# HG changeset patch
# User ballarin
# Date 1044866722 -3600
# Node ID 722593f2f06887bc53d541400a69407107aa5b00
# Parent  91713a1915ee8505869afe8c1466d8323bcaca6a
New development of algebra: Groups.

diff -r 91713a1915ee -r 722593f2f068 src/HOL/Algebra/Group.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Algebra/Group.thy	Mon Feb 10 09:45:22 2003 +0100
@@ -0,0 +1,372 @@
+(*
+  Title:  HOL/Algebra/Group.thy
+  Id:     $Id$
+  Author: Clemens Ballarin, started 4 February 2003
+
+Based on work by Florian Kammueller, L C Paulson and Markus Wenzel.
+*)
+
+header {* Algebraic Structures up to Abelian Groups *}
+
+theory Group = FuncSet:
+
+text {*
+  Definitions follow Jacobson, Basic Algebra I, Freeman, 1985; with
+  the exception of \emph{magma} which, following Bourbaki, is a set
+  together with a binary, closed operation.
+*}
+
+section {* From Magmas to Groups *}
+
+subsection {* Definitions *}
+
+record 'a magma =
+  carrier :: "'a set"
+  mult :: "['a, 'a] => 'a" (infixl "\<otimes>\<index>" 70)
+
+record 'a group = "'a magma" +
+  one :: 'a ("\<one>\<index>")
+  m_inv :: "'a => 'a" ("inv\<index> _" [81] 80)
+
+locale magma = struct G +
+  assumes m_closed [intro, simp]:
+    "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
+
+locale semigroup = magma +
+  assumes m_assoc:
+    "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
+     (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
+
+locale group = semigroup +
+  assumes one_closed [intro, simp]: "\<one> \<in> carrier G"
+    and inv_closed [intro, simp]: "x \<in> carrier G ==> inv x \<in> carrier G"
+    and l_one [simp]: "x \<in> carrier G ==> \<one> \<otimes> x = x"
+    and l_inv: "x \<in> carrier G ==> inv x \<otimes> x = \<one>"
+
+subsection {* Cancellation Laws and Basic Properties *}
+
+lemma (in group) l_cancel [simp]:
+  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
+   (x \<otimes> y = x \<otimes> z) = (y = z)"
+proof
+  assume eq: "x \<otimes> y = x \<otimes> z"
+    and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+  then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z" by (simp add: m_assoc)
+  with G show "y = z" by (simp add: l_inv)
+next
+  assume eq: "y = z"
+    and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+  then show "x \<otimes> y = x \<otimes> z" by simp
+qed
+
+lemma (in group) r_one [simp]:  
+  "x \<in> carrier G ==> x \<otimes> \<one> = x"
+proof -
+  assume x: "x \<in> carrier G"
+  then have "inv x \<otimes> (x \<otimes> \<one>) = inv x \<otimes> x"
+    by (simp add: m_assoc [symmetric] l_inv)
+  with x show ?thesis by simp 
+qed
+
+lemma (in group) r_inv:
+  "x \<in> carrier G ==> x \<otimes> inv x = \<one>"
+proof -
+  assume x: "x \<in> carrier G"
+  then have "inv x \<otimes> (x \<otimes> inv x) = inv x \<otimes> \<one>"
+    by (simp add: m_assoc [symmetric] l_inv)
+  with x show ?thesis by (simp del: r_one)
+qed
+
+lemma (in group) r_cancel [simp]:
+  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
+   (y \<otimes> x = z \<otimes> x) = (y = z)"
+proof
+  assume eq: "y \<otimes> x = z \<otimes> x"
+    and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+  then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
+    by (simp add: m_assoc [symmetric])
+  with G show "y = z" by (simp add: r_inv)
+next
+  assume eq: "y = z"
+    and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+  then show "y \<otimes> x = z \<otimes> x" by simp
+qed
+
+lemma (in group) inv_inv [simp]:
+  "x \<in> carrier G ==> inv (inv x) = x"
+proof -
+  assume x: "x \<in> carrier G"
+  then have "inv x \<otimes> inv (inv x) = inv x \<otimes> x" by (simp add: l_inv r_inv)
+  with x show ?thesis by simp
