Runtime Complexity of the query pattern
samefringe(g,g)
w.r.t. the given Prolog program could be shown to be BOUNDS(O(1), O(n^2)).

(0) Obligation:

Clauses:

gopher(nil, nil).
gopher(cons(nil, Y), cons(nil, Y)).
gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X).
samefringe(nil, nil).
samefringe(cons(U, V), cons(X, Y)) :- ','(gopher(cons(U, V), cons(U1, V1)), ','(gopher(cons(X, Y), cons(X1, Y1)), samefringe(V1, Y1))).

Query: samefringe(g,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

gopher(nil, nil).
gopher(cons(nil, Y), cons(nil, Y)).
samefringe(nil, nil).
gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X).
samefringe(cons(U, V), cons(X, Y)) :- ','(gopher(cons(U, V), cons(U1, V1)), ','(gopher(cons(X, Y), cons(X1, Y1)), samefringe(V1, Y1))).

Query: samefringe(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];
2 [label="2: samefringe(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: samefringe(T1, T2), SCOPE: 1, CLAUSE: 2 | samefringe(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | samefringe(nil, nil), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: samefringe(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nsamefringe(T1, T2) !=? samefringe(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: samefringe(nil, nil), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(gopher(cons(T11, T12), cons(X22, X23)), ','(gopher(cons(T13, T14), cons(X24, X25)), samefringe(X23, X25)))\n\nT11 is ground\nT12 is ground\nT13 is ground\nT14 is ground\nX22 is free\nX23 is free\nX24 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: gopher(cons(T11, T12), cons(X22, X23))\n\nT11 is ground\nT12 is ground\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: ','(gopher(cons(T13, T14), cons(X24, X25)), samefringe(T16, X25))\n\nT13 is ground\nT14 is ground\nT16 is ground\nX24 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: gopher(cons(T11, T12), cons(X22, X23)), SCOPE: 2, CLAUSE: 0 | gopher(cons(T11, T12), cons(X22, X23)), SCOPE: 2, CLAUSE: 1 | gopher(cons(T11, T12), cons(X22, X23)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT12 is ground\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: gopher(cons(T11, T12), cons(X22, X23)), SCOPE: 2, CLAUSE: 1 | gopher(cons(T11, T12), cons(X22, X23)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT12 is ground\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: true | gopher(cons(nil, T19), cons(X22, X23)), SCOPE: 2, CLAUSE: 3\n\nT19 is ground\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: gopher(cons(T11, T12), cons(X22, X23)), SCOPE: 2, CLAUSE: 3\n\nT11 is ground\nT12 is ground\nX22 is free\nX23 is free\nX28 is free\ngopher(cons(T11, T12), cons(X22, X23)) !=? gopher(cons(nil, X28), cons(nil, X28))", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: gopher(cons(nil, T19), cons(X22, X23)), SCOPE: 2, CLAUSE: 3\n\nT19 is ground\nX22 is free\nX23 is free", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: gopher(cons(T26, cons(T27, T28)), cons(X49, X50))\n\nT26 is ground\nT27 is ground\nT28 is ground\nX49 is free\nX50 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="43: gopher(cons(T13, T14), cons(X24, X25))\n\nT13 is ground\nT14 is ground\nX24 is free\nX25 is free", fontsize=16, style=filled, fillcolor=lightyellow];
44 [label="44: samefringe(T16, T30)\n\nT16 is ground\nT30 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 45 [arrowhead = none ];
45 -> 3;

46 [label="EVAL with clause\nsamefringe(nil, nil).\nand substitutionT1 -> nil,\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 46 [arrowhead = none ];
46 -> 5;

47 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 47 [arrowhead = none ];
47 -> 6;

48 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 48 [arrowhead = none ];
48 -> 8;

49 [label="EVAL with clause\nsamefringe(cons(X18, X19), cons(X20, X21)) :- ','(gopher(cons(X18, X19), cons(X22, X23)), ','(gopher(cons(X20, X21), cons(X24, X25)), samefringe(X23, X25))).\nand substitutionX18 -> T11,\nX19 -> T12,\nT1 -> cons(T11, T12),\nX20 -> T13,\nX21 -> T14,\nT2 -> cons(T13, T14)", fontsize=14, style = filled, fillcolor = lightblue];
6 -> 49 [arrowhead = none ];
49 -> 13;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 50 [arrowhead = none ];
50 -> 14;

51 [label="BACKTRACK\nfor clause: samefringe(cons(U, V), cons(X, Y)) :- ','(gopher(cons(U, V), cons(U1, V1)), ','(gopher(cons(X, Y), cons(X1, Y1)), samefringe(V1, Y1)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 51 [arrowhead = none ];
51 -> 11;

52 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 52 [arrowhead = none ];
52 -> 21;

53 [label="SPLIT 2\nnew knowledge:\nT11 is ground\nT12 is ground\nT15 is ground\nT16 is ground\nreplacements:X22 -> T15,\nX23 -> T16", fontsize=14, style = filled, fillcolor = violet];
13 -> 53 [arrowhead = none ];
53 -> 22;

54 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
21 -> 54 [arrowhead = none ];
54 -> 25;

55 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
22 -> 55 [arrowhead = none ];
55 -> 43;

