T38,\nT27 -> s(T38)", fontsize=14, style = filled, fillcolor = lightblue];
54 -> 86 [arrowhead = none ];
86 -> 58;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
54 -> 87 [arrowhead = none ];
87 -> 59;

88 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
55 -> 88 [arrowhead = none ];
88 -> 57;

89 [label="INSTANCE with matching:\nT26 -> T37\nT27 -> T38", fontsize=14, style = filled, fillcolor = lightgrey];
58 -> 89 [arrowhead = none , style=dashed];
89 -> 45[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f25_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f25_out1, .(z0, .(z1, z2))) → f2_out1
f45_in(0, s(z0)) → f45_out1
f45_in(s(z0), s(z1)) → U2(f45_in(z0, z1), s(z0), s(z1))
U2(f45_out1, s(z0), s(z1)) → f45_out1
f32_in(0, z0) → f32_out1
f32_in(s(z0), z1) → U3(f45_in(z0, z1), s(z0), z1)
U3(f45_out1, s(z0), z1) → f32_out1
f25_in(z0, z1, z2) → U4(f32_in(z0, z1), z0, z1, z2)
U4(f32_out1, z0, z1, z2) → U5(f2_in(.(z1, z2)), z0, z1, z2)
U5(f2_out1, z0, z1, z2) → f25_out1

Tuples:

F2_IN(.(z0, .(z1, z2))) → c2(U1'(f25_in(z0, z1, z2), .(z0, .(z1, z2))), F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(U2'(f45_in(z0, z1), s(z0), s(z1)), F45_IN(z0, z1))
F32_IN(s(z0), z1) → c8(U3'(f45_in(z0, z1), s(z0), z1), F45_IN(z0, z1))
F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(U5'(f2_in(.(z1, z2)), z0, z1, z2), F2_IN(.(z1, z2)))

S tuples:

F2_IN(.(z0, .(z1, z2))) → c2(U1'(f25_in(z0, z1, z2), .(z0, .(z1, z2))), F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(U2'(f45_in(z0, z1), s(z0), s(z1)), F45_IN(z0, z1))
F32_IN(s(z0), z1) → c8(U3'(f45_in(z0, z1), s(z0), z1), F45_IN(z0, z1))
F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(U5'(f2_in(.(z1, z2)), z0, z1, z2), F2_IN(.(z1, z2)))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f45_in, U2, f32_in, U3, f25_in, U4, U5

Defined Pair Symbols:

F2_IN, F45_IN, F32_IN, F25_IN, U4'

Compound Symbols:

c2, c5, c8, c10, c11

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

Split RHS of tuples not part of any SCC

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f25_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f25_out1, .(z0, .(z1, z2))) → f2_out1
f45_in(0, s(z0)) → f45_out1
f45_in(s(z0), s(z1)) → U2(f45_in(z0, z1), s(z0), s(z1))
U2(f45_out1, s(z0), s(z1)) → f45_out1
f32_in(0, z0) → f32_out1
f32_in(s(z0), z1) → U3(f45_in(z0, z1), s(z0), z1)
U3(f45_out1, s(z0), z1) → f32_out1
f25_in(z0, z1, z2) → U4(f32_in(z0, z1), z0, z1, z2)
U4(f32_out1, z0, z1, z2) → U5(f2_in(.(z1, z2)), z0, z1, z2)
U5(f2_out1, z0, z1, z2) → f25_out1

Tuples:

F2_IN(.(z0, .(z1, z2))) → c2(U1'(f25_in(z0, z1, z2), .(z0, .(z1, z2))), F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(U2'(f45_in(z0, z1), s(z0), s(z1)), F45_IN(z0, z1))
F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(U5'(f2_in(.(z1, z2)), z0, z1, z2), F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c(U3'(f45_in(z0, z1), s(z0), z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))

S tuples:

F2_IN(.(z0, .(z1, z2))) → c2(U1'(f25_in(z0, z1, z2), .(z0, .(z1, z2))), F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(U2'(f45_in(z0, z1), s(z0), s(z1)), F45_IN(z0, z1))
F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(U5'(f2_in(.(z1, z2)), z0, z1, z2), F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c(U3'(f45_in(z0, z1), s(z0), z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f45_in, U2, f32_in, U3, f25_in, U4, U5

