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

(0) Obligation:

Clauses:

star(X1, []).
star(.(X, U), .(X, W)) :- ','(app(U, V, W), star(.(X, U), W)).
app([], L, L).
app(.(X, L), M, .(X, N)) :- app(L, M, N).

Query: star(g,g)

(1) LPReorderTransformerProof (EQUIVALENT transformation)

Reordered facts before rules in definite LP [PROLOG].

(2) Obligation:

Clauses:

star(X1, []).
app([], L, L).
star(.(X, U), .(X, W)) :- ','(app(U, V, W), star(.(X, U), W)).
app(.(X, L), M, .(X, N)) :- app(L, M, N).

Query: star(g,g)

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

Built complexity over-approximating cdt problems from derivation graph.

digraph dp_graph {
node [outthreshold=100, inthreshold=100];
1 [label="1: star(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
4 [label="4: star(T1, T2), SCOPE: 1, CLAUSE: 0 | star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
6 [label="6: true | star(T5, []), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
7 [label="7: star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX4 is free\nstar(T1, T2) !=? star(X4, [])", fontsize=16, style=filled, fillcolor=lightyellow];
8 [label="8: star(T5, []), SCOPE: 1, CLAUSE: 2\n\nT5 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
10 [label="10: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
13 [label="13: ','(app(T15, X18, T16), star(.(T14, T15), T16))\n\nT14 is ground\nT15 is ground\nT16 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
14 [label="14: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
22 [label="22: app(T15, X18, T16)\n\nT15 is ground\nT16 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
23 [label="23: star(.(T14, T15), T16)\n\nT14 is ground\nT15 is ground\nT16 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
25 [label="25: app(T15, X18, T16), SCOPE: 2, CLAUSE: 1 | app(T15, X18, T16), SCOPE: 2, CLAUSE: 3\n\nT15 is ground\nT16 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
26 [label="26: true | app([], X18, T24), SCOPE: 2, CLAUSE: 3\n\nT24 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
27 [label="27: app(T15, X18, T16), SCOPE: 2, CLAUSE: 3\n\nT15 is ground\nT16 is ground\nX18 is free\nX31 is free\napp(T15, X18, T16) !=? app([], X31, X31)", fontsize=16, style=filled, fillcolor=lightyellow];
28 [label="28: app([], X18, T24), SCOPE: 2, CLAUSE: 3\n\nT24 is ground\nX18 is free", fontsize=16, style=filled, fillcolor=lightyellow];
29 [label="29: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
34 [label="34: app(T32, X51, T33)\n\nT32 is ground\nT33 is ground\nX51 is free", fontsize=16, style=filled, fillcolor=lightyellow];
35 [label="35: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
1 -> 36 [arrowhead = none ];
36 -> 4;

37 [label="EVAL with clause\nstar(X4, []).\nand substitutionT1 -> T5,\nX4 -> T5,\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
4 -> 37 [arrowhead = none ];
37 -> 6;

38 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
4 -> 38 [arrowhead = none ];
38 -> 7;

39 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
6 -> 39 [arrowhead = none ];
39 -> 8;

40 [label="EVAL with clause\nstar(.(X15, X16), .(X15, X17)) :- ','(app(X16, X18, X17), star(.(X15, X16), X17)).\nand substitutionX15 -> T14,\nX16 -> T15,\nT1 -> .(T14, T15),\nX17 -> T16,\nT2 -> .(T14, T16)", fontsize=14, style = filled, fillcolor = lightblue];
7 -> 40 [arrowhead = none ];
40 -> 13;

41 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
7 -> 41 [arrowhead = none ];
41 -> 14;

42 [label="BACKTRACK\nfor clause: star(.(X, U), .(X, W)) :- ','(app(U, V, W), star(.(X, U), W))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
8 -> 42 [arrowhead = none ];
42 -> 10;

43 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
13 -> 43 [arrowhead = none ];
43 -> 22;

44 [label="SPLIT 2\nnew knowledge:\nT15 is ground\nT19 is ground\nT16 is ground\nreplacements:X18 -> T19", fontsize=14, style = filled, fillcolor = violet];
13 -> 44 [arrowhead = none ];
44 -> 23;

45 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
22 -> 45 [arrowhead = none ];
45 -> 25;

46 [label="INSTANCE with matching:\nT1 -> .(T14, T15)\nT2 -> T16", fontsize=14, style = filled, fillcolor = lightgrey];
23 -> 46 [arrowhead = none , style=dashed];
46 -> 1[style=dashed];