56 [label="SPLIT 2\nnew knowledge:\nT13 is ground\nT14 is ground\nT29 is ground\nT30 is ground\nreplacements:X24 -> T29,\nX25 -> T30", fontsize=14, style = filled, fillcolor = violet];
22 -> 56 [arrowhead = none ];
56 -> 44;

57 [label="BACKTRACK\nfor clause: gopher(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 57 [arrowhead = none ];
57 -> 26;

58 [label="EVAL with clause\ngopher(cons(nil, X28), cons(nil, X28)).\nand substitutionT11 -> nil,\nT12 -> T19,\nX28 -> T19,\nX22 -> nil,\nX23 -> T19", fontsize=14, style = filled, fillcolor = lightblue];
26 -> 58 [arrowhead = none ];
58 -> 27;

59 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 59 [arrowhead = none ];
59 -> 30;

60 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 60 [arrowhead = none ];
60 -> 31;

61 [label="EVAL with clause\ngopher(cons(cons(X45, X46), X47), X48) :- gopher(cons(X45, cons(X46, X47)), X48).\nand substitutionX45 -> T26,\nX46 -> T27,\nT11 -> cons(T26, T27),\nT12 -> T28,\nX47 -> T28,\nX22 -> X49,\nX23 -> X50,\nX48 -> cons(X49, X50)", fontsize=14, style = filled, fillcolor = lightblue];
30 -> 61 [arrowhead = none ];
61 -> 39;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
30 -> 62 [arrowhead = none ];
62 -> 40;

63 [label="BACKTRACK\nfor clause: gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
31 -> 63 [arrowhead = none ];
63 -> 34;

64 [label="INSTANCE with matching:\nT11 -> T26\nT12 -> cons(T27, T28)\nX22 -> X49\nX23 -> X50", fontsize=14, style = filled, fillcolor = lightgrey];
39 -> 64 [arrowhead = none , style=dashed];
64 -> 21[style=dashed];

65 [label="INSTANCE with matching:\nT11 -> T13\nT12 -> T14\nX22 -> X24\nX23 -> X25", fontsize=14, style = filled, fillcolor = lightgrey];
43 -> 65 [arrowhead = none , style=dashed];
65 -> 21[style=dashed];

66 [label="INSTANCE with matching:\nT1 -> T16\nT2 -> T30", fontsize=14, style = filled, fillcolor = lightgrey];
44 -> 66 [arrowhead = none , style=dashed];
66 -> 2[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(nil, nil) → f2_out1
f2_in(cons(z0, z1), cons(z2, z3)) → U1(f13_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f13_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f2_out1
f21_in(nil, z0) → f21_out1(nil, z0)
f21_in(cons(z0, z1), z2) → U2(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U2(f21_out1(z0, z1), cons(z2, z3), z4) → f21_out1(z0, z1)
f13_in(z0, z1, z2, z3) → U3(f21_in(z0, z1), z0, z1, z2, z3)
U3(f21_out1(z0, z1), z2, z3, z4, z5) → U4(f22_in(z4, z5, z1), z2, z3, z4, z5, z0, z1)
U4(f22_out1(z0, z1), z2, z3, z4, z5, z6, z7) → f13_out1(z6, z7, z0, z1)
f22_in(z0, z1, z2) → U5(f21_in(z0, z1), z0, z1, z2)
U5(f21_out1(z0, z1), z2, z3, z4) → U6(f2_in(z4, z1), z2, z3, z4, z0, z1)
U6(f2_out1, z0, z1, z2, z3, z4) → f22_out1(z3, z4)

Tuples:

F2_IN(cons(z0, z1), cons(z2, z3)) → c1(U1'(f13_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3)), F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(U2'(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2), F21_IN(z0, cons(z1, z2)))
F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(U4'(f22_in(z4, z5, z1), z2, z3, z4, z5, z0, z1), F22_IN(z4, z5, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(U6'(f2_in(z4, z1), z2, z3, z4, z0, z1), F2_IN(z4, z1))

S tuples:

F2_IN(cons(z0, z1), cons(z2, z3)) → c1(U1'(f13_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3)), F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(U2'(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2), F21_IN(z0, cons(z1, z2)))
F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(U4'(f22_in(z4, z5, z1), z2, z3, z4, z5, z0, z1), F22_IN(z4, z5, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(U6'(f2_in(z4, z1), z2, z3, z4, z0, z1), F2_IN(z4, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f21_in, U2, f13_in, U3, U4, f22_in, U5, U6

Defined Pair Symbols:

F2_IN, F21_IN, F13_IN, U3', F22_IN, U5'

Compound Symbols:

c1, c4, c6, c7, c9, c10

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

Removed 4 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(nil, nil) → f2_out1
f2_in(cons(z0, z1), cons(z2, z3)) → U1(f13_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f13_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f2_out1
f21_in(nil, z0) → f21_out1(nil, z0)
f21_in(cons(z0, z1), z2) → U2(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U2(f21_out1(z0, z1), cons(z2, z3), z4) → f21_out1(z0, z1)
f13_in(z0, z1, z2, z3) → U3(f21_in(z0, z1), z0, z1, z2, z3)
U3(f21_out1(z0, z1), z2, z3, z4, z5) → U4(f22_in(z4, z5, z1), z2, z3, z4, z5, z0, z1)
U4(f22_out1(z0, z1), z2, z3, z4, z5, z6, z7) → f13_out1(z6, z7, z0, z1)
f22_in(z0, z1, z2) → U5(f21_in(z0, z1), z0, z1, z2)
U5(f21_out1(z0, z1), z2, z3, z4) → U6(f2_in(z4, z1), z2, z3, z4, z0, z1)
U6(f2_out1, z0, z1, z2, z3, z4) → f22_out1(z3, z4)

Tuples:

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

S tuples:

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, f21_in, U2, f13_in, U3, U4, f22_in, U5, U6

Defined Pair Symbols:

F13_IN, F22_IN, F2_IN, F21_IN, U3', U5'

Compound Symbols:

c6, c9, c1, c4, c7, c10

(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.

