author ballarin Wed, 11 Dec 2002 10:12:48 +0100 changeset 13745 a31e04831dd1 parent 13744 2241191a3c54 child 13746 550260ace09e
HOL/GroupTheory/Summation.thy added: summation operator for abelian groups.
 NEWS file | annotate | diff | comparison | revisions src/HOL/GroupTheory/Group.thy file | annotate | diff | comparison | revisions src/HOL/GroupTheory/README.html file | annotate | diff | comparison | revisions src/HOL/GroupTheory/ROOT.ML file | annotate | diff | comparison | revisions src/HOL/GroupTheory/Summation.thy file | annotate | diff | comparison | revisions src/HOL/IsaMakefile file | annotate | diff | comparison | revisions
```--- a/NEWS	Tue Dec 10 10:40:32 2002 +0100
+++ b/NEWS	Wed Dec 11 10:12:48 2002 +0100
@@ -90,6 +90,8 @@

*** HOL ***

+* GroupTheory: new, experimental summation operator for abelian groups.
+
* New tactic "trans_tac" and method "trans" instantiate
Provers/linorder.ML for axclasses "order" and "linorder" (predicates
"<=", "<" and "="). ```
```--- a/src/HOL/GroupTheory/Group.thy	Tue Dec 10 10:40:32 2002 +0100
+++ b/src/HOL/GroupTheory/Group.thy	Wed Dec 11 10:12:48 2002 +0100
@@ -13,8 +13,7 @@
carrier :: "'a set"
sum :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"    (infixl "\<oplus>\<index>" 65)

-locale semigroup =
-  fixes S    (structure)
+locale semigroup = struct S +
assumes sum_funcset: "sum S \<in> carrier S \<rightarrow> carrier S \<rightarrow> carrier S"
and sum_assoc:
"[|x \<in> carrier S; y \<in> carrier S; z \<in> carrier S|] ```
```--- a/src/HOL/GroupTheory/README.html	Tue Dec 10 10:40:32 2002 +0100
+++ b/src/HOL/GroupTheory/README.html	Wed Dec 11 10:12:48 2002 +0100
@@ -25,6 +25,10 @@

<LI>Theory <A HREF="Sylow.html"><CODE>Sylow</CODE></A>
contains a proof of the first Sylow theorem.
+
+<LI>Theory <A HREF="Summation.html"><CODE>Summation</CODE></A> Extends
+abelian groups by a summation operator for finite sets (provided by
+Clemens Ballarin).
</UL>

<HR>```
```--- a/src/HOL/GroupTheory/ROOT.ML	Tue Dec 10 10:40:32 2002 +0100
+++ b/src/HOL/GroupTheory/ROOT.ML	Wed Dec 11 10:12:48 2002 +0100
@@ -3,3 +3,4 @@

use_thy "Sylow";
use_thy "Module";
+use_thy "Summation";
\ No newline at end of file```
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/GroupTheory/Summation.thy	Wed Dec 11 10:12:48 2002 +0100
@@ -0,0 +1,590 @@
+(*  Title:      Summation Operator for Abelian Groups
+    ID:         \$Id\$
+    Author:     Clemens Ballarin, started 19 November 2002
+    Copyright:  TU Muenchen
+*)
+
+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
+
+subsubsection {* 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]
+
+subsubsection {* 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 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
+```
```--- a/src/HOL/IsaMakefile	Tue Dec 10 10:40:32 2002 +0100
+++ b/src/HOL/IsaMakefile	Wed Dec 11 10:12:48 2002 +0100
@@ -284,6 +284,7 @@
GroupTheory/Exponent.thy \
GroupTheory/Group.thy \
GroupTheory/Module.thy GroupTheory/Ring.thy \
+  GroupTheory/Summation.thy \
GroupTheory/Sylow.thy \
GroupTheory/ROOT.ML \
GroupTheory/document/root.tex```