Defined Pair Symbols:

F2_IN, F45_IN, F25_IN, U4', F32_IN

Compound Symbols:

c2, c5, c10, c11, c

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

Removed 4 trailing tuple parts

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f25_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f25_out1, .(z0, .(z1, z2))) → f2_out1
f45_in(0, s(z0)) → f45_out1
f45_in(s(z0), s(z1)) → U2(f45_in(z0, z1), s(z0), s(z1))
U2(f45_out1, s(z0), s(z1)) → f45_out1
f32_in(0, z0) → f32_out1
f32_in(s(z0), z1) → U3(f45_in(z0, z1), s(z0), z1)
U3(f45_out1, s(z0), z1) → f32_out1
f25_in(z0, z1, z2) → U4(f32_in(z0, z1), z0, z1, z2)
U4(f32_out1, z0, z1, z2) → U5(f2_in(.(z1, z2)), z0, z1, z2)
U5(f2_out1, z0, z1, z2) → f25_out1

Tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c

S tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c

K tuples:none
Defined Rule Symbols:

f2_in, U1, f45_in, U2, f32_in, U3, f25_in, U4, U5

Defined Pair Symbols:

F25_IN, F32_IN, F2_IN, F45_IN, U4'

Compound Symbols:

c10, c, c2, c5, c11, c

(9) 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(s(z0), z1) → c(F45_IN(z0, z1))
F32_IN(s(z0), z1) → c

We considered the (Usable) Rules:

f32_in(0, z0) → f32_out1
f32_in(s(z0), z1) → U3(f45_in(z0, z1), s(z0), z1)
f45_in(0, s(z0)) → f45_out1
f45_in(s(z0), s(z1)) → U2(f45_in(z0, z1), s(z0), s(z1))
U3(f45_out1, s(z0), z1) → f32_out1
U2(f45_out1, s(z0), s(z1)) → f45_out1

And the Tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c

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

POL(.(x₁, x₂)) = x₁ + x₂   
POL(0) = 0   
POL(F25_IN(x₁, x₂, x₃)) = x₁ + [2]x₂ + [2]x₃   
POL(F2_IN(x₁)) = [2]x₁   
POL(F32_IN(x₁, x₂)) = x₁   
POL(F45_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3(x₁, x₂, x₃)) = 0   
POL(U4'(x₁, x₂, x₃, x₄)) = [2]x₃ + [2]x₄   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(f32_in(x₁, x₂)) = 0   
POL(f32_out1) = 0   
POL(f45_in(x₁, x₂)) = [2]x₂   
POL(f45_out1) = 0   
POL(s(x₁)) = [2] + x₁   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f25_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f25_out1, .(z0, .(z1, z2))) → f2_out1
f45_in(0, s(z0)) → f45_out1
f45_in(s(z0), s(z1)) → U2(f45_in(z0, z1), s(z0), s(z1))
U2(f45_out1, s(z0), s(z1)) → f45_out1
f32_in(0, z0) → f32_out1
f32_in(s(z0), z1) → U3(f45_in(z0, z1), s(z0), z1)
U3(f45_out1, s(z0), z1) → f32_out1
f25_in(z0, z1, z2) → U4(f32_in(z0, z1), z0, z1, z2)
U4(f32_out1, z0, z1, z2) → U5(f2_in(.(z1, z2)), z0, z1, z2)
U5(f2_out1, z0, z1, z2) → f25_out1

Tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c

S tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))

K tuples:

F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F32_IN(s(z0), z1) → c

Defined Rule Symbols:

f2_in, U1, f45_in, U2, f32_in, U3, f25_in, U4, U5

Defined Pair Symbols:

F25_IN, F32_IN, F2_IN, F45_IN, U4'

Compound Symbols:

c10, c, c2, c5, c11, 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.