F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))

We considered the (Usable) Rules:

f21_in(nil, z0) → f21_out1(nil, z0)
f21_in(cons(z0, z1), z2) → U2(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U2(f21_out1(z0, z1), cons(z2, z3), z4) → f21_out1(z0, z1)

And the Tuples:

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

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

POL(F13_IN(x₁, x₂, x₃, x₄)) = x₃ + x₄   
POL(F21_IN(x₁, x₂)) = 0   
POL(F22_IN(x₁, x₂, x₃)) = x₁ + x₂   
POL(F2_IN(x₁, x₂)) = x₂   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3'(x₁, x₂, x₃, x₄, x₅)) = x₄ + x₅   
POL(U5'(x₁, x₂, x₃, x₄)) = x₁   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(cons(x₁, x₂)) = [1] + x₁ + x₂   
POL(f21_in(x₁, x₂)) = x₁ + x₂   
POL(f21_out1(x₁, x₂)) = x₂   
POL(nil) = [1]   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(nil, nil) → f2_out1
f2_in(cons(z0, z1), cons(z2, z3)) → U1(f13_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f13_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f2_out1
f21_in(nil, z0) → f21_out1(nil, z0)
f21_in(cons(z0, z1), z2) → U2(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U2(f21_out1(z0, z1), cons(z2, z3), z4) → f21_out1(z0, z1)
f13_in(z0, z1, z2, z3) → U3(f21_in(z0, z1), z0, z1, z2, z3)
U3(f21_out1(z0, z1), z2, z3, z4, z5) → U4(f22_in(z4, z5, z1), z2, z3, z4, z5, z0, z1)
U4(f22_out1(z0, z1), z2, z3, z4, z5, z6, z7) → f13_out1(z6, z7, z0, z1)
f22_in(z0, z1, z2) → U5(f21_in(z0, z1), z0, z1, z2)
U5(f21_out1(z0, z1), z2, z3, z4) → U6(f2_in(z4, z1), z2, z3, z4, z0, z1)
U6(f2_out1, z0, z1, z2, z3, z4) → f22_out1(z3, z4)

Tuples:

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

S tuples:

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

K tuples:

F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))

Defined Rule Symbols:

f2_in, U1, f21_in, U2, f13_in, U3, U4, f22_in, U5, U6

Defined Pair Symbols:

F13_IN, F22_IN, F2_IN, F21_IN, U3', U5'

Compound Symbols:

c6, c9, c1, c4, c7, c10

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))
F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(nil, nil) → f2_out1
f2_in(cons(z0, z1), cons(z2, z3)) → U1(f13_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f13_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f2_out1
f21_in(nil, z0) → f21_out1(nil, z0)
f21_in(cons(z0, z1), z2) → U2(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U2(f21_out1(z0, z1), cons(z2, z3), z4) → f21_out1(z0, z1)
f13_in(z0, z1, z2, z3) → U3(f21_in(z0, z1), z0, z1, z2, z3)
U3(f21_out1(z0, z1), z2, z3, z4, z5) → U4(f22_in(z4, z5, z1), z2, z3, z4, z5, z0, z1)
U4(f22_out1(z0, z1), z2, z3, z4, z5, z6, z7) → f13_out1(z6, z7, z0, z1)
f22_in(z0, z1, z2) → U5(f21_in(z0, z1), z0, z1, z2)
U5(f21_out1(z0, z1), z2, z3, z4) → U6(f2_in(z4, z1), z2, z3, z4, z0, z1)
U6(f2_out1, z0, z1, z2, z3, z4) → f22_out1(z3, z4)

Tuples:

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

S tuples:

F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))

K tuples:

F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

Defined Rule Symbols:

f2_in, U1, f21_in, U2, f13_in, U3, U4, f22_in, U5, U6

Defined Pair Symbols:

F13_IN, F22_IN, F2_IN, F21_IN, U3', U5'

Compound Symbols:

c6, c9, c1, c4, c7, c10

(11) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))

We considered the (Usable) Rules:

f21_in(nil, z0) → f21_out1(nil, z0)
f21_in(cons(z0, z1), z2) → U2(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U2(f21_out1(z0, z1), cons(z2, z3), z4) → f21_out1(z0, z1)

And the Tuples:

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

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

POL(F13_IN(x₁, x₂, x₃, x₄)) = [1] + x₁ + x₃ + x₂·x₄ + x₁·x₄ + x₁·x₃ + x₂·x₃   
POL(F21_IN(x₁, x₂)) = x₁   
POL(F22_IN(x₁, x₂, x₃)) = [1] + x₁ + x₂·x₃ + x₁·x₃   
POL(F2_IN(x₁, x₂)) = x₁·x₂   
POL(U2(x₁, x₂, x₃)) = x₁   
POL(U3'(x₁, x₂, x₃, x₄, x₅)) = [1] + x₁·x₅ + x₁·x₄   
POL(U5'(x₁, x₂, x₃, x₄)) = x₁·x₄   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(cons(x₁, x₂)) = [1] + x₁ + x₂   
POL(f21_in(x₁, x₂)) = x₁ + x₂   
POL(f21_out1(x₁, x₂)) = [1] + x₂   
POL(nil) = [1]   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(nil, nil) → f2_out1
f2_in(cons(z0, z1), cons(z2, z3)) → U1(f13_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f13_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f2_out1
f21_in(nil, z0) → f21_out1(nil, z0)
f21_in(cons(z0, z1), z2) → U2(f21_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U2(f21_out1(z0, z1), cons(z2, z3), z4) → f21_out1(z0, z1)
f13_in(z0, z1, z2, z3) → U3(f21_in(z0, z1), z0, z1, z2, z3)
U3(f21_out1(z0, z1), z2, z3, z4, z5) → U4(f22_in(z4, z5, z1), z2, z3, z4, z5, z0, z1)
U4(f22_out1(z0, z1), z2, z3, z4, z5, z6, z7) → f13_out1(z6, z7, z0, z1)
f22_in(z0, z1, z2) → U5(f21_in(z0, z1), z0, z1, z2)
U5(f21_out1(z0, z1), z2, z3, z4) → U6(f2_in(z4, z1), z2, z3, z4, z0, z1)
U6(f2_out1, z0, z1, z2, z3, z4) → f22_out1(z3, z4)

Tuples:

F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))

S tuples:none
K tuples:

F2_IN(cons(z0, z1), cons(z2, z3)) → c1(F13_IN(z0, z1, z2, z3))
F13_IN(z0, z1, z2, z3) → c6(U3'(f21_in(z0, z1), z0, z1, z2, z3), F21_IN(z0, z1))
U3'(f21_out1(z0, z1), z2, z3, z4, z5) → c7(F22_IN(z4, z5, z1))
F22_IN(z0, z1, z2) → c9(U5'(f21_in(z0, z1), z0, z1, z2), F21_IN(z0, z1))
U5'(f21_out1(z0, z1), z2, z3, z4) → c10(F2_IN(z4, z1))
F21_IN(cons(z0, z1), z2) → c4(F21_IN(z0, cons(z1, z2)))

Defined Rule Symbols:

f2_in, U1, f21_in, U2, f13_in, U3, U4, f22_in, U5, U6

Defined Pair Symbols:

F13_IN, F22_IN, F2_IN, F21_IN, U3', U5'

Compound Symbols:

c6, c9, c1, c4, c7, c10

(13) SIsEmptyProof (EQUIVALENT transformation)

The set S is empty

(14) BOUNDS(O(1), O(1))

(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: samefringe(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: samefringe(T1, T2), SCOPE: 1, CLAUSE: 2 | samefringe(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: true | samefringe(nil, nil), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: samefringe(T1, T2), SCOPE: 1, CLAUSE: 4\n\nT1 is ground\nT2 is ground\nsamefringe(T1, T2) !=? samefringe(nil, nil)", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: samefringe(nil, nil), SCOPE: 1, CLAUSE: 4\n\n", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17)))\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 0 | ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 1 | ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 1 | ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(T13, X17)) | ','(gopher(cons(nil, T13), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nT13 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: ','(gopher(cons(T7, T8), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 3\n\nT7 is ground\nT8 is ground\nT9 is ground\nT10 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free\nX20 is free\ngopher(cons(T7, T8), cons(X14, X15)) !=? gopher(cons(nil, X20), cons(nil, X20))", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(T13, X17))\n\nT9 is ground\nT10 is ground\nT13 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: ','(gopher(cons(nil, T13), cons(X14, X15)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X15, X17))), SCOPE: 2, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nT13 is ground\nX14 is free\nX15 is free\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: gopher(cons(T9, T10), cons(X16, X17))\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: samefringe(T13, T15)\n\nT13 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 0 | gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 1 | gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 1 | gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: true | gopher(cons(nil, T18), cons(X16, X17)), SCOPE: 3, CLAUSE: 3\n\nT18 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: gopher(cons(T9, T10), cons(X16, X17)), SCOPE: 3, CLAUSE: 3\n\nT9 is ground\nT10 is ground\nX16 is free\nX17 is free\nX23 is free\ngopher(cons(T9, T10), cons(X16, X17)) !=? gopher(cons(nil, X23), cons(nil, X23))", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: gopher(cons(nil, T18), cons(X16, X17)), SCOPE: 3, CLAUSE: 3\n\nT18 is ground\nX16 is free\nX17 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: gopher(cons(T25, cons(T26, T27)), cons(X44, X45))\n\nT25 is ground\nT26 is ground\nT27 is ground\nX44 is free\nX45 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
45 [label="45: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
46 [label="46: ','(gopher(cons(T34, cons(T35, T36)), cons(X66, X67)), ','(gopher(cons(T9, T10), cons(X16, X17)), samefringe(X67, X17)))\n\nT9 is ground\nT10 is ground\nT34 is ground\nT35 is ground\nT36 is ground\nX16 is free\nX17 is free\nX66 is free\nX67 is free", fontsize=16, style=filled, fillcolor=lightyellow];
47 [label="47: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
48 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 48 [arrowhead = none ];
48 -> 4;

