Skip to Content.
Sympa Menu

coq-club - Re: [Coq-Club] π>e

coq-club AT inria.fr

Subject: The Coq mailing list

List archive

Re: [Coq-Club] π>e


Chronological Thread 
  • From: TJ Machado <tj.machado AT gmail.com>
  • To: coq-club AT inria.fr
  • Subject: Re: [Coq-Club] π>e
  • Date: Fri, 22 Oct 2021 14:55:27 -0700
  • Authentication-results: mail2-smtp-roc.national.inria.fr; spf=None smtp.pra=tj.machado AT gmail.com; spf=Pass smtp.mailfrom=tj.machado AT gmail.com; spf=None smtp.helo=postmaster AT mail-yb1-f182.google.com
  • Ironport-data: A9a23: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
  • Ironport-hdrordr: A9a23:nd+K86qoeOVrvEfqOoBav/YaV5oSeYIsimQD101hICG9E/bo7vxG+c5w6faaskd1ZJhNo6HjBEDEewK+yXcX2+gs1NWZLW3bUQKTRekI0WKh+V3d8kbFh4lgPMlbAs5D4R7LYWSST/yW3OB1KbkdKRC8npyVuQ==
  • Ironport-phdr: A9a23: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

This seems to work if you're not opposed to using the standard library.

Require Import Reals.
Require Import Lra.

Lemma e_Rlt_PI :
  (exp 1 < PI)%R.
Proof. specialize exp_le_3; specialize PI2_3_2; lra. Qed.

On Fri, Oct 22, 2021 at 12:58 PM Agnishom Chattopadhyay <agnishom AT cmi.ac.in> wrote:
Sounds like a difficult task!

I think this is the most common approach by regular mathematics people: First, define the exponential function. Now, set e = exp(1). Next, show that the function x => exp(i x) is periodic and has a smallest period. Call this pi. Now, compute two significant digits of both and compare.

This path is extremely complex to build from the ground up. This requires defining the real numbers, defining complex differentiation and some results about numeric computation.

There are a lot of math libraries in Coq. I don't know how mature they are, but that seems like the rational place to start looking.

On Fri, Oct 22, 2021, 13:45 Alex Shkotin <alex.shkotin AT gmail.com> wrote:
Hi All!

How in Coq define π and e and prove that π>e?

Alex



Archive powered by MHonArc 2.6.19+.

Top of Page