F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))

We considered the (Usable) Rules:

f32_in(0, z0) → f32_out1
f32_in(s(z0), z1) → U3(f45_in(z0, z1), s(z0), z1)
f45_in(0, s(z0)) → f45_out1
f45_in(s(z0), s(z1)) → U2(f45_in(z0, z1), s(z0), s(z1))
U3(f45_out1, s(z0), z1) → f32_out1
U2(f45_out1, s(z0), s(z1)) → f45_out1

And the Tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(0) = 0   
POL(F25_IN(x₁, x₂, x₃)) = [1] + x₃   
POL(F2_IN(x₁)) = x₁   
POL(F32_IN(x₁, x₂)) = 0   
POL(F45_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3(x₁, x₂, x₃)) = 0   
POL(U4'(x₁, x₂, x₃, x₄)) = [1] + x₄   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c5(x₁)) = x₁   
POL(f32_in(x₁, x₂)) = 0   
POL(f32_out1) = 0   
POL(f45_in(x₁, x₂)) = 0   
POL(f45_out1) = 0   
POL(s(x₁)) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f25_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f25_out1, .(z0, .(z1, z2))) → f2_out1
f45_in(0, s(z0)) → f45_out1
f45_in(s(z0), s(z1)) → U2(f45_in(z0, z1), s(z0), s(z1))
U2(f45_out1, s(z0), s(z1)) → f45_out1
f32_in(0, z0) → f32_out1
f32_in(s(z0), z1) → U3(f45_in(z0, z1), s(z0), z1)
U3(f45_out1, s(z0), z1) → f32_out1
f25_in(z0, z1, z2) → U4(f32_in(z0, z1), z0, z1, z2)
U4(f32_out1, z0, z1, z2) → U5(f2_in(.(z1, z2)), z0, z1, z2)
U5(f2_out1, z0, z1, z2) → f25_out1

Tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c

S tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))

K tuples:

F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F32_IN(s(z0), z1) → c
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))

Defined Rule Symbols:

f2_in, U1, f45_in, U2, f32_in, U3, f25_in, U4, U5

Defined Pair Symbols:

F25_IN, F32_IN, F2_IN, F45_IN, U4'

Compound Symbols:

c10, c, c2, c5, c11, c

(13) CdtKnowledgeProof (EQUIVALENT transformation)

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

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f25_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f25_out1, .(z0, .(z1, z2))) → f2_out1
f45_in(0, s(z0)) → f45_out1
f45_in(s(z0), s(z1)) → U2(f45_in(z0, z1), s(z0), s(z1))
U2(f45_out1, s(z0), s(z1)) → f45_out1
f32_in(0, z0) → f32_out1
f32_in(s(z0), z1) → U3(f45_in(z0, z1), s(z0), z1)
U3(f45_out1, s(z0), z1) → f32_out1
f25_in(z0, z1, z2) → U4(f32_in(z0, z1), z0, z1, z2)
U4(f32_out1, z0, z1, z2) → U5(f2_in(.(z1, z2)), z0, z1, z2)
U5(f2_out1, z0, z1, z2) → f25_out1

Tuples:

F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))
F32_IN(s(z0), z1) → c

S tuples:

F45_IN(s(z0), s(z1)) → c5(F45_IN(z0, z1))

K tuples:

F32_IN(s(z0), z1) → c(F45_IN(z0, z1))
F32_IN(s(z0), z1) → c
F2_IN(.(z0, .(z1, z2))) → c2(F25_IN(z0, z1, z2))
F25_IN(z0, z1, z2) → c10(U4'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
U4'(f32_out1, z0, z1, z2) → c11(F2_IN(.(z1, z2)))

Defined Rule Symbols:

f2_in, U1, f45_in, U2, f32_in, U3, f25_in, U4, U5

Defined Pair Symbols:

F25_IN, F32_IN, F2_IN, F45_IN, U4'

Compound Symbols:

c10, c, c2, c5, c11, c

(15) 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: ordered(T1)\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: ordered(T1), SCOPE: 1, CLAUSE: 0 | ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nordered(T1) !=? ordered([])", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ordered([]), SCOPE: 1, CLAUSE: 3\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true | ordered(.(T3, [])), SCOPE: 1, CLAUSE: 3\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ordered(T1), SCOPE: 1, CLAUSE: 3\n\nT1 is ground\nX8 is free\nordered(T1) !=? ordered([])\nordered(T1) !=? ordered(.(X8, []))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ordered(.(T3, [])), SCOPE: 1, CLAUSE: 3\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(less(T8, s(T9)), ordered(.(T9, T10)))\n\nT8 is ground\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: ','(less(T8, s(T9)), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 2 | ','(less(T8, s(T9)), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 4\n\nT8 is ground\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ordered(.(T13, T10)) | ','(less(0, s(T13)), ordered(.(T13, T10))), SCOPE: 2, CLAUSE: 4\n\nT10 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(less(T8, s(T9)), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 4\n\nT8 is ground\nT9 is ground\nT10 is ground\nX20 is free\nless(T8, s(T9)) !=? less(0, s(X20))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ordered(.(T13, T10))\n\nT10 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(less(0, s(T13)), ordered(.(T13, T10))), SCOPE: 2, CLAUSE: 4\n\nT10 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: ','(less(T18, T19), ordered(.(T19, T10)))\n\nT10 is ground\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: less(T18, T19)\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: ordered(.(T19, T10))\n\nT10 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: less(T18, T19), SCOPE: 3, CLAUSE: 2 | less(T18, T19), SCOPE: 3, CLAUSE: 4\n\nT18 is ground\nT19 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: true | less(0, s(T26)), SCOPE: 3, CLAUSE: 4\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: less(T18, T19), SCOPE: 3, CLAUSE: 4\n\nT18 is ground\nT19 is ground\nX35 is free\nless(T18, T19) !=? less(0, s(X35))", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: less(0, s(T26)), SCOPE: 3, CLAUSE: 4\n\nT26 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: less(T31, T32)\n\nT31 is ground\nT32 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 57 [arrowhead = none ];
57 -> 4;

58 [label="EVAL with clause\nordered([]).\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 58 [arrowhead = none ];
58 -> 6;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 59 [arrowhead = none ];
59 -> 7;

60 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 60 [arrowhead = none ];
60 -> 10;

61 [label="EVAL with clause\nordered(.(X8, [])).\nand substitutionX8 -> T3,\nT1 -> .(T3, [])", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 61 [arrowhead = none ];
61 -> 15;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 62 [arrowhead = none ];
62 -> 17;

63 [label="BACKTRACK\nfor clause: ordered(.(X1, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 63 [arrowhead = none ];
63 -> 11;

64 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 64 [arrowhead = none ];
64 -> 13;

65 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 65 [arrowhead = none ];
65 -> 19;

66 [label="EVAL with clause\nordered(.(X15, .(X16, X17))) :- ','(less(X15, s(X16)), ordered(.(X16, X17))).\nand substitutionX15 -> T8,\nX16 -> T9,\nX17 -> T10,\nT1 -> .(T8, .(T9, T10))", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 66 [arrowhead = none ];
66 -> 23;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 67 [arrowhead = none ];
67 -> 24;

68 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(less(X, s(Y)), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 68 [arrowhead = none ];
68 -> 21;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 69 [arrowhead = none ];
69 -> 26;

70 [label="EVAL with clause\nless(0, s(X20)).\nand substitutionT8 -> 0,\nT9 -> T13,\nX20 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 70 [arrowhead = none ];
70 -> 28;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 71 [arrowhead = none ];
71 -> 29;

72 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 72 [arrowhead = none ];
72 -> 30;

73 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 73 [arrowhead = none ];
73 -> 31;

74 [label="EVAL with clause\nless(s(X27), s(X28)) :- less(X27, X28).\nand substitutionX27 -> T18,\nT8 -> s(T18),\nT9 -> T19,\nX28 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 74 [arrowhead = none ];
74 -> 35;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 75 [arrowhead = none ];
75 -> 37;

76 [label="INSTANCE with matching:\nT1 -> .(T13, T10)", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 76 [arrowhead = none , style=dashed];
76 -> 1[style=dashed];

77 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 77 [arrowhead = none ];
77 -> 34;

78 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
35 -> 78 [arrowhead = none ];
78 -> 41;

79 [label="SPLIT 2\nnew knowledge:\nT18 is ground\nT19 is ground", fontsize=14, style = filled, fillcolor = violet];
35 -> 79 [arrowhead = none ];
79 -> 43;

80 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
41 -> 80 [arrowhead = none ];
80 -> 44;

81 [label="INSTANCE with matching:\nT1 -> .(T19, T10)", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 81 [arrowhead = none , style=dashed];
81 -> 1[style=dashed];