47 [label="EVAL with clause\napp([], X31, X31).\nand substitutionT15 -> [],\nX18 -> T24,\nX31 -> T24,\nT16 -> T24,\nX32 -> T24", fontsize=14, style = filled, fillcolor = lightblue];
25 -> 47 [arrowhead = none ];
47 -> 26;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
25 -> 48 [arrowhead = none ];
48 -> 27;

49 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
26 -> 49 [arrowhead = none ];
49 -> 28;

50 [label="EVAL with clause\napp(.(X47, X48), X49, .(X47, X50)) :- app(X48, X49, X50).\nand substitutionX47 -> T31,\nX48 -> T32,\nT15 -> .(T31, T32),\nX18 -> X51,\nX49 -> X51,\nX50 -> T33,\nT16 -> .(T31, T33)", fontsize=14, style = filled, fillcolor = lightblue];
27 -> 50 [arrowhead = none ];
50 -> 34;

51 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
27 -> 51 [arrowhead = none ];
51 -> 35;

52 [label="BACKTRACK\nfor clause: app(.(X, L), M, .(X, N)) :- app(L, M, N)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
28 -> 52 [arrowhead = none ];
52 -> 29;

53 [label="INSTANCE with matching:\nT15 -> T32\nX18 -> X51\nT16 -> T33", fontsize=14, style = filled, fillcolor = lightgrey];
34 -> 53 [arrowhead = none , style=dashed];
53 -> 22[style=dashed];

}

(4) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f13_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f13_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f13_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z2), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f13_out1(z3)

Tuples:

F1_IN(.(z0, z1), .(z0, z2)) → c1(U1'(f13_in(z1, z2, z0), .(z0, z1), .(z0, z2)), F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f22_in(z1, z2), .(z0, z1), .(z0, z2)), F22_IN(z1, z2))
F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(U4'(f1_in(.(z3, z1), z2), z1, z2, z3, z0), F1_IN(.(z3, z1), z2))

S tuples:

F1_IN(.(z0, z1), .(z0, z2)) → c1(U1'(f13_in(z1, z2, z0), .(z0, z1), .(z0, z2)), F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(U2'(f22_in(z1, z2), .(z0, z1), .(z0, z2)), F22_IN(z1, z2))
F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(U4'(f1_in(.(z3, z1), z2), z1, z2, z3, z0), F1_IN(.(z3, z1), z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f22_in, U2, f13_in, U3, U4

Defined Pair Symbols:

F1_IN, F22_IN, F13_IN, U3'

Compound Symbols:

c1, c4, c6, c7

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

Removed 3 trailing tuple parts

(6) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f13_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f13_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f13_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z2), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f13_out1(z3)

Tuples:

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

S tuples:

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

K tuples:none
Defined Rule Symbols:

f1_in, U1, f22_in, U2, f13_in, U3, U4

Defined Pair Symbols:

F13_IN, F1_IN, F22_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(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.

F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))

We considered the (Usable) Rules:

f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)

And the Tuples:

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

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

POL(.(x₁, x₂)) = [2] + x₁ + x₂   
POL(F13_IN(x₁, x₂, x₃)) = x₂ + x₃   
POL(F1_IN(x₁, x₂)) = x₂   
POL(F22_IN(x₁, x₂)) = 0   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3'(x₁, x₂, x₃, x₄)) = x₃   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(f22_in(x₁, x₂)) = 0   
POL(f22_out1(x₁)) = 0   

(8) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f13_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f13_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f13_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z2), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f13_out1(z3)

Tuples:

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

S tuples:

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

K tuples:

F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))

Defined Rule Symbols:

f1_in, U1, f22_in, U2, f13_in, U3, U4

Defined Pair Symbols:

F13_IN, F1_IN, F22_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(9) CdtKnowledgeProof (EQUIVALENT transformation)

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

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))

(10) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f13_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f13_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f13_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z2), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f13_out1(z3)

Tuples:

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

S tuples:

F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))

K tuples:

F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

Defined Rule Symbols:

f1_in, U1, f22_in, U2, f13_in, U3, U4

Defined Pair Symbols:

F13_IN, F1_IN, F22_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(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.

