# HG changeset patch
# User paulson
# Date 1049885509 -7200
# Node ID eefdd6b14508407cbba9267522b2f6f73f375900
# Parent  3e496c70f2f3b2657de3fbadff7060791d568911
Removal of Summation theory

diff -r 3e496c70f2f3 -r eefdd6b14508 src/HOL/GroupTheory/ROOT.ML
--- a/src/HOL/GroupTheory/ROOT.ML	Tue Apr 08 09:44:21 2003 +0200
+++ b/src/HOL/GroupTheory/ROOT.ML	Wed Apr 09 12:51:49 2003 +0200
@@ -1,4 +1,3 @@
 no_document use_thy "FuncSet";
 
 use_thy "Bij";
-use_thy "Summation";
\ No newline at end of file
diff -r 3e496c70f2f3 -r eefdd6b14508 src/HOL/GroupTheory/Summation.thy
--- a/src/HOL/GroupTheory/Summation.thy	Tue Apr 08 09:44:21 2003 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,592 +0,0 @@
-(*  Title:      Summation Operator for Abelian Groups
-    ID:         $Id$
-    Author:     Clemens Ballarin, started 19 November 2002
-    Copyright:  TU Muenchen
-*)
-
-header {* Summation Operator *}
-
-theory Summation = Group:
-
-(* Instantiation of LC from Finite_Set.thy is not possible,
-   because here we have explicit typing rules like x : carrier G.
-   We introduce an explicit argument for the domain D *)
-
-consts
-  foldSetD :: "['a set, 'b => 'a => 'a, 'a] => ('b set * 'a) set"
-
-inductive "foldSetD D f e"
-  intros
-    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"
-
-constdefs
-  foldD :: "['a set, 'b => 'a => 'a, 'a, 'b set] => 'a"
-  "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.elims) 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: Finites)
-  case empty then show ?case by auto
-next
-  case (insert F x)
-  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
-
-subsection {* 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 : B; y : B; z : D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> 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.elims) 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: Finites)
-  case empty then show ?case by auto
-next
-  case (insert F x)
-  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 ==> ALL A x. A <= B & card A < n --> (A, x) : foldSetD D f e -->
-    (ALL y. (A, y) : foldSetD D f e --> y = x)"
-  apply (induct n)
-   apply (auto simp add: less_Suc_eq)
-  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 (simp add: less_Suc_eq_le)
-  apply (rule impI)
-  apply (rename_tac Aa xa ya Ab xb 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 : 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 (simp add: card_Suc_Diff1)
-  apply (rule_tac A1 = "Aa - {xb}" in finite_imp_foldSetD [THEN exE])
-  apply (blast intro: foldSetD_imp_finite finite_Diff)
-(* new subgoal from finite_imp_foldSetD *)
-    apply best (* blast doesn't seem to solve this *)
-   apply assumption
-  apply (frule (1) Diff1_foldSetD)
-(* new subgoal from Diff1_foldSetD *)
-    apply best
-(*
-apply (best del: foldSetD_closed elim: foldSetD_closed)
-    apply (rule f_closed) apply assumption apply (rule foldSetD_closed)
-    prefer 3 apply assumption apply (rule e_closed)
-    apply (rule f_closed) apply force apply assumption
-*)
-  apply (subgoal_tac "ya = f xb x")
-   prefer 2
-(* new subgoal to make IH applicable *) 
-  apply (subgoal_tac "Aa <= B")
-   prefer 2 apply best
-   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 (drule_tac x = xa in Diff1_foldSetD)
-     apply assumption
-     apply (rule f_closed) apply best apply (rule foldSetD_closed)
-     prefer 3 apply assumption apply (rule e_closed)
-     apply (rule f_closed) apply best apply assumption
-*)
-   apply (blast del: equalityCE dest: Diff1_foldSetD)
-   apply (simp (no_asm_simp))
-   apply (rule left_commute)
-   apply assumption apply best 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 (fastsimp dest: foldSetD_imp_finite)
-(* new subgoal by finite_imp_foldSetD *)
-    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 (simp add: foldD_insert_aux)
-  apply (rule the_equality)
-  apply (auto intro: finite_imp_foldSetD
-    cong add: conj_cong simp add: foldD_def [symmetric] foldD_equality)
-  done
-
-lemma (in LCD) foldD_closed [simp]:
-  "[| finite A; e : D; A <= B |] ==> foldD D f e A : D"
-proof (induct set: Finites)
-  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 : B; e : D; A <= B |] ==>
-   f x (foldD D f e A) = foldD D f (f x e) A"
