src/HOL/Cardinals/Cardinal_Arithmetic.thy

more antiquotations;

(*  Title:      HOL/Cardinals/Cardinal_Arithmetic.thy
    Author:     Dmitriy Traytel, TU Muenchen
    Copyright   2012

Cardinal arithmetic.
*)

header {* Cardinal Arithmetic  *}

theory Cardinal_Arithmetic
imports Cardinal_Arithmetic_FP Cardinal_Order_Relation
begin


subsection {* Binary sum *}

lemma csum_Cnotzero2:
  "Cnotzero r2 \<Longrightarrow> Cnotzero (r1 +c r2)"
unfolding csum_def
by (metis Cnotzero_imp_not_empty Field_card_of Plus_eq_empty_conv card_of_card_order_on czeroE)

lemma single_cone:
  "|{x}| =o cone"
proof -
  let ?f = "\<lambda>x. ()"
  have "bij_betw ?f {x} {()}" unfolding bij_betw_def by auto
  thus ?thesis unfolding cone_def using card_of_ordIso by blast
qed

lemma cone_Cnotzero: "Cnotzero cone"
by (simp add: cone_not_czero Card_order_cone)

lemma cone_ordLeq_ctwo: "cone \<le>o ctwo"
unfolding cone_def ctwo_def card_of_ordLeq[symmetric] by auto


subsection {* Product *}

lemma Times_cprod: "|A \<times> B| =o |A| *c |B|"
by (simp only: cprod_def Field_card_of card_of_refl)

lemma cprod_cong2: "p2 =o r2 \<Longrightarrow> q *c p2 =o q *c r2"
by (simp only: cprod_def ordIso_Times_cong2)

lemma ordLeq_cprod1: "\<lbrakk>Card_order p1; Cnotzero p2\<rbrakk> \<Longrightarrow> p1 \<le>o p1 *c p2"
unfolding cprod_def by (metis Card_order_Times1 czeroI)

lemma cprod_infinite: "Cinfinite r \<Longrightarrow> r *c r =o r"
using cprod_infinite1' Cinfinite_Cnotzero ordLeq_refl by blast


subsection {* Exponentiation *}

lemma cexp_czero: "r ^c czero =o cone"
unfolding cexp_def czero_def Field_card_of Func_empty by (rule single_cone)

lemma Pow_cexp_ctwo:
  "|Pow A| =o ctwo ^c |A|"
unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)

lemma Cnotzero_cexp:
  assumes "Cnotzero q" "Card_order r"
  shows "Cnotzero (q ^c r)"
proof (cases "r =o czero")
  case False thus ?thesis
    apply -
    apply (rule Cnotzero_mono)
    apply (rule assms(1))
    apply (rule Card_order_cexp)
    apply (rule ordLeq_cexp1)
    apply (rule conjI)
    apply (rule notI)
    apply (erule notE)
    apply (rule ordIso_transitive)
    apply assumption
    apply (rule czero_ordIso)
    apply (rule assms(2))
    apply (rule conjunct2)
    apply (rule assms(1))
  done
next
  case True thus ?thesis
    apply -
    apply (rule Cnotzero_mono)
    apply (rule cone_Cnotzero)
    apply (rule Card_order_cexp)
    apply (rule ordIso_imp_ordLeq)
    apply (rule ordIso_symmetric)
    apply (rule ordIso_transitive)
    apply (rule cexp_cong2)
    apply assumption
    apply (rule conjunct2)
    apply (rule assms(1))
    apply (rule assms(2))
    apply (rule cexp_czero)
  done
qed

lemma Cinfinite_ctwo_cexp:
  "Cinfinite r \<Longrightarrow> Cinfinite (ctwo ^c r)"
unfolding ctwo_def cexp_def cinfinite_def Field_card_of
by (rule conjI, rule infinite_Func, auto)

lemma cone_ordLeq_iff_Field:
  assumes "cone \<le>o r"
  shows "Field r \<noteq> {}"
proof (rule ccontr)
  assume "\<not> Field r \<noteq> {}"
  hence "Field r = {}" by simp
  thus False using card_of_empty3
    card_of_mono2[OF assms] Cnotzero_imp_not_empty[OF cone_Cnotzero] by auto
qed

lemma cone_ordLeq_cexp: "cone \<le>o r1 \<Longrightarrow> cone \<le>o r1 ^c r2"
by (simp add: cexp_def cone_def Func_non_emp card_of_singl_ordLeq cone_ordLeq_iff_Field)

lemma Card_order_czero: "Card_order czero"
by (simp only: card_of_Card_order czero_def)

lemma cexp_mono2'':
  assumes 2: "p2 \<le>o r2"
  and n1: "Cnotzero q"
  and n2: "Card_order p2"
  shows "q ^c p2 \<le>o q ^c r2"
proof (cases "p2 =o (czero :: 'a rel)")
  case True
  hence "q ^c p2 =o q ^c (czero :: 'a rel)" using n1 n2 cexp_cong2 Card_order_czero by blast
  also have "q ^c (czero :: 'a rel) =o cone" using cexp_czero by blast
  also have "cone \<le>o q ^c r2" using cone_ordLeq_cexp cone_ordLeq_Cnotzero n1 by blast
  finally show ?thesis .
next
  case False thus ?thesis using assms cexp_mono2' czeroI by metis
qed

lemma csum_cexp: "\<lbrakk>Cinfinite r1; Cinfinite r2; Card_order q; ctwo \<le>o q\<rbrakk> \<Longrightarrow>
  q ^c r1 +c q ^c r2 \<le>o q ^c (r1 +c r2)"
apply (rule csum_cinfinite_bound)
apply (rule cexp_mono2)
apply (rule ordLeq_csum1)
apply (erule conjunct2)
apply assumption
apply (rule notE)
apply (rule cinfinite_not_czero[of r1])
apply (erule conjunct1)
apply assumption
apply (erule conjunct2)
apply (rule cexp_mono2)
apply (rule ordLeq_csum2)
apply (erule conjunct2)
apply assumption
apply (rule notE)
apply (rule cinfinite_not_czero[of r2])
apply (erule conjunct1)
apply assumption
apply (erule conjunct2)
apply (rule Card_order_cexp)
apply (rule Card_order_cexp)
apply (rule Cinfinite_cexp)
apply assumption
apply (rule Cinfinite_csum)
by (rule disjI1)

lemma csum_cexp': "\<lbrakk>Cinfinite r; Card_order q; ctwo \<le>o q\<rbrakk> \<Longrightarrow> q +c r \<le>o q ^c r"
apply (rule csum_cinfinite_bound)
    apply (metis Cinfinite_Cnotzero ordLeq_cexp1)
   apply (metis ordLeq_cexp2)
  apply blast+
by (metis Cinfinite_cexp)

lemma card_of_Sigma_ordLeq_Cinfinite:
  "\<lbrakk>Cinfinite r; |I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r\<rbrakk> \<Longrightarrow> |SIGMA i : I. A i| \<le>o r"
unfolding cinfinite_def by (blast intro: card_of_Sigma_ordLeq_infinite_Field)


subsection {* Powerset *}

lemma Card_order_cpow: "Card_order (cpow r)"
unfolding cpow_def by (rule card_of_Card_order)

lemma cardSuc_ordLeq_cpow: "Card_order r \<Longrightarrow> cardSuc r \<le>o cpow r"
unfolding cpow_def by (metis Card_order_Pow cardSuc_ordLess_ordLeq card_of_Card_order)

lemma cpow_cexp_ctwo: "cpow r =o ctwo ^c r"
unfolding cpow_def ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)

end