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(.(X, [])).
ordered(.(X, .(Y, Xs))) :- ','(le(X, Y), ordered(.(Y, Xs))).
le(s(X), s(Y)) :- le(X, Y).
le(0, s(0)).
le(0, 0).

Query: ordered(g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

ordered([]).
ordered(.(X, [])).
le(0, s(0)).
le(0, 0).
ordered(.(X, .(Y, Xs))) :- ','(le(X, Y), ordered(.(Y, Xs))).
le(s(X), s(Y)) :- le(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: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nordered(T1) !=? ordered([])", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: ordered([]), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: true | ordered(.(T4, [])), SCOPE: 1, CLAUSE: 4\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ordered(T1), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nX7 is free\nordered(T1) !=? ordered([])\nordered(T1) !=? ordered(.(X7, []))", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ordered(.(T4, [])), SCOPE: 1, CLAUSE: 4\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(le(T12, T13), ordered(.(T13, T14)))\n\nT12 is ground\nT13 is ground\nT14 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: le(T12, 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];
37 [label="37: le(T12, T13), SCOPE: 2, CLAUSE: 2 | le(T12, T13), SCOPE: 2, CLAUSE: 3 | le(T12, T13), SCOPE: 2, CLAUSE: 5\n\nT12 is ground\nT13 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: true | le(0, s(0)), SCOPE: 2, CLAUSE: 3 | le(0, s(0)), SCOPE: 2, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: le(T12, T13), SCOPE: 2, CLAUSE: 3 | le(T12, T13), SCOPE: 2, CLAUSE: 5\n\nT12 is ground\nT13 is ground\nle(T12, T13) !=? le(0, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: le(0, s(0)), SCOPE: 2, CLAUSE: 3 | le(0, s(0)), SCOPE: 2, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: le(0, s(0)), SCOPE: 2, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: true | le(0, 0), SCOPE: 2, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="48: le(T12, T13), SCOPE: 2, CLAUSE: 5\n\nT12 is ground\nT13 is ground\nle(T12, T13) !=? le(0, s(0))\nle(T12, T13) !=? le(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
49 [label="49: le(0, 0), SCOPE: 2, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
50 [label="50: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
53 [label="53: le(T19, T20)\n\nT19 is ground\nT20 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
54 [label="54: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 55 [arrowhead = none ];
55 -> 3;

56 [label="EVAL with clause\nordered([]).\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 56 [arrowhead = none ];
56 -> 6;

57 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 57 [arrowhead = none ];
57 -> 8;

58 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 58 [arrowhead = none ];
58 -> 11;

59 [label="EVAL with clause\nordered(.(X7, [])).\nand substitutionX7 -> T4,\nT1 -> .(T4, [])", fontsize=14, style = filled, fillcolor = lightblue];
8 -> 59 [arrowhead = none ];
59 -> 16;

60 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 60 [arrowhead = none ];
60 -> 18;

61 [label="BACKTRACK\nfor clause: ordered(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 61 [arrowhead = none ];
61 -> 12;

62 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(le(X, Y), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
12 -> 62 [arrowhead = none ];
62 -> 13;

63 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
16 -> 63 [arrowhead = none ];
63 -> 20;

64 [label="EVAL with clause\nordered(.(X17, .(X18, X19))) :- ','(le(X17, X18), ordered(.(X18, X19))).\nand substitutionX17 -> T12,\nX18 -> T13,\nX19 -> T14,\nT1 -> .(T12, .(T13, T14))", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 64 [arrowhead = none ];
64 -> 24;

65 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 65 [arrowhead = none ];
65 -> 26;

66 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(le(X, Y), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 66 [arrowhead = none ];
66 -> 22;

67 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
24 -> 67 [arrowhead = none ];
67 -> 32;

68 [label="SPLIT 2\nnew knowledge:\nT12 is ground\nT13 is ground", fontsize=14, style = filled, fillcolor = violet];
24 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 69 [arrowhead = none ];
69 -> 37;

70 [label="INSTANCE with matching:\nT1 -> .(T13, T14)", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 70 [arrowhead = none , style=dashed];
70 -> 2[style=dashed];

71 [label="EVAL with clause\nle(0, s(0)).\nand substitutionT12 -> 0,\nT13 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
37 -> 71 [arrowhead = none ];
71 -> 40;

72 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 72 [arrowhead = none ];
72 -> 42;

73 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
40 -> 73 [arrowhead = none ];
73 -> 43;

74 [label="EVAL with clause\nle(0, 0).\nand substitutionT12 -> 0,\nT13 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
42 -> 74 [arrowhead = none ];
74 -> 47;

75 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
42 -> 75 [arrowhead = none ];
75 -> 48;

76 [label="BACKTRACK\nfor clause: le(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
43 -> 76 [arrowhead = none ];
76 -> 44;

77 [label="BACKTRACK\nfor clause: le(s(X), s(Y)) :- le(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
44 -> 77 [arrowhead = none ];
77 -> 45;

78 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
47 -> 78 [arrowhead = none ];
78 -> 49;

