# Theory FiniteProduct

theory FiniteProduct
imports Group
```(*  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 ‹Product Operator for Commutative Monoids›

subsubsection ‹Inductive Definition of a Relation for Products over Sets›

text ‹Instantiation of locale ‹LC› of theory ‹Finite_Set› is not
possible, because here we have explicit typing rules like
‹x ∈ carrier G›.  We introduce an explicit argument for the domain
‹D›.›

inductive_set
foldSetD :: "['a set, 'b => 'a => 'a, 'a] => ('b set * 'a) set"
for D :: "'a set" and f :: "'b => 'a => 'a" and e :: 'a
where
emptyI [intro]: "e ∈ D ==> ({}, e) ∈ foldSetD D f e"
| insertI [intro]: "[| x ~: A; f x y ∈ D; (A, y) ∈ foldSetD D f e |] ==>
(insert x A, f x y) ∈ foldSetD D f e"

inductive_cases empty_foldSetDE [elim!]: "({}, x) ∈ foldSetD D f e"

definition
foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"
where "foldD D f e A = (THE x. (A, x) ∈ foldSetD D f e)"

lemma foldSetD_closed:
"[| (A, z) ∈ foldSetD D f e ; e ∈ D; !!x y. [| x ∈ A; y ∈ D |] ==> f x y ∈ D
|] ==> z ∈ D"
by (erule foldSetD.cases) auto

lemma Diff1_foldSetD:
"[| (A - {x}, y) ∈ foldSetD D f e; x ∈ A; f x y ∈ D |] ==>
(A, f x y) ∈ foldSetD D f e"
apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
apply auto
done

lemma foldSetD_imp_finite [simp]: "(A, x) ∈ foldSetD D f e ==> finite A"
by (induct set: foldSetD) auto

lemma finite_imp_foldSetD:
"[| finite A; e ∈ D; !!x y. [| x ∈ A; y ∈ D |] ==> f x y ∈ D |] ==>
EX x. (A, x) ∈ 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) ∈ foldSetD D f e" by auto
with insert have "y ∈ D" by (auto dest: foldSetD_closed)
with y and insert have "(insert x F, f x y) ∈ foldSetD D f e"
by (intro foldSetD.intros) auto
then show ?case ..
qed

text ‹Left-Commutative Operations›

locale LCD =
fixes B :: "'b set"
and D :: "'a set"
and f :: "'b => 'a => 'a"    (infixl "⋅" 70)
assumes left_commute:
"[| x ∈ B; y ∈ B; z ∈ D |] ==> x ⋅ (y ⋅ z) = y ⋅ (x ⋅ z)"
and f_closed [simp, intro!]: "!!x y. [| x ∈ B; y ∈ D |] ==> f x y ∈ D"

lemma (in LCD) foldSetD_closed [dest]:
"(A, z) ∈ foldSetD D f e ==> z ∈ D"
by (erule foldSetD.cases) auto

lemma (in LCD) Diff1_foldSetD:
"[| (A - {x}, y) ∈ foldSetD D f e; x ∈ A; A ⊆ B |] ==>
(A, f x y) ∈ foldSetD D f e"
apply (subgoal_tac "x ∈ B")
prefer 2 apply fast
apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
apply auto
done

lemma (in LCD) foldSetD_imp_finite [simp]:
"(A, x) ∈ foldSetD D f e ==> finite A"
by (induct set: foldSetD) auto

lemma (in LCD) finite_imp_foldSetD:
"[| finite A; A ⊆ B; e ∈ D |] ==> EX x. (A, x) ∈ 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) ∈ foldSetD D f e" by auto
with insert have "y ∈ D" by auto
with y and insert have "(insert x F, f x y) ∈ foldSetD D f e"
by (intro foldSetD.intros) auto
then show ?case ..
