src/HOL/Algebra/FiniteProduct.thy
author wenzelm
Wed Jun 13 00:01:41 2007 +0200 (2007-06-13)
changeset 23350 50c5b0912a0c
parent 22265 3c5c6bdf61de
child 23746 a455e69c31cc
permissions -rw-r--r--
tuned proofs: avoid implicit prems;
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(*  Title:      HOL/Algebra/FiniteProduct.thy
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    ID:         $Id$
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    Author:     Clemens Ballarin, started 19 November 2002
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This file is largely based on HOL/Finite_Set.thy.
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*)
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theory FiniteProduct imports Group begin
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section {* Product Operator for Commutative Monoids *}
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subsection {* Inductive Definition of a Relation for Products over Sets *}
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text {* Instantiation of locale @{text LC} of theory @{text Finite_Set} is not
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  possible, because here we have explicit typing rules like 
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  @{text "x \<in> carrier G"}.  We introduce an explicit argument for the domain
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  @{text D}. *}
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consts
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  foldSetD :: "['a set, 'b => 'a => 'a, 'a] => ('b set * 'a) set"
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inductive "foldSetD D f e"
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  intros
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    emptyI [intro]: "e \<in> D ==> ({}, e) \<in> foldSetD D f e"
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    insertI [intro]: "[| x ~: A; f x y \<in> D; (A, y) \<in> foldSetD D f e |] ==>
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                      (insert x A, f x y) \<in> foldSetD D f e"
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inductive_cases empty_foldSetDE [elim!]: "({}, x) \<in> foldSetD D f e"
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constdefs
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  foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"
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  "foldD D f e A == THE x. (A, x) \<in> foldSetD D f e"
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lemma foldSetD_closed:
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  "[| (A, z) \<in> foldSetD D f e ; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D 
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      |] ==> z \<in> D";
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  by (erule foldSetD.elims) auto
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lemma Diff1_foldSetD:
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  "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; f x y \<in> D |] ==>
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   (A, f x y) \<in> foldSetD D f e"
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  apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
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    apply auto
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  done
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lemma foldSetD_imp_finite [simp]: "(A, x) \<in> foldSetD D f e ==> finite A"
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  by (induct set: foldSetD) auto
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lemma finite_imp_foldSetD:
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  "[| finite A; e \<in> D; !!x y. [| x \<in> A; y \<in> D |] ==> f x y \<in> D |] ==>
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   EX x. (A, x) \<in> foldSetD D f e"
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proof (induct set: finite)
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  case empty then show ?case by auto
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next
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  case (insert x F)
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  then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
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  with insert have "y \<in> D" by (auto dest: foldSetD_closed)
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  with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
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    by (intro foldSetD.intros) auto
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  then show ?case ..
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qed
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subsection {* Left-Commutative Operations *}
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locale LCD =
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  fixes B :: "'b set"
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  and D :: "'a set"
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  and f :: "'b => 'a => 'a"    (infixl "\<cdot>" 70)
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  assumes left_commute:
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    "[| x \<in> B; y \<in> B; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
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  and f_closed [simp, intro!]: "!!x y. [| x \<in> B; y \<in> D |] ==> f x y \<in> D"
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lemma (in LCD) foldSetD_closed [dest]:
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  "(A, z) \<in> foldSetD D f e ==> z \<in> D";
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  by (erule foldSetD.elims) auto
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lemma (in LCD) Diff1_foldSetD:
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  "[| (A - {x}, y) \<in> foldSetD D f e; x \<in> A; A \<subseteq> B |] ==>
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  (A, f x y) \<in> foldSetD D f e"
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  apply (subgoal_tac "x \<in> B")
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   prefer 2 apply fast
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  apply (erule insert_Diff [THEN subst], rule foldSetD.intros)
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    apply auto
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  done
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lemma (in LCD) foldSetD_imp_finite [simp]:
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  "(A, x) \<in> foldSetD D f e ==> finite A"
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  by (induct set: foldSetD) auto
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lemma (in LCD) finite_imp_foldSetD:
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  "[| finite A; A \<subseteq> B; e \<in> D |] ==> EX x. (A, x) \<in> foldSetD D f e"
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proof (induct set: finite)
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  case empty then show ?case by auto
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next
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  case (insert x F)
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  then obtain y where y: "(F, y) \<in> foldSetD D f e" by auto
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  with insert have "y \<in> D" by auto
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  with y and insert have "(insert x F, f x y) \<in> foldSetD D f e"
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    by (intro foldSetD.intros) auto
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  then show ?case ..