79 [label="EVAL with clause\nle(s(X28), s(X29)) :- le(X28, X29).\nand substitutionX28 -> T19,\nT12 -> s(T19),\nX29 -> T20,\nT13 -> s(T20)", fontsize=14, style = filled, fillcolor = lightblue];
48 -> 79 [arrowhead = none ];
79 -> 53;

80 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
48 -> 80 [arrowhead = none ];
80 -> 54;

81 [label="BACKTRACK\nfor clause: le(s(X), s(Y)) :- le(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
49 -> 81 [arrowhead = none ];
81 -> 50;

82 [label="INSTANCE with matching:\nT12 -> T19\nT13 -> T20", fontsize=14, style = filled, fillcolor = lightgrey];
53 -> 82 [arrowhead = none , style=dashed];
82 -> 32[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f24_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f24_out1, .(z0, .(z1, z2))) → f2_out1
f32_in(0, s(0)) → f32_out1
f32_in(0, 0) → f32_out1
f32_in(s(z0), s(z1)) → U2(f32_in(z0, z1), s(z0), s(z1))
U2(f32_out1, s(z0), s(z1)) → f32_out1
f24_in(z0, z1, z2) → U3(f32_in(z0, z1), z0, z1, z2)
U3(f32_out1, z0, z1, z2) → U4(f2_in(.(z1, z2)), z0, z1, z2)
U4(f2_out1, z0, z1, z2) → f24_out1

Tuples:

F2_IN(.(z0, .(z1, z2))) → c2(U1'(f24_in(z0, z1, z2), .(z0, .(z1, z2))), F24_IN(z0, z1, z2))
F32_IN(s(z0), s(z1)) → c6(U2'(f32_in(z0, z1), s(z0), s(z1)), F32_IN(z0, z1))
F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
U3'(f32_out1, z0, z1, z2) → c9(U4'(f2_in(.(z1, z2)), z0, z1, z2), F2_IN(.(z1, z2)))

S tuples:

F2_IN(.(z0, .(z1, z2))) → c2(U1'(f24_in(z0, z1, z2), .(z0, .(z1, z2))), F24_IN(z0, z1, z2))
F32_IN(s(z0), s(z1)) → c6(U2'(f32_in(z0, z1), s(z0), s(z1)), F32_IN(z0, z1))
F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
U3'(f32_out1, z0, z1, z2) → c9(U4'(f2_in(.(z1, z2)), z0, z1, z2), F2_IN(.(z1, z2)))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f32_in, U2, f24_in, U3, U4

Defined Pair Symbols:

F2_IN, F32_IN, F24_IN, U3'

Compound Symbols:

c2, c6, c8, c9

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f24_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f24_out1, .(z0, .(z1, z2))) → f2_out1
f32_in(0, s(0)) → f32_out1
f32_in(0, 0) → f32_out1
f32_in(s(z0), s(z1)) → U2(f32_in(z0, z1), s(z0), s(z1))
U2(f32_out1, s(z0), s(z1)) → f32_out1
f24_in(z0, z1, z2) → U3(f32_in(z0, z1), z0, z1, z2)
U3(f32_out1, z0, z1, z2) → U4(f2_in(.(z1, z2)), z0, z1, z2)
U4(f2_out1, z0, z1, z2) → f24_out1

Tuples:

F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
F32_IN(s(z0), s(z1)) → c6(F32_IN(z0, z1))
U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))

S tuples:

F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
F32_IN(s(z0), s(z1)) → c6(F32_IN(z0, z1))
U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f32_in, U2, f24_in, U3, U4

Defined Pair Symbols:

F24_IN, F2_IN, F32_IN, U3'

Compound Symbols:

c8, c2, c6, c9

(7) 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), s(z1)) → c6(F32_IN(z0, z1))

We considered the (Usable) Rules:

f32_in(0, s(0)) → f32_out1
f32_in(0, 0) → f32_out1
f32_in(s(z0), s(z1)) → U2(f32_in(z0, z1), s(z0), s(z1))
U2(f32_out1, s(z0), s(z1)) → f32_out1

And the Tuples:

F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
F32_IN(s(z0), s(z1)) → c6(F32_IN(z0, z1))
U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))

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

POL(.(x₁, x₂)) = x₁ + x₂   
POL(0) = 0   
POL(F24_IN(x₁, x₂, x₃)) = x₁ + x₂ + x₃   
POL(F2_IN(x₁)) = x₁   
POL(F32_IN(x₁, x₂)) = x₁   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3'(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f32_in(x₁, x₂)) = 0   
POL(f32_out1) = 0   
POL(s(x₁)) = [1] + x₁   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f24_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f24_out1, .(z0, .(z1, z2))) → f2_out1
f32_in(0, s(0)) → f32_out1
f32_in(0, 0) → f32_out1
f32_in(s(z0), s(z1)) → U2(f32_in(z0, z1), s(z0), s(z1))
U2(f32_out1, s(z0), s(z1)) → f32_out1
f24_in(z0, z1, z2) → U3(f32_in(z0, z1), z0, z1, z2)
U3(f32_out1, z0, z1, z2) → U4(f2_in(.(z1, z2)), z0, z1, z2)
U4(f2_out1, z0, z1, z2) → f24_out1

Tuples:

F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
F32_IN(s(z0), s(z1)) → c6(F32_IN(z0, z1))
U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))

S tuples:

F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))

K tuples:

F32_IN(s(z0), s(z1)) → c6(F32_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f32_in, U2, f24_in, U3, U4

Defined Pair Symbols:

F24_IN, F2_IN, F32_IN, U3'

Compound Symbols:

c8, c2, c6, c9

(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.

