src/HOL/Algebra/FiniteProduct.thy
 author nipkow Fri, 13 Nov 2009 14:14:04 +0100 changeset 33657 a4179bf442d1 parent 32960 69916a850301 child 35054 a5db9779b026 permissions -rw-r--r--
renamed lemmas "anti_sym" -> "antisym"
```
(*  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 @{text LC} of theory @{text Finite_Set} is not
possible, because here we have explicit typing rules like
@{text "x \<in> carrier G"}.  We introduce an explicit argument for the domain
@{text 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 \<in> D ==> ({}, e) \<in> foldSetD D f e"
| insertI [intro]: "[| x ~: A; f x y \<in> D; (A, y) \<in> foldSetD D f e |] ==>
(insert x A, f x y) \<in> foldSetD D f e"

inductive_cases empty_foldSetDE [elim!]: "({}, x) \<in> foldSetD D f e"

constdefs
foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"
"foldD D f e A == THE x. (A, x) \<in> foldSetD D f e"

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

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

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

lemma finite_imp_foldSetD:
"[| finite A; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D |] ==>
EX 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

text {* Left-Commutative Operations *}

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

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

lemma (in LCD) Diff1_foldSetD:
"[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B |] ==>
(A, f x y) \<in> foldSetD D f e"
apply (subgoal_tac "x \<in> 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) \<in> foldSetD D f e ==> finite A"
by (induct set: foldSetD) auto

lemma (in LCD) finite_imp_foldSetD:
"[| finite A; A \<subseteq> B; e \<in> D |] ==> EX 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:
"e \<in> D ==> \<forall>A x. A \<subseteq> B & card A < n --> (A, x) \<in> foldSetD D f e -->
(\<forall>y. (A, y) \<in> 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 @{text "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 \<notin> xb"}. *}
apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb \<in> Aa & xa \<in> 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 \<le> 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 finite_Diff)
apply best
apply assumption
apply (frule (1) Diff1_foldSetD)
apply best
apply (subgoal_tac "ya = f xb x")
prefer 2
apply (subgoal_tac "Aa \<subseteq> B")
prefer 2 apply best (* slow *)
apply (blast del: equalityCE)
apply (subgoal_tac "(Ab - {xa}, x) \<in> 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) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |]
==> y = x"
by (blast intro: foldSetD_determ_aux [rule_format])

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

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

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

lemma (in LCD) foldD_insert:
"[| finite A; x ~: A; x \<in> B; e \<in> D; A \<subseteq> 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 \<in> D; A \<subseteq> B |] ==> foldD D f e A \<in> D"
proof (induct set: finite)
case empty then show ?case by (simp add: foldD_empty)
next
case insert then show ?case by (simp add: foldD_insert)
qed

lemma (in LCD) foldD_commute:
"[| finite A; x \<in> B; e \<in> D; A \<subseteq> 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 \<subseteq> C; B \<subseteq> C |] ==> A Int B \<subseteq> C"
by blast

lemma (in LCD) foldD_nest_Un_Int:
"[| finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> 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 \<in> D; A \<subseteq> B; C \<subseteq> B |]
==> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"

-- {* Delete rules to do with @{text 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 @{text "f :: 'a => 'a => 'a"}
instead of @{text "'b => 'a => 'a"}.