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qed
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lemma (in LCD) foldSetD_determ_aux:
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  "e \<in> D ==> \<forall>A x. A \<subseteq> B & card A < n --> (A, x) \<in> foldSetD D f e -->
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    (\<forall>y. (A, y) \<in> foldSetD D f e --> y = x)"
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  apply (induct n)
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   apply (auto simp add: less_Suc_eq) (* slow *)
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  apply (erule foldSetD.cases)
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   apply blast
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  apply (erule foldSetD.cases)
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   apply blast
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  apply clarify
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  txt {* force simplification of @{text "card A < card (insert ...)"}. *}
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  apply (erule rev_mp)
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  apply (simp add: less_Suc_eq_le)
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  apply (rule impI)
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  apply (rename_tac Aa xa ya Ab xb yb, case_tac "xa = xb")
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   apply (subgoal_tac "Aa = Ab")
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    prefer 2 apply (blast elim!: equalityE)
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   apply blast
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  txt {* case @{prop "xa \<notin> xb"}. *}
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  apply (subgoal_tac "Aa - {xb} = Ab - {xa} & xb \<in> Aa & xa \<in> Ab")
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   prefer 2 apply (blast elim!: equalityE)
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  apply clarify
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  apply (subgoal_tac "Aa = insert xb Ab - {xa}")
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   prefer 2 apply blast
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  apply (subgoal_tac "card Aa \<le> card Ab")
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   prefer 2
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   apply (rule Suc_le_mono [THEN subst])
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   apply (simp add: card_Suc_Diff1)
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  apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
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     apply (blast intro: foldSetD_imp_finite finite_Diff)
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    apply best
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   apply assumption
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  apply (frule (1) Diff1_foldSetD)
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   apply best
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  apply (subgoal_tac "ya = f xb x")
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   prefer 2
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   apply (subgoal_tac "Aa \<subseteq> B")
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    prefer 2 apply best (* slow *)
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   apply (blast del: equalityCE)
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  apply (subgoal_tac "(Ab - {xa}, x) \<in> foldSetD D f e")
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   prefer 2 apply simp
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  apply (subgoal_tac "yb = f xa x")
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   prefer 2 
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   apply (blast del: equalityCE dest: Diff1_foldSetD)
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  apply (simp (no_asm_simp))
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  apply (rule left_commute)
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    apply assumption
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   apply best (* slow *)
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  apply best
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  done
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lemma (in LCD) foldSetD_determ:
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  "[| (A, x) \<in> foldSetD D f e; (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |]
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  ==> y = x"
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  by (blast intro: foldSetD_determ_aux [rule_format])
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lemma (in LCD) foldD_equality:
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  "[| (A, y) \<in> foldSetD D f e; e \<in> D; A \<subseteq> B |] ==> foldD D f e A = y"
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  by (unfold foldD_def) (blast intro: foldSetD_determ)
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lemma foldD_empty [simp]:
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  "e \<in> D ==> foldD D f e {} = e"
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  by (unfold foldD_def) blast
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lemma (in LCD) foldD_insert_aux:
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  "[| x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
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    ((insert x A, v) \<in> foldSetD D f e) =
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    (EX y. (A, y) \<in> foldSetD D f e & v = f x y)"
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  apply auto
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  apply (rule_tac A1 = A in finite_imp_foldSetD [THEN exE])
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     apply (fastsimp dest: foldSetD_imp_finite)
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    apply assumption
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   apply assumption
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  apply (blast intro: foldSetD_determ)
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  done
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lemma (in LCD) foldD_insert:
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    "[| finite A; x ~: A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
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     foldD D f e (insert x A) = f x (foldD D f e A)"
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  apply (unfold foldD_def)
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  apply (simp add: foldD_insert_aux)
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  apply (rule the_equality)
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   apply (auto intro: finite_imp_foldSetD
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     cong add: conj_cong simp add: foldD_def [symmetric] foldD_equality)
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  done
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lemma (in LCD) foldD_closed [simp]:
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  "[| finite A; e \<in> D; A \<subseteq> B |] ==> foldD D f e A \<in> D"
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proof (induct set: finite)
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  case empty then show ?case by (simp add: foldD_empty)
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next
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  case insert then show ?case by (simp add: foldD_insert)
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qed
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lemma (in LCD) foldD_commute:
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  "[| finite A; x \<in> B; e \<in> D; A \<subseteq> B |] ==>
