(* Title: HOL/Algebra/FiniteProduct.thy
Author: Clemens Ballarin, started 19 November 2002
This file is largely based on HOL/Finite_Set.thy.
*)
theory FiniteProduct
imports Group
begin
subsection \<open>Product Operator for Commutative Monoids\<close>
subsubsection \<open>Inductive Definition of a Relation for Products over Sets\<close>
text \<open>Instantiation of locale \<open>LC\<close> of theory \<open>Finite_Set\<close> is not
possible, because here we have explicit typing rules like
\<open>x \<in> carrier G\<close>. We introduce an explicit argument for the domain
\<open>D\<close>.\<close>
inductive_set
foldSetD :: "['a set, 'b \<Rightarrow> 'a \<Rightarrow> 'a, 'a] \<Rightarrow> ('b set * 'a) set"
for D :: "'a set" and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" and e :: 'a
where
emptyI [intro]: "e \<in> D \<Longrightarrow> ({}, e) \<in> foldSetD D f e"
| insertI [intro]: "\<lbrakk>x \<notin> A; f x y \<in> D; (A, y) \<in> foldSetD D f e\<rbrakk> \<Longrightarrow>
(insert x A, f x y) \<in> foldSetD D f e"
inductive_cases empty_foldSetDE [elim!]: "({}, x) \<in> foldSetD D f e"
definition
foldD :: "['a set, 'b \<Rightarrow> 'a \<Rightarrow> 'a, 'a, 'b set] \<Rightarrow> 'a"
where "foldD D f e A = (THE x. (A, x) \<in> foldSetD D f e)"
lemma foldSetD_closed: "(A, z) \<in> foldSetD D f e \<Longrightarrow> z \<in> D"
by (erule foldSetD.cases) auto
lemma Diff1_foldSetD:
"\<lbrakk>(A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D\<rbrakk> \<Longrightarrow>
(A, f x y) \<in> foldSetD D f e"
by (metis Diff_insert_absorb foldSetD.insertI mk_disjoint_insert)
lemma foldSetD_imp_finite [simp]: "(A, x) \<in> foldSetD D f e \<Longrightarrow> finite A"
by (induct set: foldSetD) auto
lemma finite_imp_foldSetD:
"\<lbrakk>finite A; e \<in> D; \<And>x y. \<lbrakk>x \<in> A; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D\<rbrakk>
\<Longrightarrow> \<exists>x. (A, x) \<in> foldSetD D f e"
proof (induct set: finite)
case empty then show ?case by auto
next
case (insert x F)
then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
with insert have "y \<in> D" by (auto dest: foldSetD_closed)
with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
by (intro foldSetD.intros) auto
then show ?case ..
qed
lemma foldSetD_backwards:
assumes "A \<noteq> {}" "(A, z) \<in> foldSetD D f e"
shows "\<exists>x y. x \<in> A \<and> (A - { x }, y) \<in> foldSetD D f e \<and> z = f x y"
using assms(2) by (cases) (simp add: assms(1), metis Diff_insert_absorb insertI1)
subsubsection \<open>Left-Commutative Operations\<close>
locale LCD =
fixes B :: "'b set"
and D :: "'a set"
and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<cdot>" 70)
assumes left_commute:
"\<lbrakk>x \<in> B; y \<in> B; z \<in> D\<rbrakk> \<Longrightarrow> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
and f_closed [simp, intro!]: "!!x y. \<lbrakk>x \<in> B; y \<in> D\<rbrakk> \<Longrightarrow> f x y \<in> D"
lemma (in LCD) foldSetD_closed [dest]: "(A, z) \<in> foldSetD D f e \<Longrightarrow> z \<in> D"
by (erule foldSetD.cases) auto
lemma (in LCD) Diff1_foldSetD:
"\<lbrakk>(A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B\<rbrakk> \<Longrightarrow>
(A, f x y) \<in> foldSetD D f e"
by (meson Diff1_foldSetD f_closed local.foldSetD_closed subsetCE)
lemma (in LCD) finite_imp_foldSetD:
"\<lbrakk>finite A; A \<subseteq> B; e \<in> D\<rbrakk> \<Longrightarrow> \<exists>x. (A, x) \<in> foldSetD D f e"
proof (induct set: finite)
case empty then show ?case by auto
next
case (insert x F)
then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
with insert have "y \<in> D" by auto
with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
by (intro foldSetD.intros) auto
then show ?case ..
