(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
d(x) → e(u(x))
d(u(x)) → c(x)
c(u(x)) → b(x)
v(e(x)) → x
b(u(x)) → a(e(x))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
d(z0) → e(u(z0))
d(u(z0)) → c(z0)
c(u(z0)) → b(z0)
v(e(z0)) → z0
b(u(z0)) → a(e(z0))
Tuples:
D(z0) → c1
D(u(z0)) → c2(C(z0))
C(u(z0)) → c3(B(z0))
V(e(z0)) → c4
B(u(z0)) → c5
S tuples:
D(z0) → c1
D(u(z0)) → c2(C(z0))
C(u(z0)) → c3(B(z0))
V(e(z0)) → c4
B(u(z0)) → c5
K tuples:none
Defined Rule Symbols:
d, c, v, b
Defined Pair Symbols:
D, C, V, B
Compound Symbols:
c1, c2, c3, c4, c5
(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 5 trailing nodes:
D(u(z0)) → c2(C(z0))
C(u(z0)) → c3(B(z0))
D(z0) → c1
V(e(z0)) → c4
B(u(z0)) → c5
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
d(z0) → e(u(z0))
d(u(z0)) → c(z0)
c(u(z0)) → b(z0)
v(e(z0)) → z0
b(u(z0)) → a(e(z0))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
d, c, v, b
Defined Pair Symbols:none
Compound Symbols:none
(5) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(6) BOUNDS(1, 1)