-  apply (induct set: Finites)
-   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: Finites)
-   apply simp
-  apply (simp add: foldD_insert foldD_commute Int_insert_left insert_absorb
-    Int_mono2 Un_subset_iff)
-  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"
-  by (simp add: foldD_nest_Un_Int)
-
--- {* 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]
-
-subsection {* 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 : D ==> x \<cdot> e = x"
-    and commute: "[| x : D; y : D |] ==> x \<cdot> y = y \<cdot> x"
-    and assoc: "[| x : D; y : D; z : D |] ==> (x \<cdot> y) \<cdot> z = x \<cdot> (y \<cdot> z)"
-    and e_closed [simp]: "e : D"
-    and f_closed [simp]: "[| x : D; y : D |] ==> x \<cdot> y : D"
-
-lemma (in ACeD) left_commute:
-  "[| x : D; y : D; z : D |] ==> x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)"
-proof -
-  assume D: "x : D" "y : D" "z : 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 : D ==> e \<cdot> x = x"
-proof -
-  assume D: "x : D"
-  have "x \<cdot> e = x" by (rule ident)
-  with 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 \<cdot> foldD D f e B =
-    foldD D f e (A Un B) \<cdot> foldD D f e (A Int B)"
-  apply (induct set: Finites)
-   apply (simp add: left_commute LCD.foldD_closed [OF LCD.intro [of D]])
-(* left_commute is required to show premise of LCD.intro *)
-  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 Un_subset_iff)
-  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 \<cdot> foldD D f e B"
-  by (simp add: foldD_Un_Int
-    left_commute LCD.foldD_closed [OF LCD.intro [of D]] Un_subset_iff)
-
-subsection {* Abelian groups with summation operator *}
-
-lemma (in abelian_group) sum_lcomm:
-  "[| x : carrier G; y : carrier G; z : carrier G |] ==>
-   x \<oplus> (y \<oplus> z) = y \<oplus> (x \<oplus> z)"
-proof -
-  assume "x : carrier G" "y : carrier G" "z : carrier G"
-  then have "x \<oplus> (y \<oplus> z) = (x \<oplus> y) \<oplus> z" by (simp add: sum_assoc)
-  also from prems have "... = (y \<oplus> x) \<oplus> z" by (simp add: sum_commute)
-  also from prems have "... = y \<oplus> (x \<oplus> z)" by (simp add: sum_assoc)
-  finally show ?thesis .
-qed
-
-lemmas (in abelian_group) AC = sum_assoc sum_commute sum_lcomm
-
-record ('a, 'b) group_with_sum = "'a group" +
-  setSum :: "['b => 'a, 'b set] => 'a"
-
-(* TODO: nice syntax for the summation operator inside the locale
-   like \<Oplus>\<index> i\<in>A. f i, probably needs hand-coded translation *)
-
-locale agroup_with_sum = abelian_group +
-  assumes setSum_def:
-  "setSum G f A = (if finite A then foldD (carrier G) (op \<oplus> o f) \<zero> A else \<zero>)"
-
-ML_setup {* 
-
-Context.>> (fn thy => (simpset_ref_of thy :=
-  simpset_of thy setsubgoaler asm_full_simp_tac; thy)) *}
-
-lemma (in agroup_with_sum) setSum_empty [simp]: 
-  "setSum G f {} = \<zero>"
-  by (simp add: setSum_def)
-
-ML_setup {* 
-
-Context.>> (fn thy => (simpset_ref_of thy :=
-  simpset_of thy setsubgoaler asm_simp_tac; thy)) *}
-
-lemma insert_conj:
-  "[| a = b; a : B |] ==> a : insert b B"
-  by blast
-
-declare funcsetI [intro]
-  funcset_mem [dest]
-
-lemma (in agroup_with_sum) setSum_insert [simp]:
-  "[| finite F; a \<notin> F; f : F -> carrier G; f a : carrier G |] ==>
-   setSum G f (insert a F) = f a \<oplus> setSum G f F"
-  apply (rule trans)
-  apply (simp add: setSum_def)
-  apply (rule trans)
-  apply (rule LCD.foldD_insert [OF LCD.intro [of "insert a F"]])
-    apply simp
-    apply (rule sum_lcomm)
-      apply fast apply fast apply assumption
-    apply (fastsimp intro: sum_closed)
-    apply simp+ apply fast
-  apply (auto simp add: setSum_def)
-  done
-
-lemma (in agroup_with_sum) setSum_0:
-  "setSum G (%i. \<zero>) A = \<zero>"
-(*  apply (case_tac "finite A")
-   prefer 2 apply (simp add: setSum_def) *)
-proof (cases "finite A")
-  case True then show ?thesis
-  proof (induct set: Finites)
-    case empty show ?case by simp
-  next
-    case (insert A a)
-    have "(%i. \<zero>) : A -> carrier G" by auto
-    with insert show ?case by simp
-  qed
-next
-  case False then show ?thesis by (simp add: setSum_def)
-qed
-
-(*
-lemma setSum_eq_0_iff [simp]:
-    "finite F ==> (setSum f F = 0) = (ALL a:F. f a = (0::nat))"
-  by (induct set: Finites) auto
