Theory Coset

theory Coset
imports Group
(*  Title:      HOL/Algebra/Coset.thy
    Author:     Florian Kammueller
    Author:     L C Paulson
    Author:     Stephan Hohe
*)

theory Coset
imports Group
begin

section ‹Cosets and Quotient Groups›

definition
  r_coset    :: "[_, 'a set, 'a] ⇒ 'a set"    (infixl "#>ı" 60)
  where "H #>G a = (⋃h∈H. {h ⊗G a})"

definition
  l_coset    :: "[_, 'a, 'a set] ⇒ 'a set"    (infixl "<#ı" 60)
  where "a <#G H = (⋃h∈H. {a ⊗G h})"

definition
  RCOSETS  :: "[_, 'a set] ⇒ ('a set)set"   ("rcosetsı _" [81] 80)
  where "rcosetsG H = (⋃a∈carrier G. {H #>G a})"

definition
  set_mult  :: "[_, 'a set ,'a set] ⇒ 'a set" (infixl "<#>ı" 60)
  where "H <#>G K = (⋃h∈H. ⋃k∈K. {h ⊗G k})"

definition
  SET_INV :: "[_,'a set] ⇒ 'a set"  ("set'_invı _" [81] 80)
  where "set_invG H = (⋃h∈H. {invG h})"


locale normal = subgroup + group +
  assumes coset_eq: "(∀x ∈ carrier G. H #> x = x <# H)"

abbreviation
  normal_rel :: "['a set, ('a, 'b) monoid_scheme] ⇒ bool"  (infixl "⊲" 60) where
  "H ⊲ G ≡ normal H G"


subsection ‹Basic Properties of Cosets›

lemma (in group) coset_mult_assoc:
     "[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |]
      ==> (M #> g) #> h = M #> (g ⊗ h)"
by (force simp add: r_coset_def m_assoc)

lemma (in group) coset_mult_one [simp]: "M ⊆ carrier G ==> M #> 𝟭 = M"
by (force simp add: r_coset_def)

lemma (in group) coset_mult_inv1:
     "[| M #> (x ⊗ (inv y)) = M;  x ∈ carrier G ; y ∈ carrier G;
         M ⊆ carrier G |] ==> M #> x = M #> y"
apply (erule subst [of concl: "%z. M #> x = z #> y"])
apply (simp add: coset_mult_assoc m_assoc)
done

lemma (in group) coset_mult_inv2:
     "[| M #> x = M #> y;  x ∈ carrier G;  y ∈ carrier G;  M ⊆ carrier G |]
      ==> M #> (x ⊗ (inv y)) = M "
apply (simp add: coset_mult_assoc [symmetric])
apply (simp add: coset_mult_assoc)
done

lemma (in group) coset_join1:
     "[| H #> x = H;  x ∈ carrier G;  subgroup H G |] ==> x ∈ H"
apply (erule subst)
apply (simp add: r_coset_def)
apply (blast intro: l_one subgroup.one_closed sym)
done

lemma (in group) solve_equation:
    "⟦subgroup H G; x ∈ H; y ∈ H⟧ ⟹ ∃h∈H. y = h ⊗ x"
apply (rule bexI [of _ "y ⊗ (inv x)"])
apply (auto simp add: subgroup.m_closed subgroup.m_inv_closed m_assoc
                      subgroup.subset [THEN subsetD])
done

lemma (in group) repr_independence:
     "⟦y ∈ H #> x;  x ∈ carrier G; subgroup H G⟧ ⟹ H #> x = H #> y"
by (auto simp add: r_coset_def m_assoc [symmetric]
                   subgroup.subset [THEN subsetD]
                   subgroup.m_closed solve_equation)

lemma (in group) coset_join2:
     "⟦x ∈ carrier G;  subgroup H G;  x∈H⟧ ⟹ H #> x = H"
  --‹Alternative proof is to put @{term "x=𝟭"} in @{text repr_independence}.›
by (force simp add: subgroup.m_closed r_coset_def solve_equation)

lemma (in monoid) r_coset_subset_G:
     "[| H ⊆ carrier G; x ∈ carrier G |] ==> H #> x ⊆ carrier G"
by (auto simp add: r_coset_def)

lemma (in group) rcosI:
     "[| h ∈ H; H ⊆ carrier G; x ∈ carrier G|] ==> h ⊗ x ∈ H #> x"
by (auto simp add: r_coset_def)

lemma (in group) rcosetsI:
     "⟦H ⊆ carrier G; x ∈ carrier G⟧ ⟹ H #> x ∈ rcosets H"
by (auto simp add: RCOSETS_def)

text‹Really needed?›
lemma (in group) transpose_inv:
     "[| x ⊗ y = z;  x ∈ carrier G;  y ∈ carrier G;  z ∈ carrier G |]
      ==> (inv x) ⊗ z = y"
by (force simp add: m_assoc [symmetric])

lemma (in group) rcos_self: "[| x ∈ carrier G; subgroup H G |] ==> x ∈ H #> x"
apply (simp add: r_coset_def)
apply (blast intro: sym l_one subgroup.subset [THEN subsetD]
                    subgroup.one_closed)
done

text (in group) ‹Opposite of @{thm [source] "repr_independence"}›
lemma (in group) repr_independenceD:
  assumes "subgroup H G"
  assumes ycarr: "y ∈ carrier G"
      and repr:  "H #> x = H #> y"
  shows "y ∈ H #> x"
proof -
  interpret subgroup H G by fact
  show ?thesis  apply (subst repr)
  apply (intro rcos_self)
   apply (rule ycarr)
   apply (rule is_subgroup)
  done
qed