qed
lemma (in LCD) foldSetD_determ_aux:
assumes "e \<in> D" and A: "card A < n" "A \<subseteq> B" "(A, x) \<in> foldSetD D f e" "(A, y) \<in> foldSetD D f e"
shows "y = x"
using A
proof (induction n arbitrary: A x y)
case 0
then show ?case
by auto
next
case (Suc n)
then consider "card A = n" | "card A < n"
by linarith
then show ?case
proof cases
case 1
show ?thesis
using foldSetD.cases [OF \<open>(A,x) \<in> foldSetD D (\<cdot>) e\<close>]
proof cases
case 1
then show ?thesis
using \<open>(A,y) \<in> foldSetD D (\<cdot>) e\<close> by auto
next
case (2 x' A' y')
note A' = this
show ?thesis
using foldSetD.cases [OF \<open>(A,y) \<in> foldSetD D (\<cdot>) e\<close>]
proof cases
case 1
then show ?thesis
using \<open>(A,x) \<in> foldSetD D (\<cdot>) e\<close> by auto
next
case (2 x'' A'' y'')
note A'' = this
show ?thesis
proof (cases "x' = x''")
case True
show ?thesis
proof (cases "y' = y''")
case True
then show ?thesis
using A' A'' \<open>x' = x''\<close> by (blast elim!: equalityE)
next
case False
then show ?thesis
using A' A'' \<open>x' = x''\<close>
by (metis \<open>card A = n\<close> Suc.IH Suc.prems(2) card_insert_disjoint foldSetD_imp_finite insert_eq_iff insert_subset lessI)
qed
next
case False
then have *: "A' - {x''} = A'' - {x'}" "x'' \<in> A'" "x' \<in> A''"
using A' A'' by fastforce+
then have "A' = insert x'' A'' - {x'}"
using \<open>x' \<notin> A'\<close> by blast
then have card: "card A' \<le> card A''"
using A' A'' * by (metis card_Suc_Diff1 eq_refl foldSetD_imp_finite)
obtain u where u: "(A' - {x''}, u) \<in> foldSetD D (\<cdot>) e"
using finite_imp_foldSetD [of "A' - {x''}"] A' Diff_insert \<open>A \<subseteq> B\<close> \<open>e \<in> D\<close> by fastforce
have "y' = f x'' u"
using Diff1_foldSetD [OF u] \<open>x'' \<in> A'\<close> \<open>card A = n\<close> A' Suc.IH \<open>A \<subseteq> B\<close> by auto
then have "(A'' - {x'}, u) \<in> foldSetD D f e"
using "*"(1) u by auto
then have "y'' = f x' u"
using A'' by (metis * \<open>card A = n\<close> A'(1) Diff1_foldSetD Suc.IH \<open>A \<subseteq> B\<close>
card card_Suc_Diff1 card_insert_disjoint foldSetD_imp_finite insert_subset le_imp_less_Suc)
then show ?thesis
using A' A''
by (metis \<open>A \<subseteq> B\<close> \<open>y' = x'' \<cdot> u\<close> insert_subset left_commute local.foldSetD_closed u)
qed
qed
qed
next
case 2 with Suc show ?thesis by blast
qed
qed
lemma (in LCD) foldSetD_determ:
"\<lbrakk>(A, x) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B\<rbrakk>
\<Longrightarrow> y = x"
by (blast intro: foldSetD_determ_aux [rule_format])
lemma (in LCD) foldD_equality:
"\<lbrakk>(A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B\<rbrakk> \<Longrightarrow> foldD D f e A = y"
by (unfold foldD_def) (blast intro: foldSetD_determ)
lemma foldD_empty [simp]:
"e \<in> D \<Longrightarrow> foldD D f e {} = e"
by (unfold foldD_def) blast
lemma (in LCD) foldD_insert_aux:
"\<lbrakk>x \<notin> A; x \<in> B; e \<in> D; A \<subseteq> B\<rbrakk>
\<Longrightarrow> ((insert x A, v) \<in> foldSetD D f e) \<longleftrightarrow> (\<exists>y. (A, y) \<in> foldSetD D f e \<and> v = f x y)"
apply auto