F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))

We considered the (Usable) Rules:

f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)

And the Tuples:

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F13_IN(x₁, x₂, x₃)) = x₂ + x₂²   
POL(F1_IN(x₁, x₂)) = x₂²   
POL(F22_IN(x₁, x₂)) = x₂   
POL(U2(x₁, x₂, x₃)) = 0   
POL(U3'(x₁, x₂, x₃, x₄)) = x₃²   
POL([]) = [1]   
POL(c1(x₁)) = x₁   
POL(c4(x₁)) = x₁   
POL(c6(x₁, x₂)) = x₁ + x₂   
POL(c7(x₁)) = x₁   
POL(f22_in(x₁, x₂)) = x₁ + x₁²   
POL(f22_out1(x₁)) = 0   

(12) Obligation:

Complexity Dependency Tuples Problem
Rules:

f1_in(z0, []) → f1_out1
f1_in(.(z0, z1), .(z0, z2)) → U1(f13_in(z1, z2, z0), .(z0, z1), .(z0, z2))
U1(f13_out1(z0), .(z1, z2), .(z1, z3)) → f1_out1
f22_in([], z0) → f22_out1(z0)
f22_in(.(z0, z1), .(z0, z2)) → U2(f22_in(z1, z2), .(z0, z1), .(z0, z2))
U2(f22_out1(z0), .(z1, z2), .(z1, z3)) → f22_out1(z0)
f13_in(z0, z1, z2) → U3(f22_in(z0, z1), z0, z1, z2)
U3(f22_out1(z0), z1, z2, z3) → U4(f1_in(.(z3, z1), z2), z1, z2, z3, z0)
U4(f1_out1, z0, z1, z2, z3) → f13_out1(z3)

Tuples:

F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))

S tuples:none
K tuples:

F1_IN(.(z0, z1), .(z0, z2)) → c1(F13_IN(z1, z2, z0))
F13_IN(z0, z1, z2) → c6(U3'(f22_in(z0, z1), z0, z1, z2), F22_IN(z0, z1))
U3'(f22_out1(z0), z1, z2, z3) → c7(F1_IN(.(z3, z1), z2))
F22_IN(.(z0, z1), .(z0, z2)) → c4(F22_IN(z1, z2))

Defined Rule Symbols:

f1_in, U1, f22_in, U2, f13_in, U3, U4

Defined Pair Symbols:

F13_IN, F1_IN, F22_IN, U3'

Compound Symbols:

c6, c1, c4, c7

(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];
2 [label="2: star(T1, T2)\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow, color=red];
3 [label="3: star(T1, T2), SCOPE: 1, CLAUSE: 0 | star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
5 [label="5: true | star(T4, []), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
9 [label="9: star(T1, T2), SCOPE: 1, CLAUSE: 2\n\nT1 is ground\nT2 is ground\nX3 is free\nstar(T1, T2) !=? star(X3, [])", fontsize=16, style=filled, fillcolor=lightyellow];
11 [label="11: star(T4, []), SCOPE: 1, CLAUSE: 2\n\nT4 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
12 [label="12: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
15 [label="15: ','(app(T11, X13, T12), star(.(T10, T11), T12))\n\nT10 is ground\nT11 is ground\nT12 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
16 [label="16: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
17 [label="17: ','(app(T11, X13, T12), star(.(T10, T11), T12)), SCOPE: 2, CLAUSE: 1 | ','(app(T11, X13, T12), star(.(T10, T11), T12)), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT11 is ground\nT12 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
18 [label="18: star(.(T10, []), T15) | ','(app([], X13, T15), star(.(T10, []), T15)), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT15 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
19 [label="19: ','(app(T11, X13, T12), star(.(T10, T11), T12)), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT11 is ground\nT12 is ground\nX13 is free\nX18 is free\napp(T11, X13, T12) !=? app([], X18, X18)", fontsize=16, style=filled, fillcolor=lightyellow];
20 [label="20: star(.(T10, []), T15)\n\nT10 is ground\nT15 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
21 [label="21: ','(app([], X13, T15), star(.(T10, []), T15)), SCOPE: 2, CLAUSE: 3\n\nT10 is ground\nT15 is ground\nX13 is free", fontsize=16, style=filled, fillcolor=lightyellow];
24 [label="24: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
30 [label="30: ','(app(T27, X42, T28), star(.(T10, .(T26, T27)), .(T26, T28)))\n\nT10 is ground\nT26 is ground\nT27 is ground\nT28 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
31 [label="31: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
32 [label="32: app(T27, X42, T28)\n\nT27 is ground\nT28 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
33 [label="33: star(.(T10, .(T26, T27)), .(T26, T28))\n\nT10 is ground\nT26 is ground\nT27 is ground\nT28 is ground", fontsize=16, style=filled, fillcolor=lightyellow];
36 [label="36: app(T27, X42, T28), SCOPE: 3, CLAUSE: 1 | app(T27, X42, T28), SCOPE: 3, CLAUSE: 3\n\nT27 is ground\nT28 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
37 [label="37: true | app([], X42, T36), SCOPE: 3, CLAUSE: 3\n\nT36 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
38 [label="38: app(T27, X42, T28), SCOPE: 3, CLAUSE: 3\n\nT27 is ground\nT28 is ground\nX42 is free\nX55 is free\napp(T27, X42, T28) !=? app([], X55, X55)", fontsize=16, style=filled, fillcolor=lightyellow];
39 [label="39: app([], X42, T36), SCOPE: 3, CLAUSE: 3\n\nT36 is ground\nX42 is free", fontsize=16, style=filled, fillcolor=lightyellow];
40 [label="40: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
41 [label="41: app(T44, X75, T45)\n\nT44 is ground\nT45 is ground\nX75 is free", fontsize=16, style=filled, fillcolor=lightyellow];
42 [label="42: \n\n", fontsize=16, style=filled, fillcolor=lightyellow];
43 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
2 -> 43 [arrowhead = none ];
43 -> 3;

