(* Title: HOL/BNF_Cardinal_Arithmetic.thy
Author: Dmitriy Traytel, TU Muenchen
Copyright 2012
Cardinal arithmetic as needed by bounded natural functors.
*)
header {* Cardinal Arithmetic as Needed by Bounded Natural Functors *}
theory BNF_Cardinal_Arithmetic
imports BNF_Cardinal_Order_Relation
begin
lemma dir_image: "\<lbrakk>\<And>x y. (f x = f y) = (x = y); Card_order r\<rbrakk> \<Longrightarrow> r =o dir_image r f"
by (rule dir_image_ordIso) (auto simp add: inj_on_def card_order_on_def)
(*should supersede a weaker lemma from the library*)
lemma dir_image_Field: "Field (dir_image r f) = f ` Field r"
unfolding dir_image_def Field_def Range_def Domain_def by fast
lemma card_order_dir_image:
assumes bij: "bij f" and co: "card_order r"
shows "card_order (dir_image r f)"
proof -
from assms have "Field (dir_image r f) = UNIV"
using card_order_on_Card_order[of UNIV r] unfolding bij_def dir_image_Field by auto
moreover from bij have "\<And>x y. (f x = f y) = (x = y)" unfolding bij_def inj_on_def by auto
with co have "Card_order (dir_image r f)"
using card_order_on_Card_order[of UNIV r] Card_order_ordIso2[OF _ dir_image] by blast
ultimately show ?thesis by auto
qed
lemma ordIso_refl: "Card_order r \<Longrightarrow> r =o r"
by (rule card_order_on_ordIso)
lemma ordLeq_refl: "Card_order r \<Longrightarrow> r \<le>o r"
by (rule ordIso_imp_ordLeq, rule card_order_on_ordIso)
lemma card_of_ordIso_subst: "A = B \<Longrightarrow> |A| =o |B|"
by (simp only: ordIso_refl card_of_Card_order)
lemma Field_card_order: "card_order r \<Longrightarrow> Field r = UNIV"
using card_order_on_Card_order[of UNIV r] by simp
lemma card_of_Times_Plus_distrib:
"|A <*> (B <+> C)| =o |A <*> B <+> A <*> C|" (is "|?RHS| =o |?LHS|")
proof -
let ?f = "\<lambda>(a, bc). case bc of Inl b \<Rightarrow> Inl (a, b) | Inr c \<Rightarrow> Inr (a, c)"
have "bij_betw ?f ?RHS ?LHS" unfolding bij_betw_def inj_on_def by force
thus ?thesis using card_of_ordIso by blast
qed
lemma Func_Times_Range:
"|Func A (B <*> C)| =o |Func A B <*> Func A C|" (is "|?LHS| =o |?RHS|")
proof -
let ?F = "\<lambda>fg. (\<lambda>x. if x \<in> A then fst (fg x) else undefined,
\<lambda>x. if x \<in> A then snd (fg x) else undefined)"
let ?G = "\<lambda>(f, g) x. if x \<in> A then (f x, g x) else undefined"
have "bij_betw ?F ?LHS ?RHS" unfolding bij_betw_def inj_on_def
apply safe
apply (simp add: Func_def fun_eq_iff)
apply (metis (no_types) pair_collapse)
apply (auto simp: Func_def fun_eq_iff)[2]
proof -
fix f g assume "f \<in> Func A B" "g \<in> Func A C"
thus "(f, g) \<in> ?F ` Func A (B \<times> C)"
by (intro image_eqI[of _ _ "?G (f, g)"]) (auto simp: Func_def)
qed
thus ?thesis using card_of_ordIso by blast
qed
subsection {* Zero *}
definition czero where
"czero = card_of {}"
lemma czero_ordIso:
"czero =o czero"
using card_of_empty_ordIso by (simp add: czero_def)
lemma card_of_ordIso_czero_iff_empty:
"|A| =o (czero :: 'b rel) \<longleftrightarrow> A = ({} :: 'a set)"