qed

lemma (in LCD) foldSetD_determ_aux:
"e ∈ D ==> ∀A x. A ⊆ B & card A < n --> (A, x) ∈ foldSetD D f e -->
(∀y. (A, y) ∈ foldSetD D f e --> y = x)"
apply (induct n)
apply (auto simp add: less_Suc_eq) (* slow *)
apply (erule foldSetD.cases)
apply blast
apply (erule foldSetD.cases)
apply blast
apply clarify
txt ‹force simplification of ‹card A < card (insert ...)›.›
apply (erule rev_mp)
apply (rule impI)
apply (rename_tac xa Aa ya xb Ab yb, case_tac "xa = xb")
apply (subgoal_tac "Aa = Ab")
prefer 2 apply (blast elim!: equalityE)
apply blast
txt ‹case @{prop "xa ∉ xb"}.›
apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb ∈ Aa & xa ∈ Ab")
prefer 2 apply (blast elim!: equalityE)
apply clarify
apply (subgoal_tac "Aa = insert xb Ab - {xa}")
prefer 2 apply blast
apply (subgoal_tac "card Aa ≤ card Ab")
prefer 2
apply (rule Suc_le_mono [THEN subst])
apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
apply (blast intro: foldSetD_imp_finite)
apply best
apply assumption
apply (frule (1) Diff1_foldSetD)
apply best
apply (subgoal_tac "ya = f xb x")
prefer 2
apply (subgoal_tac "Aa ⊆ B")
prefer 2 apply best (* slow *)
apply (blast del: equalityCE)
apply (subgoal_tac "(Ab - {xa}, x) ∈ foldSetD D f e")
prefer 2 apply simp
apply (subgoal_tac "yb = f xa x")
prefer 2
apply (blast del: equalityCE dest: Diff1_foldSetD)
apply (simp (no_asm_simp))
apply (rule left_commute)
apply assumption
apply best (* slow *)
apply best
done

lemma (in LCD) foldSetD_determ:
"[| (A, x) ∈ foldSetD D f e; (A, y) ∈ foldSetD D f e; e ∈ D; A ⊆ B |]
==> y = x"
by (blast intro: foldSetD_determ_aux [rule_format])

lemma (in LCD) foldD_equality:
"[| (A, y) ∈ foldSetD D f e; e ∈ D; A ⊆ B |] ==> foldD D f e A = y"
by (unfold foldD_def) (blast intro: foldSetD_determ)

lemma foldD_empty [simp]:
"e ∈ D ==> foldD D f e {} = e"
by (unfold foldD_def) blast

lemma (in LCD) foldD_insert_aux:
"[| x ~: A; x ∈ B; e ∈ D; A ⊆ B |] ==>
((insert x A, v) ∈ foldSetD D f e) =
(EX y. (A, y) ∈ foldSetD D f e & v = f x y)"
apply auto
apply (rule_tac A1 = A in finite_imp_foldSetD [THEN exE])
apply (fastforce dest: foldSetD_imp_finite)
apply assumption
apply assumption
apply (blast intro: foldSetD_determ)
done

lemma (in LCD) foldD_insert:
"[| finite A; x ~: A; x ∈ B; e ∈ D; A ⊆ B |] ==>
foldD D f e (insert x A) = f x (foldD D f e A)"
apply (unfold foldD_def)
apply (rule the_equality)
apply (auto intro: finite_imp_foldSetD
done

lemma (in LCD) foldD_closed [simp]:
"[| finite A; e ∈ D; A ⊆ B |] ==> foldD D f e A ∈ 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:
"[| finite A; x ∈ B; e ∈ D; A ⊆ B |] ==>
f x (foldD D f e A) = foldD D f (f x e) A"
apply (induct set: finite)
apply simp
apply (auto simp add: left_commute foldD_insert)
done

lemma Int_mono2:
"[| A ⊆ C; B ⊆ C |] ==> A Int B ⊆ C"
by blast

lemma (in LCD) foldD_nest_Un_Int:
"[| finite A; finite C; e ∈ D; A ⊆ B; C ⊆ B |] ==>
foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
apply (induct set: finite)
apply simp
apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
Int_mono2)
done

lemma (in LCD) foldD_nest_Un_disjoint:
"[| finite A; finite B; A Int B = {}; e ∈ D; A ⊆ B; C ⊆ B |]
==> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"

― ‹Delete rules to do with ‹foldSetD› relation.›

declare foldSetD_imp_finite [simp del]
empty_foldSetDE [rule del]
foldSetD.intros [rule del]
declare (in LCD)
foldSetD_closed [rule del]

text ‹Commutative Monoids›

text ‹
We enter a more restrictive context, with ‹f :: 'a => 'a => 'a›
instead of ‹'b => 'a => 'a›.