F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))

We considered the (Usable) Rules:

f32_in(0, s(0)) → f32_out1
f32_in(0, 0) → f32_out1
f32_in(s(z0), s(z1)) → U2(f32_in(z0, z1), s(z0), s(z1))
U2(f32_out1, s(z0), s(z1)) → f32_out1

And the Tuples:

F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
F32_IN(s(z0), s(z1)) → c6(F32_IN(z0, z1))
U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(0) = 0   
POL(F24_IN(x₁, x₂, x₃)) = [2] + x₃   
POL(F2_IN(x₁)) = x₁   
POL(F32_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3'(x₁, x₂, x₃, x₄)) = [1] + x₄   
POL(c2(x₁)) = x₁   
POL(c6(x₁)) = x₁   
POL(c8(x₁, x₂)) = x₁ + x₂   
POL(c9(x₁)) = x₁   
POL(f32_in(x₁, x₂)) = 0   
POL(f32_out1) = 0   
POL(s(x₁)) = 0   

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in([]) → f2_out1
f2_in(.(z0, [])) → f2_out1
f2_in(.(z0, .(z1, z2))) → U1(f24_in(z0, z1, z2), .(z0, .(z1, z2)))
U1(f24_out1, .(z0, .(z1, z2))) → f2_out1
f32_in(0, s(0)) → f32_out1
f32_in(0, 0) → f32_out1
f32_in(s(z0), s(z1)) → U2(f32_in(z0, z1), s(z0), s(z1))
U2(f32_out1, s(z0), s(z1)) → f32_out1
f24_in(z0, z1, z2) → U3(f32_in(z0, z1), z0, z1, z2)
U3(f32_out1, z0, z1, z2) → U4(f2_in(.(z1, z2)), z0, z1, z2)
U4(f2_out1, z0, z1, z2) → f24_out1

Tuples:

F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))
F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
F32_IN(s(z0), s(z1)) → c6(F32_IN(z0, z1))
U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))

S tuples:

F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))

K tuples:

F32_IN(s(z0), s(z1)) → c6(F32_IN(z0, z1))
F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))

Defined Rule Symbols:

f2_in, U1, f32_in, U2, f24_in, U3, U4

Defined Pair Symbols:

F24_IN, F2_IN, F32_IN, U3'

Compound Symbols:

c8, c2, c6, c9

(11) CdtKnowledgeProof (EQUIVALENT transformation)

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

U3'(f32_out1, z0, z1, z2) → c9(F2_IN(.(z1, z2)))
F2_IN(.(z0, .(z1, z2))) → c2(F24_IN(z0, z1, z2))
F24_IN(z0, z1, z2) → c8(U3'(f32_in(z0, z1), z0, z1, z2), F32_IN(z0, z1))

Now S is empty

(12) BOUNDS(O(1), O(1))