text ‹Elements of a right coset are in the carrier›
lemma (in subgroup) elemrcos_carrier:
  assumes "group G"
  assumes acarr: "a ∈ carrier G"
    and a': "a' ∈ H #> a"
  shows "a' ∈ carrier G"
proof -
  interpret group G by fact
  from subset and acarr
  have "H #> a ⊆ carrier G" by (rule r_coset_subset_G)
  from this and a'
  show "a' ∈ carrier G"
    by fast
qed

lemma (in subgroup) rcos_const:
  assumes "group G"
  assumes hH: "h ∈ H"
  shows "H #> h = H"
proof -
  interpret group G by fact
  show ?thesis apply (unfold r_coset_def)
    apply rule
    apply rule
    apply clarsimp
    apply (intro subgroup.m_closed)
    apply (rule is_subgroup)
    apply assumption
    apply (rule hH)
    apply rule
    apply simp
  proof -
    fix h'
    assume h'H: "h' ∈ H"
    note carr = hH[THEN mem_carrier] h'H[THEN mem_carrier]
    from carr
    have a: "h' = (h' ⊗ inv h) ⊗ h" by (simp add: m_assoc)
    from h'H hH
    have "h' ⊗ inv h ∈ H" by simp
    from this and a
    show "∃x∈H. h' = x ⊗ h" by fast
  qed
qed

text ‹Step one for lemma @{text "rcos_module"}›
lemma (in subgroup) rcos_module_imp:
  assumes "group G"
  assumes xcarr: "x ∈ carrier G"
      and x'cos: "x' ∈ H #> x"
  shows "(x' ⊗ inv x) ∈ H"
proof -
  interpret group G by fact
  from xcarr x'cos
      have x'carr: "x' ∈ carrier G"
      by (rule elemrcos_carrier[OF is_group])
  from xcarr
      have ixcarr: "inv x ∈ carrier G"
      by simp
  from x'cos
      have "∃h∈H. x' = h ⊗ x"
      unfolding r_coset_def
      by fast
  from this
      obtain h
        where hH: "h ∈ H"
        and x': "x' = h ⊗ x"
      by auto
  from hH and subset
      have hcarr: "h ∈ carrier G" by fast
  note carr = xcarr x'carr hcarr
  from x' and carr
      have "x' ⊗ (inv x) = (h ⊗ x) ⊗ (inv x)" by fast
  also from carr
      have "… = h ⊗ (x ⊗ inv x)" by (simp add: m_assoc)
  also from carr
      have "… = h ⊗ 𝟭" by simp
  also from carr
      have "… = h" by simp
  finally
      have "x' ⊗ (inv x) = h" by simp
  from hH this
      show "x' ⊗ (inv x) ∈ H" by simp
qed

text ‹Step two for lemma @{text "rcos_module"}›
lemma (in subgroup) rcos_module_rev:
  assumes "group G"
  assumes carr: "x ∈ carrier G" "x' ∈ carrier G"
      and xixH: "(x' ⊗ inv x) ∈ H"
  shows "x' ∈ H #> x"
proof -
  interpret group G by fact
  from xixH
      have "∃h∈H. x' ⊗ (inv x) = h" by fast
  from this
      obtain h
        where hH: "h ∈ H"
        and hsym: "x' ⊗ (inv x) = h"
      by fast
  from hH subset have hcarr: "h ∈ carrier G" by simp
  note carr = carr hcarr
  from hsym[symmetric] have "h ⊗ x = x' ⊗ (inv x) ⊗ x" by fast
  also from carr
      have "… = x' ⊗ ((inv x) ⊗ x)" by (simp add: m_assoc)
  also from carr
      have "… = x' ⊗ 𝟭" by simp
  also from carr
      have "… = x'" by simp
  finally
      have "h ⊗ x = x'" by simp
  from this[symmetric] and hH
      show "x' ∈ H #> x"
      unfolding r_coset_def
      by fast
qed

text ‹Module property of right cosets›
lemma (in subgroup) rcos_module:
  assumes "group G"
  assumes carr: "x ∈ carrier G" "x' ∈ carrier G"
  shows "(x' ∈ H #> x) = (x' ⊗ inv x ∈ H)"
proof -
  interpret group G by fact
  show ?thesis proof  assume "x' ∈ H #> x"
    from this and carr
    show "x' ⊗ inv x ∈ H"
      by (intro rcos_module_imp[OF is_group])
  next
    assume "x' ⊗ inv x ∈ H"
    from this and carr
    show "x' ∈ H #> x"
      by (intro rcos_module_rev[OF is_group])
  qed
qed

text ‹Right cosets are subsets of the carrier.› 
lemma (in subgroup) rcosets_carrier:
  assumes "group G"
  assumes XH: "X ∈ rcosets H"
  shows "X ⊆ carrier G"
proof -
  interpret group G by fact
  from XH have "∃x∈ carrier G. X = H #> x"
      unfolding RCOSETS_def
      by fast
  from this
      obtain x
        where xcarr: "x∈ carrier G"
        and X: "X = H #> x"
      by fast
  from subset and xcarr
      show "X ⊆ carrier G"
      unfolding X
      by (rule r_coset_subset_G)
qed

text ‹Multiplication of general subsets›
lemma (in monoid) set_mult_closed:
  assumes Acarr: "A ⊆ carrier G"
      and Bcarr: "B ⊆ carrier G"
  shows "A <#> B ⊆ carrier G"
apply rule apply (simp add: set_mult_def, clarsimp)
proof -
  fix a b
  assume "a ∈ A"
  from this and Acarr
      have acarr: "a ∈ carrier G" by fast

