(* Title: MiscAlgebra.thy
Author: Jeremy Avigad
These are things that can be added to the Algebra library.
*)
theory MiscAlgebra
imports
"~~/src/HOL/Algebra/Ring"
"~~/src/HOL/Algebra/FiniteProduct"
begin;
(* finiteness stuff *)
lemma bounded_set1_int [intro]: "finite {(x::int). a < x & x < b & P x}"
apply (subgoal_tac "{x. a < x & x < b & P x} <= {a<..<b}")
apply (erule finite_subset)
apply auto
done
(* The rest is for the algebra libraries *)
(* These go in Group.thy. *)
(*
Show that the units in any monoid give rise to a group.
The file Residues.thy provides some infrastructure to use
facts about the unit group within the ring locale.
*)
definition units_of :: "('a, 'b) monoid_scheme => 'a monoid" where
"units_of G == (| carrier = Units G,
Group.monoid.mult = Group.monoid.mult G,
one = one G |)";
(*
lemma (in monoid) Units_mult_closed [intro]:
"x : Units G ==> y : Units G ==> x \<otimes> y : Units G"
apply (unfold Units_def)
apply (clarsimp)
apply (rule_tac x = "xaa \<otimes> xa" in bexI)
apply auto
apply (subst m_assoc)
apply auto
apply (subst (2) m_assoc [symmetric])
apply auto
apply (subst m_assoc)
apply auto
apply (subst (2) m_assoc [symmetric])
apply auto
done
*)
lemma (in monoid) units_group: "group(units_of G)"
apply (unfold units_of_def)
apply (rule groupI)
apply auto
apply (subst m_assoc)
apply auto
apply (rule_tac x = "inv x" in bexI)
apply auto
done
lemma (in comm_monoid) units_comm_group: "comm_group(units_of G)"
apply (rule group.group_comm_groupI)
apply (rule units_group)
apply (insert prems)
apply (unfold units_of_def Units_def comm_monoid_def comm_monoid_axioms_def)
apply auto;
done;
lemma units_of_carrier: "carrier (units_of G) = Units G"
by (unfold units_of_def, auto)
lemma units_of_mult: "mult(units_of G) = mult G"
by (unfold units_of_def, auto)
lemma units_of_one: "one(units_of G) = one G"
by (unfold units_of_def, auto)
lemma (in monoid) units_of_inv: "x : Units G ==>
m_inv (units_of G) x = m_inv G x"
apply (rule sym)
apply (subst m_inv_def)
apply (rule the1_equality)
apply (rule ex_ex1I)
apply (subst (asm) Units_def)
apply auto
apply (erule inv_unique)
apply auto
apply (rule Units_closed)
apply (simp_all only: units_of_carrier [symmetric])
apply (insert units_group)
apply auto
apply (subst units_of_mult [symmetric])
apply (subst units_of_one [symmetric])
apply (erule group.r_inv, assumption)
apply (subst units_of_mult [symmetric])
apply (subst units_of_one [symmetric])
apply (erule group.l_inv, assumption)
done
lemma (in group) inj_on_const_mult: "a: (carrier G) ==>
inj_on (%x. a \<otimes> x) (carrier G)"
by (unfold inj_on_def, auto)
lemma (in group) surj_const_mult: "a : (carrier G) ==>
(%x. a \<otimes> x) ` (carrier G) = (carrier G)"
apply (auto simp add: image_def)
apply (rule_tac x = "(m_inv G a) \<otimes> x" in bexI)
apply auto
(* auto should get this. I suppose we need "comm_monoid_simprules"
for mult_ac rewriting. *)
apply (subst m_assoc [symmetric])
apply auto
done
lemma (in group) l_cancel_one [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow>
(x \<otimes> a = x) = (a = one G)"
apply auto
apply (subst l_cancel [symmetric])
prefer 4
apply (erule ssubst)
apply auto
done
lemma (in group) r_cancel_one [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow>
(a \<otimes> x = x) = (a = one G)"
apply auto
apply (subst r_cancel [symmetric])
prefer 4
apply (erule ssubst)