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   f x (foldD D f e A) = foldD D f (f x e) A"
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  apply (induct set: finite)
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   apply simp
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  apply (auto simp add: left_commute foldD_insert)
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  done
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lemma Int_mono2:
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  "[| A \<subseteq> C; B \<subseteq> C |] ==> A Int B \<subseteq> C"
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  by blast
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lemma (in LCD) foldD_nest_Un_Int:
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  "[| finite A; finite C; e \<in> D; A \<subseteq> B; C \<subseteq> B |] ==>
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   foldD D f (foldD D f e C) A = foldD D f (foldD D f e (A Int C)) (A Un C)"
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  apply (induct set: finite)
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   apply simp
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  apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
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    Int_mono2 Un_subset_iff)
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  done
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lemma (in LCD) foldD_nest_Un_disjoint:
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  "[| finite A; finite B; A Int B = {}; e \<in> D; A \<subseteq> B; C \<subseteq> B |]
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    ==> foldD D f e (A Un B) = foldD D f (foldD D f e B) A"
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  by (simp add: foldD_nest_Un_Int)
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-- {* Delete rules to do with @{text foldSetD} relation. *}
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declare foldSetD_imp_finite [simp del]
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  empty_foldSetDE [rule del]
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  foldSetD.intros [rule del]
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declare (in LCD)
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  foldSetD_closed [rule del]
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subsection {* Commutative Monoids *}
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text {*
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  We enter a more restrictive context, with @{text "f :: 'a => 'a => 'a"}
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  instead of @{text "'b => 'a => 'a"}.
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*}
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locale ACeD =
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  fixes D :: "'a set"
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    and f :: "'a => 'a => 'a"    (infixl "\<cdot>" 70)
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    and e :: 'a
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  assumes ident [simp]: "x \<in> D ==> x \<cdot> e = x"
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    and commute: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y = y \<cdot> x"
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    and assoc: "[| x \<in> D; y \<in> D; z \<in> D |] ==> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
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    and e_closed [simp]: "e \<in> D"
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    and f_closed [simp]: "[| x \<in> D; y \<in> D |] ==> x \<cdot> y \<in> D"
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lemma (in ACeD) left_commute:
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  "[| x \<in> D; y \<in> D; z \<in> D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
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proof -
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  assume D: "x \<in> D" "y \<in> D" "z \<in> D"
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  then have "x \<cdot> (y \<cdot> z) = (y \<cdot> z) \<cdot> x" by (simp add: commute)
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  also from D have "... = y \<cdot> (z \<cdot> x)" by (simp add: assoc)
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  also from D have "z \<cdot> x = x \<cdot> z" by (simp add: commute)
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  finally show ?thesis .
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qed
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lemmas (in ACeD) AC = assoc commute left_commute
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lemma (in ACeD) left_ident [simp]: "x \<in> D ==> e \<cdot> x = x"
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proof -
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  assume "x \<in> D"
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  then have "x \<cdot> e = x" by (rule ident)
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  with `x \<in> D` show ?thesis by (simp add: commute)
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qed
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lemma (in ACeD) foldD_Un_Int:
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  "[| finite A; finite B; A \<subseteq> D; B \<subseteq> D |] ==>
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    foldD D f e A \<cdot> foldD D f e B =
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    foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
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  apply (induct set: finite)
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   apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
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  apply (simp add: AC insert_absorb Int_insert_left
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    LCD.foldD_insert [OF LCD.intro [of D]]
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    LCD.foldD_closed [OF LCD.intro [of D]]
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    Int_mono2 Un_subset_iff)
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  done
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lemma (in ACeD) foldD_Un_disjoint:
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  "[| finite A; finite B; A Int B = {}; A \<subseteq> D; B \<subseteq> D |] ==>
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    foldD D f e (A Un B) = foldD D f e A \<cdot> foldD D f e B"
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  by (simp add: foldD_Un_Int
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    left_commute LCD.foldD_closed [OF LCD.intro [of D]] Un_subset_iff)
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subsection {* Products over Finite Sets *}
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constdefs (structure G)
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  finprod :: "[('b, 'm) monoid_scheme, 'a => 'b, 'a set] => 'b"
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  "finprod G f A == if finite A
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      then foldD (carrier G) (mult G o f) \<one> A
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      else arbitrary"
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syntax
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  "_finprod" :: "index => idt => 'a set => 'b => 'b"
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      ("(3\<Otimes>__:_. _)" [1000, 0, 51, 10] 10)
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syntax (xsymbols)
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  "_finprod" :: "index => idt => 'a set => 'b => 'b"
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      ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
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syntax (HTML output)
