(0) Obligation:
Clauses:
append(.(H, X), Y, .(X, Z)) :- append(X, Y, Z).
append([], Y, Y).
Query: append(g,g,a)
(1) LPReorderTransformerProof (EQUIVALENT transformation)
Reordered facts before rules in definite LP [PROLOG].
(2) Obligation:
Clauses:
append([], Y, Y).
append(.(H, X), Y, .(X, Z)) :- append(X, Y, Z).
Query: append(g,g,a)
(3) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)
Built complexity over-approximating cdt problems from derivation graph.
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in([], z0) → f2_out1(z0)
f2_in(.(z0, z1), z2) → U1(f2_in(z1, z2), .(z0, z1), z2)
U1(f2_out1(z0), .(z1, z2), z3) → f2_out1(.(z2, z0))
Tuples:
F2_IN(.(z0, z1), z2) → c1(U1'(f2_in(z1, z2), .(z0, z1), z2), F2_IN(z1, z2))
S tuples:
F2_IN(.(z0, z1), z2) → c1(U1'(f2_in(z1, z2), .(z0, z1), z2), F2_IN(z1, z2))
K tuples:none
Defined Rule Symbols:
f2_in, U1
Defined Pair Symbols:
F2_IN
Compound Symbols:
c1
(5) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in([], z0) → f2_out1(z0)
f2_in(.(z0, z1), z2) → U1(f2_in(z1, z2), .(z0, z1), z2)
U1(f2_out1(z0), .(z1, z2), z3) → f2_out1(.(z2, z0))
Tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, z2))
S tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, z2))
K tuples:none
Defined Rule Symbols:
f2_in, U1
Defined Pair Symbols:
F2_IN
Compound Symbols:
c1
(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(.(z0, z1), z2) → c1(F2_IN(z1, z2))
We considered the (Usable) Rules:none
And the Tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, z2))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [1] + x2
POL(F2_IN(x1, x2)) = x1
POL(c1(x1)) = x1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in([], z0) → f2_out1(z0)
f2_in(.(z0, z1), z2) → U1(f2_in(z1, z2), .(z0, z1), z2)
U1(f2_out1(z0), .(z1, z2), z3) → f2_out1(.(z2, z0))
Tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, z2))
S tuples:none
K tuples:
F2_IN(.(z0, z1), z2) → c1(F2_IN(z1, z2))
Defined Rule Symbols:
f2_in, U1
Defined Pair Symbols:
F2_IN
Compound Symbols:
c1
(9) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(10) BOUNDS(O(1), O(1))
(11) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)
Built complexity over-approximating cdt problems from derivation graph.
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in([], z0) → f1_out1(z0)
f1_in(.(z0, []), z1) → f1_out1(.([], z1))
f1_in(.(z0, .(z1, z2)), z3) → U1(f1_in(z2, z3), .(z0, .(z1, z2)), z3)
U1(f1_out1(z0), .(z1, .(z2, z3)), z4) → f1_out1(.(.(z2, z3), .(z3, z0)))
Tuples:
F1_IN(.(z0, .(z1, z2)), z3) → c2(U1'(f1_in(z2, z3), .(z0, .(z1, z2)), z3), F1_IN(z2, z3))
S tuples:
F1_IN(.(z0, .(z1, z2)), z3) → c2(U1'(f1_in(z2, z3), .(z0, .(z1, z2)), z3), F1_IN(z2, z3))
K tuples:none
Defined Rule Symbols:
f1_in, U1
Defined Pair Symbols:
F1_IN
Compound Symbols:
c2
(13) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in([], z0) → f1_out1(z0)
f1_in(.(z0, []), z1) → f1_out1(.([], z1))
f1_in(.(z0, .(z1, z2)), z3) → U1(f1_in(z2, z3), .(z0, .(z1, z2)), z3)
U1(f1_out1(z0), .(z1, .(z2, z3)), z4) → f1_out1(.(.(z2, z3), .(z3, z0)))
Tuples:
F1_IN(.(z0, .(z1, z2)), z3) → c2(F1_IN(z2, z3))
S tuples:
F1_IN(.(z0, .(z1, z2)), z3) → c2(F1_IN(z2, z3))
K tuples:none
Defined Rule Symbols:
f1_in, U1
Defined Pair Symbols:
F1_IN
Compound Symbols:
c2
(15) 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)), z3) → c2(F1_IN(z2, z3))
We considered the (Usable) Rules:none
And the Tuples:
F1_IN(.(z0, .(z1, z2)), z3) → c2(F1_IN(z2, z3))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(.(x1, x2)) = [1] + x2
POL(F1_IN(x1, x2)) = [2]x1
POL(c2(x1)) = x1
(16) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in([], z0) → f1_out1(z0)
f1_in(.(z0, []), z1) → f1_out1(.([], z1))
f1_in(.(z0, .(z1, z2)), z3) → U1(f1_in(z2, z3), .(z0, .(z1, z2)), z3)
U1(f1_out1(z0), .(z1, .(z2, z3)), z4) → f1_out1(.(.(z2, z3), .(z3, z0)))
Tuples:
F1_IN(.(z0, .(z1, z2)), z3) → c2(F1_IN(z2, z3))
S tuples:none
K tuples:
F1_IN(.(z0, .(z1, z2)), z3) → c2(F1_IN(z2, z3))
Defined Rule Symbols:
f1_in, U1
Defined Pair Symbols:
F1_IN
Compound Symbols:
c2