WORST_CASE(?,O(1))
proof of input_G8GyN1HV5r.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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)


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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:

   terms(N) -> cons(recip(sqr(N)))
   sqr(0) -> 0
   sqr(s) -> s
   dbl(0) -> 0
   dbl(s) -> s
   add(0, X) -> X
   add(s, Y) -> s
   first(0, X) -> nil
   first(s, cons(Y)) -> cons(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:

first(s, cons(x0)) ->^* cons(x0)

first(0, x0) ->^* nil

add(s, x0) ->^* s

dbl(s) ->^* s

dbl(0) ->^* 0

sqr(s) ->^* s

sqr(0) ->^* 0

terms(s) ->^* cons(recip(s))

terms(0) ->^* cons(recip(0))




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(2)
BOUNDS(1, 1)