unfolding czero_def by (rule iffI[OF card_of_empty2]) (auto simp: card_of_refl card_of_empty_ordIso)
(* A "not czero" Cardinal predicate *)
abbreviation Cnotzero where
"Cnotzero (r :: 'a rel) \<equiv> \<not>(r =o (czero :: 'a rel)) \<and> Card_order r"
(*helper*)
lemma Cnotzero_imp_not_empty: "Cnotzero r \<Longrightarrow> Field r \<noteq> {}"
by (metis Card_order_iff_ordIso_card_of czero_def)
lemma czeroI:
"\<lbrakk>Card_order r; Field r = {}\<rbrakk> \<Longrightarrow> r =o czero"
using Cnotzero_imp_not_empty ordIso_transitive[OF _ czero_ordIso] by blast
lemma czeroE:
"r =o czero \<Longrightarrow> Field r = {}"
unfolding czero_def
by (drule card_of_cong) (simp only: Field_card_of card_of_empty2)
lemma Cnotzero_mono:
"\<lbrakk>Cnotzero r; Card_order q; r \<le>o q\<rbrakk> \<Longrightarrow> Cnotzero q"
apply (rule ccontr)
apply auto
apply (drule czeroE)
apply (erule notE)
apply (erule czeroI)
apply (drule card_of_mono2)
apply (simp only: card_of_empty3)
done
subsection {* (In)finite cardinals *}
definition cinfinite where
"cinfinite r = (\<not> finite (Field r))"
abbreviation Cinfinite where
"Cinfinite r \<equiv> cinfinite r \<and> Card_order r"
definition cfinite where
"cfinite r = finite (Field r)"
abbreviation Cfinite where
"Cfinite r \<equiv> cfinite r \<and> Card_order r"
lemma Cfinite_ordLess_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r <o s"
unfolding cfinite_def cinfinite_def
by (metis card_order_on_well_order_on finite_ordLess_infinite)
lemmas natLeq_card_order = natLeq_Card_order[unfolded Field_natLeq]
lemma natLeq_cinfinite: "cinfinite natLeq"
unfolding cinfinite_def Field_natLeq by (metis infinite_UNIV_nat)
lemma natLeq_ordLeq_cinfinite:
assumes inf: "Cinfinite r"
shows "natLeq \<le>o r"
proof -
from inf have "natLeq \<le>o |Field r|" by (metis cinfinite_def infinite_iff_natLeq_ordLeq)
also from inf have "|Field r| =o r" by (simp add: card_of_unique ordIso_symmetric)
finally show ?thesis .
qed
lemma cinfinite_not_czero: "cinfinite r \<Longrightarrow> \<not> (r =o (czero :: 'a rel))"
unfolding cinfinite_def by (metis czeroE finite.emptyI)
lemma Cinfinite_Cnotzero: "Cinfinite r \<Longrightarrow> Cnotzero r"
by (metis cinfinite_not_czero)
lemma Cinfinite_cong: "\<lbrakk>r1 =o r2; Cinfinite r1\<rbrakk> \<Longrightarrow> Cinfinite r2"
by (metis Card_order_ordIso2 card_of_mono2 card_of_ordLeq_infinite cinfinite_def ordIso_iff_ordLeq)
lemma cinfinite_mono: "\<lbrakk>r1 \<le>o r2; cinfinite r1\<rbrakk> \<Longrightarrow> cinfinite r2"
by (metis card_of_mono2 card_of_ordLeq_infinite cinfinite_def)
subsection {* Binary sum *}
definition csum (infixr "+c" 65) where
"r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
lemma Field_csum: "Field (r +c s) = Inl ` Field r \<union> Inr ` Field s"
unfolding csum_def Field_card_of by auto
lemma Card_order_csum:
"Card_order (r1 +c r2)"
unfolding csum_def by (simp add: card_of_Card_order)
lemma csum_Cnotzero1:
"Cnotzero r1 \<Longrightarrow> Cnotzero (r1 +c r2)"
