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Re: [Axiom-developer] [Proving Axiom Correct] Bootstrapping a library

From: Renaud Rioboo
Subject: Re: [Axiom-developer] [Proving Axiom Correct] Bootstrapping a library
Date: Fri, 10 Feb 2017 11:13:31 +0100
User-agent: Mozilla/5.0 (X11; Linux x86_64; rv:45.0) Gecko/20100101 Thunderbird/45.6.0

Dear Axiom gurus,

Axiom's NNI inherits from a dozen Category objects, one of which
is BasicType which contains two signatures:

 ?=?: (%,%) -> Boolean       ?~=?: (%,%) -> Boolean

In the ideal case we would decorate BasicType with the existing
definitions of = and ~= so we could create a new library structure
for the logic system. So BasicType would contain

theorem = (a, b : Type) : Boolean := .....
theorem ~= (a, b : Type) : Boolean := ....

Since BasicType is not an implementation you need to write a specification for equal and different. These specifictions should be inherited and proved at the domain level. You can see the standard library of FoCaLiZe for details.

In practice you need a language for writing logical statements and a language to prove these statements. Again see the FoCaLiZe library (for instance lattices) to see how a statement can be used in a proof.

Unfortunately it appears the Coq and Lean will not take kindly to
removing the existing libraries and replacing them with a new version
that only contains a limited number of theorems. I'm not yet sure about
FoCaL but I suspect it has the same bootstrap problem.

Inheritance is managed by the FoCaLiZe compiler together with dependencies which enables to have statements and proofs in a coherent way.

Renaud Rioboo

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