apply auto
done
(* Is there a better way to do this? *)
lemma (in group) l_cancel_one' [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow>
(x = x \<otimes> a) = (a = one G)"
by (subst eq_commute, simp)
lemma (in group) r_cancel_one' [simp]: "x : carrier G \<Longrightarrow> a : carrier G \<Longrightarrow>
(x = a \<otimes> x) = (a = one G)"
by (subst eq_commute, simp)
(* This should be generalized to arbitrary groups, not just commutative
ones, using Lagrange's theorem. *)
lemma (in comm_group) power_order_eq_one:
assumes fin [simp]: "finite (carrier G)"
and a [simp]: "a : carrier G"
shows "a (^) card(carrier G) = one G"
proof -
have "(\<Otimes>x:carrier G. x) = (\<Otimes>x:carrier G. a \<otimes> x)"
by (subst (2) finprod_reindex [symmetric],
auto simp add: Pi_def inj_on_const_mult surj_const_mult)
also have "\<dots> = (\<Otimes>x:carrier G. a) \<otimes> (\<Otimes>x:carrier G. x)"
by (auto simp add: finprod_multf Pi_def)
also have "(\<Otimes>x:carrier G. a) = a (^) card(carrier G)"
by (auto simp add: finprod_const)
finally show ?thesis
(* uses the preceeding lemma *)
by auto
qed
(* Miscellaneous *)
lemma (in cring) field_intro2: "\<zero>\<^bsub>R\<^esub> ~= \<one>\<^bsub>R\<^esub> \<Longrightarrow> ALL x : carrier R - {\<zero>\<^bsub>R\<^esub>}.
x : Units R \<Longrightarrow> field R"
apply (unfold_locales)
apply (insert prems, auto)
apply (rule trans)
apply (subgoal_tac "a = (a \<otimes> b) \<otimes> inv b")
apply assumption
apply (subst m_assoc)
apply (auto simp add: Units_r_inv)
apply (unfold Units_def)
apply auto
done
lemma (in monoid) inv_char: "x : carrier G \<Longrightarrow> y : carrier G \<Longrightarrow>
x \<otimes> y = \<one> \<Longrightarrow> y \<otimes> x = \<one> \<Longrightarrow> inv x = y"
apply (subgoal_tac "x : Units G")
apply (subgoal_tac "y = inv x \<otimes> \<one>")
apply simp
apply (erule subst)
apply (subst m_assoc [symmetric])
apply auto
apply (unfold Units_def)
apply auto
done
lemma (in comm_monoid) comm_inv_char: "x : carrier G \<Longrightarrow> y : carrier G \<Longrightarrow>
x \<otimes> y = \<one> \<Longrightarrow> inv x = y"
apply (rule inv_char)
apply auto
apply (subst m_comm, auto)
done
lemma (in ring) inv_neg_one [simp]: "inv (\<ominus> \<one>) = \<ominus> \<one>"
apply (rule inv_char)
apply (auto simp add: l_minus r_minus)
done
lemma (in monoid) inv_eq_imp_eq: "x : Units G \<Longrightarrow> y : Units G \<Longrightarrow>
inv x = inv y \<Longrightarrow> x = y"
apply (subgoal_tac "inv(inv x) = inv(inv y)")
apply (subst (asm) Units_inv_inv)+
apply auto
done
lemma (in ring) Units_minus_one_closed [intro]: "\<ominus> \<one> : Units R"
apply (unfold Units_def)
apply auto
apply (rule_tac x = "\<ominus> \<one>" in bexI)
apply auto
apply (simp add: l_minus r_minus)
done
lemma (in monoid) inv_one [simp]: "inv \<one> = \<one>"
apply (rule inv_char)
apply auto
done
lemma (in ring) inv_eq_neg_one_eq: "x : Units R \<Longrightarrow> (inv x = \<ominus> \<one>) = (x = \<ominus> \<one>)"
apply auto
apply (subst Units_inv_inv [symmetric])
apply auto
done
lemma (in monoid) inv_eq_one_eq: "x : Units G \<Longrightarrow> (inv x = \<one>) = (x = \<one>)"
apply auto
apply (subst Units_inv_inv [symmetric])
apply auto
done
(* This goes in FiniteProduct *)
lemma (in comm_monoid) finprod_UN_disjoint:
"finite I \<Longrightarrow> (ALL i:I. finite (A i)) \<longrightarrow> (ALL i:I. ALL j:I. i ~= j \<longrightarrow>
