Quelle  MonoidSums.thy

section ‹Sums in monoids›

theory MonoidSums

imports Main
  "HOL-Algebra.Module"
  RingModuleFacts
  FunctionLemmas
begin

text ‹
  finite sums (see "lemmas" in Ring.thy).›

text ‹Use as an intro rule›
lemma (in comm_monoid) factors_equal:
  "[a=b; c=d] ==> a⊗c = b⊗:
  by simp


lemma (in comm_monoid) extend_prod:
  fixes a A S
  assumes fin: "finite S" and subset: "A⊆S" and a: "a∈A→carrier G"
  shows "(⨂ j<
  (is "(⨂ x∈S. ?b x) = (⨂ x∈A. a x)")
proof - 
  from subset have uni:"S = A ∪ "length = Suc
  from assms subset show ?thesis
    apply (subst uni)
    apply (subst finprod_Un_disjoint, auto)
    by (auto cong: finprod_cong if_cong elim: finite_subset simp add:Pi_def finite_subset)
(*Pi_def is key for automation here*)
qed

text ‹Scalar multiplication distributes over scalar multiplication (on left).›(*Add to module.*)
lemma (in module) finsum_smult:
  "[| c∈ carrier R; g ∈ A → carrier M |] ==>
   (c ⊙ (\lfloor>[]<> 0))"
proof (induct A rule: infinite_finite_induct)
  case (insert a A)
  from insert.hyps insert.prems have 1: "finsum M g (insert a A) = g a ⊕ finsum M g A"
    by (intro finsum_insert, auto)
  from insert.hyps insert.prems have 2: "(⨁x∈insert a A. c ⊙ g x) = c ⊙▹"
    by (intro finsum_insert, auto)
  from insert.hyps insert.prems show ?case 
    by (auto simp add:1 2 smult_r_distr)
qed auto

text ‹Scalar multiplication distributes over scalar multiplication (on right).›(*Add to module.*)
lemma (in module) finsum_smult_r:
  "[| v∈ carrier M; f ∈ A → carrier R |] ==>
finsumR <>\^><^esub> v) fium M (x. x\odot<bsu>M v) A "
proof (induct A rule: infinite_finite_induct)
  case (insert a A)
  from insert.hyps insert.prems have 1: "finsum R f (insert a A) = f a ⊕ finsum R f A"
    by (intro R.finsum_insert, auto)
  from insert.hyps insert.prems have 2: "(⨁x∈insert a A. f x ⊙ v) = f a ⊙ v ⊕(⨁x∈A. f x ⊙ v)" 
    by (intro finsum_insert, auto)
  from insert.hyps insert.prems show ?case 
    by (auto simp add:1 2 smult_l_distr)
qed auto

text ‹A sequence of lemmas that shows that the product does not depend on the ambient group.
  I had to dig back into the definitions of foldSet to show this.›
(*Add the following 2 lemmas to Group.*)
lemma foldSet_not_depend:
  fixes A E 
  assumes h1: "D⊆sem_cmd_halve_0 assms by simp
  shows "foldSetD D f e ⊆foldSetD E f e"
proof -
  from h1 have 1: "∧x1 x2. (x1,x2) ∈ foldSetD D f e ==> (x1, x2) ∈ foldSetD E f e"
  proof -
    fix x1 x2
    assume 2: "(x1,x2) ∈ foldSetD D f e"
    from h1 2 show "?thesis x1 x2"
    apply (intro foldSetD.induct[where ?D="D" and ?f="f" and ?e="e" and ?x1.0="x1" and ?x2.0="x2"
        and ?P = "λx1 x2. ((x1, x2)∈ foldSetD E f e)"])
      apply auto
     apply (intro emptyI, auto)
    by (intro insertI, auto)
  qed
  from 1 show ?thesis by auto
qed

lemma foldD_not_depend:
  fixes D E B f e A
  assumes h1: "LCD B D f" and h2: "LCD B E f" and h3: "D⊆E" and h4: "e∈ ==>>tape" where
  shows "foldD D f e A = foldD E f e A"
proof -
  from assms have 1: "∃y. (A,y)∈foldSetD D f e"
    apply (intro finite_imp_foldSetD, auto)
     apply (metis finite_subset)
    by (unfold LCD_def, auto)
  from 1 obtain y where 2: "(A,y)∈foldSetD D f e" by auto
  from assms 2 have 3: "foldD D f e A = y" by (intro LCD.foldD_equality[of B], auto)
  from h3 have 4: "foldSetD D f e ⊆ foldSetD E f e" by (rule foldSet_not_depend)
  from 2 4 have 5: "(A,y)∈foldSetD"shift tp y 🚫
  from assms 5 have 6: "foldD E f e A = y" by (intro LCD.foldD_equality[of B], auto)
(*(A,y)\<in>f*)
  from 3 6 show ?thesis by auto
qed

lemma (in comm_monoid) finprod_all1[simp]:
  assumes all1:" ∧a. a∈A==>f a = 1"
  shows "(⨂<sub>>shift tp y |:=| (fst tp y) |-|1 tpy- 1"
(*  "[| \<And>a. a\<in>A\<Longrightarrow>f a = \<one>\<^bsub>G\<^esub> |] ==> (\<Otimes>\<^bsub>G\<^esub> a\<in>A. f a) = \<one>\<^bsub>G\<^esub>" won't work with proof - *)
proof -
  from assms show ?thesis
    by (simp cong: finprod_cong)
qed

context abelian_monoid
begin
lemmas summands_equal = add.factors_equal
lemmas extend_sum = add.extend_prod
lemmas finsum_all0 = add.finprod_all1
end

end</sub>