  assume "b ∈ B"
  from this and Bcarr
      have bcarr: "b ∈ carrier G" by fast

  from acarr bcarr
      show "a ⊗ b ∈ carrier G" by (rule m_closed)
qed

lemma (in comm_group) mult_subgroups:
  assumes subH: "subgroup H G"
      and subK: "subgroup K G"
  shows "subgroup (H <#> K) G"
apply (rule subgroup.intro)
   apply (intro set_mult_closed subgroup.subset[OF subH] subgroup.subset[OF subK])
  apply (simp add: set_mult_def) apply clarsimp defer 1
  apply (simp add: set_mult_def) defer 1
  apply (simp add: set_mult_def, clarsimp) defer 1
proof -
  fix ha hb ka kb
  assume haH: "ha ∈ H" and hbH: "hb ∈ H" and kaK: "ka ∈ K" and kbK: "kb ∈ K"
  note carr = haH[THEN subgroup.mem_carrier[OF subH]] hbH[THEN subgroup.mem_carrier[OF subH]]
              kaK[THEN subgroup.mem_carrier[OF subK]] kbK[THEN subgroup.mem_carrier[OF subK]]
  from carr
      have "(ha ⊗ ka) ⊗ (hb ⊗ kb) = ha ⊗ (ka ⊗ hb) ⊗ kb" by (simp add: m_assoc)
  also from carr
      have "… = ha ⊗ (hb ⊗ ka) ⊗ kb" by (simp add: m_comm)
  also from carr
      have "… = (ha ⊗ hb) ⊗ (ka ⊗ kb)" by (simp add: m_assoc)
  finally
      have eq: "(ha ⊗ ka) ⊗ (hb ⊗ kb) = (ha ⊗ hb) ⊗ (ka ⊗ kb)" .

  from haH hbH have hH: "ha ⊗ hb ∈ H" by (simp add: subgroup.m_closed[OF subH])
  from kaK kbK have kK: "ka ⊗ kb ∈ K" by (simp add: subgroup.m_closed[OF subK])
  
  from hH and kK and eq
      show "∃h'∈H. ∃k'∈K. (ha ⊗ ka) ⊗ (hb ⊗ kb) = h' ⊗ k'" by fast
next
  have "𝟭 = 𝟭 ⊗ 𝟭" by simp
  from subgroup.one_closed[OF subH] subgroup.one_closed[OF subK] this
      show "∃h∈H. ∃k∈K. 𝟭 = h ⊗ k" by fast
next
  fix h k
  assume hH: "h ∈ H"
     and kK: "k ∈ K"

  from hH[THEN subgroup.mem_carrier[OF subH]] kK[THEN subgroup.mem_carrier[OF subK]]
      have "inv (h ⊗ k) = inv h ⊗ inv k" by (simp add: inv_mult_group m_comm)

  from subgroup.m_inv_closed[OF subH hH] and subgroup.m_inv_closed[OF subK kK] and this
      show "∃ha∈H. ∃ka∈K. inv (h ⊗ k) = ha ⊗ ka" by fast
qed

lemma (in subgroup) lcos_module_rev:
  assumes "group G"
  assumes carr: "x ∈ carrier G" "x' ∈ carrier G"
      and xixH: "(inv x ⊗ x') ∈ H"
  shows "x' ∈ x <# H"
proof -
  interpret group G by fact
  from xixH
      have "∃h∈H. (inv x) ⊗ x' = h" by fast
  from this
      obtain h
        where hH: "h ∈ H"
        and hsym: "(inv x) ⊗ x' = h"
      by fast

  from hH subset have hcarr: "h ∈ carrier G" by simp
  note carr = carr hcarr
  from hsym[symmetric] have "x ⊗ h = x ⊗ ((inv x) ⊗ x')" by fast
  also from carr
      have "… = (x ⊗ (inv x)) ⊗ x'" by (simp add: m_assoc[symmetric])
  also from carr
      have "… = 𝟭 ⊗ x'" by simp
  also from carr
      have "… = x'" by simp
  finally
      have "x ⊗ h = x'" by simp

  from this[symmetric] and hH
      show "x' ∈ x <# H"
      unfolding l_coset_def
      by fast
qed


subsection ‹Normal subgroups›

lemma normal_imp_subgroup: "H ⊲ G ⟹ subgroup H G"
  by (simp add: normal_def subgroup_def)

lemma (in group) normalI: 
  "subgroup H G ⟹ (∀x ∈ carrier G. H #> x = x <# H) ⟹ H ⊲ G"
  by (simp add: normal_def normal_axioms_def is_group)

lemma (in normal) inv_op_closed1:
     "⟦x ∈ carrier G; h ∈ H⟧ ⟹ (inv x) ⊗ h ⊗ x ∈ H"
apply (insert coset_eq) 
apply (auto simp add: l_coset_def r_coset_def)
apply (drule bspec, assumption)
apply (drule equalityD1 [THEN subsetD], blast, clarify)
apply (simp add: m_assoc)
apply (simp add: m_assoc [symmetric])
done

lemma (in normal) inv_op_closed2:
     "⟦x ∈ carrier G; h ∈ H⟧ ⟹ x ⊗ h ⊗ (inv x) ∈ H"
apply (subgoal_tac "inv (inv x) ⊗ h ⊗ (inv x) ∈ H") 
apply (simp add: ) 
apply (blast intro: inv_op_closed1) 
done

