(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:
f(f(a)) → f(g(n__f(a)))
f(X) → n__f(X)
activate(n__f(X)) → f(X)
activate(X) → X
Rewrite Strategy: INNERMOST
(1) NestedDefinedSymbolProof (BOTH BOUNDS(ID, ID) transformation)
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
f(f(a)) → f(g(n__f(a)))
(2) 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:
activate(n__f(X)) → f(X)
f(X) → n__f(X)
activate(X) → X
Rewrite Strategy: INNERMOST
(3) 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 2.
The certificate found is represented by the following graph.
Start state: 3
Accept states: [4]
Transitions:
3→4[activate_1|0, f_1|0, f_1|1, n__f_1|1, n__f_1|2]
4→4[n__f_1|0]
(4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
Tuples:
ACTIVATE(n__f(z0)) → c(F(z0))
ACTIVATE(z0) → c1
F(z0) → c2
S tuples:
ACTIVATE(n__f(z0)) → c(F(z0))
ACTIVATE(z0) → c1
F(z0) → c2
K tuples:none
Defined Rule Symbols:
activate, f
Defined Pair Symbols:
ACTIVATE, F
Compound Symbols:
c, c1, c2
(7) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 3 trailing nodes:
ACTIVATE(n__f(z0)) → c(F(z0))
F(z0) → c2
ACTIVATE(z0) → c1
(8) Obligation:
Complexity Dependency Tuples Problem
Rules:
activate(n__f(z0)) → f(z0)
activate(z0) → z0
f(z0) → n__f(z0)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
activate, f
Defined Pair Symbols:none
Compound Symbols:none
(9) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(10) BOUNDS(1, 1)