49 [label="EVAL with clause\nsamefringe(nil, nil).\nand substitutionT1 -> nil,\nT2 -> nil", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 49 [arrowhead = none ];
49 -> 7;

50 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 50 [arrowhead = none ];
50 -> 9;

51 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 51 [arrowhead = none ];
51 -> 10;

52 [label="EVAL with clause\nsamefringe(cons(X10, X11), cons(X12, X13)) :- ','(gopher(cons(X10, X11), cons(X14, X15)), ','(gopher(cons(X12, X13), cons(X16, X17)), samefringe(X15, X17))).\nand substitutionX10 -> T7,\nX11 -> T8,\nT1 -> cons(T7, T8),\nX12 -> T9,\nX13 -> T10,\nT2 -> cons(T9, T10)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 52 [arrowhead = none ];
52 -> 15;

53 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 53 [arrowhead = none ];
53 -> 16;

54 [label="BACKTRACK\nfor clause: samefringe(cons(U, V), cons(X, Y)) :- ','(gopher(cons(U, V), cons(U1, V1)), ','(gopher(cons(X, Y), cons(X1, Y1)), samefringe(V1, Y1)))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
10 -> 54 [arrowhead = none ];
54 -> 12;

55 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 55 [arrowhead = none ];
55 -> 17;

56 [label="BACKTRACK\nfor clause: gopher(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 56 [arrowhead = none ];
56 -> 18;

57 [label="EVAL with clause\ngopher(cons(nil, X20), cons(nil, X20)).\nand substitutionT7 -> nil,\nT8 -> T13,\nX20 -> T13,\nX14 -> nil,\nX15 -> T13", fontsize=14, style = filled, fillcolor = lightblue];
18 -> 57 [arrowhead = none ];
57 -> 19;

58 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
18 -> 58 [arrowhead = none ];
58 -> 20;

59 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 59 [arrowhead = none ];
59 -> 23;

60 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
19 -> 60 [arrowhead = none ];
60 -> 24;

61 [label="EVAL with clause\ngopher(cons(cons(X62, X63), X64), X65) :- gopher(cons(X62, cons(X63, X64)), X65).\nand substitutionX62 -> T34,\nX63 -> T35,\nT7 -> cons(T34, T35),\nT8 -> T36,\nX64 -> T36,\nX14 -> X66,\nX15 -> X67,\nX65 -> cons(X66, X67)", fontsize=14, style = filled, fillcolor = lightblue];
20 -> 61 [arrowhead = none ];
61 -> 46;

62 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
20 -> 62 [arrowhead = none ];
62 -> 47;

63 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
23 -> 63 [arrowhead = none ];
63 -> 28;

64 [label="SPLIT 2\nnew knowledge:\nT9 is ground\nT10 is ground\nT14 is ground\nT15 is ground\nreplacements:X16 -> T14,\nX17 -> T15", fontsize=14, style = filled, fillcolor = violet];
23 -> 64 [arrowhead = none ];
64 -> 29;

65 [label="BACKTRACK\nfor clause: gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
24 -> 65 [arrowhead = none ];
65 -> 45;

66 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
28 -> 66 [arrowhead = none ];
66 -> 32;

67 [label="INSTANCE with matching:\nT1 -> T13\nT2 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
29 -> 67 [arrowhead = none , style=dashed];
67 -> 1[style=dashed];

68 [label="BACKTRACK\nfor clause: gopher(nil, nil)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
32 -> 68 [arrowhead = none ];
68 -> 33;

69 [label="EVAL with clause\ngopher(cons(nil, X23), cons(nil, X23)).\nand substitutionT9 -> nil,\nT10 -> T18,\nX23 -> T18,\nX16 -> nil,\nX17 -> T18", fontsize=14, style = filled, fillcolor = lightblue];
33 -> 69 [arrowhead = none ];
69 -> 35;

70 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
33 -> 70 [arrowhead = none ];
70 -> 36;

71 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
35 -> 71 [arrowhead = none ];
71 -> 37;

72 [label="EVAL with clause\ngopher(cons(cons(X40, X41), X42), X43) :- gopher(cons(X40, cons(X41, X42)), X43).\nand substitutionX40 -> T25,\nX41 -> T26,\nT9 -> cons(T25, T26),\nT10 -> T27,\nX42 -> T27,\nX16 -> X44,\nX17 -> X45,\nX43 -> cons(X44, X45)", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 72 [arrowhead = none ];
72 -> 41;

73 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 73 [arrowhead = none ];
73 -> 42;

74 [label="BACKTRACK\nfor clause: gopher(cons(cons(U, V), W), X) :- gopher(cons(U, cons(V, W)), X)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 74 [arrowhead = none ];
74 -> 38;

75 [label="INSTANCE with matching:\nT9 -> T25\nT10 -> cons(T26, T27)\nX16 -> X44\nX17 -> X45", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 75 [arrowhead = none , style=dashed];
75 -> 28[style=dashed];