(13) 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: 4\n\nT1 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | ordered([]), SCOPE: 1, CLAUSE: 1 | ordered([]), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: ordered(T1), SCOPE: 1, CLAUSE: 1 | ordered(T1), SCOPE: 1, CLAUSE: 4\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: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: ordered([]), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: true | ordered(.(T3, [])), SCOPE: 1, CLAUSE: 4\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ordered(T1), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nX6 is free\nordered(T1) !=? ordered([])\nordered(T1) !=? ordered(.(X6, []))", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ordered(.(T3, [])), SCOPE: 1, CLAUSE: 4\n\nT3 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(le(T8, T9), ordered(.(T9, T10)))\n\nT8 is ground\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: ','(le(T8, T9), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 2 | ','(le(T8, T9), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 3 | ','(le(T8, T9), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 5\n\nT8 is ground\nT9 is ground\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: ordered(.(s(0), T10)) | ','(le(0, s(0)), ordered(.(s(0), T10))), SCOPE: 2, CLAUSE: 3 | ','(le(0, s(0)), ordered(.(s(0), T10))), SCOPE: 2, CLAUSE: 5\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: ','(le(T8, T9), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 3 | ','(le(T8, T9), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 5\n\nT8 is ground\nT9 is ground\nT10 is ground\nle(T8, T9) !=? le(0, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ordered(.(s(0), T10))\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: ','(le(0, s(0)), ordered(.(s(0), T10))), SCOPE: 2, CLAUSE: 3 | ','(le(0, s(0)), ordered(.(s(0), T10))), SCOPE: 2, CLAUSE: 5\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: ','(le(0, s(0)), ordered(.(s(0), T10))), SCOPE: 2, CLAUSE: 5\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: ordered(.(0, T10)) | ','(le(0, 0), ordered(.(0, T10))), SCOPE: 2, CLAUSE: 5\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: ','(le(T8, T9), ordered(.(T9, T10))), SCOPE: 2, CLAUSE: 5\n\nT8 is ground\nT9 is ground\nT10 is ground\nle(T8, T9) !=? le(0, s(0))\nle(T8, T9) !=? le(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: ordered(.(0, T10))\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: ','(le(0, 0), ordered(.(0, T10))), SCOPE: 2, CLAUSE: 5\n\nT10 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
51 [label="51: ','(le(T15, T16), ordered(.(s(T16), T10)))\n\nT10 is ground\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
52 [label="52: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
55 [label="55: le(T15, T16)\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
56 [label="56: ordered(.(s(T16), T10))\n\nT10 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
57 [label="57: le(T15, T16), SCOPE: 3, CLAUSE: 2 | le(T15, T16), SCOPE: 3, CLAUSE: 3 | le(T15, T16), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
58 [label="58: true | le(0, s(0)), SCOPE: 3, CLAUSE: 3 | le(0, s(0)), SCOPE: 3, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
59 [label="59: le(T15, T16), SCOPE: 3, CLAUSE: 3 | le(T15, T16), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nle(T15, T16) !=? le(0, s(0))", fontsize=16, style=filled, fillcolor=lightyellow];
60 [label="60: le(0, s(0)), SCOPE: 3, CLAUSE: 3 | le(0, s(0)), SCOPE: 3, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
61 [label="61: le(0, s(0)), SCOPE: 3, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
62 [label="62: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
63 [label="63: true | le(0, 0), SCOPE: 3, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
64 [label="64: le(T15, T16), SCOPE: 3, CLAUSE: 5\n\nT15 is ground\nT16 is ground\nle(T15, T16) !=? le(0, s(0))\nle(T15, T16) !=? le(0, 0)", fontsize=16, style=filled, fillcolor=lightyellow];
65 [label="65: le(0, 0), SCOPE: 3, CLAUSE: 5\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
66 [label="66: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
67 [label="67: le(T21, T22)\n\nT21 is ground\nT22 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
68 [label="68: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
69 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 69 [arrowhead = none ];
69 -> 4;

70 [label="EVAL with clause\nordered([]).\nand substitutionT1 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 70 [arrowhead = none ];
70 -> 5;

71 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 71 [arrowhead = none ];
71 -> 7;

72 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 72 [arrowhead = none ];
72 -> 9;

73 [label="EVAL with clause\nordered(.(X6, [])).\nand substitutionX6 -> T3,\nT1 -> .(T3, [])", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 73 [arrowhead = none ];
73 -> 15;

74 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 74 [arrowhead = none ];
74 -> 17;

75 [label="BACKTRACK\nfor clause: ordered(.(X, []))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 75 [arrowhead = none ];
75 -> 10;

76 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(le(X, Y), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 76 [arrowhead = none ];
76 -> 14;

77 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
15 -> 77 [arrowhead = none ];
77 -> 19;

78 [label="EVAL with clause\nordered(.(X13, .(X14, X15))) :- ','(le(X13, X14), ordered(.(X14, X15))).\nand substitutionX13 -> T8,\nX14 -> T9,\nX15 -> T10,\nT1 -> .(T8, .(T9, T10))", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 78 [arrowhead = none ];
78 -> 23;

79 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 79 [arrowhead = none ];
79 -> 25;

80 [label="BACKTRACK\nfor clause: ordered(.(X, .(Y, Xs))) :- ','(le(X, Y), ordered(.(Y, Xs)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 80 [arrowhead = none ];
80 -> 21;

81 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
23 -> 81 [arrowhead = none ];
81 -> 27;

82 [label="EVAL with clause\nle(0, s(0)).\nand substitutionT8 -> 0,\nT9 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 82 [arrowhead = none ];
82 -> 28;

83 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 83 [arrowhead = none ];
83 -> 29;

84 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 84 [arrowhead = none ];
84 -> 30;

85 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
28 -> 85 [arrowhead = none ];
85 -> 31;

86 [label="EVAL with clause\nle(0, 0).\nand substitutionT8 -> 0,\nT9 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
29 -> 86 [arrowhead = none ];
86 -> 36;

87 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
29 -> 87 [arrowhead = none ];
87 -> 38;

88 [label="INSTANCE with matching:\nT1 -> .(s(0), T10)", fontsize=14, style = filled, fillcolor = lightgrey];
30 -> 88 [arrowhead = none , style=dashed];
88 -> 1[style=dashed];

89 [label="BACKTRACK\nfor clause: le(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 89 [arrowhead = none ];
89 -> 34;

90 [label="BACKTRACK\nfor clause: le(s(X), s(Y)) :- le(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
34 -> 90 [arrowhead = none ];
90 -> 35;

91 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 91 [arrowhead = none ];
91 -> 39;

92 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
36 -> 92 [arrowhead = none ];
92 -> 41;

93 [label="EVAL with clause\nle(s(X24), s(X25)) :- le(X24, X25).\nand substitutionX24 -> T15,\nT8 -> s(T15),\nX25 -> T16,\nT9 -> s(T16)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 93 [arrowhead = none ];
93 -> 51;

94 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 94 [arrowhead = none ];
94 -> 52;