-
-lemma setSum_SucD: "setSum f A = Suc n ==> EX a:A. 0 < f a"
-  apply (case_tac "finite A")
-   prefer 2 apply (simp add: setSum_def)
-  apply (erule rev_mp)
-  apply (erule finite_induct)
-   apply auto
-  done
-
-lemma card_eq_setSum: "finite A ==> card A = setSum (\<lambda>x. 1) A"
-*)  -- {* Could allow many @{text "card"} proofs to be simplified. *}
-(*
-  by (induct set: Finites) auto
-*)
-
-lemma (in agroup_with_sum) setSum_closed:
-  "[| finite A; f : A -> carrier G |] ==> setSum G f A : carrier G"
-proof (induct set: Finites)
-  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
-(*
-lemma (in agroup_with_sum) setSum_closed:
-  "[| finite A; f``A <= carrier G |] ==> setSum G f A : carrier G"
-
-lemma (in agroup_with_sum) setSum_closed:
-  "[| finite A; !!i. i : A ==> f i : carrier G |] ==>
-   setSum G f A : carrier G"
-*)
-
-lemma funcset_Int_left [simp, intro]:
-  "[| f : A -> C; f : B -> C |] ==> f : A Int B -> C"
-  by fast
-
-lemma funcset_Int_right:
-  "(f : A -> B Int C) = (f : A -> B & f : A -> C)"
-  by fast
-
-lemma funcset_Un_right:
-  "[| f : A -> B; f : A -> C |] ==> f : A -> B Un C"
-  by fast
-
-lemma funcset_Un_left [iff]:
-  "(f : A Un B -> C) = (f : A -> C & f : B -> C)"
-  by fast
-
-lemma (in agroup_with_sum) setSum_Un_Int:
-  "[| finite A; finite B; g : A -> carrier G; g : B -> carrier G |] ==>
-   setSum G g (A Un B) \<oplus> setSum G g (A Int B) = setSum G g A \<oplus> setSum G g B"
-  -- {* The reversed orientation looks more natural, but LOOPS as a simprule! *}
-proof (induct set: Finites)
-  case empty then show ?case by (simp add: setSum_closed)
-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: AC Int_insert_left insert_absorb setSum_closed
-          Int_mono2 Un_subset_iff) 
-qed
-
-lemma (in agroup_with_sum) setSum_Un_disjoint:
-  "[| finite A; finite B; A Int B = {};
-      g : A -> carrier G; g : B -> carrier G |]
-   ==> setSum G g (A Un B) = setSum G g A \<oplus> setSum G g B"
-  apply (subst setSum_Un_Int [symmetric])
-    apply (auto simp add: setSum_closed)
-  done
-
-(*
-lemma setSum_UN_disjoint:
-  fixes f :: "'a => 'b::plus_ac0"
-  shows
-    "finite I ==> (ALL i:I. finite (A i)) ==>
-        (ALL i:I. ALL j:I. i \<noteq> j --> A i Int A j = {}) ==>
-      setSum f (UNION I A) = setSum (\<lambda>i. setSum f (A i)) I"
-  apply (induct set: Finites)
-   apply simp
-  apply atomize
-  apply (subgoal_tac "ALL i:F. x \<noteq> i")
-   prefer 2 apply blast
-  apply (subgoal_tac "A x Int UNION F A = {}")
-   prefer 2 apply blast
-  apply (simp add: setSum_Un_disjoint)
-  done
-*)
-lemma (in agroup_with_sum) setSum_addf:
-  "[| finite A; f : A -> carrier G; g : A -> carrier G |] ==>
-   setSum G (%x. f x \<oplus> g x) A = (setSum G f A \<oplus> setSum G g A)"
-proof (induct set: Finites)
-  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 \<oplus> g x) a : carrier G" by (simp add: Pi_def)
-  from insert have fgA: "(%x. f x \<oplus> g x) : A -> carrier G"
-    by (simp add: Pi_def)
-  show ?case  (* check if all simps are really necessary *)
-    by (simp add: insert fA fa gA ga fgA fga AC setSum_closed Int_insert_left insert_absorb Int_mono2 Un_subset_iff)
-qed
-
-(*
-lemma setSum_Un: "finite A ==> finite B ==>
-    (setSum f (A Un B) :: nat) = setSum f A + setSum f B - setSum f (A Int B)"
-  -- {* For the natural numbers, we have subtraction. *}
-  apply (subst setSum_Un_Int [symmetric])
-    apply auto
-  done
-
-lemma setSum_diff1: "(setSum f (A - {a}) :: nat) =
-    (if a:A then setSum f A - f a else setSum f A)"
-  apply (case_tac "finite A")
-   prefer 2 apply (simp add: setSum_def)
-  apply (erule finite_induct)
-   apply (auto simp add: insert_Diff_if)
-  apply (drule_tac a = a in mk_disjoint_insert)
-  apply auto
-  done
-*)
-
-lemma (in agroup_with_sum) setSum_cong:
-  "[| A = B; g : B -> carrier G;
-      !!i. i : B ==> f i = g i |] ==> setSum G f A = setSum 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 |] ==> setSum G f A = setSum G g B"
-    proof induct
-      case empty thus ?case by simp
-    next
-      case (insert B x)
-      then have "setSum G f A = setSum G f (insert x B)" by simp
-      also from insert have "... = f x \<oplus> setSum G f B"
-      proof (intro setSum_insert)
-	show "finite B" .