*}

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

lemma (in ACeD) left_commute:
"[| x \<in> D; y \<in> D; z \<in> D |] ==> 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 ==> e \<cdot> x = x"
proof -
assume "x \<in> D"
then have "x \<cdot> e = x" by (rule ident)
with `x \<in> D` show ?thesis by (simp add: commute)
qed

lemma (in ACeD) foldD_Un_Int:
"[| finite A; finite B; A \<subseteq> D; B \<subseteq> D |] ==>
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)"
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 \<subseteq> D; B \<subseteq> D |] ==>
foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
left_commute LCD.foldD_closed [OF LCD.intro [of D]])

subsubsection {* Products over Finite Sets *}

constdefs (structure G)
finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b"
"finprod G f A == if finite A
then foldD (carrier G) (mult G o f) \<one> A
else undefined"

syntax
"_finprod" :: "index => idt => 'a set => 'b => 'b"
("(3\<Otimes>__:_. _)" [1000, 0, 51, 10] 10)
syntax (xsymbols)
"_finprod" :: "index => idt => 'a set => 'b => 'b"
("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
syntax (HTML output)
"_finprod" :: "index => idt => 'a set => 'b => 'b"
("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
translations
"\<Otimes>\<index>i:A. b" == "finprod \<struct>\<index> (%i. b) A"
-- {* Beware of argument permutation! *}

lemma (in comm_monoid) finprod_empty [simp]:
"finprod G f {} = \<one>"

declare funcsetI [intro]
funcset_mem [dest]

context comm_monoid begin

lemma finprod_insert [simp]:
"[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |] ==>
finprod G f (insert a F) = f a \<otimes> 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 (fastsimp intro: m_closed)
apply simp+
apply fast
done

lemma finprod_one [simp]:
"finite A ==> (\<Otimes>i:A. \<one>) = \<one>"
proof (induct set: finite)
case empty show ?case by simp
next
case (insert a A)
have "(%i. \<one>) \<in> A -> carrier G" by auto
with insert show ?case by simp
qed

lemma finprod_closed [simp]:
fixes A
assumes fin: "finite A" and f: "f \<in> A -> carrier G"
shows "finprod G f A \<in> carrier G"
using fin f
proof 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 -> carrier G" by fast
from insert A a show ?case by simp
qed

lemma funcset_Int_left [simp, intro]:
"[| f \<in> A -> C; f \<in> B -> C |] ==> f \<in> A Int B -> C"
by fast

lemma funcset_Un_left [iff]:
"(f \<in> A Un B -> C) = (f \<in> A -> C & f \<in> B -> C)"
by fast

lemma finprod_Un_Int:
"[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
finprod G g (A Un B) \<otimes> finprod G g (A Int B) =
finprod G g A \<otimes> 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 add: finprod_closed)
next
case (insert a A)
then have a: "g a \<in> carrier G" by fast
from insert have A: "g \<in> A -> carrier G" by fast
from insert A a show ?case
by (simp add: m_ac Int_insert_left insert_absorb finprod_closed
Int_mono2)
qed

lemma finprod_Un_disjoint:
"[| finite A; finite B; A Int B = {};
g \<in> A -> carrier G; g \<in> B -> carrier G |]
==> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"
apply (subst finprod_Un_Int [symmetric])
done

lemma finprod_multf:
"[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
finprod G (%x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"
proof (induct set: finite)
case empty show ?case by simp
next
case (insert a A) then
have fA: "f \<in> A -> carrier G" by fast
from insert have fa: "f a \<in> carrier G" by fast
from insert have gA: "g \<in> A -> 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 -> carrier G"
show ?case
by (simp add: insert fA fa gA ga fgA m_ac)
qed

lemma finprod_cong':
"[| A = B; g \<in> B -> carrier G;
!!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
proof -
assume prems: "A = B" "g \<in> B -> carrier G"
"!!i. i \<in> B ==> f i = g i"
show ?thesis
proof (cases "finite B")
case True
then have "!!A. [| A = B; g \<in> B -> carrier G;
!!i. i \<in> 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 \<otimes> 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 \<in> insert x B \<Longrightarrow> f i = g i"
"g \<in> insert x B \<rightarrow> carrier G"
thus "f \<in> B -> carrier G" by fastsimp
next
assume "x ~: 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 fastsimp
qed
also from insert have "... = g x \<otimes> finprod G g B" by fastsimp
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 add: finprod_def)
qed
qed

lemma finprod_cong:
"[| A = B; f \<in> B -> carrier G = True;
!!i. i \<in> B ==> 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') force+

text {*Usually, if this rule causes a failed congruence proof error,
the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
Adding @{thm [source] Pi_def} to the simpset is often useful.
For this reason, @{thm [source] comm_monoid.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 \<in> {0::nat} -> carrier G ==> finprod G f {..0} = f 0"

lemma finprod_Suc [simp]:
"f \<in> {..Suc n} -> carrier G ==>
finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"

lemma finprod_Suc2:
"f \<in> {..Suc n} -> carrier G ==>
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_mult [simp]:
"[| f \<in> {..n} -> carrier G; g \<in> {..n} -> carrier G |] ==>
finprod G (%i. f i \<otimes> g i) {..n::nat} =
finprod G f {..n} \<otimes> 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:
assumes fin: "finite A"
shows "f : (h ` A) \<rightarrow> carrier G \<Longrightarrow>
inj_on h A ==> finprod G f (h ` A) = finprod G (%x. f (h x)) A"
using fin apply induct
apply (auto simp add: finprod_insert Pi_def)
done

lemma finprod_const:
assumes fin [simp]: "finite A"
and a [simp]: "a : carrier G"
shows "finprod G (%x. a) A = a (^) card A"
using fin apply induct
apply force
apply (subst finprod_insert)
apply auto
apply (subst m_comm)
apply auto
done

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

lemma finprod_singleton:
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)

end

end
```