44 [label="EVAL with clause\nstar(X3, []).\nand substitutionT1 -> T4,\nX3 -> T4,\nT2 -> []", fontsize=14, style = filled, fillcolor = lightblue];
3 -> 44 [arrowhead = none ];
44 -> 5;

45 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
3 -> 45 [arrowhead = none ];
45 -> 9;

46 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
5 -> 46 [arrowhead = none ];
46 -> 11;

47 [label="EVAL with clause\nstar(.(X10, X11), .(X10, X12)) :- ','(app(X11, X13, X12), star(.(X10, X11), X12)).\nand substitutionX10 -> T10,\nX11 -> T11,\nT1 -> .(T10, T11),\nX12 -> T12,\nT2 -> .(T10, T12)", fontsize=14, style = filled, fillcolor = lightblue];
9 -> 47 [arrowhead = none ];
47 -> 15;

48 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
9 -> 48 [arrowhead = none ];
48 -> 16;

49 [label="BACKTRACK\nfor clause: star(.(X, U), .(X, W)) :- ','(app(U, V, W), star(.(X, U), W))because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
11 -> 49 [arrowhead = none ];
49 -> 12;

50 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
15 -> 50 [arrowhead = none ];
50 -> 17;

51 [label="EVAL with clause\napp([], X18, X18).\nand substitutionT11 -> [],\nX13 -> T15,\nX18 -> T15,\nT12 -> T15,\nX19 -> T15", fontsize=14, style = filled, fillcolor = lightblue];
17 -> 51 [arrowhead = none ];
51 -> 18;

52 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
17 -> 52 [arrowhead = none ];
52 -> 19;

53 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 53 [arrowhead = none ];
53 -> 20;

54 [label="PARALLEL", fontsize=14, style = filled, fillcolor = violet];
18 -> 54 [arrowhead = none ];
54 -> 21;

55 [label="EVAL with clause\napp(.(X38, X39), X40, .(X38, X41)) :- app(X39, X40, X41).\nand substitutionX38 -> T26,\nX39 -> T27,\nT11 -> .(T26, T27),\nX13 -> X42,\nX40 -> X42,\nX41 -> T28,\nT12 -> .(T26, T28)", fontsize=14, style = filled, fillcolor = lightblue];
19 -> 55 [arrowhead = none ];
55 -> 30;

56 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
19 -> 56 [arrowhead = none ];
56 -> 31;

57 [label="INSTANCE with matching:\nT1 -> .(T10, [])\nT2 -> T15", fontsize=14, style = filled, fillcolor = lightgrey];
20 -> 57 [arrowhead = none , style=dashed];
57 -> 2[style=dashed];

58 [label="BACKTRACK\nfor clause: app(.(X, L), M, .(X, N)) :- app(L, M, N)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
21 -> 58 [arrowhead = none ];
58 -> 24;