text‹Alternative characterization of normal subgroups›
lemma (in group) normal_inv_iff:
     "(N ⊲ G) = 
      (subgroup N G & (∀x ∈ carrier G. ∀h ∈ N. x ⊗ h ⊗ (inv x) ∈ N))"
      (is "_ = ?rhs")
proof
  assume N: "N ⊲ G"
  show ?rhs
    by (blast intro: N normal.inv_op_closed2 normal_imp_subgroup) 
next
  assume ?rhs
  hence sg: "subgroup N G" 
    and closed: "⋀x. x∈carrier G ⟹ ∀h∈N. x ⊗ h ⊗ inv x ∈ N" by auto
  hence sb: "N ⊆ carrier G" by (simp add: subgroup.subset) 
  show "N ⊲ G"
  proof (intro normalI [OF sg], simp add: l_coset_def r_coset_def, clarify)
    fix x
    assume x: "x ∈ carrier G"
    show "(⋃h∈N. {h ⊗ x}) = (⋃h∈N. {x ⊗ h})"
    proof
      show "(⋃h∈N. {h ⊗ x}) ⊆ (⋃h∈N. {x ⊗ h})"
      proof clarify
        fix n
        assume n: "n ∈ N" 
        show "n ⊗ x ∈ (⋃h∈N. {x ⊗ h})"
        proof 
          from closed [of "inv x"]
          show "inv x ⊗ n ⊗ x ∈ N" by (simp add: x n)
          show "n ⊗ x ∈ {x ⊗ (inv x ⊗ n ⊗ x)}"
            by (simp add: x n m_assoc [symmetric] sb [THEN subsetD])
        qed
      qed
    next
      show "(⋃h∈N. {x ⊗ h}) ⊆ (⋃h∈N. {h ⊗ x})"
      proof clarify
        fix n
        assume n: "n ∈ N" 
        show "x ⊗ n ∈ (⋃h∈N. {h ⊗ x})"
        proof 
          show "x ⊗ n ⊗ inv x ∈ N" by (simp add: x n closed)
          show "x ⊗ n ∈ {x ⊗ n ⊗ inv x ⊗ x}"
            by (simp add: x n m_assoc sb [THEN subsetD])
        qed
      qed
    qed
  qed
qed


subsection‹More Properties of Cosets›

lemma (in group) lcos_m_assoc:
     "[| M ⊆ carrier G; g ∈ carrier G; h ∈ carrier G |]
      ==> g <# (h <# M) = (g ⊗ h) <# M"
by (force simp add: l_coset_def m_assoc)

lemma (in group) lcos_mult_one: "M ⊆ carrier G ==> 𝟭 <# M = M"
by (force simp add: l_coset_def)

lemma (in group) l_coset_subset_G:
     "[| H ⊆ carrier G; x ∈ carrier G |] ==> x <# H ⊆ carrier G"
by (auto simp add: l_coset_def subsetD)

lemma (in group) l_coset_swap:
     "⟦y ∈ x <# H;  x ∈ carrier G;  subgroup H G⟧ ⟹ x ∈ y <# H"
proof (simp add: l_coset_def)
  assume "∃h∈H. y = x ⊗ h"
    and x: "x ∈ carrier G"
    and sb: "subgroup H G"
  then obtain h' where h': "h' ∈ H & x ⊗ h' = y" by blast
  show "∃h∈H. x = y ⊗ h"
  proof
    show "x = y ⊗ inv h'" using h' x sb
      by (auto simp add: m_assoc subgroup.subset [THEN subsetD])
    show "inv h' ∈ H" using h' sb
      by (auto simp add: subgroup.subset [THEN subsetD] subgroup.m_inv_closed)
  qed
qed

lemma (in group) l_coset_carrier:
     "[| y ∈ x <# H;  x ∈ carrier G;  subgroup H G |] ==> y ∈ carrier G"
by (auto simp add: l_coset_def m_assoc
                   subgroup.subset [THEN subsetD] subgroup.m_closed)

lemma (in group) l_repr_imp_subset:
  assumes y: "y ∈ x <# H" and x: "x ∈ carrier G" and sb: "subgroup H G"
  shows "y <# H ⊆ x <# H"
proof -
  from y
  obtain h' where "h' ∈ H" "x ⊗ h' = y" by (auto simp add: l_coset_def)
  thus ?thesis using x sb
    by (auto simp add: l_coset_def m_assoc
                       subgroup.subset [THEN subsetD] subgroup.m_closed)
qed

lemma (in group) l_repr_independence:
  assumes y: "y ∈ x <# H" and x: "x ∈ carrier G" and sb: "subgroup H G"
  shows "x <# H = y <# H"
proof
  show "x <# H ⊆ y <# H"
    by (rule l_repr_imp_subset,
        (blast intro: l_coset_swap l_coset_carrier y x sb)+)
  show "y <# H ⊆ x <# H" by (rule l_repr_imp_subset [OF y x sb])
qed

lemma (in group) setmult_subset_G:
     "⟦H ⊆ carrier G; K ⊆ carrier G⟧ ⟹ H <#> K ⊆ carrier G"
by (auto simp add: set_mult_def subsetD)

lemma (in group) subgroup_mult_id: "subgroup H G ⟹ H <#> H = H"
apply (auto simp add: subgroup.m_closed set_mult_def Sigma_def image_def)
apply (rule_tac x = x in bexI)
apply (rule bexI [of _ "𝟭"])
apply (auto simp add: subgroup.one_closed subgroup.subset [THEN subsetD])
done