by (metis Diff_insert_absorb f_closed finite_Diff foldSetD.insertI foldSetD_determ foldSetD_imp_finite insert_subset local.finite_imp_foldSetD local.foldSetD_closed)
lemma (in LCD) foldD_insert:
assumes "finite A" "x \<notin> A" "x \<in> B" "e \<in> D" "A \<subseteq> B"
shows "foldD D f e (insert x A) = f x (foldD D f e A)"
proof -
have "(THE v. \<exists>y. (A, y) \<in> foldSetD D (\<cdot>) e \<and> v = x \<cdot> y) = x \<cdot> (THE y. (A, y) \<in> foldSetD D (\<cdot>) e)"
by (rule the_equality) (use assms foldD_def foldD_equality foldD_def finite_imp_foldSetD in \<open>metis+\<close>)
then show ?thesis
unfolding foldD_def using assms by (simp add: foldD_insert_aux)
qed
lemma (in LCD) foldD_closed [simp]:
"\<lbrakk>finite A; e \<in> D; A \<subseteq> B\<rbrakk> \<Longrightarrow> foldD D f e A \<in> D"
proof (induct set: finite)
case empty then show ?case by simp
next
case insert then show ?case by (simp add: foldD_insert)
qed
lemma (in LCD) foldD_commute:
"\<lbrakk>finite A; x \<in> B; e \<in> D; A \<subseteq> B\<rbrakk> \<Longrightarrow>
f x (foldD D f e A) = foldD D f (f x e) A"
by (induct set: finite) (auto simp add: left_commute foldD_insert)
lemma Int_mono2:
"\<lbrakk>A \<subseteq> C; B \<subseteq> C\<rbrakk> \<Longrightarrow> A Int B \<subseteq> C"
by blast
lemma (in LCD) foldD_nest_Un_Int:
"\<lbrakk>finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> B\<rbrakk> \<Longrightarrow>
foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
proof (induction set: finite)
case (insert x F)
then show ?case
by (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb Int_mono2)
qed simp
lemma (in LCD) foldD_nest_Un_disjoint:
"\<lbrakk>finite A; finite B; A Int B = {}; e \<in> D; A \<subseteq> B; C \<subseteq> B\<rbrakk>
\<Longrightarrow> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
by (simp add: foldD_nest_Un_Int)
\<comment> \<open>Delete rules to do with \<open>foldSetD\<close> relation.\<close>
declare foldSetD_imp_finite [simp del]
empty_foldSetDE [rule del]
foldSetD.intros [rule del]
declare (in LCD)
foldSetD_closed [rule del]
text \<open>Commutative Monoids\<close>
text \<open>
We enter a more restrictive context, with \<open>f :: 'a \<Rightarrow> 'a \<Rightarrow> 'a\<close>
instead of \<open>'b \<Rightarrow> 'a \<Rightarrow> 'a\<close>.
\<close>
locale ACeD =
fixes D :: "'a set"
and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<cdot>" 70)
and e :: 'a
assumes ident [simp]: "x \<in> D \<Longrightarrow> x \<cdot> e = x"
and commute: "\<lbrakk>x \<in> D; y \<in> D\<rbrakk> \<Longrightarrow> x \<cdot> y = y \<cdot> x"
and assoc: "\<lbrakk>x \<in> D; y \<in> D; z \<in> D\<rbrakk> \<Longrightarrow> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
and e_closed [simp]: "e \<in> D"
and f_closed [simp]: "\<lbrakk>x \<in> D; y \<in> D\<rbrakk> \<Longrightarrow> x \<cdot> y \<in> D"
lemma (in ACeD) left_commute:
"\<lbrakk>x \<in> D; y \<in> D; z \<in> D\<rbrakk> \<Longrightarrow> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
proof -
assume D: "x \<in> D" "y \<in> D" "z \<in> D"
then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)
also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)
also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)
finally show ?thesis .