82 [label="EVAL with clause\nless(0, s(X35)).\nand substitutionT18 -> 0,\nX35 -> T26,\nT19 -> s(T26)", fontsize=14, style = filled, fillcolor = lightblue];
44 -> 82 [arrowhead = none ];
82 -> 46;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 83 [arrowhead = none ];
83 -> 48;

84 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
46 -> 84 [arrowhead = none ];
84 -> 49;

85 [label="EVAL with clause\nless(s(X42), s(X43)) :- less(X42, X43).\nand substitutionX42 -> T31,\nT18 -> s(T31),\nX43 -> T32,\nT19 -> s(T32)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 85 [arrowhead = none ];
85 -> 53;

86 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 86 [arrowhead = none ];
86 -> 56;

87 [label="BACKTRACK\nfor clause: less(s(X), s(Y)) :- less(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 87 [arrowhead = none ];
87 -> 50;

88 [label="INSTANCE with matching:\nT18 -> T31\nT19 -> T32", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 88 [arrowhead = none , style=dashed];
88 -> 41[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(z0, z1))) → U1(f28_in(z0, z1), .(0, .(z0, z1)))
f1_in(.(s(z0), .(z1, z2))) → U2(f35_in(z0, z1, z2), .(s(z0), .(z1, z2)))
U1(f28_out1, .(0, .(z0, z1))) → f1_out1
U1(f28_out2, .(0, .(z0, z1))) → f1_out1
U2(f35_out1, .(s(z0), .(z1, z2))) → f1_out1
f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1
f35_in(z0, z1, z2) → U4(f41_in(z0, z1), z0, z1, z2)
U4(f41_out1, z0, z1, z2) → U5(f1_in(.(z1, z2)), z0, z1, z2)
U5(f1_out1, z0, z1, z2) → f35_out1
f28_in(z0, z1) → U6(f1_in(.(z0, z1)), f31_in(z0, z1), z0, z1)
U6(f1_out1, z0, z1, z2) → f28_out1
U6(z0, f31_out1, z1, z2) → f28_out2

Tuples:

F1_IN(.(0, .(z0, z1))) → c2(U1'(f28_in(z0, z1), .(0, .(z0, z1))), F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(U2'(f35_in(z0, z1, z2), .(s(z0), .(z1, z2))), F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(U3'(f41_in(z0, z1), s(z0), s(z1)), F41_IN(z0, z1))
F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(U5'(f1_in(.(z1, z2)), z0, z1, z2), F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(U6'(f1_in(.(z0, z1)), f31_in(z0, z1), z0, z1), F1_IN(.(z0, z1)))

S tuples:

F1_IN(.(0, .(z0, z1))) → c2(U1'(f28_in(z0, z1), .(0, .(z0, z1))), F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(U2'(f35_in(z0, z1, z2), .(s(z0), .(z1, z2))), F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(U3'(f41_in(z0, z1), s(z0), s(z1)), F41_IN(z0, z1))
F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(U5'(f1_in(.(z1, z2)), z0, z1, z2), F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(U6'(f1_in(.(z0, z1)), f31_in(z0, z1), z0, z1), F1_IN(.(z0, z1)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f41_in, U3, f35_in, U4, U5, f28_in, U6

Defined Pair Symbols:

F1_IN, F41_IN, F35_IN, U4', F28_IN

Compound Symbols:

c2, c3, c8, c10, c11, c13

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

Removed 5 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(z0, z1))) → U1(f28_in(z0, z1), .(0, .(z0, z1)))
f1_in(.(s(z0), .(z1, z2))) → U2(f35_in(z0, z1, z2), .(s(z0), .(z1, z2)))
U1(f28_out1, .(0, .(z0, z1))) → f1_out1
U1(f28_out2, .(0, .(z0, z1))) → f1_out1
U2(f35_out1, .(s(z0), .(z1, z2))) → f1_out1
f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1
f35_in(z0, z1, z2) → U4(f41_in(z0, z1), z0, z1, z2)
U4(f41_out1, z0, z1, z2) → U5(f1_in(.(z1, z2)), z0, z1, z2)
U5(f1_out1, z0, z1, z2) → f35_out1
f28_in(z0, z1) → U6(f1_in(.(z0, z1)), f31_in(z0, z1), z0, z1)
U6(f1_out1, z0, z1, z2) → f28_out1
U6(z0, f31_out1, z1, z2) → f28_out2

Tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

S tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, f41_in, U3, f35_in, U4, U5, f28_in, U6

Defined Pair Symbols:

F35_IN, F1_IN, F41_IN, U4', F28_IN

Compound Symbols:

c10, c2, c3, c8, c11, c13

(19) 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.