›

locale ACeD =
fixes D :: "'a set"
and f :: "'a => 'a => 'a"    (infixl "⋅" 70)
and e :: 'a
assumes ident [simp]: "x ∈ D ==> x ⋅ e = x"
and commute: "[| x ∈ D; y ∈ D |] ==> x ⋅ y = y ⋅ x"
and assoc: "[| x ∈ D; y ∈ D; z ∈ D |] ==> (x ⋅ y) ⋅ z = x ⋅ (y ⋅ z)"
and e_closed [simp]: "e ∈ D"
and f_closed [simp]: "[| x ∈ D; y ∈ D |] ==> x ⋅ y ∈ D"

lemma (in ACeD) left_commute:
"[| x ∈ D; y ∈ D; z ∈ D |] ==> x ⋅ (y ⋅ z) = y ⋅ (x ⋅ z)"
proof -
assume D: "x ∈ D" "y ∈ D" "z ∈ D"
then have "x ⋅ (y ⋅ z) = (y ⋅ z) ⋅ x" by (simp add: commute)
also from D have "... = y ⋅ (z ⋅ x)" by (simp add: assoc)
also from D have "z ⋅ x = x ⋅ z" by (simp add: commute)
finally show ?thesis .
qed

lemmas (in ACeD) AC = assoc commute left_commute

lemma (in ACeD) left_ident [simp]: "x ∈ D ==> e ⋅ x = x"
proof -
assume "x ∈ D"
then have "x ⋅ e = x" by (rule ident)
with ‹x ∈ D› show ?thesis by (simp add: commute)
qed

lemma (in ACeD) foldD_Un_Int:
"[| finite A; finite B; A ⊆ D; B ⊆ D |] ==>
foldD D f e A ⋅ foldD D f e B =
foldD D f e (A Un B) ⋅ foldD D f e (A Int B)"
apply (induct set: finite)
apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
apply (simp add: AC insert_absorb Int_insert_left
LCD.foldD_insert [OF LCD.intro [of D]]
LCD.foldD_closed [OF LCD.intro [of D]]
Int_mono2)
done

lemma (in ACeD) foldD_Un_disjoint:
"[| finite A; finite B; A Int B = {}; A ⊆ D; B ⊆ D |] ==>
foldD D f e (A Un B) = foldD D f e A ⋅ foldD D f e B"
left_commute LCD.foldD_closed [OF LCD.intro [of D]])

subsubsection ‹Products over Finite Sets›

definition
finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b"
where "finprod G f A =
(if finite A
then foldD (carrier G) (mult G o f) 𝟭⇘G⇙ A
else 𝟭⇘G⇙)"

syntax
"_finprod" :: "index => idt => 'a set => 'b => 'b"
("(3⨂__∈_. _)" [1000, 0, 51, 10] 10)
translations
"⨂⇘G⇙i∈A. b" ⇌ "CONST finprod G (%i. b) A"
― ‹Beware of argument permutation!›

lemma (in comm_monoid) finprod_empty [simp]:
"finprod G f {} = 𝟭"

lemma (in comm_monoid) finprod_infinite[simp]:
"¬ finite A ⟹ finprod G f A = 𝟭"

declare funcsetI [intro]
funcset_mem [dest]

context comm_monoid begin

lemma finprod_insert [simp]:
"[| finite F; a ∉ F; f ∈ F → carrier G; f a ∈ carrier G |] ==>
finprod G f (insert a F) = f a ⊗ finprod G f F"
apply (rule trans)
apply (rule trans)
apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
apply simp
apply (rule m_lcomm)
apply fast
apply fast
apply assumption
apply fastforce
apply simp+
apply fast
done

lemma finprod_one [simp]: "(⨂i∈A. 𝟭) = 𝟭"
proof (induct A rule: infinite_finite_induct)
case empty show ?case by simp
next
case (insert a A)
have "(%i. 𝟭) ∈ A → carrier G" by auto
with insert show ?case by simp
qed simp

lemma finprod_closed [simp]:
fixes A
assumes f: "f ∈ A → carrier G"
shows "finprod G f A ∈ 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 ∈ carrier G" by fast
from insert have A: "f ∈ A → carrier G" by fast
from insert A a show ?case by simp
qed simp

lemma funcset_Int_left [simp, intro]:
"[| f ∈ A → C; f ∈ B → C |] ==> f ∈ A Int B → C"
by fast

lemma funcset_Un_left [iff]:
"(f ∈ A Un B → C) = (f ∈ A → C & f ∈ B → C)"
by fast

lemma finprod_Un_Int:
"[| finite A; finite B; g ∈ A → carrier G; g ∈ B → carrier G |] ==>
finprod G g (A Un B) ⊗ finprod G g (A Int B) =
finprod G g A ⊗ finprod G g B"
― ‹The reversed orientation looks more natural, but LOOPS as a simprule!›
proof (induct set: finite)
case empty then show ?case by simp
next
case (insert a A)
then have a: "g a ∈ carrier G" by fast