95 [label="INSTANCE with matching:\nT1 -> .(0, T10)", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 95 [arrowhead = none , style=dashed];
95 -> 1[style=dashed];

96 [label="BACKTRACK\nfor clause: le(s(X), s(Y)) :- le(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
41 -> 96 [arrowhead = none ];
96 -> 46;

97 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
51 -> 97 [arrowhead = none ];
97 -> 55;

98 [label="SPLIT 2\nnew knowledge:\nT15 is ground\nT16 is ground", fontsize=14, style = filled, fillcolor = violet];
51 -> 98 [arrowhead = none ];
98 -> 56;

99 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
55 -> 99 [arrowhead = none ];
99 -> 57;

100 [label="INSTANCE with matching:\nT1 -> .(s(T16), T10)", fontsize=14, style = filled, fillcolor = lightgrey];
56 -> 100 [arrowhead = none , style=dashed];
100 -> 1[style=dashed];

101 [label="EVAL with clause\nle(0, s(0)).\nand substitutionT15 -> 0,\nT16 -> s(0)", fontsize=14, style = filled, fillcolor = lightblue];
57 -> 101 [arrowhead = none ];
101 -> 58;

102 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
57 -> 102 [arrowhead = none ];
102 -> 59;

103 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
58 -> 103 [arrowhead = none ];
103 -> 60;

104 [label="EVAL with clause\nle(0, 0).\nand substitutionT15 -> 0,\nT16 -> 0", fontsize=14, style = filled, fillcolor = lightblue];
59 -> 104 [arrowhead = none ];
104 -> 63;

105 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
59 -> 105 [arrowhead = none ];
105 -> 64;

106 [label="BACKTRACK\nfor clause: le(0, 0)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
60 -> 106 [arrowhead = none ];
106 -> 61;

107 [label="BACKTRACK\nfor clause: le(s(X), s(Y)) :- le(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
61 -> 107 [arrowhead = none ];
107 -> 62;

108 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
63 -> 108 [arrowhead = none ];
108 -> 65;

109 [label="EVAL with clause\nle(s(X34), s(X35)) :- le(X34, X35).\nand substitutionX34 -> T21,\nT15 -> s(T21),\nX35 -> T22,\nT16 -> s(T22)", fontsize=14, style = filled, fillcolor = lightblue];
64 -> 109 [arrowhead = none ];
109 -> 67;

110 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
64 -> 110 [arrowhead = none ];
110 -> 68;

111 [label="BACKTRACK\nfor clause: le(s(X), s(Y)) :- le(X, Y)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
65 -> 111 [arrowhead = none ];
111 -> 66;

112 [label="INSTANCE with matching:\nT15 -> T21\nT16 -> T22", fontsize=14, style = filled, fillcolor = lightgrey];
67 -> 112 [arrowhead = none , style=dashed];
112 -> 55[style=dashed];

}

(14) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(s(0), z0))) → U1(f28_in(z0), .(0, .(s(0), z0)))
f1_in(.(0, .(0, z0))) → U2(f36_in(z0), .(0, .(0, z0)))
f1_in(.(s(z0), .(s(z1), z2))) → U3(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2)))
U1(f28_out1, .(0, .(s(0), z0))) → f1_out1
U1(f28_out2, .(0, .(s(0), z0))) → f1_out1
U1(f28_out3, .(0, .(s(0), z0))) → f1_out1
U2(f36_out1, .(0, .(0, z0))) → f1_out1
U2(f36_out2, .(0, .(0, z0))) → f1_out1
U3(f51_out1, .(s(z0), .(s(z1), z2))) → f1_out1
f55_in(0, s(0)) → f55_out1
f55_in(0, 0) → f55_out1
f55_in(s(z0), s(z1)) → U4(f55_in(z0, z1), s(z0), s(z1))
U4(f55_out1, s(z0), s(z1)) → f55_out1
f51_in(z0, z1, z2) → U5(f55_in(z0, z1), z0, z1, z2)
U5(f55_out1, z0, z1, z2) → U6(f1_in(.(s(z1), z2)), z0, z1, z2)
U6(f1_out1, z0, z1, z2) → f51_out1
f28_in(z0) → U7(f1_in(.(s(0), z0)), f31_in(z0), z0)
U7(f1_out1, z0, z1) → f28_out1
U7(z0, f31_out1, z1) → f28_out2
U7(z0, f31_out2, z1) → f28_out3
f36_in(z0) → U8(f1_in(.(0, z0)), f41_in(z0), z0)
U8(f1_out1, z0, z1) → f36_out1
U8(z0, f41_out1, z1) → f36_out2

Tuples:

F1_IN(.(0, .(s(0), z0))) → c2(U1'(f28_in(z0), .(0, .(s(0), z0))), F28_IN(z0))
F1_IN(.(0, .(0, z0))) → c3(U2'(f36_in(z0), .(0, .(0, z0))), F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(U3'(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2))), F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(U4'(f55_in(z0, z1), s(z0), s(z1)), F55_IN(z0, z1))
F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(U6'(f1_in(.(s(z1), z2)), z0, z1, z2), F1_IN(.(s(z1), z2)))
F28_IN(z0) → c18(U7'(f1_in(.(s(0), z0)), f31_in(z0), z0), F1_IN(.(s(0), z0)))
F36_IN(z0) → c22(U8'(f1_in(.(0, z0)), f41_in(z0), z0), F1_IN(.(0, z0)))