F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

We considered the (Usable) Rules:

f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1

And the Tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

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

POL(.(x₁, x₂)) = x₁ + x₂   
POL(0) = [1]   
POL(F1_IN(x₁)) = x₁   
POL(F28_IN(x₁, x₂)) = [1] + x₁ + x₂   
POL(F35_IN(x₁, x₂, x₃)) = x₂ + x₃   
POL(F41_IN(x₁, x₂)) = 0   
POL(U3(x₁, x₂, x₃)) = 0   
POL(U4'(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(f41_in(x₁, x₂)) = 0   
POL(f41_out1) = 0   
POL(s(x₁)) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(z0, z1))) → U1(f28_in(z0, z1), .(0, .(z0, z1)))
f1_in(.(s(z0), .(z1, z2))) → U2(f35_in(z0, z1, z2), .(s(z0), .(z1, z2)))
U1(f28_out1, .(0, .(z0, z1))) → f1_out1
U1(f28_out2, .(0, .(z0, z1))) → f1_out1
U2(f35_out1, .(s(z0), .(z1, z2))) → f1_out1
f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1
f35_in(z0, z1, z2) → U4(f41_in(z0, z1), z0, z1, z2)
U4(f41_out1, z0, z1, z2) → U5(f1_in(.(z1, z2)), z0, z1, z2)
U5(f1_out1, z0, z1, z2) → f35_out1
f28_in(z0, z1) → U6(f1_in(.(z0, z1)), f31_in(z0, z1), z0, z1)
U6(f1_out1, z0, z1, z2) → f28_out1
U6(z0, f31_out1, z1, z2) → f28_out2

Tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

S tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))

K tuples:

F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

Defined Rule Symbols:

f1_in, U1, U2, f41_in, U3, f35_in, U4, U5, f28_in, U6

Defined Pair Symbols:

F35_IN, F1_IN, F41_IN, U4', F28_IN

Compound Symbols:

c10, c2, c3, c8, c11, c13

(21) 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.

F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))

We considered the (Usable) Rules:

f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1

And the Tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(0) = 0   
POL(F1_IN(x₁)) = x₁   
POL(F28_IN(x₁, x₂)) = [1] + x₂   
POL(F35_IN(x₁, x₂, x₃)) = [1] + x₃   
POL(F41_IN(x₁, x₂)) = 0   
POL(U3(x₁, x₂, x₃)) = 0   
POL(U4'(x₁, x₂, x₃, x₄)) = [1] + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(f41_in(x₁, x₂)) = 0   
POL(f41_out1) = 0   
POL(s(x₁)) = 0   

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(z0, z1))) → U1(f28_in(z0, z1), .(0, .(z0, z1)))
f1_in(.(s(z0), .(z1, z2))) → U2(f35_in(z0, z1, z2), .(s(z0), .(z1, z2)))
U1(f28_out1, .(0, .(z0, z1))) → f1_out1
U1(f28_out2, .(0, .(z0, z1))) → f1_out1
U2(f35_out1, .(s(z0), .(z1, z2))) → f1_out1
f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1
f35_in(z0, z1, z2) → U4(f41_in(z0, z1), z0, z1, z2)
U4(f41_out1, z0, z1, z2) → U5(f1_in(.(z1, z2)), z0, z1, z2)
U5(f1_out1, z0, z1, z2) → f35_out1
f28_in(z0, z1) → U6(f1_in(.(z0, z1)), f31_in(z0, z1), z0, z1)
U6(f1_out1, z0, z1, z2) → f28_out1
U6(z0, f31_out1, z1, z2) → f28_out2

Tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

S tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))

K tuples:

F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))

Defined Rule Symbols:

f1_in, U1, U2, f41_in, U3, f35_in, U4, U5, f28_in, U6

Defined Pair Symbols:

F35_IN, F1_IN, F41_IN, U4', F28_IN

Compound Symbols:

c10, c2, c3, c8, c11, c13

(23) CdtKnowledgeProof (EQUIVALENT transformation)