76 [label="INSTANCE with matching:\nT7 -> T34\nT8 -> cons(T35, T36)\nX14 -> X66\nX15 -> X67", fontsize=14, style = filled, fillcolor = lightgrey];
46 -> 76 [arrowhead = none , style=dashed];
76 -> 15[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(nil, nil) → f1_out1
f1_in(cons(z0, z1), cons(z2, z3)) → U1(f15_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f15_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f1_out1
f15_in(nil, z0, z1, z2) → U2(f19_in(z1, z2, z0), nil, z0, z1, z2)
f15_in(cons(z0, z1), z2, z3, z4) → U3(f15_in(z0, cons(z1, z2), z3, z4), cons(z0, z1), z2, z3, z4)
U2(f19_out1(z0, z1), nil, z2, z3, z4) → f15_out1(nil, z2, z0, z1)
U2(f19_out2(z0, z1, z2, z3), nil, z4, z5, z6) → f15_out1(z0, z1, z2, z3)
U3(f15_out1(z0, z1, z2, z3), cons(z4, z5), z6, z7, z8) → f15_out1(z0, z1, z2, z3)
f28_in(nil, z0) → f28_out1(nil, z0)
f28_in(cons(z0, z1), z2) → U4(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U4(f28_out1(z0, z1), cons(z2, z3), z4) → f28_out1(z0, z1)
f23_in(z0, z1, z2) → U5(f28_in(z0, z1), z0, z1, z2)
U5(f28_out1(z0, z1), z2, z3, z4) → U6(f1_in(z4, z1), z2, z3, z4, z0, z1)
U6(f1_out1, z0, z1, z2, z3, z4) → f23_out1(z3, z4)
f19_in(z0, z1, z2) → U7(f23_in(z0, z1, z2), f24_in(z2, z0, z1), z0, z1, z2)
U7(f23_out1(z0, z1), z2, z3, z4, z5) → f19_out1(z0, z1)
U7(z0, f24_out1(z1, z2, z3, z4), z5, z6, z7) → f19_out2(z1, z2, z3, z4)

Tuples:

F1_IN(cons(z0, z1), cons(z2, z3)) → c1(U1'(f15_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3)), F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(U2'(f19_in(z1, z2, z0), nil, z0, z1, z2), F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(U3'(f15_in(z0, cons(z1, z2), z3, z4), cons(z0, z1), z2, z3, z4), F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(U4'(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2), F28_IN(z0, cons(z1, z2)))
F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(U6'(f1_in(z4, z1), z2, z3, z4, z0, z1), F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(U7'(f23_in(z0, z1, z2), f24_in(z2, z0, z1), z0, z1, z2), F23_IN(z0, z1, z2))

S tuples:

F1_IN(cons(z0, z1), cons(z2, z3)) → c1(U1'(f15_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3)), F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(U2'(f19_in(z1, z2, z0), nil, z0, z1, z2), F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(U3'(f15_in(z0, cons(z1, z2), z3, z4), cons(z0, z1), z2, z3, z4), F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(U4'(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2), F28_IN(z0, cons(z1, z2)))
F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(U6'(f1_in(z4, z1), z2, z3, z4, z0, z1), F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(U7'(f23_in(z0, z1, z2), f24_in(z2, z0, z1), z0, z1, z2), F23_IN(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f15_in, U2, U3, f28_in, U4, f23_in, U5, U6, f19_in, U7

Defined Pair Symbols:

F1_IN, F15_IN, F28_IN, F23_IN, U5', F19_IN

Compound Symbols:

c1, c3, c4, c9, c11, c12, c14

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

Removed 6 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(nil, nil) → f1_out1
f1_in(cons(z0, z1), cons(z2, z3)) → U1(f15_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f15_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f1_out1
f15_in(nil, z0, z1, z2) → U2(f19_in(z1, z2, z0), nil, z0, z1, z2)
f15_in(cons(z0, z1), z2, z3, z4) → U3(f15_in(z0, cons(z1, z2), z3, z4), cons(z0, z1), z2, z3, z4)
U2(f19_out1(z0, z1), nil, z2, z3, z4) → f15_out1(nil, z2, z0, z1)
U2(f19_out2(z0, z1, z2, z3), nil, z4, z5, z6) → f15_out1(z0, z1, z2, z3)
U3(f15_out1(z0, z1, z2, z3), cons(z4, z5), z6, z7, z8) → f15_out1(z0, z1, z2, z3)
f28_in(nil, z0) → f28_out1(nil, z0)
f28_in(cons(z0, z1), z2) → U4(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U4(f28_out1(z0, z1), cons(z2, z3), z4) → f28_out1(z0, z1)
f23_in(z0, z1, z2) → U5(f28_in(z0, z1), z0, z1, z2)
U5(f28_out1(z0, z1), z2, z3, z4) → U6(f1_in(z4, z1), z2, z3, z4, z0, z1)
U6(f1_out1, z0, z1, z2, z3, z4) → f23_out1(z3, z4)
f19_in(z0, z1, z2) → U7(f23_in(z0, z1, z2), f24_in(z2, z0, z1), z0, z1, z2)
U7(f23_out1(z0, z1), z2, z3, z4, z5) → f19_out1(z0, z1)
U7(z0, f24_out1(z1, z2, z3, z4), z5, z6, z7) → f19_out2(z1, z2, z3, z4)

Tuples:

F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))

S tuples:

F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f15_in, U2, U3, f28_in, U4, f23_in, U5, U6, f19_in, U7

Defined Pair Symbols:

F23_IN, F1_IN, F15_IN, F28_IN, U5', F19_IN

Compound Symbols:

c11, c1, c3, c4, c9, c12, c14

(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.