S tuples:

F1_IN(.(0, .(s(0), z0))) → c2(U1'(f28_in(z0), .(0, .(s(0), z0))), F28_IN(z0))
F1_IN(.(0, .(0, z0))) → c3(U2'(f36_in(z0), .(0, .(0, z0))), F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(U3'(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2))), F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(U4'(f55_in(z0, z1), s(z0), s(z1)), F55_IN(z0, z1))
F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(U6'(f1_in(.(s(z1), z2)), z0, z1, z2), F1_IN(.(s(z1), z2)))
F28_IN(z0) → c18(U7'(f1_in(.(s(0), z0)), f31_in(z0), z0), F1_IN(.(s(0), z0)))
F36_IN(z0) → c22(U8'(f1_in(.(0, z0)), f41_in(z0), z0), F1_IN(.(0, z0)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, f55_in, U4, f51_in, U5, U6, f28_in, U7, f36_in, U8

Defined Pair Symbols:

F1_IN, F55_IN, F51_IN, U5', F28_IN, F36_IN

Compound Symbols:

c2, c3, c4, c13, c15, c16, c18, c22

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

Split RHS of tuples not part of any SCC

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(s(0), z0))) → U1(f28_in(z0), .(0, .(s(0), z0)))
f1_in(.(0, .(0, z0))) → U2(f36_in(z0), .(0, .(0, z0)))
f1_in(.(s(z0), .(s(z1), z2))) → U3(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2)))
U1(f28_out1, .(0, .(s(0), z0))) → f1_out1
U1(f28_out2, .(0, .(s(0), z0))) → f1_out1
U1(f28_out3, .(0, .(s(0), z0))) → f1_out1
U2(f36_out1, .(0, .(0, z0))) → f1_out1
U2(f36_out2, .(0, .(0, z0))) → f1_out1
U3(f51_out1, .(s(z0), .(s(z1), z2))) → f1_out1
f55_in(0, s(0)) → f55_out1
f55_in(0, 0) → f55_out1
f55_in(s(z0), s(z1)) → U4(f55_in(z0, z1), s(z0), s(z1))
U4(f55_out1, s(z0), s(z1)) → f55_out1
f51_in(z0, z1, z2) → U5(f55_in(z0, z1), z0, z1, z2)
U5(f55_out1, z0, z1, z2) → U6(f1_in(.(s(z1), z2)), z0, z1, z2)
U6(f1_out1, z0, z1, z2) → f51_out1
f28_in(z0) → U7(f1_in(.(s(0), z0)), f31_in(z0), z0)
U7(f1_out1, z0, z1) → f28_out1
U7(z0, f31_out1, z1) → f28_out2
U7(z0, f31_out2, z1) → f28_out3
f36_in(z0) → U8(f1_in(.(0, z0)), f41_in(z0), z0)
U8(f1_out1, z0, z1) → f36_out1
U8(z0, f41_out1, z1) → f36_out2

Tuples:

F1_IN(.(0, .(0, z0))) → c3(U2'(f36_in(z0), .(0, .(0, z0))), F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(U3'(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2))), F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(U4'(f55_in(z0, z1), s(z0), s(z1)), F55_IN(z0, z1))
F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(U6'(f1_in(.(s(z1), z2)), z0, z1, z2), F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(U8'(f1_in(.(0, z0)), f41_in(z0), z0), F1_IN(.(0, z0)))
F1_IN(.(0, .(s(0), z0))) → c(U1'(f28_in(z0), .(0, .(s(0), z0))))
F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F28_IN(z0) → c(U7'(f1_in(.(s(0), z0)), f31_in(z0), z0))
F28_IN(z0) → c(F1_IN(.(s(0), z0)))

S tuples:

F1_IN(.(0, .(0, z0))) → c3(U2'(f36_in(z0), .(0, .(0, z0))), F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(U3'(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2))), F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(U4'(f55_in(z0, z1), s(z0), s(z1)), F55_IN(z0, z1))
F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(U6'(f1_in(.(s(z1), z2)), z0, z1, z2), F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(U8'(f1_in(.(0, z0)), f41_in(z0), z0), F1_IN(.(0, z0)))
F1_IN(.(0, .(s(0), z0))) → c(U1'(f28_in(z0), .(0, .(s(0), z0))))
F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F28_IN(z0) → c(U7'(f1_in(.(s(0), z0)), f31_in(z0), z0))
F28_IN(z0) → c(F1_IN(.(s(0), z0)))

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, f55_in, U4, f51_in, U5, U6, f28_in, U7, f36_in, U8

Defined Pair Symbols:

F1_IN, F55_IN, F51_IN, U5', F36_IN, F28_IN

Compound Symbols:

c3, c4, c13, c15, c16, c22, c

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

Removed 7 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(s(0), z0))) → U1(f28_in(z0), .(0, .(s(0), z0)))
f1_in(.(0, .(0, z0))) → U2(f36_in(z0), .(0, .(0, z0)))
f1_in(.(s(z0), .(s(z1), z2))) → U3(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2)))
U1(f28_out1, .(0, .(s(0), z0))) → f1_out1
U1(f28_out2, .(0, .(s(0), z0))) → f1_out1
U1(f28_out3, .(0, .(s(0), z0))) → f1_out1
U2(f36_out1, .(0, .(0, z0))) → f1_out1
U2(f36_out2, .(0, .(0, z0))) → f1_out1
U3(f51_out1, .(s(z0), .(s(z1), z2))) → f1_out1
f55_in(0, s(0)) → f55_out1
f55_in(0, 0) → f55_out1
f55_in(s(z0), s(z1)) → U4(f55_in(z0, z1), s(z0), s(z1))
U4(f55_out1, s(z0), s(z1)) → f55_out1
f51_in(z0, z1, z2) → U5(f55_in(z0, z1), z0, z1, z2)
U5(f55_out1, z0, z1, z2) → U6(f1_in(.(s(z1), z2)), z0, z1, z2)
U6(f1_out1, z0, z1, z2) → f51_out1
f28_in(z0) → U7(f1_in(.(s(0), z0)), f31_in(z0), z0)
U7(f1_out1, z0, z1) → f28_out1
U7(z0, f31_out1, z1) → f28_out2
U7(z0, f31_out2, z1) → f28_out3
f36_in(z0) → U8(f1_in(.(0, z0)), f41_in(z0), z0)
U8(f1_out1, z0, z1) → f36_out1
U8(z0, f41_out1, z1) → f36_out2

Tuples:

F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F28_IN(z0) → c(F1_IN(.(s(0), z0)))
F1_IN(.(0, .(0, z0))) → c3(F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(F1_IN(.(0, z0)))
F1_IN(.(0, .(s(0), z0))) → c
F28_IN(z0) → c

S tuples:

F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F28_IN(z0) → c(F1_IN(.(s(0), z0)))
F1_IN(.(0, .(0, z0))) → c3(F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(F1_IN(.(0, z0)))
F1_IN(.(0, .(s(0), z0))) → c
F28_IN(z0) → c

K tuples:none
Defined Rule Symbols:

f1_in, U1, U2, U3, f55_in, U4, f51_in, U5, U6, f28_in, U7, f36_in, U8

Defined Pair Symbols:

F51_IN, F1_IN, F28_IN, F55_IN, U5', F36_IN

Compound Symbols:

c15, c, c3, c4, c13, c16, c22, c

(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.

F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F1_IN(.(0, .(s(0), z0))) → c

We considered the (Usable) Rules:

f55_in(0, s(0)) → f55_out1
f55_in(0, 0) → f55_out1
f55_in(s(z0), s(z1)) → U4(f55_in(z0, z1), s(z0), s(z1))
U4(f55_out1, s(z0), s(z1)) → f55_out1

And the Tuples:

F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F28_IN(z0) → c(F1_IN(.(s(0), z0)))
F1_IN(.(0, .(0, z0))) → c3(F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(F1_IN(.(0, z0)))
F1_IN(.(0, .(s(0), z0))) → c
F28_IN(z0) → c

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

POL(.(x₁, x₂)) = x₁   
POL(0) = [2]   
POL(F1_IN(x₁)) = x₁   
POL(F28_IN(x₁)) = 0   
POL(F36_IN(x₁)) = [2]   
POL(F51_IN(x₁, x₂, x₃)) = 0   
POL(F55_IN(x₁, x₂)) = 0   
POL(U4(x₁, x₂, x₃)) = 0   
POL(U5'(x₁, x₂, x₃, x₄)) = 0   
POL(c) = 0   
POL(c(x₁)) = x₁   
POL(c13(x₁)) = x₁   
POL(c15(x₁, x₂)) = x₁ + x₂   
POL(c16(x₁)) = x₁   
POL(c22(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(f55_in(x₁, x₂)) = 0   
POL(f55_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, .(s(0), z0))) → U1(f28_in(z0), .(0, .(s(0), z0)))
f1_in(.(0, .(0, z0))) → U2(f36_in(z0), .(0, .(0, z0)))
f1_in(.(s(z0), .(s(z1), z2))) → U3(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2)))
U1(f28_out1, .(0, .(s(0), z0))) → f1_out1
U1(f28_out2, .(0, .(s(0), z0))) → f1_out1
U1(f28_out3, .(0, .(s(0), z0))) → f1_out1
U2(f36_out1, .(0, .(0, z0))) → f1_out1
U2(f36_out2, .(0, .(0, z0))) → f1_out1
U3(f51_out1, .(s(z0), .(s(z1), z2))) → f1_out1
f55_in(0, s(0)) → f55_out1
f55_in(0, 0) → f55_out1
f55_in(s(z0), s(z1)) → U4(f55_in(z0, z1), s(z0), s(z1))
U4(f55_out1, s(z0), s(z1)) → f55_out1
f51_in(z0, z1, z2) → U5(f55_in(z0, z1), z0, z1, z2)
U5(f55_out1, z0, z1, z2) → U6(f1_in(.(s(z1), z2)), z0, z1, z2)
U6(f1_out1, z0, z1, z2) → f51_out1
f28_in(z0) → U7(f1_in(.(s(0), z0)), f31_in(z0), z0)
U7(f1_out1, z0, z1) → f28_out1
U7(z0, f31_out1, z1) → f28_out2
U7(z0, f31_out2, z1) → f28_out3
f36_in(z0) → U8(f1_in(.(0, z0)), f41_in(z0), z0)
U8(f1_out1, z0, z1) → f36_out1
U8(z0, f41_out1, z1) → f36_out2