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  "_finprod" :: "index => idt => 'a set => 'b => 'b"
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      ("(3\<Otimes>__\<in>_. _)" [1000, 0, 51, 10] 10)
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translations
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  "\<Otimes>\<index>i:A. b" == "finprod \<struct>\<index> (%i. b) A"
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  -- {* Beware of argument permutation! *}
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lemma (in comm_monoid) finprod_empty [simp]: 
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  "finprod G f {} = \<one>"
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  by (simp add: finprod_def)
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declare funcsetI [intro]
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  funcset_mem [dest]
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lemma (in comm_monoid) finprod_insert [simp]:
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  "[| finite F; a \<notin> F; f \<in> F -> carrier G; f a \<in> carrier G |] ==>
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   finprod G f (insert a F) = f a \<otimes> finprod G f F"
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  apply (rule trans)
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   apply (simp add: finprod_def)
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  apply (rule trans)
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   apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
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         apply simp
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         apply (rule m_lcomm)
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           apply fast
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          apply fast
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         apply assumption
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        apply (fastsimp intro: m_closed)
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       apply simp+
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   apply fast
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  apply (auto simp add: finprod_def)
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  done
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lemma (in comm_monoid) finprod_one [simp]:
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  "finite A ==> (\<Otimes>i:A. \<one>) = \<one>"
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proof (induct set: finite)
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  case empty show ?case by simp
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next
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  case (insert a A)
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  have "(%i. \<one>) \<in> A -> carrier G" by auto
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  with insert show ?case by simp
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qed
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lemma (in comm_monoid) finprod_closed [simp]:
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  fixes A
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  assumes fin: "finite A" and f: "f \<in> A -> carrier G" 
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  shows "finprod G f A \<in> carrier G"
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using fin f
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proof induct
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  case empty show ?case by simp
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next
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  case (insert a A)
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  then have a: "f a \<in> carrier G" by fast
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  from insert have A: "f \<in> A -> carrier G" by fast
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  from insert A a show ?case by simp
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qed
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lemma funcset_Int_left [simp, intro]:
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  "[| f \<in> A -> C; f \<in> B -> C |] ==> f \<in> A Int B -> C"
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  by fast
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lemma funcset_Un_left [iff]:
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  "(f \<in> A Un B -> C) = (f \<in> A -> C & f \<in> B -> C)"
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  by fast
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lemma (in comm_monoid) finprod_Un_Int:
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  "[| finite A; finite B; g \<in> A -> carrier G; g \<in> B -> carrier G |] ==>
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     finprod G g (A Un B) \<otimes> finprod G g (A Int B) =
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     finprod G g A \<otimes> finprod G g B"
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-- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
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proof (induct set: finite)
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  case empty then show ?case by (simp add: finprod_closed)
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next
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  case (insert a A)
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  then have a: "g a \<in> carrier G" by fast
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  from insert have A: "g \<in> A -> carrier G" by fast
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  from insert A a show ?case
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    by (simp add: m_ac Int_insert_left insert_absorb finprod_closed
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          Int_mono2 Un_subset_iff) 
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qed
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lemma (in comm_monoid) finprod_Un_disjoint:
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  "[| finite A; finite B; A Int B = {};
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      g \<in> A -> carrier G; g \<in> B -> carrier G |]
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   ==> finprod G g (A Un B) = finprod G g A \<otimes> finprod G g B"
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  apply (subst finprod_Un_Int [symmetric])
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      apply (auto simp add: finprod_closed)
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  done
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lemma (in comm_monoid) finprod_multf:
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  "[| finite A; f \<in> A -> carrier G; g \<in> A -> carrier G |] ==>
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   finprod G (%x. f x \<otimes> g x) A = (finprod G f A \<otimes> finprod G g A)"
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proof (induct set: finite)
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  case empty show ?case by simp
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next
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  case (insert a A) then
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  have fA: "f \<in> A -> carrier G" by fast
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  from insert have fa: "f a \<in> carrier G" by fast
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  from insert have gA: "g \<in> A -> carrier G" by fast
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  from insert have ga: "g a \<in> carrier G" by fast
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  from insert have fgA: "(%x. f x \<otimes> g x) \<in> A -> carrier G"
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    by (simp add: Pi_def)