+qed
+
+lemma (in group) inv_mult:
+  "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
+proof -
+  assume G: "x \<in> carrier G" "y \<in> carrier G"
+  then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
+    by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric] l_inv)
+  with G show ?thesis by simp
+qed
+
+subsection {* Substructures *}
+
+locale submagma = var H + struct G +
+  assumes subset [intro, simp]: "H \<subseteq> carrier G"
+    and m_closed [intro, simp]: "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
+
+declare (in submagma) magma.intro [intro] semigroup.intro [intro]
+
+(*
+alternative definition of submagma
+
+locale submagma = var H + struct G +
+  assumes subset [intro, simp]: "carrier H \<subseteq> carrier G"
+    and m_equal [simp]: "mult H = mult G"
+    and m_closed [intro, simp]:
+      "[| x \<in> carrier H; y \<in> carrier H |] ==> x \<otimes> y \<in> carrier H"
+*)
+
+lemma submagma_imp_subset:
+  "submagma H G ==> H \<subseteq> carrier G"
+  by (rule submagma.subset)
+
+lemma (in submagma) subsetD [dest, simp]:
+  "x \<in> H ==> x \<in> carrier G"
+  using subset by blast
+
+lemma (in submagma) magmaI [intro]:
+  includes magma G
+  shows "magma (G(| carrier := H |))"
+  by rule simp
+
+lemma (in submagma) semigroup_axiomsI [intro]:
+  includes semigroup G
+  shows "semigroup_axioms (G(| carrier := H |))"
+    by rule (simp add: m_assoc)
+
+lemma (in submagma) semigroupI [intro]:
+  includes semigroup G
+  shows "semigroup (G(| carrier := H |))"
+  using prems by fast
+
+locale subgroup = submagma H G +
+  assumes one_closed [intro, simp]: "\<one> \<in> H"
+    and m_inv_closed [intro, simp]: "x \<in> H ==> inv x \<in> H"
+
+declare (in subgroup) group.intro [intro]
+
+lemma (in subgroup) group_axiomsI [intro]:
+  includes group G
+  shows "group_axioms (G(| carrier := H |))"
+  by rule (simp_all add: l_inv)
+
+lemma (in subgroup) groupI [intro]:
+  includes group G
+  shows "group (G(| carrier := H |))"
+  using prems by fast
+
+text {*
+  Since @{term H} is nonempty, it contains some element @{term x}.  Since
+  it is closed under inverse, it contains @{text "inv x"}.  Since
+  it is closed under product, it contains @{text "x \<otimes> inv x = \<one>"}.
+*}
+
+lemma (in group) one_in_subset:
+  "[| H \<subseteq> carrier G; H \<noteq> {}; \<forall>a \<in> H. inv a \<in> H; \<forall>a\<in>H. \<forall>b\<in>H. a \<otimes> b \<in> H |]
+   ==> \<one> \<in> H"
+by (force simp add: l_inv)
+
+text {* A characterization of subgroups: closed, non-empty subset. *}
+
+lemma (in group) subgroupI:
+  assumes subset: "H \<subseteq> carrier G" and non_empty: "H \<noteq> {}"
+    and inv: "!!a. a \<in> H ==> inv a \<in> H"
+    and mult: "!!a b. [|a \<in> H; b \<in> H|] ==> a \<otimes> b \<in> H"
+  shows "subgroup H G"
+proof
+  from subset and mult show "submagma H G" ..
+next
+  have "\<one> \<in> H" by (rule one_in_subset) (auto simp only: prems)
+  with inv show "subgroup_axioms H G"
+    by (intro subgroup_axioms.intro) simp_all
+qed
+
+text {*
+  Repeat facts of submagmas for subgroups.  Necessary???
+*}
+
+lemma (in subgroup) subset:
+  "H \<subseteq> carrier G"
+  ..
+
+lemma (in subgroup) m_closed:
+  "[| x \<in> H; y \<in> H |] ==> x \<otimes> y \<in> H"
+  ..
+
+declare magma.m_closed [simp]
+
+declare group.one_closed [iff] group.inv_closed [simp]
+  group.l_one [simp] group.r_one [simp] group.inv_inv [simp]
+
+lemma subgroup_nonempty:
+  "~ subgroup {} G"
+  by (blast dest: subgroup.one_closed)
+
+lemma (in subgroup) finite_imp_card_positive:
+  "finite (carrier G) ==> 0 < card H"
+proof (rule classical)
+  have sub: "subgroup H G" using prems ..