59 [label="SPLIT 1", fontsize=14, style = filled, fillcolor = violet];
30 -> 59 [arrowhead = none ];
59 -> 32;

60 [label="SPLIT 2\nnew knowledge:\nT27 is ground\nT31 is ground\nT28 is ground\nreplacements:X42 -> T31", fontsize=14, style = filled, fillcolor = violet];
30 -> 60 [arrowhead = none ];
60 -> 33;

61 [label="CASE", fontsize=14, style = filled, fillcolor = lightcyan];
32 -> 61 [arrowhead = none ];
61 -> 36;

62 [label="INSTANCE with matching:\nT1 -> .(T10, .(T26, T27))\nT2 -> .(T26, T28)", fontsize=14, style = filled, fillcolor = lightgrey];
33 -> 62 [arrowhead = none , style=dashed];
62 -> 2[style=dashed];

63 [label="EVAL with clause\napp([], X55, X55).\nand substitutionT27 -> [],\nX42 -> T36,\nX55 -> T36,\nT28 -> T36,\nX56 -> T36", fontsize=14, style = filled, fillcolor = lightblue];
36 -> 63 [arrowhead = none ];
63 -> 37;

64 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
36 -> 64 [arrowhead = none ];
64 -> 38;

65 [label="SUCCESS", fontsize=14, style = filled, fillcolor = lightgreen];
37 -> 65 [arrowhead = none ];
65 -> 39;

66 [label="EVAL with clause\napp(.(X71, X72), X73, .(X71, X74)) :- app(X72, X73, X74).\nand substitutionX71 -> T43,\nX72 -> T44,\nT27 -> .(T43, T44),\nX42 -> X75,\nX73 -> X75,\nX74 -> T45,\nT28 -> .(T43, T45)", fontsize=14, style = filled, fillcolor = lightblue];
38 -> 66 [arrowhead = none ];
66 -> 41;

67 [label="EVAL-BACKTRACK", fontsize=14, style = filled, fillcolor = lightgreen];
38 -> 67 [arrowhead = none ];
67 -> 42;

68 [label="BACKTRACK\nfor clause: app(.(X, L), M, .(X, N)) :- app(L, M, N)because of non-unification", fontsize=14, style = filled, fillcolor = lightgreen];
39 -> 68 [arrowhead = none ];
68 -> 40;

69 [label="INSTANCE with matching:\nT27 -> T44\nX42 -> X75\nT28 -> T45", fontsize=14, style = filled, fillcolor = lightgrey];
41 -> 69 [arrowhead = none , style=dashed];
69 -> 32[style=dashed];

}

(16) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(.(z0, []), .(z0, z1)) → U1(f18_in(z0, z1), .(z0, []), .(z0, z1))
f2_in(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → U2(f30_in(z2, z3, z0, z1), .(z0, .(z1, z2)), .(z0, .(z1, z3)))
U1(f18_out1, .(z0, []), .(z0, z1)) → f2_out1
U1(f18_out2(z0), .(z1, []), .(z1, z2)) → f2_out1
U2(f30_out1(z0), .(z1, .(z2, z3)), .(z1, .(z2, z4))) → f2_out1
f32_in([], z0) → f32_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f32_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f32_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f30_in(z0, z1, z2, z3) → U4(f32_in(z0, z1), z0, z1, z2, z3)
U4(f32_out1(z0), z1, z2, z3, z4) → U5(f2_in(.(z3, .(z4, z1)), .(z4, z2)), z1, z2, z3, z4, z0)
U5(f2_out1, z0, z1, z2, z3, z4) → f30_out1(z4)
f18_in(z0, z1) → U6(f2_in(.(z0, []), z1), f21_in(z1, z0), z0, z1)
U6(f2_out1, z0, z1, z2) → f18_out1
U6(z0, f21_out1(z1), z2, z3) → f18_out2(z1)

Tuples:

F2_IN(.(z0, []), .(z0, z1)) → c1(U1'(f18_in(z0, z1), .(z0, []), .(z0, z1)), F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(U2'(f30_in(z2, z3, z0, z1), .(z0, .(z1, z2)), .(z0, .(z1, z3))), F30_IN(z2, z3, z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c7(U3'(f32_in(z1, z2), .(z0, z1), .(z0, z2)), F32_IN(z1, z2))
F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(U5'(f2_in(.(z3, .(z4, z1)), .(z4, z2)), z1, z2, z3, z4, z0), F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(U6'(f2_in(.(z0, []), z1), f21_in(z1, z0), z0, z1), F2_IN(.(z0, []), z1))