unfolding csum_def
by (metis Cnotzero_imp_not_empty Plus_eq_empty_conv card_of_Card_order card_of_ordIso_czero_iff_empty)
lemma card_order_csum:
assumes "card_order r1" "card_order r2"
shows "card_order (r1 +c r2)"
proof -
have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
thus ?thesis unfolding csum_def by (auto simp: card_of_card_order_on)
qed
lemma cinfinite_csum:
"cinfinite r1 \<or> cinfinite r2 \<Longrightarrow> cinfinite (r1 +c r2)"
unfolding cinfinite_def csum_def by (auto simp: Field_card_of)
lemma Cinfinite_csum1:
"Cinfinite r1 \<Longrightarrow> Cinfinite (r1 +c r2)"
unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
lemma Cinfinite_csum:
"Cinfinite r1 \<or> Cinfinite r2 \<Longrightarrow> Cinfinite (r1 +c r2)"
unfolding cinfinite_def csum_def by (metis Field_card_of card_of_Card_order finite_Plus_iff)
lemma Cinfinite_csum_strong:
"\<lbrakk>Cinfinite r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 +c r2)"
by (metis Cinfinite_csum)
lemma csum_cong: "\<lbrakk>p1 =o r1; p2 =o r2\<rbrakk> \<Longrightarrow> p1 +c p2 =o r1 +c r2"
by (simp only: csum_def ordIso_Plus_cong)
lemma csum_cong1: "p1 =o r1 \<Longrightarrow> p1 +c q =o r1 +c q"
by (simp only: csum_def ordIso_Plus_cong1)
lemma csum_cong2: "p2 =o r2 \<Longrightarrow> q +c p2 =o q +c r2"
by (simp only: csum_def ordIso_Plus_cong2)
lemma csum_mono: "\<lbrakk>p1 \<le>o r1; p2 \<le>o r2\<rbrakk> \<Longrightarrow> p1 +c p2 \<le>o r1 +c r2"
by (simp only: csum_def ordLeq_Plus_mono)
lemma csum_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 +c q \<le>o r1 +c q"
by (simp only: csum_def ordLeq_Plus_mono1)
lemma csum_mono2: "p2 \<le>o r2 \<Longrightarrow> q +c p2 \<le>o q +c r2"
by (simp only: csum_def ordLeq_Plus_mono2)
lemma ordLeq_csum1: "Card_order p1 \<Longrightarrow> p1 \<le>o p1 +c p2"
by (simp only: csum_def Card_order_Plus1)
lemma ordLeq_csum2: "Card_order p2 \<Longrightarrow> p2 \<le>o p1 +c p2"
by (simp only: csum_def Card_order_Plus2)
lemma csum_com: "p1 +c p2 =o p2 +c p1"
by (simp only: csum_def card_of_Plus_commute)
lemma csum_assoc: "(p1 +c p2) +c p3 =o p1 +c p2 +c p3"
by (simp only: csum_def Field_card_of card_of_Plus_assoc)
lemma Cfinite_csum: "\<lbrakk>Cfinite r; Cfinite s\<rbrakk> \<Longrightarrow> Cfinite (r +c s)"
unfolding cfinite_def csum_def Field_card_of using card_of_card_order_on by simp
lemma csum_csum: "(r1 +c r2) +c (r3 +c r4) =o (r1 +c r3) +c (r2 +c r4)"
proof -
have "(r1 +c r2) +c (r3 +c r4) =o r1 +c r2 +c (r3 +c r4)"
by (metis csum_assoc)
also have "r1 +c r2 +c (r3 +c r4) =o r1 +c (r2 +c r3) +c r4"
by (metis csum_assoc csum_cong2 ordIso_symmetric)
also have "r1 +c (r2 +c r3) +c r4 =o r1 +c (r3 +c r2) +c r4"
by (metis csum_com csum_cong1 csum_cong2)
also have "r1 +c (r3 +c r2) +c r4 =o r1 +c r3 +c r2 +c r4"
by (metis csum_assoc csum_cong2 ordIso_symmetric)
also have "r1 +c r3 +c r2 +c r4 =o (r1 +c r3) +c (r2 +c r4)"
by (metis csum_assoc ordIso_symmetric)
finally show ?thesis .