qed
lemmas (in ACeD) AC = assoc commute left_commute
lemma (in ACeD) left_ident [simp]: "x \<in> D \<Longrightarrow> e \<cdot> x = x"
proof -
assume "x \<in> D"
then have "x \<cdot> e = x" by (rule ident)
with \<open>x \<in> D\<close> show ?thesis by (simp add: commute)
qed
lemma (in ACeD) foldD_Un_Int:
"\<lbrakk>finite A; finite B; A \<subseteq> D; B \<subseteq> D\<rbrakk> \<Longrightarrow>
foldD D f e A \<cdot> foldD D f e B =
foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
proof (induction set: finite)
case empty
then show ?case
by(simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
next
case (insert x F)
then show ?case
by(simp add: AC insert_absorb Int_insert_left Int_mono2
LCD.foldD_insert [OF LCD.intro [of D]]
LCD.foldD_closed [OF LCD.intro [of D]])
qed
lemma (in ACeD) foldD_Un_disjoint:
"\<lbrakk>finite A; finite B; A Int B = {}; A \<subseteq> D; B \<subseteq> D\<rbrakk> \<Longrightarrow>
foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
by (simp add: foldD_Un_Int
left_commute LCD.foldD_closed [OF LCD.intro [of D]])
subsubsection \<open>Products over Finite Sets\<close>
definition
finprod :: "[('b, 'm) monoid_scheme, 'a \<Rightarrow> 'b, 'a set] \<Rightarrow> 'b"
where "finprod G f A =
(if finite A
then foldD (carrier G) (mult G \<circ> f) \<one>\<^bsub>G\<^esub> A
else \<one>\<^bsub>G\<^esub>)"
syntax
"_finprod" :: "index \<Rightarrow> idt \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> 'b"
("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
translations
"\<Otimes>\<^bsub>G\<^esub>i\<in>A. b" \<rightleftharpoons> "CONST finprod G (%i. b) A"
\<comment> \<open>Beware of argument permutation!\<close>
lemma (in comm_monoid) finprod_empty [simp]:
"finprod G f {} = \<one>"
by (simp add: finprod_def)
lemma (in comm_monoid) finprod_infinite[simp]:
"\<not> finite A \<Longrightarrow> finprod G f A = \<one>"
by (simp add: finprod_def)
declare funcsetI [intro]
funcset_mem [dest]
context comm_monoid begin
lemma finprod_insert [simp]:
assumes "finite F" "a \<notin> F" "f \<in> F \<rightarrow> carrier G" "f a \<in> carrier G"
shows "finprod G f (insert a F) = f a \<otimes> finprod G f F"
proof -
have "finprod G f (insert a F) = foldD (carrier G) ((\<otimes>) \<circ> f) \<one> (insert a F)"
by (simp add: finprod_def assms)
also have "... = ((\<otimes>) \<circ> f) a (foldD (carrier G) ((\<otimes>) \<circ> f) \<one> F)"
by (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
(use assms in \<open>auto simp: m_lcomm Pi_iff\<close>)
also have "... = f a \<otimes> finprod G f F"
using \<open>finite F\<close> by (auto simp add: finprod_def)
finally show ?thesis .
qed
lemma finprod_one_eqI: "(\<And>x. x \<in> A \<Longrightarrow> f x = \<one>) \<Longrightarrow> finprod G f A = \<one>"
proof (induct A rule: infinite_finite_induct)
case empty show ?case by simp
next
case (insert a A)
have "(\<lambda>i. \<one>) \<in> A \<rightarrow> carrier G" by auto
with insert show ?case by simp
qed simp
lemma finprod_one [simp]: "(\<Otimes>i\<in>A. \<one>) = \<one>"
by (simp add: finprod_one_eqI)
lemma finprod_closed [simp]:
fixes A
assumes f: "f \<in> A \<rightarrow> carrier G"
shows "finprod G f A \<in> carrier G"
using f
proof (induct A rule: infinite_finite_induct)
case empty show ?case by simp
next
case (insert a A)
then have a: "f a \<in> carrier G" by fast
from insert have A: "f \<in> A \<rightarrow> carrier G" by fast
from insert A a show ?case by simp
qed simp
lemma funcset_Int_left [simp, intro]:
"\<lbrakk>f \<in> A \<rightarrow> C; f \<in> B \<rightarrow> C\<rbrakk> \<Longrightarrow> f \<in> A Int B \<rightarrow> C"
by fast
lemma funcset_Un_left [iff]:
"(f \<in> A Un B \<rightarrow> C) = (f \<in> A \<rightarrow> C \<and> f \<in> B \<rightarrow> C)"
