WORST_CASE(?,O(1))
proof of input_Td2UGc4ypJ.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).

(0) CpxTRS
(1) NarrowingOnBasicTermsTerminatesProof [FINISHED, 0 ms]
(2) BOUNDS(1, 1)


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, 1).


The TRS R consists of the following rules:

   f(n__f(n__a)) -> f(n__g(f(n__a)))
   f(X) -> n__f(X)
   a -> n__a
   g(X) -> n__g(X)
   activate(n__f(X)) -> f(X)
   activate(n__a) -> a
   activate(n__g(X)) -> g(X)
   activate(X) -> X

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) NarrowingOnBasicTermsTerminatesProof (FINISHED)
Constant runtime complexity proven by termination of constructor-based narrowing.

The maximal most general narrowing sequences give rise to the following rewrite sequences:

activate(n__g(x0)) ->^* n__g(x0)

activate(n__a) ->^* n__a

activate(n__f(x0)) ->^* n__f(x0)

activate(n__f(n__f(n__a))) ->^* n__f(n__g(n__f(n__a)))

g(x0) ->^* n__g(x0)

a ->^* n__a

f(x0) ->^* n__f(x0)

f(n__f(n__a)) ->^* n__f(n__g(n__f(n__a)))




----------------------------------------

(2)
BOUNDS(1, 1)