subsubsection ‹Set of Inverses of an @{text r_coset}.›

lemma (in normal) rcos_inv:
  assumes x:     "x ∈ carrier G"
  shows "set_inv (H #> x) = H #> (inv x)" 
proof (simp add: r_coset_def SET_INV_def x inv_mult_group, safe)
  fix h
  assume h: "h ∈ H"
  show "inv x ⊗ inv h ∈ (⋃j∈H. {j ⊗ inv x})"
  proof
    show "inv x ⊗ inv h ⊗ x ∈ H"
      by (simp add: inv_op_closed1 h x)
    show "inv x ⊗ inv h ∈ {inv x ⊗ inv h ⊗ x ⊗ inv x}"
      by (simp add: h x m_assoc)
  qed
  show "h ⊗ inv x ∈ (⋃j∈H. {inv x ⊗ inv j})"
  proof
    show "x ⊗ inv h ⊗ inv x ∈ H"
      by (simp add: inv_op_closed2 h x)
    show "h ⊗ inv x ∈ {inv x ⊗ inv (x ⊗ inv h ⊗ inv x)}"
      by (simp add: h x m_assoc [symmetric] inv_mult_group)
  qed
qed


subsubsection ‹Theorems for @{text "<#>"} with @{text "#>"} or @{text "<#"}.›

lemma (in group) setmult_rcos_assoc:
     "⟦H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G⟧
      ⟹ H <#> (K #> x) = (H <#> K) #> x"
by (force simp add: r_coset_def set_mult_def m_assoc)

lemma (in group) rcos_assoc_lcos:
     "⟦H ⊆ carrier G; K ⊆ carrier G; x ∈ carrier G⟧
      ⟹ (H #> x) <#> K = H <#> (x <# K)"
by (force simp add: r_coset_def l_coset_def set_mult_def m_assoc)

lemma (in normal) rcos_mult_step1:
     "⟦x ∈ carrier G; y ∈ carrier G⟧
      ⟹ (H #> x) <#> (H #> y) = (H <#> (x <# H)) #> y"
by (simp add: setmult_rcos_assoc subset
              r_coset_subset_G l_coset_subset_G rcos_assoc_lcos)

lemma (in normal) rcos_mult_step2:
     "⟦x ∈ carrier G; y ∈ carrier G⟧
      ⟹ (H <#> (x <# H)) #> y = (H <#> (H #> x)) #> y"
by (insert coset_eq, simp add: normal_def)

lemma (in normal) rcos_mult_step3:
     "⟦x ∈ carrier G; y ∈ carrier G⟧
      ⟹ (H <#> (H #> x)) #> y = H #> (x ⊗ y)"
by (simp add: setmult_rcos_assoc coset_mult_assoc
              subgroup_mult_id normal.axioms subset normal_axioms)

lemma (in normal) rcos_sum:
     "⟦x ∈ carrier G; y ∈ carrier G⟧
      ⟹ (H #> x) <#> (H #> y) = H #> (x ⊗ y)"
by (simp add: rcos_mult_step1 rcos_mult_step2 rcos_mult_step3)

lemma (in normal) rcosets_mult_eq: "M ∈ rcosets H ⟹ H <#> M = M"
  -- ‹generalizes @{text subgroup_mult_id}›
  by (auto simp add: RCOSETS_def subset
        setmult_rcos_assoc subgroup_mult_id normal.axioms normal_axioms)


subsubsection‹An Equivalence Relation›

definition
  r_congruent :: "[('a,'b)monoid_scheme, 'a set] ⇒ ('a*'a)set"  ("rcongı _")
  where "rcongG H = {(x,y). x ∈ carrier G & y ∈ carrier G & invG x ⊗G y ∈ H}"


lemma (in subgroup) equiv_rcong:
   assumes "group G"
   shows "equiv (carrier G) (rcong H)"
proof -
  interpret group G by fact
  show ?thesis
  proof (intro equivI)
    show "refl_on (carrier G) (rcong H)"
      by (auto simp add: r_congruent_def refl_on_def) 
  next
    show "sym (rcong H)"
    proof (simp add: r_congruent_def sym_def, clarify)
      fix x y
      assume [simp]: "x ∈ carrier G" "y ∈ carrier G" 
         and "inv x ⊗ y ∈ H"
      hence "inv (inv x ⊗ y) ∈ H" by simp
      thus "inv y ⊗ x ∈ H" by (simp add: inv_mult_group)
    qed
  next
    show "trans (rcong H)"
    proof (simp add: r_congruent_def trans_def, clarify)
      fix x y z
      assume [simp]: "x ∈ carrier G" "y ∈ carrier G" "z ∈ carrier G"
         and "inv x ⊗ y ∈ H" and "inv y ⊗ z ∈ H"
      hence "(inv x ⊗ y) ⊗ (inv y ⊗ z) ∈ H" by simp
      hence "inv x ⊗ (y ⊗ inv y) ⊗ z ∈ H"
        by (simp add: m_assoc del: r_inv Units_r_inv) 
      thus "inv x ⊗ z ∈ H" by simp
    qed
  qed
qed

text‹Equivalence classes of @{text rcong} correspond to left cosets.
  Was there a mistake in the definitions? I'd have expected them to
  correspond to right cosets.›

(* CB: This is correct, but subtle.
   We call H #> a the right coset of a relative to H.  According to
   Jacobson, this is what the majority of group theory literature does.
   He then defines the notion of congruence relation ~ over monoids as
   equivalence relation with a ~ a' & b ~ b' ⟹ a*b ~ a'*b'.
   Our notion of right congruence induced by K: rcong K appears only in
   the context where K is a normal subgroup.  Jacobson doesn't name it.
   But in this context left and right cosets are identical.
*)

lemma (in subgroup) l_coset_eq_rcong:
  assumes "group G"
  assumes a: "a ∈ carrier G"
  shows "a <# H = rcong H `` {a}"
proof -
  interpret group G by fact
  show ?thesis by (force simp add: r_congruent_def l_coset_def m_assoc [symmetric] a ) 
qed