by fast
lemma finprod_Un_Int:
"\<lbrakk>finite A; finite B; g \<in> A \<rightarrow> carrier G; g \<in> B \<rightarrow> carrier G\<rbrakk> \<Longrightarrow>
finprod G g (A Un B) \<otimes> finprod G g (A Int B) =
finprod G g A \<otimes> finprod G g B"
\<comment> \<open>The reversed orientation looks more natural, but LOOPS as a simprule!\<close>
proof (induct set: finite)
case empty then show ?case by simp
next
case (insert a A)
then have a: "g a \<in> carrier G" by fast
from insert have A: "g \<in> A \<rightarrow> carrier G" by fast
from insert A a show ?case
by (simp add: m_ac Int_insert_left insert_absorb Int_mono2)
qed
lemma finprod_Un_disjoint:
"\<lbrakk>finite A; finite B; A Int B = {};
g \<in> A \<rightarrow> carrier G; g \<in> B \<rightarrow> carrier G\<rbrakk>
\<Longrightarrow> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"
by (metis Pi_split_domain finprod_Un_Int finprod_closed finprod_empty r_one)
lemma finprod_multf [simp]:
"\<lbrakk>f \<in> A \<rightarrow> carrier G; g \<in> A \<rightarrow> carrier G\<rbrakk> \<Longrightarrow>
finprod G (\<lambda>x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"
proof (induct A rule: infinite_finite_induct)
case empty show ?case by simp
next
case (insert a A) then
have fA: "f \<in> A \<rightarrow> carrier G" by fast
from insert have fa: "f a \<in> carrier G" by fast
from insert have gA: "g \<in> A \<rightarrow> carrier G" by fast
from insert have ga: "g a \<in> carrier G" by fast
from insert have fgA: "(%x. f x \<otimes> g x) \<in> A \<rightarrow> carrier G"
by (simp add: Pi_def)
show ?case
by (simp add: insert fA fa gA ga fgA m_ac)
qed simp
lemma finprod_cong':
"\<lbrakk>A = B; g \<in> B \<rightarrow> carrier G;
!!i. i \<in> B \<Longrightarrow> f i = g i\<rbrakk> \<Longrightarrow> finprod G f A = finprod G g B"
proof -
assume prems: "A = B" "g \<in> B \<rightarrow> carrier G"
"!!i. i \<in> B \<Longrightarrow> f i = g i"
show ?thesis
proof (cases "finite B")
case True
then have "!!A. \<lbrakk>A = B; g \<in> B \<rightarrow> carrier G;
!!i. i \<in> B \<Longrightarrow> f i = g i\<rbrakk> \<Longrightarrow> finprod G f A = finprod G g B"
proof induct
case empty thus ?case by simp
next
case (insert x B)
then have "finprod G f A = finprod G f (insert x B)" by simp
also from insert have "... = f x \<otimes> finprod G f B"
proof (intro finprod_insert)
show "finite B" by fact
next
show "x \<notin> B" by fact
next
assume "x \<notin> B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
"g \<in> insert x B \<rightarrow> carrier G"
thus "f \<in> B \<rightarrow> carrier G" by fastforce
next
assume "x \<notin> B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
"g \<in> insert x B \<rightarrow> carrier G"
thus "f x \<in> carrier G" by fastforce
qed
also from insert have "... = g x \<otimes> finprod G g B" by fastforce
also from insert have "... = finprod G g (insert x B)"
by (intro finprod_insert [THEN sym]) auto
finally show ?case .
qed
with prems show ?thesis by simp
next
case False with prems show ?thesis by simp
qed
qed
lemma finprod_cong:
"\<lbrakk>A = B; f \<in> B \<rightarrow> carrier G = True;
\<And>i. i \<in> B =simp=> f i = g i\<rbrakk> \<Longrightarrow> finprod G f A = finprod G g B"
(* This order of prems is slightly faster (3%) than the last two swapped. *)
by (rule finprod_cong') (auto simp add: simp_implies_def)
text \<open>Usually, if this rule causes a failed congruence proof error,
the reason is that the premise \<open>g \<in> B \<rightarrow> carrier G\<close> cannot be shown.
Adding @{thm [source] Pi_def} to the simpset is often useful.
For this reason, @{thm [source] finprod_cong}
is not added to the simpset by default.
