(0) Obligation:
Clauses:
less(0, s(X1)).
less(s(X), s(Y)) :- less(X, Y).
Query: less(a,g)
(1) PrologToCdtProblemTransformerProof (UPPER BOUND (ID) transformation)
Built complexity over-approximating cdt problems from derivation graph.
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(s(z0)) → f1_out1(0)
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
f1_in(s(z0)) → U2(f1_in(z0), s(z0))
U1(f1_out1(z0), s(z1)) → f1_out1(s(z0))
U2(f1_out1(z0), s(z1)) → f1_out1(s(z0))
Tuples:
F1_IN(s(z0)) → c1(U1'(f1_in(z0), s(z0)), F1_IN(z0))
F1_IN(s(z0)) → c2(U2'(f1_in(z0), s(z0)), F1_IN(z0))
S tuples:
F1_IN(s(z0)) → c1(U1'(f1_in(z0), s(z0)), F1_IN(z0))
F1_IN(s(z0)) → c2(U2'(f1_in(z0), s(z0)), F1_IN(z0))
K tuples:none
Defined Rule Symbols:
f1_in, U1, U2
Defined Pair Symbols:
F1_IN
Compound Symbols:
c1, c2
(3) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 2 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(s(z0)) → f1_out1(0)
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
f1_in(s(z0)) → U2(f1_in(z0), s(z0))
U1(f1_out1(z0), s(z1)) → f1_out1(s(z0))
U2(f1_out1(z0), s(z1)) → f1_out1(s(z0))
Tuples:
F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))
S tuples:
F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))
K tuples:none
Defined Rule Symbols:
f1_in, U1, U2
Defined Pair Symbols:
F1_IN
Compound Symbols:
c1, c2
(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(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))
We considered the (Usable) Rules:none
And the Tuples:
F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F1_IN(x1)) = x1
POL(c1(x1)) = x1
POL(c2(x1)) = x1
POL(s(x1)) = [2] + x1
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f1_in(s(z0)) → f1_out1(0)
f1_in(s(z0)) → U1(f1_in(z0), s(z0))
f1_in(s(z0)) → U2(f1_in(z0), s(z0))
U1(f1_out1(z0), s(z1)) → f1_out1(s(z0))
U2(f1_out1(z0), s(z1)) → f1_out1(s(z0))
Tuples:
F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))
S tuples:none
K tuples:
F1_IN(s(z0)) → c1(F1_IN(z0))
F1_IN(s(z0)) → c2(F1_IN(z0))
Defined Rule Symbols:
f1_in, U1, U2
Defined Pair Symbols:
F1_IN
Compound Symbols:
c1, c2
(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(s(z0)) → f2_out1(0)
f2_in(s(s(z0))) → f2_out1(s(0))
f2_in(s(s(z0))) → U1(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U3(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U4(f2_in(z0), s(s(z0)))
U1(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U2(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U3(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U4(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
Tuples:
F2_IN(s(s(z0))) → c2(U1'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c3(U2'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c4(U3'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c5(U4'(f2_in(z0), s(s(z0))), F2_IN(z0))
S tuples:
F2_IN(s(s(z0))) → c2(U1'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c3(U2'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c4(U3'(f2_in(z0), s(s(z0))), F2_IN(z0))
F2_IN(s(s(z0))) → c5(U4'(f2_in(z0), s(s(z0))), F2_IN(z0))
K tuples:none
Defined Rule Symbols:
f2_in, U1, U2, U3, U4
Defined Pair Symbols:
F2_IN
Compound Symbols:
c2, c3, c4, c5
(11) CdtGraphRemoveTrailingTuplepartsProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing tuple parts
(12) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(s(z0)) → f2_out1(0)
f2_in(s(s(z0))) → f2_out1(s(0))
f2_in(s(s(z0))) → U1(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U3(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U4(f2_in(z0), s(s(z0)))
U1(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U2(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U3(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U4(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
Tuples:
F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))
S tuples:
F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))
K tuples:none
Defined Rule Symbols:
f2_in, U1, U2, U3, U4
Defined Pair Symbols:
F2_IN
Compound Symbols:
c2, c3, c4, c5
(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(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))
We considered the (Usable) Rules:none
And the Tuples:
F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(F2_IN(x1)) = x1
POL(c2(x1)) = x1
POL(c3(x1)) = x1
POL(c4(x1)) = x1
POL(c5(x1)) = x1
POL(s(x1)) = [1] + x1
(14) Obligation:
Complexity Dependency Tuples Problem
Rules:
f2_in(s(z0)) → f2_out1(0)
f2_in(s(s(z0))) → f2_out1(s(0))
f2_in(s(s(z0))) → U1(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U2(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U3(f2_in(z0), s(s(z0)))
f2_in(s(s(z0))) → U4(f2_in(z0), s(s(z0)))
U1(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U2(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U3(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
U4(f2_out1(z0), s(s(z1))) → f2_out1(s(s(z0)))
Tuples:
F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))
S tuples:none
K tuples:
F2_IN(s(s(z0))) → c2(F2_IN(z0))
F2_IN(s(s(z0))) → c3(F2_IN(z0))
F2_IN(s(s(z0))) → c4(F2_IN(z0))
F2_IN(s(s(z0))) → c5(F2_IN(z0))
Defined Rule Symbols:
f2_in, U1, U2, U3, U4
Defined Pair Symbols:
F2_IN
Compound Symbols:
c2, c3, c4, c5