subsubsection‹Two Distinct Right Cosets are Disjoint›

lemma (in group) rcos_equation:
  assumes "subgroup H G"
  assumes p: "ha ⊗ a = h ⊗ b" "a ∈ carrier G" "b ∈ carrier G" "h ∈ H" "ha ∈ H" "hb ∈ H"
  shows "hb ⊗ a ∈ (⋃h∈H. {h ⊗ b})"
proof -
  interpret subgroup H G by fact
  from p show ?thesis apply (rule_tac UN_I [of "hb ⊗ ((inv ha) ⊗ h)"])
    apply (simp add: )
    apply (simp add: m_assoc transpose_inv)
    done
qed

lemma (in group) rcos_disjoint:
  assumes "subgroup H G"
  assumes p: "a ∈ rcosets H" "b ∈ rcosets H" "a≠b"
  shows "a ∩ b = {}"
proof -
  interpret subgroup H G by fact
  from p show ?thesis
    apply (simp add: RCOSETS_def r_coset_def)
    apply (blast intro: rcos_equation assms sym)
    done
qed


subsection ‹Further lemmas for @{text "r_congruent"}›

text ‹The relation is a congruence›

lemma (in normal) congruent_rcong:
  shows "congruent2 (rcong H) (rcong H) (λa b. a ⊗ b <# H)"
proof (intro congruent2I[of "carrier G" _ "carrier G" _] equiv_rcong is_group)
  fix a b c
  assume abrcong: "(a, b) ∈ rcong H"
    and ccarr: "c ∈ carrier G"

  from abrcong
      have acarr: "a ∈ carrier G"
        and bcarr: "b ∈ carrier G"
        and abH: "inv a ⊗ b ∈ H"
      unfolding r_congruent_def
      by fast+

  note carr = acarr bcarr ccarr

  from ccarr and abH
      have "inv c ⊗ (inv a ⊗ b) ⊗ c ∈ H" by (rule inv_op_closed1)
  moreover
      from carr and inv_closed
      have "inv c ⊗ (inv a ⊗ b) ⊗ c = (inv c ⊗ inv a) ⊗ (b ⊗ c)" 
      by (force cong: m_assoc)
  moreover 
      from carr and inv_closed
      have "… = (inv (a ⊗ c)) ⊗ (b ⊗ c)"
      by (simp add: inv_mult_group)
  ultimately
      have "(inv (a ⊗ c)) ⊗ (b ⊗ c) ∈ H" by simp
  from carr and this
     have "(b ⊗ c) ∈ (a ⊗ c) <# H"
     by (simp add: lcos_module_rev[OF is_group])
  from carr and this and is_subgroup
     show "(a ⊗ c) <# H = (b ⊗ c) <# H" by (intro l_repr_independence, simp+)
next
  fix a b c
  assume abrcong: "(a, b) ∈ rcong H"
    and ccarr: "c ∈ carrier G"

  from ccarr have "c ∈ Units G" by simp
  hence cinvc_one: "inv c ⊗ c = 𝟭" by (rule Units_l_inv)

  from abrcong
      have acarr: "a ∈ carrier G"
       and bcarr: "b ∈ carrier G"
       and abH: "inv a ⊗ b ∈ H"
      by (unfold r_congruent_def, fast+)

  note carr = acarr bcarr ccarr

  from carr and inv_closed
     have "inv a ⊗ b = inv a ⊗ (𝟭 ⊗ b)" by simp
  also from carr and inv_closed
      have "… = inv a ⊗ (inv c ⊗ c) ⊗ b" by simp
  also from carr and inv_closed
      have "… = (inv a ⊗ inv c) ⊗ (c ⊗ b)" by (force cong: m_assoc)
  also from carr and inv_closed
      have "… = inv (c ⊗ a) ⊗ (c ⊗ b)" by (simp add: inv_mult_group)
  finally
      have "inv a ⊗ b = inv (c ⊗ a) ⊗ (c ⊗ b)" .
  from abH and this
      have "inv (c ⊗ a) ⊗ (c ⊗ b) ∈ H" by simp

  from carr and this
     have "(c ⊗ b) ∈ (c ⊗ a) <# H"
     by (simp add: lcos_module_rev[OF is_group])
  from carr and this and is_subgroup
     show "(c ⊗ a) <# H = (c ⊗ b) <# H" by (intro l_repr_independence, simp+)
qed


subsection ‹Order of a Group and Lagrange's Theorem›

definition
  order :: "('a, 'b) monoid_scheme ⇒ nat"
  where "order S = card (carrier S)"

lemma (in monoid) order_gt_0_iff_finite: "0 < order G ⟷ finite (carrier G)"
by(auto simp add: order_def card_gt_0_iff)

lemma (in group) rcosets_part_G:
  assumes "subgroup H G"
  shows "⋃(rcosets H) = carrier G"
proof -
  interpret subgroup H G by fact
  show ?thesis
    apply (rule equalityI)
    apply (force simp add: RCOSETS_def r_coset_def)
    apply (auto simp add: RCOSETS_def intro: rcos_self assms)
    done
qed

lemma (in group) cosets_finite:
     "⟦c ∈ rcosets H;  H ⊆ carrier G;  finite (carrier G)⟧ ⟹ finite c"
apply (auto simp add: RCOSETS_def)
apply (simp add: r_coset_subset_G [THEN finite_subset])
done

text‹The next two lemmas support the proof of @{text card_cosets_equal}.›
lemma (in group) inj_on_f:
    "⟦H ⊆ carrier G;  a ∈ carrier G⟧ ⟹ inj_on (λy. y ⊗ inv a) (H #> a)"
apply (rule inj_onI)
apply (subgoal_tac "x ∈ carrier G & y ∈ carrier G")
 prefer 2 apply (blast intro: r_coset_subset_G [THEN subsetD])
apply (simp add: subsetD)
done