S tuples:

F2_IN(.(z0, []), .(z0, z1)) → c1(U1'(f18_in(z0, z1), .(z0, []), .(z0, z1)), F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(U2'(f30_in(z2, z3, z0, z1), .(z0, .(z1, z2)), .(z0, .(z1, z3))), F30_IN(z2, z3, z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c7(U3'(f32_in(z1, z2), .(z0, z1), .(z0, z2)), F32_IN(z1, z2))
F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(U5'(f2_in(.(z3, .(z4, z1)), .(z4, z2)), z1, z2, z3, z4, z0), F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(U6'(f2_in(.(z0, []), z1), f21_in(z1, z0), z0, z1), F2_IN(.(z0, []), z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f32_in, U3, f30_in, U4, U5, f18_in, U6

Defined Pair Symbols:

F2_IN, F32_IN, F30_IN, U4', F18_IN

Compound Symbols:

c1, c2, c7, c9, c10, c12

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

Removed 5 trailing tuple parts

(18) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(.(z0, []), .(z0, z1)) → U1(f18_in(z0, z1), .(z0, []), .(z0, z1))
f2_in(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → U2(f30_in(z2, z3, z0, z1), .(z0, .(z1, z2)), .(z0, .(z1, z3)))
U1(f18_out1, .(z0, []), .(z0, z1)) → f2_out1
U1(f18_out2(z0), .(z1, []), .(z1, z2)) → f2_out1
U2(f30_out1(z0), .(z1, .(z2, z3)), .(z1, .(z2, z4))) → f2_out1
f32_in([], z0) → f32_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f32_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f32_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f30_in(z0, z1, z2, z3) → U4(f32_in(z0, z1), z0, z1, z2, z3)
U4(f32_out1(z0), z1, z2, z3, z4) → U5(f2_in(.(z3, .(z4, z1)), .(z4, z2)), z1, z2, z3, z4, z0)
U5(f2_out1, z0, z1, z2, z3, z4) → f30_out1(z4)
f18_in(z0, z1) → U6(f2_in(.(z0, []), z1), f21_in(z1, z0), z0, z1)
U6(f2_out1, z0, z1, z2) → f18_out1
U6(z0, f21_out1(z1), z2, z3) → f18_out2(z1)

Tuples:

F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c7(F32_IN(z1, z2))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(F2_IN(.(z0, []), z1))

S tuples:

F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c7(F32_IN(z1, z2))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(F2_IN(.(z0, []), z1))

K tuples:none
Defined Rule Symbols:

f2_in, U1, U2, f32_in, U3, f30_in, U4, U5, f18_in, U6

Defined Pair Symbols:

F30_IN, F2_IN, F32_IN, U4', F18_IN

Compound Symbols:

c9, c1, c2, c7, c10, c12

(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.

F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))

We considered the (Usable) Rules:

f32_in([], z0) → f32_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f32_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f32_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)

And the Tuples:

F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c7(F32_IN(z1, z2))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(F2_IN(.(z0, []), z1))

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

POL(.(x₁, x₂)) = [1] + x₂   
POL(F18_IN(x₁, x₂)) = x₂   
POL(F2_IN(x₁, x₂)) = x₂   
POL(F30_IN(x₁, x₂, x₃, x₄)) = [1] + x₂   
POL(F32_IN(x₁, x₂)) = 0   
POL(U3(x₁, x₂, x₃)) = 0   
POL(U4'(x₁, x₂, x₃, x₄, x₅)) = [1] + x₃   
POL([]) = 0   
POL(c1(x₁)) = x₁   
POL(c10(x₁)) = x₁   
POL(c12(x₁)) = x₁   
POL(c2(x₁)) = x₁   
POL(c7(x₁)) = x₁   
POL(c9(x₁, x₂)) = x₁ + x₂   
POL(f32_in(x₁, x₂)) = 0   
POL(f32_out1(x₁)) = 0   

