(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
a(a(x)) → c
b(u(x)) → b(d(x))
d(a(x)) → a(d(x))
d(b(x)) → u(a(b(x)))
a(u(x)) → u(a(x))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:
A(u(z0)) → c2(A(z0))
B(u(z0)) → c3(B(d(z0)), D(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(A(b(z0)), B(z0))
S tuples:
A(u(z0)) → c2(A(z0))
B(u(z0)) → c3(B(d(z0)), D(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(A(b(z0)), B(z0))
K tuples:none
Defined Rule Symbols:
a, b, d
Defined Pair Symbols:
A, B, D
Compound Symbols:
c2, c3, c4, c5
(3) CdtGraphRemoveTrailingProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing tuple parts
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:
A(u(z0)) → c2(A(z0))
B(u(z0)) → c3(B(d(z0)), D(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
S tuples:
A(u(z0)) → c2(A(z0))
B(u(z0)) → c3(B(d(z0)), D(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
K tuples:none
Defined Rule Symbols:
a, b, d
Defined Pair Symbols:
A, B, D
Compound Symbols:
c2, c3, c4, c5
(5) CdtNarrowingProof (BOTH BOUNDS(ID, ID) transformation)
Use narrowing to replace
B(
u(
z0)) →
c3(
B(
d(
z0)),
D(
z0)) by
B(u(a(z0))) → c3(B(a(d(z0))), D(a(z0)))
B(u(b(z0))) → c3(B(u(a(b(z0)))), D(b(z0)))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:
A(u(z0)) → c2(A(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
B(u(a(z0))) → c3(B(a(d(z0))), D(a(z0)))
B(u(b(z0))) → c3(B(u(a(b(z0)))), D(b(z0)))
S tuples:
A(u(z0)) → c2(A(z0))
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
B(u(a(z0))) → c3(B(a(d(z0))), D(a(z0)))
B(u(b(z0))) → c3(B(u(a(b(z0)))), D(b(z0)))
K tuples:none
Defined Rule Symbols:
a, b, d
Defined Pair Symbols:
A, D, B
Compound Symbols:
c2, c4, c5, c3
(7) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
D(a(z0)) → c4(A(d(z0)), D(z0))
D(b(z0)) → c5(B(z0))
B(u(a(z0))) → c3(B(a(d(z0))), D(a(z0)))
B(u(b(z0))) → c3(B(u(a(b(z0)))), D(b(z0)))
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:
A(u(z0)) → c2(A(z0))
S tuples:
A(u(z0)) → c2(A(z0))
K tuples:none
Defined Rule Symbols:
a, b, d
Defined Pair Symbols:
A
Compound Symbols:
c2
(9) 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.
A(u(z0)) → c2(A(z0))
We considered the (Usable) Rules:none
And the Tuples:
A(u(z0)) → c2(A(z0))
The order we found is given by the following interpretation:
Polynomial interpretation :
POL(A(x1)) = x1
POL(c2(x1)) = x1
POL(u(x1)) = [2] + x1
(10) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(a(z0)) → c
a(u(z0)) → u(a(z0))
b(u(z0)) → b(d(z0))
d(a(z0)) → a(d(z0))
d(b(z0)) → u(a(b(z0)))
Tuples:
A(u(z0)) → c2(A(z0))
S tuples:none
K tuples:
A(u(z0)) → c2(A(z0))
Defined Rule Symbols:
a, b, d
Defined Pair Symbols:
A
Compound Symbols:
c2
(11) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(12) BOUNDS(O(1), O(1))