(A i) Int (A j) = {}) \<longrightarrow>
(ALL i:I. ALL x: (A i). g x : carrier G) \<longrightarrow>
finprod G g (UNION I A) = finprod G (%i. finprod G g (A i)) I"
apply (induct set: finite)
apply force
apply clarsimp
apply (subst finprod_Un_disjoint)
apply blast
apply (erule finite_UN_I)
apply blast
apply (fastsimp)
apply (auto intro!: funcsetI finprod_closed)
done
lemma (in comm_monoid) finprod_Union_disjoint:
"[| finite C; (ALL A:C. finite A & (ALL x:A. f x : carrier G));
(ALL A:C. ALL B:C. A ~= B --> A Int B = {}) |]
==> finprod G f (Union C) = finprod G (finprod G f) C"
apply (frule finprod_UN_disjoint [of C id f])
apply (unfold Union_def id_def, auto)
done
lemma (in comm_monoid) finprod_one [rule_format]:
"finite A \<Longrightarrow> (ALL x:A. f x = \<one>) \<longrightarrow>
finprod G f A = \<one>"
by (induct set: finite) auto
(* need better simplification rules for rings *)
(* the next one holds more generally for abelian groups *)
lemma (in cring) sum_zero_eq_neg:
"x : carrier R \<Longrightarrow> y : carrier R \<Longrightarrow> x \<oplus> y = \<zero> \<Longrightarrow> x = \<ominus> y"
apply (subgoal_tac "\<ominus> y = \<zero> \<oplus> \<ominus> y")
apply (erule ssubst)back
apply (erule subst)
apply (simp add: ring_simprules)+
done
(* there's a name conflict -- maybe "domain" should be
"integral_domain" *)
lemma (in Ring.domain) square_eq_one:
fixes x
assumes [simp]: "x : carrier R" and
"x \<otimes> x = \<one>"
shows "x = \<one> | x = \<ominus>\<one>"
proof -
have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = x \<otimes> x \<oplus> \<ominus> \<one>"
by (simp add: ring_simprules)
also with `x \<otimes> x = \<one>` have "\<dots> = \<zero>"
by (simp add: ring_simprules)
finally have "(x \<oplus> \<one>) \<otimes> (x \<oplus> \<ominus> \<one>) = \<zero>" .
hence "(x \<oplus> \<one>) = \<zero> | (x \<oplus> \<ominus> \<one>) = \<zero>"
by (intro integral, auto)
thus ?thesis
apply auto
apply (erule notE)
apply (rule sum_zero_eq_neg)
apply auto
apply (subgoal_tac "x = \<ominus> (\<ominus> \<one>)")
apply (simp add: ring_simprules)
apply (rule sum_zero_eq_neg)
apply auto
done
qed
lemma (in Ring.domain) inv_eq_self: "x : Units R \<Longrightarrow>
x = inv x \<Longrightarrow> x = \<one> | x = \<ominus> \<one>"
apply (rule square_eq_one)
apply auto
apply (erule ssubst)back
apply (erule Units_r_inv)
done
(*
The following translates theorems about groups to the facts about
the units of a ring. (The list should be expanded as more things are
needed.)
*)
lemma (in ring) finite_ring_finite_units [intro]: "finite (carrier R) \<Longrightarrow>
finite (Units R)"
by (rule finite_subset, auto)
(* this belongs with MiscAlgebra.thy *)
lemma (in monoid) units_of_pow:
"x : Units G \<Longrightarrow> x (^)\<^bsub>units_of G\<^esub> (n::nat) = x (^)\<^bsub>G\<^esub> n"
apply (induct n)
apply (auto simp add: units_group group.is_monoid
monoid.nat_pow_0 monoid.nat_pow_Suc units_of_one units_of_mult
One_nat_def)
done
lemma (in cring) units_power_order_eq_one: "finite (Units R) \<Longrightarrow> a : Units R
\<Longrightarrow> a (^) card(Units R) = \<one>"
apply (subst units_of_carrier [symmetric])
apply (subst units_of_one [symmetric])
apply (subst units_of_pow [symmetric])
apply assumption
apply (rule comm_group.power_order_eq_one)
apply (rule units_comm_group)
apply (unfold units_of_def, auto)
done
end