lemma (in group) inj_on_g:
    "⟦H ⊆ carrier G;  a ∈ carrier G⟧ ⟹ inj_on (λy. y ⊗ a) H"
by (force simp add: inj_on_def subsetD)

lemma (in group) card_cosets_equal:
     "⟦c ∈ rcosets H;  H ⊆ carrier G; finite(carrier G)⟧
      ⟹ card c = card H"
apply (auto simp add: RCOSETS_def)
apply (rule card_bij_eq)
     apply (rule inj_on_f, assumption+)
    apply (force simp add: m_assoc subsetD r_coset_def)
   apply (rule inj_on_g, assumption+)
  apply (force simp add: m_assoc subsetD r_coset_def)
 txt‹The sets @{term "H #> a"} and @{term "H"} are finite.›
 apply (simp add: r_coset_subset_G [THEN finite_subset])
apply (blast intro: finite_subset)
done

lemma (in group) rcosets_subset_PowG:
     "subgroup H G  ⟹ rcosets H ⊆ Pow(carrier G)"
apply (simp add: RCOSETS_def)
apply (blast dest: r_coset_subset_G subgroup.subset)
done


theorem (in group) lagrange:
     "⟦finite(carrier G); subgroup H G⟧
      ⟹ card(rcosets H) * card(H) = order(G)"
apply (simp (no_asm_simp) add: order_def rcosets_part_G [symmetric])
apply (subst mult.commute)
apply (rule card_partition)
   apply (simp add: rcosets_subset_PowG [THEN finite_subset])
  apply (simp add: rcosets_part_G)
 apply (simp add: card_cosets_equal subgroup.subset)
apply (simp add: rcos_disjoint)
done


subsection ‹Quotient Groups: Factorization of a Group›

definition
  FactGroup :: "[('a,'b) monoid_scheme, 'a set] ⇒ ('a set) monoid" (infixl "Mod" 65)
    --‹Actually defined for groups rather than monoids›
   where "FactGroup G H = ⦇carrier = rcosetsG H, mult = set_mult G, one = H⦈"

lemma (in normal) setmult_closed:
     "⟦K1 ∈ rcosets H; K2 ∈ rcosets H⟧ ⟹ K1 <#> K2 ∈ rcosets H"
by (auto simp add: rcos_sum RCOSETS_def)

lemma (in normal) setinv_closed:
     "K ∈ rcosets H ⟹ set_inv K ∈ rcosets H"
by (auto simp add: rcos_inv RCOSETS_def)

lemma (in normal) rcosets_assoc:
     "⟦M1 ∈ rcosets H; M2 ∈ rcosets H; M3 ∈ rcosets H⟧
      ⟹ M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)"
by (auto simp add: RCOSETS_def rcos_sum m_assoc)

lemma (in subgroup) subgroup_in_rcosets:
  assumes "group G"
  shows "H ∈ rcosets H"
proof -
  interpret group G by fact
  from _ subgroup_axioms have "H #> 𝟭 = H"
    by (rule coset_join2) auto
  then show ?thesis
    by (auto simp add: RCOSETS_def)
qed

lemma (in normal) rcosets_inv_mult_group_eq:
     "M ∈ rcosets H ⟹ set_inv M <#> M = H"
by (auto simp add: RCOSETS_def rcos_inv rcos_sum subgroup.subset normal.axioms normal_axioms)

theorem (in normal) factorgroup_is_group:
  "group (G Mod H)"
apply (simp add: FactGroup_def)
apply (rule groupI)
    apply (simp add: setmult_closed)
   apply (simp add: normal_imp_subgroup subgroup_in_rcosets [OF is_group])
  apply (simp add: restrictI setmult_closed rcosets_assoc)
 apply (simp add: normal_imp_subgroup
                  subgroup_in_rcosets rcosets_mult_eq)
apply (auto dest: rcosets_inv_mult_group_eq simp add: setinv_closed)
done

lemma mult_FactGroup [simp]: "X ⊗(G Mod H) X' = X <#>G X'"
  by (simp add: FactGroup_def) 

lemma (in normal) inv_FactGroup:
     "X ∈ carrier (G Mod H) ⟹ invG Mod H X = set_inv X"
apply (rule group.inv_equality [OF factorgroup_is_group]) 
apply (simp_all add: FactGroup_def setinv_closed rcosets_inv_mult_group_eq)
done

text‹The coset map is a homomorphism from @{term G} to the quotient group
  @{term "G Mod H"}›
lemma (in normal) r_coset_hom_Mod:
  "(λa. H #> a) ∈ hom G (G Mod H)"
  by (auto simp add: FactGroup_def RCOSETS_def Pi_def hom_def rcos_sum)

 
subsection‹The First Isomorphism Theorem›

text‹The quotient by the kernel of a homomorphism is isomorphic to the 
  range of that homomorphism.›

definition
  kernel :: "('a, 'm) monoid_scheme ⇒ ('b, 'n) monoid_scheme ⇒  ('a ⇒ 'b) ⇒ 'a set"
    --‹the kernel of a homomorphism›
  where "kernel G H h = {x. x ∈ carrier G & h x = 𝟭H}"

lemma (in group_hom) subgroup_kernel: "subgroup (kernel G H h) G"
apply (rule subgroup.intro) 
apply (auto simp add: kernel_def group.intro is_group) 
done

text‹The kernel of a homomorphism is a normal subgroup›
lemma (in group_hom) normal_kernel: "(kernel G H h) ⊲ G"
apply (simp add: G.normal_inv_iff subgroup_kernel)
apply (simp add: kernel_def)
done