+  assume fin: "finite (carrier G)"
+    and zero: "~ 0 < card H"
+  then have "finite H" by (blast intro: finite_subset dest: subset)
+  with zero sub have "subgroup {} G" by simp
+  with subgroup_nonempty show ?thesis by contradiction
+qed
+
+subsection {* Direct Products *}
+
+constdefs
+  DirProdMagma ::
+    "[('a, 'c) magma_scheme, ('b, 'd) magma_scheme] => ('a \<times> 'b) magma"
+    (infixr "\<times>\<^sub>m" 80)
+  "G \<times>\<^sub>m H == (| carrier = carrier G \<times> carrier H,
+    mult = (%(xg, xh) (yg, yh). (mult G xg yg, mult H xh yh)) |)"
+
+  DirProdGroup ::
+    "[('a, 'c) group_scheme, ('b, 'd) group_scheme] => ('a \<times> 'b) group"
+    (infixr "\<times>\<^sub>g" 80)
+  "G \<times>\<^sub>g H == (| carrier = carrier (G \<times>\<^sub>m H),
+    mult = mult (G \<times>\<^sub>m H),
+    one = (one G, one H),
+    m_inv = (%(g, h). (m_inv G g, m_inv H h)) |)"
+
+lemma DirProdMagma_magma:
+  includes magma G + magma H
+  shows "magma (G \<times>\<^sub>m H)"
+  by rule (auto simp add: DirProdMagma_def)
+
+lemma DirProdMagma_semigroup_axioms:
+  includes semigroup G + semigroup H
+  shows "semigroup_axioms (G \<times>\<^sub>m H)"
+  by rule (auto simp add: DirProdMagma_def G.m_assoc H.m_assoc)
+
+lemma DirProdMagma_semigroup:
+  includes semigroup G + semigroup H
+  shows "semigroup (G \<times>\<^sub>m H)"
+  using prems
+  by (fast intro: semigroup.intro
+    DirProdMagma_magma DirProdMagma_semigroup_axioms)
+
+lemma DirProdGroup_magma:
+  includes magma G + magma H
+  shows "magma (G \<times>\<^sub>g H)"
+  by rule (auto simp add: DirProdGroup_def DirProdMagma_def)
+
+lemma DirProdGroup_semigroup_axioms:
+  includes semigroup G + semigroup H
+  shows "semigroup_axioms (G \<times>\<^sub>g H)"
+  by rule
+    (auto simp add: DirProdGroup_def DirProdMagma_def G.m_assoc H.m_assoc)
+
+lemma DirProdGroup_semigroup:
+  includes semigroup G + semigroup H
+  shows "semigroup (G \<times>\<^sub>g H)"
+  using prems
+  by (fast intro: semigroup.intro
+    DirProdGroup_magma DirProdGroup_semigroup_axioms)
+
+(* ... and further lemmas for group ... *)
+
+lemma
+  includes group G + group H
+  shows "group (G \<times>\<^sub>g H)"
+by rule
+  (auto intro: magma.intro semigroup_axioms.intro group_axioms.intro
+    simp add: DirProdGroup_def DirProdMagma_def
+      G.m_assoc H.m_assoc G.l_inv H.l_inv)
+
+subsection {* Homomorphisms *}
+
+constdefs
+  hom :: "[('a, 'c) magma_scheme, ('b, 'd) magma_scheme] => ('a => 'b)set"
+  "hom G H ==
+    {h. h \<in> carrier G -> carrier H &
+      (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (mult G x y) = mult H (h x) (h y))}"
+
+lemma (in semigroup) hom:
+  includes semigroup G
+  shows "semigroup (| carrier = hom G G, mult = op o |)"
+proof
+  show "magma (| carrier = hom G G, mult = op o |)"
+    by rule (simp add: Pi_def hom_def)
+next
+  show "semigroup_axioms (| carrier = hom G G, mult = op o |)"
+    by rule (simp add: o_assoc)
+qed
+
+lemma hom_mult:
+  "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |] 
+   ==> h (mult G x y) = mult H (h x) (h y)"
+  by (simp add: hom_def) 
+
+lemma hom_closed:
+  "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
+  by (auto simp add: hom_def funcset_mem)
+
+locale group_hom = group G + group H + var h +
+  assumes homh: "h \<in> hom G H"
+  notes hom_mult [simp] = hom_mult [OF homh]
+    and hom_closed [simp] = hom_closed [OF homh]
+
+lemma (in group_hom) one_closed [simp]:
+  "h \<one> \<in> carrier H"
+  by simp
+
+lemma (in group_hom) hom_one [simp]:
+  "h \<one> = \<one>\<^sub>2"
+proof -
+  have "h \<one> \<otimes>\<^sub>2 \<one>\<^sub>2 = h \<one> \<otimes>\<^sub>2 h \<one>"
+    by (simp add: hom_mult [symmetric] del: hom_mult)
+  then show ?thesis by (simp del: r_one)
+qed
+
+lemma (in group_hom) inv_closed [simp]:
+  "x \<in> carrier G ==> h (inv x) \<in> carrier H"
+  by simp
+
+lemma (in group_hom) hom_inv [simp]:
+  "x \<in> carrier G ==> h (inv x) = inv\<^sub>2 (h x)"
+proof -
+  assume x: "x \<in> carrier G"
+  then have "h x \<otimes>\<^sub>2 h (inv x) = \<one>\<^sub>2"
+    by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)
+  also from x have "... = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)"
+    by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)
+  finally have "h x \<otimes>\<^sub>2 h (inv x) = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" .
+  with x show ?thesis by simp
+qed
+
+section {* Abelian Structures *}
+
+subsection {* Definition *}
+
+locale abelian_semigroup = semigroup +
+  assumes m_comm: "[| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
+
+lemma (in abelian_semigroup) m_lcomm:
+  "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
+   x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
+proof -
+  assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+  from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
+  also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
+  also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
+  finally show ?thesis .
+qed
+
+lemmas (in abelian_semigroup) ac = m_assoc m_comm m_lcomm
+
+locale abelian_group = abelian_semigroup + group
+
+end