(20) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(.(z0, []), .(z0, z1)) → U1(f18_in(z0, z1), .(z0, []), .(z0, z1))
f2_in(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → U2(f30_in(z2, z3, z0, z1), .(z0, .(z1, z2)), .(z0, .(z1, z3)))
U1(f18_out1, .(z0, []), .(z0, z1)) → f2_out1
U1(f18_out2(z0), .(z1, []), .(z1, z2)) → f2_out1
U2(f30_out1(z0), .(z1, .(z2, z3)), .(z1, .(z2, z4))) → f2_out1
f32_in([], z0) → f32_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f32_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f32_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f30_in(z0, z1, z2, z3) → U4(f32_in(z0, z1), z0, z1, z2, z3)
U4(f32_out1(z0), z1, z2, z3, z4) → U5(f2_in(.(z3, .(z4, z1)), .(z4, z2)), z1, z2, z3, z4, z0)
U5(f2_out1, z0, z1, z2, z3, z4) → f30_out1(z4)
f18_in(z0, z1) → U6(f2_in(.(z0, []), z1), f21_in(z1, z0), z0, z1)
U6(f2_out1, z0, z1, z2) → f18_out1
U6(z0, f21_out1(z1), z2, z3) → f18_out2(z1)

Tuples:

F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c7(F32_IN(z1, z2))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(F2_IN(.(z0, []), z1))

S tuples:

F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c7(F32_IN(z1, z2))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(F2_IN(.(z0, []), z1))

K tuples:

F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))

Defined Rule Symbols:

f2_in, U1, U2, f32_in, U3, f30_in, U4, U5, f18_in, U6

Defined Pair Symbols:

F30_IN, F2_IN, F32_IN, U4', F18_IN

Compound Symbols:

c9, c1, c2, c7, c10, c12

(21) CdtKnowledgeProof (EQUIVALENT transformation)

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

F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(F2_IN(.(z0, []), z1))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))
F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))

(22) Obligation:

Complexity Dependency Tuples Problem
Rules:

f2_in(z0, []) → f2_out1
f2_in(.(z0, []), .(z0, z1)) → U1(f18_in(z0, z1), .(z0, []), .(z0, z1))
f2_in(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → U2(f30_in(z2, z3, z0, z1), .(z0, .(z1, z2)), .(z0, .(z1, z3)))
U1(f18_out1, .(z0, []), .(z0, z1)) → f2_out1
U1(f18_out2(z0), .(z1, []), .(z1, z2)) → f2_out1
U2(f30_out1(z0), .(z1, .(z2, z3)), .(z1, .(z2, z4))) → f2_out1
f32_in([], z0) → f32_out1(z0)
f32_in(.(z0, z1), .(z0, z2)) → U3(f32_in(z1, z2), .(z0, z1), .(z0, z2))
U3(f32_out1(z0), .(z1, z2), .(z1, z3)) → f32_out1(z0)
f30_in(z0, z1, z2, z3) → U4(f32_in(z0, z1), z0, z1, z2, z3)
U4(f32_out1(z0), z1, z2, z3, z4) → U5(f2_in(.(z3, .(z4, z1)), .(z4, z2)), z1, z2, z3, z4, z0)
U5(f2_out1, z0, z1, z2, z3, z4) → f30_out1(z4)
f18_in(z0, z1) → U6(f2_in(.(z0, []), z1), f21_in(z1, z0), z0, z1)
U6(f2_out1, z0, z1, z2) → f18_out1
U6(z0, f21_out1(z1), z2, z3) → f18_out2(z1)

Tuples:

F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))
F32_IN(.(z0, z1), .(z0, z2)) → c7(F32_IN(z1, z2))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(F2_IN(.(z0, []), z1))

S tuples:

F32_IN(.(z0, z1), .(z0, z2)) → c7(F32_IN(z1, z2))

K tuples:

F2_IN(.(z0, []), .(z0, z1)) → c1(F18_IN(z0, z1))
F2_IN(.(z0, .(z1, z2)), .(z0, .(z1, z3))) → c2(F30_IN(z2, z3, z0, z1))
F30_IN(z0, z1, z2, z3) → c9(U4'(f32_in(z0, z1), z0, z1, z2, z3), F32_IN(z0, z1))
U4'(f32_out1(z0), z1, z2, z3, z4) → c10(F2_IN(.(z3, .(z4, z1)), .(z4, z2)))
F18_IN(z0, z1) → c12(F2_IN(.(z0, []), z1))

Defined Rule Symbols:

f2_in, U1, U2, f32_in, U3, f30_in, U4, U5, f18_in, U6

Defined Pair Symbols:

F30_IN, F2_IN, F32_IN, U4', F18_IN

Compound Symbols:

c9, c1, c2, c7, c10, c12