README.md

Fol-archived

A New Coq Formalisation of Classical First-Order Logic with Proofs of the Soundness and Completeness Theorems.

Contents

1. How to build

2. Overview of files

3. References

1. How to build

1. Install opam.

2. Open this repository in your terminal.

3. Type the following script in the terminal:

opam switch create . ocaml.5.1.1
eval $(opam env)
opam pin add coq 8.18.0 -y
coq_makefile -f _CoqProject -o Makefile
make -j4 -k

The expected output

@HilbertCalculus_sound
     : forall (L : language) (Gamma : ensemble (frm L)) (C : frm L),
       Gamma ⊢ C -> Gamma ⊨ C
Axioms:
classic : forall P : Prop, P \/ ~ P
@HilbertCalculus_complete
     : forall L : language,
       isCountable (function_symbols L) ->
       isCountable (constant_symbols L) ->
       isCountable (relation_symbols L) ->
       forall (X : ensemble (frm L)) (b : frm L), X ⊨ b -> X ⊢ b
Axioms:
classic : forall P : Prop, P \/ ~ P

2. Overview of files

Data

• Vector.v replaces Coq.Vectors.Vector.

Index

• Index.v accumulates all source code and checks for consistency.

Logic

• BasicFol.v contains basic definitions of first-order logic.

• BasicFol2.v contains extra definitions of first-order logic.

• ClassicalFol.v formalises the meta-theory on first-order logic, using the axiom classic : forall P : Prop, P \/ ~ P.

• HilbertFol.v contains basic facts about a Hilbert calculus for first-order logic.

• HilbertFol2.v contains advanced facts about the Hilbert calculus for first-order logic.

Math

• BooleanAlgebra.v formalises a basic theory on Boolean algebras.

• ThN.v contains basic facts about the natural numbers.

Prelude

• ClassicalFacts.v contains facts about CIC + (classic : forall P : Prop, P \/ ~ P).

• ConstructiveFacts.v contains facts about CIC.

• Notations.v reserves all notations to avoid the conflict.

• SfLib.v is a copy of snu-sf/sflib.v.

• Prelude.v is the prelude of this repository.

3. References

1. sflib

2. stdpp

3. Constructive Completeness Proofs and Delimited Control

4. CoqGym/goedel

5. formalmetatheory-stoughton

6. Constructive Analysis of First-Order Completeness