F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))

We considered the (Usable) Rules:

f28_in(nil, z0) → f28_out1(nil, z0)
f28_in(cons(z0, z1), z2) → U4(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U4(f28_out1(z0, z1), cons(z2, z3), z4) → f28_out1(z0, z1)

And the Tuples:

F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))

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

POL(F15_IN(x₁, x₂, x₃, x₄)) = x₁ + x₂   
POL(F19_IN(x₁, x₂, x₃)) = x₃   
POL(F1_IN(x₁, x₂)) = x₁   
POL(F23_IN(x₁, x₂, x₃)) = x₃   
POL(F28_IN(x₁, x₂)) = 0   
POL(U4(x₁, x₂, x₃)) = 0   
POL(U5'(x₁, x₂, x₃, x₄)) = x₄   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(cons(x₁, x₂)) = x₁ + x₂   
POL(f28_in(x₁, x₂)) = [2]x₁   
POL(f28_out1(x₁, x₂)) = 0   
POL(nil) = [2]   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(nil, nil) → f1_out1
f1_in(cons(z0, z1), cons(z2, z3)) → U1(f15_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f15_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f1_out1
f15_in(nil, z0, z1, z2) → U2(f19_in(z1, z2, z0), nil, z0, z1, z2)
f15_in(cons(z0, z1), z2, z3, z4) → U3(f15_in(z0, cons(z1, z2), z3, z4), cons(z0, z1), z2, z3, z4)
U2(f19_out1(z0, z1), nil, z2, z3, z4) → f15_out1(nil, z2, z0, z1)
U2(f19_out2(z0, z1, z2, z3), nil, z4, z5, z6) → f15_out1(z0, z1, z2, z3)
U3(f15_out1(z0, z1, z2, z3), cons(z4, z5), z6, z7, z8) → f15_out1(z0, z1, z2, z3)
f28_in(nil, z0) → f28_out1(nil, z0)
f28_in(cons(z0, z1), z2) → U4(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U4(f28_out1(z0, z1), cons(z2, z3), z4) → f28_out1(z0, z1)
f23_in(z0, z1, z2) → U5(f28_in(z0, z1), z0, z1, z2)
U5(f28_out1(z0, z1), z2, z3, z4) → U6(f1_in(z4, z1), z2, z3, z4, z0, z1)
U6(f1_out1, z0, z1, z2, z3, z4) → f23_out1(z3, z4)
f19_in(z0, z1, z2) → U7(f23_in(z0, z1, z2), f24_in(z2, z0, z1), z0, z1, z2)
U7(f23_out1(z0, z1), z2, z3, z4, z5) → f19_out1(z0, z1)
U7(z0, f24_out1(z1, z2, z3, z4), z5, z6, z7) → f19_out2(z1, z2, z3, z4)

Tuples:

F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))

S tuples:

F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))

K tuples:

F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))

Defined Rule Symbols:

f1_in, U1, f15_in, U2, U3, f28_in, U4, f23_in, U5, U6, f19_in, U7

Defined Pair Symbols:

F23_IN, F1_IN, F15_IN, F28_IN, U5', F19_IN

Compound Symbols:

c11, c1, c3, c4, c9, c12, c14

(21) CdtKnowledgeProof (EQUIVALENT transformation)

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

F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))
F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(nil, nil) → f1_out1
f1_in(cons(z0, z1), cons(z2, z3)) → U1(f15_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f15_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f1_out1
f15_in(nil, z0, z1, z2) → U2(f19_in(z1, z2, z0), nil, z0, z1, z2)
f15_in(cons(z0, z1), z2, z3, z4) → U3(f15_in(z0, cons(z1, z2), z3, z4), cons(z0, z1), z2, z3, z4)
U2(f19_out1(z0, z1), nil, z2, z3, z4) → f15_out1(nil, z2, z0, z1)
U2(f19_out2(z0, z1, z2, z3), nil, z4, z5, z6) → f15_out1(z0, z1, z2, z3)
U3(f15_out1(z0, z1, z2, z3), cons(z4, z5), z6, z7, z8) → f15_out1(z0, z1, z2, z3)
f28_in(nil, z0) → f28_out1(nil, z0)
f28_in(cons(z0, z1), z2) → U4(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U4(f28_out1(z0, z1), cons(z2, z3), z4) → f28_out1(z0, z1)
f23_in(z0, z1, z2) → U5(f28_in(z0, z1), z0, z1, z2)
U5(f28_out1(z0, z1), z2, z3, z4) → U6(f1_in(z4, z1), z2, z3, z4, z0, z1)
U6(f1_out1, z0, z1, z2, z3, z4) → f23_out1(z3, z4)
f19_in(z0, z1, z2) → U7(f23_in(z0, z1, z2), f24_in(z2, z0, z1), z0, z1, z2)
U7(f23_out1(z0, z1), z2, z3, z4, z5) → f19_out1(z0, z1)
U7(z0, f24_out1(z1, z2, z3, z4), z5, z6, z7) → f19_out2(z1, z2, z3, z4)

