(0) Obligation:
Clauses:
f([], RES, RES).
f(.(Head, Tail), X, RES) :- g(Tail, X, .(Head, Tail), RES).
g(A, B, C, RES) :- f(A, .(B, C), RES).
Query: f(g,g,a)
(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)
Built complexity over-approximating cdt problems from derivation graph.
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in([], z0) → f1_out1(z0)
f1_in(.(z0, z1), z2) → U1(f1_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2)
U1(f1_out1(z0), .(z1, z2), z3) → f1_out1(z0)
Tuples:
F1_IN(.(z0, z1), z2) → c1(U1'(f1_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2), F1_IN(z1, .(z2, .(z0, z1))))
S tuples:
F1_IN(.(z0, z1), z2) → c1(U1'(f1_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2), F1_IN(z1, .(z2, .(z0, z1))))
K tuples:none
Defined Rule Symbols:
f1_in, U1
Defined Pair Symbols:
F1_IN
Compound Symbols:
c1
(3) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in([], z0) → f1_out1(z0)
f1_in(.(z0, z1), z2) → U1(f1_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2)
U1(f1_out1(z0), .(z1, z2), z3) → f1_out1(z0)
Tuples:
F1_IN(.(z0, z1), z2) → c1(F1_IN(z1, .(z2, .(z0, z1))))
S tuples:
F1_IN(.(z0, z1), z2) → c1(F1_IN(z1, .(z2, .(z0, z1))))
K tuples:none
Defined Rule Symbols:
f1_in, U1
Defined Pair Symbols:
F1_IN
Compound Symbols:
c1
(5) 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), z2) → c1(F1_IN(z1, .(z2, .(z0, z1))))
We considered the (Usable) Rules:none
And the Tuples:
F1_IN(.(z0, z1), z2) → c1(F1_IN(z1, .(z2, .(z0, z1))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [3] + x2
POL(F1_IN(x1, x2)) = [3]x1
POL(c1(x1)) = x1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in([], z0) → f1_out1(z0)
f1_in(.(z0, z1), z2) → U1(f1_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2)
U1(f1_out1(z0), .(z1, z2), z3) → f1_out1(z0)
Tuples:
F1_IN(.(z0, z1), z2) → c1(F1_IN(z1, .(z2, .(z0, z1))))
S tuples:none
K tuples:
F1_IN(.(z0, z1), z2) → c1(F1_IN(z1, .(z2, .(z0, z1))))
Defined Rule Symbols:
f1_in, U1
Defined Pair Symbols:
F1_IN
Compound Symbols:
c1
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))
(9) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)
Built complexity over-approximating cdt problems from derivation graph.
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in([], z0) → f2_out1(z0)
f2_in(.(z0, z1), z2) → U1(f2_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2)
U1(f2_out1(z0), .(z1, z2), z3) → f2_out1(z0)
Tuples:
F2_IN(.(z0, z1), z2) → c1(U1'(f2_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2), F2_IN(z1, .(z2, .(z0, z1))))
S tuples:
F2_IN(.(z0, z1), z2) → c1(U1'(f2_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2), F2_IN(z1, .(z2, .(z0, z1))))
K tuples:none
Defined Rule Symbols:
f2_in, U1
Defined Pair Symbols:
F2_IN
Compound Symbols:
c1
(11) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in([], z0) → f2_out1(z0)
f2_in(.(z0, z1), z2) → U1(f2_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2)
U1(f2_out1(z0), .(z1, z2), z3) → f2_out1(z0)
Tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, .(z2, .(z0, z1))))
S tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, .(z2, .(z0, z1))))
K tuples:none
Defined Rule Symbols:
f2_in, U1
Defined Pair Symbols:
F2_IN
Compound Symbols:
c1
(13) CdtPolyRedPairProof (UPPER BOUND (ADD(O(n^1))) transformation)
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, .(z2, .(z0, z1))))
We considered the (Usable) Rules:none
And the Tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, .(z2, .(z0, z1))))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [3] + x2
POL(F2_IN(x1, x2)) = [3]x1
POL(c1(x1)) = x1
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in([], z0) → f2_out1(z0)
f2_in(.(z0, z1), z2) → U1(f2_in(z1, .(z2, .(z0, z1))), .(z0, z1), z2)
U1(f2_out1(z0), .(z1, z2), z3) → f2_out1(z0)
Tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, .(z2, .(z0, z1))))
S tuples:none
K tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, .(z2, .(z0, z1))))
Defined Rule Symbols:
f2_in, U1
Defined Pair Symbols:
F2_IN
Compound Symbols:
c1