Tuples:

F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F28_IN(z0) → c(F1_IN(.(s(0), z0)))
F1_IN(.(0, .(0, z0))) → c3(F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(F1_IN(.(0, z0)))
F1_IN(.(0, .(s(0), z0))) → c
F28_IN(z0) → c

S tuples:

F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
F28_IN(z0) → c(F1_IN(.(s(0), z0)))
F1_IN(.(0, .(0, z0))) → c3(F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(F1_IN(.(0, z0)))
F28_IN(z0) → c

K tuples:

F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F1_IN(.(0, .(s(0), z0))) → c

Defined Rule Symbols:

f1_in, U1, U2, U3, f55_in, U4, f51_in, U5, U6, f28_in, U7, f36_in, U8

Defined Pair Symbols:

F51_IN, F1_IN, F28_IN, F55_IN, U5', F36_IN

Compound Symbols:

c15, c, c3, c4, c13, c16, c22, c

(21) CdtKnowledgeProof (EQUIVALENT transformation)

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

F28_IN(z0) → c(F1_IN(.(s(0), z0)))
F28_IN(z0) → c

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in([]) → f1_out1
f1_in(.(z0, [])) → f1_out1
f1_in(.(0, .(s(0), z0))) → U1(f28_in(z0), .(0, .(s(0), z0)))
f1_in(.(0, .(0, z0))) → U2(f36_in(z0), .(0, .(0, z0)))
f1_in(.(s(z0), .(s(z1), z2))) → U3(f51_in(z0, z1, z2), .(s(z0), .(s(z1), z2)))
U1(f28_out1, .(0, .(s(0), z0))) → f1_out1
U1(f28_out2, .(0, .(s(0), z0))) → f1_out1
U1(f28_out3, .(0, .(s(0), z0))) → f1_out1
U2(f36_out1, .(0, .(0, z0))) → f1_out1
U2(f36_out2, .(0, .(0, z0))) → f1_out1
U3(f51_out1, .(s(z0), .(s(z1), z2))) → f1_out1
f55_in(0, s(0)) → f55_out1
f55_in(0, 0) → f55_out1
f55_in(s(z0), s(z1)) → U4(f55_in(z0, z1), s(z0), s(z1))
U4(f55_out1, s(z0), s(z1)) → f55_out1
f51_in(z0, z1, z2) → U5(f55_in(z0, z1), z0, z1, z2)
U5(f55_out1, z0, z1, z2) → U6(f1_in(.(s(z1), z2)), z0, z1, z2)
U6(f1_out1, z0, z1, z2) → f51_out1
f28_in(z0) → U7(f1_in(.(s(0), z0)), f31_in(z0), z0)
U7(f1_out1, z0, z1) → f28_out1
U7(z0, f31_out1, z1) → f28_out2
U7(z0, f31_out2, z1) → f28_out3
f36_in(z0) → U8(f1_in(.(0, z0)), f41_in(z0), z0)
U8(f1_out1, z0, z1) → f36_out1
U8(z0, f41_out1, z1) → f36_out2

Tuples:

F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F28_IN(z0) → c(F1_IN(.(s(0), z0)))
F1_IN(.(0, .(0, z0))) → c3(F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(F1_IN(.(0, z0)))
F1_IN(.(0, .(s(0), z0))) → c
F28_IN(z0) → c

S tuples:

F51_IN(z0, z1, z2) → c15(U5'(f55_in(z0, z1), z0, z1, z2), F55_IN(z0, z1))
F1_IN(.(0, .(0, z0))) → c3(F36_IN(z0))
F1_IN(.(s(z0), .(s(z1), z2))) → c4(F51_IN(z0, z1, z2))
F55_IN(s(z0), s(z1)) → c13(F55_IN(z0, z1))
U5'(f55_out1, z0, z1, z2) → c16(F1_IN(.(s(z1), z2)))
F36_IN(z0) → c22(F1_IN(.(0, z0)))

K tuples:

F1_IN(.(0, .(s(0), z0))) → c(F28_IN(z0))
F1_IN(.(0, .(s(0), z0))) → c
F28_IN(z0) → c(F1_IN(.(s(0), z0)))
F28_IN(z0) → c

Defined Rule Symbols:

f1_in, U1, U2, U3, f55_in, U4, f51_in, U5, U6, f28_in, U7, f36_in, U8

Defined Pair Symbols:

F51_IN, F1_IN, F28_IN, F55_IN, U5', F36_IN

Compound Symbols:

c15, c, c3, c4, c13, c16, c22, c