From ecb86a5f01b65de47dec543d6b9f12fe2f9ad1c3 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Beno=C3=AEt=20Legat?= <benoit.legat@gmail.com>
Date: Mon, 2 Sep 2024 11:49:44 +0200
Subject: [PATCH] Add programmatic construction to README

---
 README.md | 49 ++++++++++++++++++++++++++++++++++---------------
 1 file changed, 34 insertions(+), 15 deletions(-)

diff --git a/README.md b/README.md
index 88e08ec..c9c65a2 100644
--- a/README.md
+++ b/README.md
@@ -8,24 +8,43 @@
 Extension of MultivariatePolynomials to semialgebraic sets, i.e. sets defined by inequalities and equalities between polynomials.
 The following example shows how to build an [algebraic set/algebraic variety](https://en.wikipedia.org/wiki/Algebraic_variety)
 ```julia
-using TypedPolynomials
-@polyvar x y z
-# Algebraic variety https://en.wikipedia.org/wiki/Algebraic_variety#/media/File:Elliptic_curve2.png
-@set y^2 == x^3 - x
-@set x^3 == 2x*y && x^2*y == 2y^2 - x
-@set x*y^2 == x*z - y && x*y == z^2 && x == y*z^4
-@set x^4*y^2 == z^5 && x^3*y^3 == 1 && x^2*y^4 == 2z
-@set x == z^2 && y == z^3
+julia> using SemialgebraicSets, TypedPolynomials
+
+julia> @polyvar x y z;
+
+julia> @set x*y^2 == x*z - y && x*y == z^2 && x == y*z^4
+Algebraic Set defined by 3 equalities
+ y - x*z + x*y^2 = 0
+ -z^2 + x*y = 0
+ x - y*z^4 = 0
+
+julia> algebraic_set([x*y^2 - x*z - y, x*y - z^2, x - y*z^4])
+Algebraic Set defined by 3 equalities
+ -y - x*z + x*y^2 = 0
+ -z^2 + x*y = 0
+ x - y*z^4 = 0
 ```
 
-The following example shows how to build an [basic semialgebraic set](http://www.mit.edu/~parrilo/cdc03_workshop/10_positivstellensatz_2003_12_07_02_screen.pdf)
+The following example shows how to build an [basic semialgebraic set](http://www.mit.edu/~parrilo/cdc03_workshop/10_positivstellensatz_2003_12_07_02_screen.pdf).
 ```julia
-using TypedPolynomials
-@polyvar x y
-@set x^2 + y^2 <= 1 # Euclidean ball
-# Cutting the algebraic variety https://en.wikipedia.org/wiki/Algebraic_variety#/media/File:Elliptic_curve2.png
-@set y^2 == x^3 - x && x <= 0
-@set y^2 == x^3 - x && x >= 1
+julia> using SemialgebraicSets, TypedPolynomials
+
+julia> @polyvar x y;
+
+julia> @set x^2 + y^2 <= 1 # Euclidean ball
+
+julia> @set y^2 == x^3 - x && x <= 0 # Cutting the algebraic variety https://en.wikipedia.org/wiki/Algebraic_variety#/media/File:Elliptic_curve2.png
+Basic semialgebraic Set defined by 1 equalitty
+ x + y^2 - x^3 = 0
+1 inequalitty
+ -x ≥ 0
+
+
+julia> basic_semialgebraic_set(algebraic_set([y^2- x^3 - x]), [-x])
+Basic semialgebraic Set defined by 1 equalitty
+ -x + y^2 - x^3 = 0
+1 inequalitty
+ -x ≥ 0
 ```
 
 ## Solving systems of algebraic equations