lemma (in group_hom) FactGroup_nonempty:
  assumes X: "X ∈ carrier (G Mod kernel G H h)"
  shows "X ≠ {}"
proof -
  from X
  obtain g where "g ∈ carrier G" 
             and "X = kernel G H h #> g"
    by (auto simp add: FactGroup_def RCOSETS_def)
  thus ?thesis 
   by (auto simp add: kernel_def r_coset_def image_def intro: hom_one)
qed


lemma (in group_hom) FactGroup_the_elem_mem:
  assumes X: "X ∈ carrier (G Mod (kernel G H h))"
  shows "the_elem (h`X) ∈ carrier H"
proof -
  from X
  obtain g where g: "g ∈ carrier G" 
             and "X = kernel G H h #> g"
    by (auto simp add: FactGroup_def RCOSETS_def)
  hence "h ` X = {h g}" by (auto simp add: kernel_def r_coset_def image_def g)
  thus ?thesis by (auto simp add: g)
qed

lemma (in group_hom) FactGroup_hom:
     "(λX. the_elem (h`X)) ∈ hom (G Mod (kernel G H h)) H"
apply (simp add: hom_def FactGroup_the_elem_mem normal.factorgroup_is_group [OF normal_kernel] group.axioms monoid.m_closed)
proof (intro ballI)
  fix X and X'
  assume X:  "X  ∈ carrier (G Mod kernel G H h)"
     and X': "X' ∈ carrier (G Mod kernel G H h)"
  then
  obtain g and g'
           where "g ∈ carrier G" and "g' ∈ carrier G" 
             and "X = kernel G H h #> g" and "X' = kernel G H h #> g'"
    by (auto simp add: FactGroup_def RCOSETS_def)
  hence all: "∀x∈X. h x = h g" "∀x∈X'. h x = h g'" 
    and Xsub: "X ⊆ carrier G" and X'sub: "X' ⊆ carrier G"
    by (force simp add: kernel_def r_coset_def image_def)+
  hence "h ` (X <#> X') = {h g ⊗H h g'}" using X X'
    by (auto dest!: FactGroup_nonempty
             simp add: set_mult_def image_eq_UN 
                       subsetD [OF Xsub] subsetD [OF X'sub]) 
  thus "the_elem (h ` (X <#> X')) = the_elem (h ` X) ⊗H the_elem (h ` X')"
    by (simp add: all image_eq_UN FactGroup_nonempty X X')
qed


text‹Lemma for the following injectivity result›
lemma (in group_hom) FactGroup_subset:
     "⟦g ∈ carrier G; g' ∈ carrier G; h g = h g'⟧
      ⟹  kernel G H h #> g ⊆ kernel G H h #> g'"
apply (clarsimp simp add: kernel_def r_coset_def image_def)
apply (rename_tac y)  
apply (rule_tac x="y ⊗ g ⊗ inv g'" in exI) 
apply (simp add: G.m_assoc) 
done

lemma (in group_hom) FactGroup_inj_on:
     "inj_on (λX. the_elem (h ` X)) (carrier (G Mod kernel G H h))"
proof (simp add: inj_on_def, clarify) 
  fix X and X'
  assume X:  "X  ∈ carrier (G Mod kernel G H h)"
     and X': "X' ∈ carrier (G Mod kernel G H h)"
  then
  obtain g and g'
           where gX: "g ∈ carrier G"  "g' ∈ carrier G" 
              "X = kernel G H h #> g" "X' = kernel G H h #> g'"
    by (auto simp add: FactGroup_def RCOSETS_def)
  hence all: "∀x∈X. h x = h g" "∀x∈X'. h x = h g'" 
    by (force simp add: kernel_def r_coset_def image_def)+
  assume "the_elem (h ` X) = the_elem (h ` X')"
  hence h: "h g = h g'"
    by (simp add: image_eq_UN all FactGroup_nonempty X X') 
  show "X=X'" by (rule equalityI) (simp_all add: FactGroup_subset h gX) 
qed

text‹If the homomorphism @{term h} is onto @{term H}, then so is the
homomorphism from the quotient group›
lemma (in group_hom) FactGroup_onto:
  assumes h: "h ` carrier G = carrier H"
  shows "(λX. the_elem (h ` X)) ` carrier (G Mod kernel G H h) = carrier H"
proof
  show "(λX. the_elem (h ` X)) ` carrier (G Mod kernel G H h) ⊆ carrier H"
    by (auto simp add: FactGroup_the_elem_mem)
  show "carrier H ⊆ (λX. the_elem (h ` X)) ` carrier (G Mod kernel G H h)"
  proof
    fix y
    assume y: "y ∈ carrier H"
    with h obtain g where g: "g ∈ carrier G" "h g = y"
      by (blast elim: equalityE) 
    hence "(⋃x∈kernel G H h #> g. {h x}) = {y}" 
      by (auto simp add: y kernel_def r_coset_def) 
    with g show "y ∈ (λX. the_elem (h ` X)) ` carrier (G Mod kernel G H h)" 
      by (auto intro!: bexI simp add: FactGroup_def RCOSETS_def image_eq_UN)
  qed
qed


text‹If @{term h} is a homomorphism from @{term G} onto @{term H}, then the
 quotient group @{term "G Mod (kernel G H h)"} is isomorphic to @{term H}.›
theorem (in group_hom) FactGroup_iso:
  "h ` carrier G = carrier H
   ⟹ (λX. the_elem (h`X)) ∈ (G Mod (kernel G H h)) ≅ H"
by (simp add: iso_def FactGroup_hom FactGroup_inj_on bij_betw_def 
              FactGroup_onto) 


end