-      next
-	show "x ~: B" .
-      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 : 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 \<oplus> setSum G g B" by fastsimp
-      also from insert have "... = setSum G g (insert x B)"
-      by (intro setSum_insert [THEN sym]) auto
-      finally show ?case .
-    qed
-    with prems show ?thesis by simp
-  next
-    case False with prems show ?thesis by (simp add: setSum_def)
-  qed
-qed
-
-lemma (in agroup_with_sum) setSum_cong1 [cong]:
-  "[| A = B; !!i. i : B ==> f i = g i;
-      g : B -> carrier G = True |] ==> setSum G f A = setSum G g B"
-  by (rule setSum_cong) fast+
-
-text {*Usually, if this rule causes a failed congruence proof error,
-   the reason is that the premise @{text "g : B -> carrier G"} cannot be shown.
-   Adding @{thm [source] Pi_def} to the simpset is often useful. *}
-
-declare funcsetI [rule del]
-  funcset_mem [rule del]
-
-(*** Examples --- Summation over the integer interval {..n} ***)
-
-(* New locale where index set is restricted to nat *)
-
-locale agroup_with_natsum = agroup_with_sum +
-  assumes "False ==> setSum G f (A::nat set) = setSum G f A"
-
-lemma (in agroup_with_natsum) natSum_0 [simp]:
-  "f : {0::nat} -> carrier G ==> setSum G f {..0} = f 0"
-by (simp add: Pi_def)
-
-lemma (in agroup_with_natsum) natsum_Suc [simp]:
-  "f : {..Suc n} -> carrier G ==>
-   setSum G f {..Suc n} = (f (Suc n) \<oplus> setSum G f {..n})"
-by (simp add: Pi_def atMost_Suc)
-
-lemma (in agroup_with_natsum) natsum_Suc2:
-  "f : {..Suc n} -> carrier G ==>
-   setSum G f {..Suc n} = (setSum G (%i. f (Suc i)) {..n} \<oplus> f 0)"
-proof (induct n)
-  case 0 thus ?case by (simp add: Pi_def)
-next
-  case Suc thus ?case by (simp add: sum_assoc Pi_def setSum_closed)
-qed
-
-lemma (in agroup_with_natsum) natsum_zero [simp]:
-  "setSum G (%i. \<zero>) {..n::nat} = \<zero>"
-by (induct n) (simp_all add: Pi_def)
-
-lemma (in agroup_with_natsum) natsum_add [simp]:
-  "[| f : {..n} -> carrier G; g : {..n} -> carrier G |] ==>
-   setSum G (%i. f i \<oplus> g i) {..n::nat} = setSum G f {..n} \<oplus> setSum G g {..n}"
-by (induct n) (simp_all add: AC Pi_def setSum_closed)
-
-end
-
diff -r 3e496c70f2f3 -r eefdd6b14508 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Tue Apr 08 09:44:21 2003 +0200
+++ b/src/HOL/IsaMakefile	Wed Apr 09 12:51:49 2003 +0200
@@ -286,7 +286,6 @@
   Library/Primes.thy Library/FuncSet.thy \
   GroupTheory/Bij.thy \
   GroupTheory/Group.thy \
-  GroupTheory/Summation.thy \
   GroupTheory/ROOT.ML \
   GroupTheory/document/root.tex
 	@$(ISATOOL) usedir -g true $(OUT)/HOL GroupTheory