\<close>
end
declare funcsetI [rule del]
funcset_mem [rule del]
context comm_monoid begin
lemma finprod_0 [simp]:
"f \<in> {0::nat} \<rightarrow> carrier G \<Longrightarrow> finprod G f {..0} = f 0"
by (simp add: Pi_def)
lemma finprod_0':
"f \<in> {..n} \<rightarrow> carrier G \<Longrightarrow> (f 0) \<otimes> finprod G f {Suc 0..n} = finprod G f {..n}"
proof -
assume A: "f \<in> {.. n} \<rightarrow> carrier G"
hence "(f 0) \<otimes> finprod G f {Suc 0..n} = finprod G f {..0} \<otimes> finprod G f {Suc 0..n}"
using finprod_0[of f] by (simp add: funcset_mem)
also have " ... = finprod G f ({..0} \<union> {Suc 0..n})"
using finprod_Un_disjoint[of "{..0}" "{Suc 0..n}" f] A by (simp add: funcset_mem)
also have " ... = finprod G f {..n}"
by (simp add: atLeastAtMost_insertL atMost_atLeast0)
finally show ?thesis .
qed
lemma finprod_Suc [simp]:
"f \<in> {..Suc n} \<rightarrow> carrier G \<Longrightarrow>
finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"
by (simp add: Pi_def atMost_Suc)
lemma finprod_Suc2:
"f \<in> {..Suc n} \<rightarrow> carrier G \<Longrightarrow>
finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \<otimes> f 0)"
proof (induct n)
case 0 thus ?case by (simp add: Pi_def)
next
case Suc thus ?case by (simp add: m_assoc Pi_def)
qed
lemma finprod_Suc3:
assumes "f \<in> {..n :: nat} \<rightarrow> carrier G"
shows "finprod G f {.. n} = (f n) \<otimes> finprod G f {..< n}"
proof (cases "n = 0")
case True thus ?thesis
using assms atMost_Suc by simp
next
case False
then obtain k where "n = Suc k"
using not0_implies_Suc by blast
thus ?thesis
using finprod_Suc[of f k] assms atMost_Suc lessThan_Suc_atMost by simp
qed
lemma finprod_reindex: \<^marker>\<open>contributor \<open>Jeremy Avigad\<close>\<close>
"f \<in> (h ` A) \<rightarrow> carrier G \<Longrightarrow>
inj_on h A \<Longrightarrow> finprod G f (h ` A) = finprod G (\<lambda>x. f (h x)) A"
proof (induct A rule: infinite_finite_induct)
case (infinite A)
hence "\<not> finite (h ` A)"
using finite_imageD by blast
with \<open>\<not> finite A\<close> show ?case by simp
qed (auto simp add: Pi_def)
lemma finprod_const: \<^marker>\<open>contributor \<open>Jeremy Avigad\<close>\<close>
assumes a [simp]: "a \<in> carrier G"
shows "finprod G (\<lambda>x. a) A = a [^] card A"
proof (induct A rule: infinite_finite_induct)
case (insert b A)
show ?case
proof (subst finprod_insert[OF insert(1-2)])
show "a \<otimes> (\<Otimes>x\<in>A. a) = a [^] card (insert b A)"
by (insert insert, auto, subst m_comm, auto)
qed auto
qed auto
lemma finprod_singleton: \<^marker>\<open>contributor \<open>Jesus Aransay\<close>\<close>
assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
shows "(\<Otimes>j\<in>A. if i = j then f j else \<one>) = f i"
using i_in_A finprod_insert [of "A - {i}" i "(\<lambda>j. if i = j then f j else \<one>)"]
fin_A f_Pi finprod_one [of "A - {i}"]
finprod_cong [of "A - {i}" "A - {i}" "(\<lambda>j. if i = j then f j else \<one>)" "(\<lambda>i. \<one>)"]
unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)
lemma finprod_singleton_swap:
assumes i_in_A: "i \<in> A" and fin_A: "finite A" and f_Pi: "f \<in> A \<rightarrow> carrier G"
shows "(\<Otimes>j\<in>A. if j = i then f j else \<one>) = f i"
using finprod_singleton [OF assms] by (simp add: eq_commute)
lemma finprod_mono_neutral_cong_left:
assumes "finite B"
and "A \<subseteq> B"
and 1: "\<And>i. i \<in> B - A \<Longrightarrow> h i = \<one>"
and gh: "\<And>x. x \<in> A \<Longrightarrow> g x = h x"