qed
lemma Plus_csum: "|A <+> B| =o |A| +c |B|"
by (simp only: csum_def Field_card_of card_of_refl)
lemma Un_csum: "|A \<union> B| \<le>o |A| +c |B|"
using ordLeq_ordIso_trans[OF card_of_Un_Plus_ordLeq Plus_csum] by blast
subsection {* One *}
definition cone where
"cone = card_of {()}"
lemma Card_order_cone: "Card_order cone"
unfolding cone_def by (rule card_of_Card_order)
lemma Cfinite_cone: "Cfinite cone"
unfolding cfinite_def by (simp add: Card_order_cone)
lemma cone_not_czero: "\<not> (cone =o czero)"
unfolding czero_def cone_def by (metis empty_not_insert card_of_empty3[of "{()}"] ordIso_iff_ordLeq)
lemma cone_ordLeq_Cnotzero: "Cnotzero r \<Longrightarrow> cone \<le>o r"
unfolding cone_def by (metis Card_order_singl_ordLeq czeroI)
subsection {* Two *}
definition ctwo where
"ctwo = |UNIV :: bool set|"
lemma Card_order_ctwo: "Card_order ctwo"
unfolding ctwo_def by (rule card_of_Card_order)
lemma ctwo_not_czero: "\<not> (ctwo =o czero)"
using card_of_empty3[of "UNIV :: bool set"] ordIso_iff_ordLeq
unfolding czero_def ctwo_def by (metis UNIV_not_empty)
lemma ctwo_Cnotzero: "Cnotzero ctwo"
by (simp add: ctwo_not_czero Card_order_ctwo)
subsection {* Family sum *}
definition Csum where
"Csum r rs \<equiv> |SIGMA i : Field r. Field (rs i)|"
(* Similar setup to the one for SIGMA from theory Big_Operators: *)
syntax "_Csum" ::
"pttrn => ('a * 'a) set => 'b * 'b set => (('a * 'b) * ('a * 'b)) set"
("(3CSUM _:_. _)" [0, 51, 10] 10)
translations
"CSUM i:r. rs" == "CONST Csum r (%i. rs)"
lemma SIGMA_CSUM: "|SIGMA i : I. As i| = (CSUM i : |I|. |As i| )"
by (auto simp: Csum_def Field_card_of)
(* NB: Always, under the cardinal operator,
operations on sets are reduced automatically to operations on cardinals.
This should make cardinal reasoning more direct and natural. *)
subsection {* Product *}
definition cprod (infixr "*c" 80) where
"r1 *c r2 = |Field r1 <*> Field r2|"
lemma card_order_cprod:
assumes "card_order r1" "card_order r2"
shows "card_order (r1 *c r2)"
proof -
have "Field r1 = UNIV" "Field r2 = UNIV" using assms card_order_on_Card_order by auto
thus ?thesis by (auto simp: cprod_def card_of_card_order_on)
qed
lemma Card_order_cprod: "Card_order (r1 *c r2)"
by (simp only: cprod_def Field_card_of card_of_card_order_on)
lemma cprod_mono1: "p1 \<le>o r1 \<Longrightarrow> p1 *c q \<le>o r1 *c q"
by (simp only: cprod_def ordLeq_Times_mono1)
lemma cprod_mono2: "p2 \<le>o r2 \<Longrightarrow> q *c p2 \<le>o q *c r2"
by (simp only: cprod_def ordLeq_Times_mono2)
lemma ordLeq_cprod2: "\<lbrakk>Cnotzero p1; Card_order p2\<rbrakk> \<Longrightarrow> p2 \<le>o p1 *c p2"
unfolding cprod_def by (metis Card_order_Times2 czeroI)
lemma cinfinite_cprod: "\<lbrakk>cinfinite r1; cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
by (simp add: cinfinite_def cprod_def Field_card_of infinite_cartesian_product)
lemma cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> cinfinite (r1 *c r2)"
by (metis cinfinite_mono ordLeq_cprod2)
lemma Cinfinite_cprod2: "\<lbrakk>Cnotzero r1; Cinfinite r2\<rbrakk> \<Longrightarrow> Cinfinite (r1 *c r2)"
by (blast intro: cinfinite_cprod2 Card_order_cprod)
lemma cprod_com: "p1 *c p2 =o p2 *c p1"
by (simp only: cprod_def card_of_Times_commute)
lemma card_of_Csum_Times:
"\<forall>i \<in> I. |A i| \<le>o |B| \<Longrightarrow> (CSUM i : |I|. |A i| ) \<le>o |I| *c |B|"
by (simp only: Csum_def cprod_def Field_card_of card_of_Sigma_Times)
lemma card_of_Csum_Times':
assumes "Card_order r" "\<forall>i \<in> I. |A i| \<le>o r"
shows "(CSUM i : |I|. |A i| ) \<le>o |I| *c r"
proof -
from assms(1) have *: "r =o |Field r|" by (simp add: card_of_unique)
with assms(2) have "\<forall>i \<in> I. |A i| \<le>o |Field r|" by (blast intro: ordLeq_ordIso_trans)
hence "(CSUM i : |I|. |A i| ) \<le>o |I| *c |Field r|" by (simp only: card_of_Csum_Times)
also from * have "|I| *c |Field r| \<le>o |I| *c r"
by (simp only: Field_card_of card_of_refl cprod_def ordIso_imp_ordLeq)
finally show ?thesis .
