(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:
a(b(x)) → a(c(b(x)))
Rewrite Strategy: INNERMOST
(1) CpxTrsMatchBoundsProof (EQUIVALENT transformation)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1.
The certificate found is represented by the following graph.
Start state: 1
Accept states: [2]
Transitions:
1→2[a_1|0]
1→3[a_1|1]
2→2[b_1|0, c_1|0]
3→4[c_1|1]
4→2[b_1|1]
(2) BOUNDS(1, n^1)
(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(b(z0)) → a(c(b(z0)))
Tuples:
A(b(z0)) → c1(A(c(b(z0))))
S tuples:
A(b(z0)) → c1(A(c(b(z0))))
K tuples:none
Defined Rule Symbols:
a
Defined Pair Symbols:
A
Compound Symbols:
c1
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 1 trailing nodes:
A(b(z0)) → c1(A(c(b(z0))))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
a(b(z0)) → a(c(b(z0)))
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
a
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)