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

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(z0, z1))) → U1(f28_in(z0, z1), .(0, .(z0, z1)))
f1_in(.(s(z0), .(z1, z2))) → U2(f35_in(z0, z1, z2), .(s(z0), .(z1, z2)))
U1(f28_out1, .(0, .(z0, z1))) → f1_out1
U1(f28_out2, .(0, .(z0, z1))) → f1_out1
U2(f35_out1, .(s(z0), .(z1, z2))) → f1_out1
f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1
f35_in(z0, z1, z2) → U4(f41_in(z0, z1), z0, z1, z2)
U4(f41_out1, z0, z1, z2) → U5(f1_in(.(z1, z2)), z0, z1, z2)
U5(f1_out1, z0, z1, z2) → f35_out1
f28_in(z0, z1) → U6(f1_in(.(z0, z1)), f31_in(z0, z1), z0, z1)
U6(f1_out1, z0, z1, z2) → f28_out1
U6(z0, f31_out1, z1, z2) → f28_out2

Tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

S tuples:

F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))

K tuples:

F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))

Defined Rule Symbols:

f1_in, U1, U2, f41_in, U3, f35_in, U4, U5, f28_in, U6

Defined Pair Symbols:

F35_IN, F1_IN, F41_IN, U4', F28_IN

Compound Symbols:

c10, c2, c3, c8, c11, c13

(25) 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.

F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))

We considered the (Usable) Rules:

f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1

And the Tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

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

POL(.(x₁, x₂)) = x₁ + x₂   
POL(0) = 0   
POL(F1_IN(x₁)) = x₁   
POL(F28_IN(x₁, x₂)) = x₁ + x₂   
POL(F35_IN(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(F41_IN(x₁, x₂)) = x₁   
POL(U3(x₁, x₂, x₃)) = 0   
POL(U4'(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(c10(x₁, x₂)) = x₁ + x₂   
POL(c11(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c8(x₁)) = x₁   
POL(f41_in(x₁, x₂)) = 0   
POL(f41_out1) = 0   
POL(s(x₁)) = [2] + x₁   

(26) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(z0, z1))) → U1(f28_in(z0, z1), .(0, .(z0, z1)))
f1_in(.(s(z0), .(z1, z2))) → U2(f35_in(z0, z1, z2), .(s(z0), .(z1, z2)))
U1(f28_out1, .(0, .(z0, z1))) → f1_out1
U1(f28_out2, .(0, .(z0, z1))) → f1_out1
U2(f35_out1, .(s(z0), .(z1, z2))) → f1_out1
f41_in(0, s(z0)) → f41_out1
f41_in(s(z0), s(z1)) → U3(f41_in(z0, z1), s(z0), s(z1))
U3(f41_out1, s(z0), s(z1)) → f41_out1
f35_in(z0, z1, z2) → U4(f41_in(z0, z1), z0, z1, z2)
U4(f41_out1, z0, z1, z2) → U5(f1_in(.(z1, z2)), z0, z1, z2)
U5(f1_out1, z0, z1, z2) → f35_out1
f28_in(z0, z1) → U6(f1_in(.(z0, z1)), f31_in(z0, z1), z0, z1)
U6(f1_out1, z0, z1, z2) → f28_out1
U6(z0, f31_out1, z1, z2) → f28_out2

Tuples:

F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))

S tuples:none
K tuples:

F28_IN(z0, z1) → c13(F1_IN(.(z0, z1)))
F1_IN(.(0, .(z0, z1))) → c2(F28_IN(z0, z1))
F1_IN(.(s(z0), .(z1, z2))) → c3(F35_IN(z0, z1, z2))
F35_IN(z0, z1, z2) → c10(U4'(f41_in(z0, z1), z0, z1, z2), F41_IN(z0, z1))
U4'(f41_out1, z0, z1, z2) → c11(F1_IN(.(z1, z2)))
F41_IN(s(z0), s(z1)) → c8(F41_IN(z0, z1))

Defined Rule Symbols:

f1_in, U1, U2, f41_in, U3, f35_in, U4, U5, f28_in, U6

Defined Pair Symbols:

F35_IN, F1_IN, F41_IN, U4', F28_IN

Compound Symbols:

c10, c2, c3, c8, c11, c13

(27) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(28) BOUNDS(O(1), O(1))