src/HOL/Algebra/README.thy

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+theory README imports Main
+begin
+
+section \<open>Algebra --- Classical Algebra, using Explicit Structures and Locales\<close>
+
+text \<open>
+  This directory contains proofs in classical algebra. It is intended as a
+  base for any algebraic development in Isabelle. Emphasis is on reusability.
+  This is achieved by modelling algebraic structures as first-class citizens
+  of the logic (not axiomatic type classes, say). The library is expected to
+  grow in future releases of Isabelle. Contributions are welcome.
+\<close>
+
+subsection \<open>GroupTheory, including Sylow's Theorem\<close>
+
+text \<open>
+  These proofs are mainly by Florian Kammüller. (Later, Larry Paulson
+  simplified some of the proofs.) These theories were indeed the original
+  motivation for locales.
+
+  Here is an outline of the directory's contents:
+
+  \<^item> Theory \<^file>\<open>Group.thy\<close> defines semigroups, monoids, groups, commutative
+    monoids, commutative groups, homomorphisms and the subgroup relation. It
+    also defines the product of two groups (This theory was reimplemented by
+    Clemens Ballarin).
+
+  \<^item> Theory \<^file>\<open>FiniteProduct.thy\<close> extends commutative groups by a product
+    operator for finite sets (provided by Clemens Ballarin).
+
+  \<^item> Theory \<^file>\<open>Coset.thy\<close> defines the factorization of a group and shows that
+    the factorization a normal subgroup is a group.
+
+  \<^item> Theory \<^file>\<open>Bij.thy\<close> defines bijections over sets and operations on them and
+    shows that they are a group. It shows that automorphisms form a group.
+
+  \<^item> Theory \<^file>\<open>Exponent.thy\<close> the combinatorial argument underlying Sylow's
+    first theorem.
+
+  \<^item> Theory \<^file>\<open>Sylow.thy\<close> contains a proof of the first Sylow theorem.
+\<close>
+
+
+subsection \<open>Rings and Polynomials\<close>
+
+text \<open>
+  \<^item> Theory \<^file>\<open>Ring.thy\<close> defines Abelian monoids and groups. The difference to
+    commutative structures is merely notational: the binary operation is
+    addition rather than multiplication. Commutative rings are obtained by
+    inheriting properties from Abelian groups and commutative monoids. Further
+    structures in the algebraic hierarchy of rings: integral domain.
+
+  \<^item> Theory \<^file>\<open>Module.thy\<close> introduces the notion of a R-left-module over an
+    Abelian group, where R is a ring.
+
+  \<^item> Theory \<^file>\<open>UnivPoly.thy\<close> constructs univariate polynomials over rings and
+    integral domains. Degree function. Universal Property.
+\<close>
+
+
+subsection \<open>Development of Polynomials using Type Classes\<close>
+
+text \<open>
+  A development of univariate polynomials for HOL's ring classes is available
+  at \<^file>\<open>~~/src/HOL/Computational_Algebra/Polynomial.thy\<close>.
+
+  [Jacobson1985] Nathan Jacobson, Basic Algebra I, Freeman, 1985.
+
+  [Ballarin1999] Clemens Ballarin, Computer Algebra and Theorem Proving,
+  Author's PhD thesis, 1999. Also University of Cambridge, Computer Laboratory
+  Technical Report number 473.
+\<close>
+
+end