from insert have A: "g ∈ A → 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:
"[| finite A; finite B; A Int B = {};
g ∈ A → carrier G; g ∈ B → carrier G |]
==> finprod G g (A Un B) = finprod G g A ⊗ finprod G g B"
apply (subst finprod_Un_Int [symmetric])
apply auto
done

lemma finprod_multf:
"[| f ∈ A → carrier G; g ∈ A → carrier G |] ==>
finprod G (%x. f x ⊗ g x) A = (finprod G f A ⊗ 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 ∈ A → carrier G" by fast
from insert have fa: "f a ∈ carrier G" by fast
from insert have gA: "g ∈ A → carrier G" by fast
from insert have ga: "g a ∈ carrier G" by fast
from insert have fgA: "(%x. f x ⊗ g x) ∈ A → carrier G"
show ?case
by (simp add: insert fA fa gA ga fgA m_ac)
qed simp

lemma finprod_cong':
"[| A = B; g ∈ B → carrier G;
!!i. i ∈ B ==> f i = g i |] ==> finprod G f A = finprod G g B"
proof -
assume prems: "A = B" "g ∈ B → carrier G"
"!!i. i ∈ B ==> f i = g i"
show ?thesis
proof (cases "finite B")
case True
then have "!!A. [| A = B; g ∈ B → carrier G;
!!i. i ∈ B ==> f i = g i |] ==> 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 ⊗ finprod G f B"
proof (intro finprod_insert)
show "finite B" by fact
next
show "x ~: B" by fact
next
assume "x ~: B" "!!i. i ∈ insert x B ⟹ f i = g i"
"g ∈ insert x B → carrier G"
thus "f ∈ B → carrier G" by fastforce
next
assume "x ~: B" "!!i. i ∈ insert x B ⟹ f i = g i"
"g ∈ insert x B → carrier G"
thus "f x ∈ carrier G" by fastforce
qed
also from insert have "... = g x ⊗ 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:
"[| A = B; f ∈ B → carrier G = True;
!!i. i ∈ B =simp=> f i = g i |] ==> 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 ‹Usually, if this rule causes a failed congruence proof error,
the reason is that the premise ‹g ∈ B → carrier G› 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.
›

end

declare funcsetI [rule del]
funcset_mem [rule del]

context comm_monoid begin

lemma finprod_0 [simp]:
"f ∈ {0::nat} → carrier G ==> finprod G f {..0} = f 0"

lemma finprod_Suc [simp]:
"f ∈ {..Suc n} → carrier G ==>
finprod G f {..Suc n} = (f (Suc n) ⊗ finprod G f {..n})"

lemma finprod_Suc2:
"f ∈ {..Suc n} → carrier G ==>
finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} ⊗ 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_mult [simp]:
"[| f ∈ {..n} → carrier G; g ∈ {..n} → carrier G |] ==>
finprod G (%i. f i ⊗ g i) {..n::nat} =
finprod G f {..n} ⊗ finprod G g {..n}"
by (induct n) (simp_all add: m_ac Pi_def)

(* The following two were contributed by Jeremy Avigad. *)

lemma finprod_reindex:
"f : (h ` A) → carrier G ⟹
inj_on h A ==> finprod G f (h ` A) = finprod G (%x. f (h x)) A"
proof (induct A rule: infinite_finite_induct)
case (infinite A)
hence "¬ finite (h ` A)"
using finite_imageD by blast
with ‹¬ finite A› show ?case by simp

lemma finprod_const:
assumes a [simp]: "a : carrier G"
shows "finprod G (%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 ⊗ (⨂x∈A. a) = a (^) card (insert b A)"
by (insert insert, auto, subst m_comm, auto)
qed auto
qed auto

(* The following lemma was contributed by Jesus Aransay. *)

lemma finprod_singleton:
assumes i_in_A: "i ∈ A" and fin_A: "finite A" and f_Pi: "f ∈ A → carrier G"
shows "(⨂j∈A. if i = j then f j else 𝟭) = f i"
using i_in_A finprod_insert [of "A - {i}" i "(λj. if i = j then f j else 𝟭)"]
fin_A f_Pi finprod_one [of "A - {i}"]
finprod_cong [of "A - {i}" "A - {i}" "(λj. if i = j then f j else 𝟭)" "(λi. 𝟭)"]
unfolding Pi_def simp_implies_def by (force simp add: insert_absorb)

end

end
```