Tuples:

F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))

S tuples:

F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))

K tuples:

F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))
F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))

Defined Rule Symbols:

f1_in, U1, f15_in, U2, U3, f28_in, U4, f23_in, U5, U6, f19_in, U7

Defined Pair Symbols:

F23_IN, F1_IN, F15_IN, F28_IN, U5', F19_IN

Compound Symbols:

c11, c1, c3, c4, c9, c12, c14

(23) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^2))) transformation)

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

F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))

We considered the (Usable) Rules:

f28_in(nil, z0) → f28_out1(nil, z0)
f28_in(cons(z0, z1), z2) → U4(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U4(f28_out1(z0, z1), cons(z2, z3), z4) → f28_out1(z0, z1)

And the Tuples:

F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))

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

POL(F15_IN(x₁, x₂, x₃, x₄)) = [1] + x₂·x₄ + x₁·x₄ + x₁·x₃ + x₂·x₃   
POL(F19_IN(x₁, x₂, x₃)) = [1] + x₁ + x₂·x₃ + x₁·x₃   
POL(F1_IN(x₁, x₂)) = [1] + x₁ + x₁·x₂   
POL(F23_IN(x₁, x₂, x₃)) = [1] + x₁ + x₂·x₃ + x₁·x₃   
POL(F28_IN(x₁, x₂)) = x₁   
POL(U4(x₁, x₂, x₃)) = x₁   
POL(U5'(x₁, x₂, x₃, x₄)) = [1] + x₁·x₄   
POL(c1(x₁)) = x₁   
POL(c11(x₁, x₂)) = x₁ + x₂   
POL(c12(x₁)) = x₁   
POL(c14(x₁)) = x₁   
POL(c3(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c9(x₁)) = x₁   
POL(cons(x₁, x₂)) = [1] + x₁ + x₂   
POL(f28_in(x₁, x₂)) = x₁ + x₂   
POL(f28_out1(x₁, x₂)) = [1] + x₂   
POL(nil) = [1]   

(24) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(nil, nil) → f1_out1
f1_in(cons(z0, z1), cons(z2, z3)) → U1(f15_in(z0, z1, z2, z3), cons(z0, z1), cons(z2, z3))
U1(f15_out1(z0, z1, z2, z3), cons(z4, z5), cons(z6, z7)) → f1_out1
f15_in(nil, z0, z1, z2) → U2(f19_in(z1, z2, z0), nil, z0, z1, z2)
f15_in(cons(z0, z1), z2, z3, z4) → U3(f15_in(z0, cons(z1, z2), z3, z4), cons(z0, z1), z2, z3, z4)
U2(f19_out1(z0, z1), nil, z2, z3, z4) → f15_out1(nil, z2, z0, z1)
U2(f19_out2(z0, z1, z2, z3), nil, z4, z5, z6) → f15_out1(z0, z1, z2, z3)
U3(f15_out1(z0, z1, z2, z3), cons(z4, z5), z6, z7, z8) → f15_out1(z0, z1, z2, z3)
f28_in(nil, z0) → f28_out1(nil, z0)
f28_in(cons(z0, z1), z2) → U4(f28_in(z0, cons(z1, z2)), cons(z0, z1), z2)
U4(f28_out1(z0, z1), cons(z2, z3), z4) → f28_out1(z0, z1)
f23_in(z0, z1, z2) → U5(f28_in(z0, z1), z0, z1, z2)
U5(f28_out1(z0, z1), z2, z3, z4) → U6(f1_in(z4, z1), z2, z3, z4, z0, z1)
U6(f1_out1, z0, z1, z2, z3, z4) → f23_out1(z3, z4)
f19_in(z0, z1, z2) → U7(f23_in(z0, z1, z2), f24_in(z2, z0, z1), z0, z1, z2)
U7(f23_out1(z0, z1), z2, z3, z4, z5) → f19_out1(z0, z1)
U7(z0, f24_out1(z1, z2, z3, z4), z5, z6, z7) → f19_out2(z1, z2, z3, z4)

Tuples:

F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))

S tuples:

F15_IN(cons(z0, z1), z2, z3, z4) → c4(F15_IN(z0, cons(z1, z2), z3, z4))

K tuples:

F15_IN(nil, z0, z1, z2) → c3(F19_IN(z1, z2, z0))
F19_IN(z0, z1, z2) → c14(F23_IN(z0, z1, z2))
F23_IN(z0, z1, z2) → c11(U5'(f28_in(z0, z1), z0, z1, z2), F28_IN(z0, z1))
U5'(f28_out1(z0, z1), z2, z3, z4) → c12(F1_IN(z4, z1))
F1_IN(cons(z0, z1), cons(z2, z3)) → c1(F15_IN(z0, z1, z2, z3))
F28_IN(cons(z0, z1), z2) → c9(F28_IN(z0, cons(z1, z2)))

Defined Rule Symbols:

f1_in, U1, f15_in, U2, U3, f28_in, U4, f23_in, U5, U6, f19_in, U7

Defined Pair Symbols:

F23_IN, F1_IN, F15_IN, F28_IN, U5', F19_IN

Compound Symbols:

c11, c1, c3, c4, c9, c12, c14