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  show ?case
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    by (simp add: insert fA fa gA ga fgA m_ac)
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qed
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   408
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lemma (in comm_monoid) finprod_cong':
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  "[| A = B; g \<in> B -> carrier G;
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      !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
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proof -
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  assume prems: "A = B" "g \<in> B -> carrier G"
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   414
    "!!i. i \<in> B ==> f i = g i"
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   415
  show ?thesis
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   416
  proof (cases "finite B")
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   417
    case True
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    then have "!!A. [| A = B; g \<in> B -> carrier G;
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      !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
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    proof induct
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   421
      case empty thus ?case by simp
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   422
    next
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   423
      case (insert x B)
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   424
      then have "finprod G f A = finprod G f (insert x B)" by simp
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   425
      also from insert have "... = f x \<otimes> finprod G f B"
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   426
      proof (intro finprod_insert)
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   427
	show "finite B" by fact
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      next
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   429
	show "x ~: B" by fact
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   430
      next
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   431
	assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
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	  "g \<in> insert x B \<rightarrow> carrier G"
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   433
	thus "f \<in> B -> carrier G" by fastsimp
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   434
      next
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   435
	assume "x ~: B" "!!i. i \<in> insert x B \<Longrightarrow> f i = g i"
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   436
	  "g \<in> insert x B \<rightarrow> carrier G"
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   437
	thus "f x \<in> carrier G" by fastsimp
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   438
      qed
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   439
      also from insert have "... = g x \<otimes> finprod G g B" by fastsimp
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   440
      also from insert have "... = finprod G g (insert x B)"
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   441
      by (intro finprod_insert [THEN sym]) auto
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   442
      finally show ?case .
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   443
    qed
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   444
    with prems show ?thesis by simp
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   445
  next
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   446
    case False with prems show ?thesis by (simp add: finprod_def)
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   447
  qed
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   448
qed
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   449
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   450
lemma (in comm_monoid) finprod_cong:
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   451
  "[| A = B; f \<in> B -> carrier G = True;
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   452
      !!i. i \<in> B ==> f i = g i |] ==> finprod G f A = finprod G g B"
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   453
  (* This order of prems is slightly faster (3%) than the last two swapped. *)
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   454
  by (rule finprod_cong') force+
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   455
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   456
text {*Usually, if this rule causes a failed congruence proof error,
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   457
  the reason is that the premise @{text "g \<in> B -> carrier G"} cannot be shown.
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   458
  Adding @{thm [source] Pi_def} to the simpset is often useful.
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   459
  For this reason, @{thm [source] comm_monoid.finprod_cong}
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   460
  is not added to the simpset by default.
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   461
*}
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   462
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   463
declare funcsetI [rule del]
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   464
  funcset_mem [rule del]
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   465
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   466
lemma (in comm_monoid) finprod_0 [simp]:
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   467
  "f \<in> {0::nat} -> carrier G ==> finprod G f {..0} = f 0"
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   468
by (simp add: Pi_def)
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   469
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   470
lemma (in comm_monoid) finprod_Suc [simp]:
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   471
  "f \<in> {..Suc n} -> carrier G ==>
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   472
   finprod G f {..Suc n} = (f (Suc n) \<otimes> finprod G f {..n})"
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   473
by (simp add: Pi_def atMost_Suc)
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   474
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   475
lemma (in comm_monoid) finprod_Suc2:
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   476
  "f \<in> {..Suc n} -> carrier G ==>
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   477
   finprod G f {..Suc n} = (finprod G (%i. f (Suc i)) {..n} \<otimes> f 0)"
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   478
proof (induct n)
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   479
  case 0 thus ?case by (simp add: Pi_def)
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   480
next
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   481
  case Suc thus ?case by (simp add: m_assoc Pi_def)
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   482
qed
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   483
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   484
lemma (in comm_monoid) finprod_mult [simp]:
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   485
  "[| f \<in> {..n} -> carrier G; g \<in> {..n} -> carrier G |] ==>
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   486
     finprod G (%i. f i \<otimes> g i) {..n::nat} =
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   487
     finprod G f {..n} \<otimes> finprod G g {..n}"
ballarin@13936
   488
  by (induct n) (simp_all add: m_ac Pi_def)
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   489
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   490
end
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   491