WORST_CASE(?,O(1))
proof of input_9IJnf2uzda.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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)


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(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(0, 1, x) -> f(h(x), h(x), x)
   h(0) -> 0
   h(g(x, y)) -> y

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(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:

h(0) ->^* 0

f(0, 1, g(x0, x1)) ->^* f(x1, x1, g(x0, x1))

f(0, 1, 0) ->^* f(0, 0, 0)

f(0, 1, g(x0, x1)) ->^* f(x1, x1, g(x0, x1))

f(0, 1, 0) ->^* f(0, 0, 0)




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(2)
BOUNDS(1, 1)