and h: "h \<in> B \<rightarrow> carrier G"
shows "finprod G g A = finprod G h B"
proof-
have eq: "A \<union> (B - A) = B" using \<open>A \<subseteq> B\<close> by blast
have d: "A \<inter> (B - A) = {}" using \<open>A \<subseteq> B\<close> by blast
from \<open>finite B\<close> \<open>A \<subseteq> B\<close> have f: "finite A" "finite (B - A)"
by (auto intro: finite_subset)
have "h \<in> A \<rightarrow> carrier G" "h \<in> B - A \<rightarrow> carrier G"
using assms by (auto simp: image_subset_iff_funcset)
moreover have "finprod G g A = finprod G h A \<otimes> finprod G h (B - A)"
proof -
have "finprod G h (B - A) = \<one>"
using "1" finprod_one_eqI by blast
moreover have "finprod G g A = finprod G h A"
using \<open>h \<in> A \<rightarrow> carrier G\<close> finprod_cong' gh by blast
ultimately show ?thesis
by (simp add: \<open>h \<in> A \<rightarrow> carrier G\<close>)
qed
ultimately show ?thesis
by (simp add: finprod_Un_disjoint [OF f d, unfolded eq])
qed
lemma finprod_mono_neutral_cong_right:
assumes "finite B"
and "A \<subseteq> B" "\<And>i. i \<in> B - A \<Longrightarrow> g i = \<one>" "\<And>x. x \<in> A \<Longrightarrow> g x = h x" "g \<in> B \<rightarrow> carrier G"
shows "finprod G g B = finprod G h A"
using assms by (auto intro!: finprod_mono_neutral_cong_left [symmetric])
lemma finprod_mono_neutral_cong:
assumes [simp]: "finite B" "finite A"
and *: "\<And>i. i \<in> B - A \<Longrightarrow> h i = \<one>" "\<And>i. i \<in> A - B \<Longrightarrow> g i = \<one>"
and gh: "\<And>x. x \<in> A \<inter> B \<Longrightarrow> g x = h x"
and g: "g \<in> A \<rightarrow> carrier G"
and h: "h \<in> B \<rightarrow> carrier G"
shows "finprod G g A = finprod G h B"
proof-
have "finprod G g A = finprod G g (A \<inter> B)"
by (rule finprod_mono_neutral_cong_right) (use assms in auto)
also have "\<dots> = finprod G h (A \<inter> B)"
by (rule finprod_cong) (use assms in auto)
also have "\<dots> = finprod G h B"
by (rule finprod_mono_neutral_cong_left) (use assms in auto)
finally show ?thesis .
qed
end
(* Jeremy Avigad. 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 \<in> carrier G"
shows "a [^] card(carrier G) = one G"
proof -
have "(\<Otimes>x\<in>carrier G. x) = (\<Otimes>x\<in>carrier G. a \<otimes> x)"
by (subst (2) finprod_reindex [symmetric],
auto simp add: Pi_def inj_on_cmult surj_const_mult)
also have "\<dots> = (\<Otimes>x\<in>carrier G. a) \<otimes> (\<Otimes>x\<in>carrier G. x)"
by (auto simp add: finprod_multf Pi_def)
also have "(\<Otimes>x\<in>carrier G. a) = a [^] card(carrier G)"
by (auto simp add: finprod_const)
finally show ?thesis
by auto
qed
lemma (in comm_monoid) finprod_UN_disjoint:
assumes
"finite I" "\<And>i. i \<in> I \<Longrightarrow> finite (A i)" "pairwise (\<lambda>i j. disjnt (A i) (A j)) I"
"\<And>i x. i \<in> I \<Longrightarrow> x \<in> A i \<Longrightarrow> g x \<in> carrier G"
shows "finprod G g (\<Union>(A ` I)) = finprod G (\<lambda>i. finprod G g (A i)) I"
using assms
proof (induction set: finite)
case empty
then show ?case
by force
next
case (insert i I)
then show ?case
unfolding pairwise_def disjnt_def
apply clarsimp
apply (subst finprod_Un_disjoint)
apply (fastforce intro!: funcsetI finprod_closed)+
done
qed
lemma (in comm_monoid) finprod_Union_disjoint:
"\<lbrakk>finite C; \<And>A. A \<in> C \<Longrightarrow> finite A \<and> (\<forall>x\<in>A. f x \<in> carrier G); pairwise disjnt C\<rbrakk> \<Longrightarrow>
finprod G f (\<Union>C) = finprod G (finprod G f) C"
by (frule finprod_UN_disjoint [of C id f]) auto
end