qed
lemma cprod_csum_distrib1: "r1 *c r2 +c r1 *c r3 =o r1 *c (r2 +c r3)"
unfolding csum_def cprod_def by (simp add: Field_card_of card_of_Times_Plus_distrib ordIso_symmetric)
lemma csum_absorb2': "\<lbrakk>Card_order r2; r1 \<le>o r2; cinfinite r1 \<or> cinfinite r2\<rbrakk> \<Longrightarrow> r1 +c r2 =o r2"
unfolding csum_def by (metis Card_order_Plus_infinite cinfinite_def cinfinite_mono)
lemma csum_absorb1':
assumes card: "Card_order r2"
and r12: "r1 \<le>o r2" and cr12: "cinfinite r1 \<or> cinfinite r2"
shows "r2 +c r1 =o r2"
by (rule ordIso_transitive, rule csum_com, rule csum_absorb2', (simp only: assms)+)
lemma csum_absorb1: "\<lbrakk>Cinfinite r2; r1 \<le>o r2\<rbrakk> \<Longrightarrow> r2 +c r1 =o r2"
by (rule csum_absorb1') auto
subsection {* Exponentiation *}
definition cexp (infixr "^c" 90) where
"r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
lemma Card_order_cexp: "Card_order (r1 ^c r2)"
unfolding cexp_def by (rule card_of_Card_order)
lemma cexp_mono':
assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
shows "p1 ^c p2 \<le>o r1 ^c r2"
proof(cases "Field p1 = {}")
case True
hence "|Field |Func (Field p2) (Field p1)|| \<le>o cone"
unfolding cone_def Field_card_of
by (cases "Field p2 = {}", auto intro: card_of_ordLeqI2 simp: Func_empty)
(metis Func_is_emp card_of_empty ex_in_conv)
hence "|Func (Field p2) (Field p1)| \<le>o cone" by (simp add: Field_card_of cexp_def)
hence "p1 ^c p2 \<le>o cone" unfolding cexp_def .
thus ?thesis
proof (cases "Field p2 = {}")
case True
with n have "Field r2 = {}" .
hence "cone \<le>o r1 ^c r2" unfolding cone_def cexp_def Func_def
by (auto intro: card_of_ordLeqI[where f="\<lambda>_ _. undefined"])
thus ?thesis using `p1 ^c p2 \<le>o cone` ordLeq_transitive by auto
next
case False with True have "|Field (p1 ^c p2)| =o czero"
unfolding card_of_ordIso_czero_iff_empty cexp_def Field_card_of Func_def by auto
thus ?thesis unfolding cexp_def card_of_ordIso_czero_iff_empty Field_card_of
by (simp add: card_of_empty)
qed
next
case False
have 1: "|Field p1| \<le>o |Field r1|" and 2: "|Field p2| \<le>o |Field r2|"
using 1 2 by (auto simp: card_of_mono2)
obtain f1 where f1: "f1 ` Field r1 = Field p1"
using 1 unfolding card_of_ordLeq2[OF False, symmetric] by auto
obtain f2 where f2: "inj_on f2 (Field p2)" "f2 ` Field p2 \<subseteq> Field r2"
using 2 unfolding card_of_ordLeq[symmetric] by blast
have 0: "Func_map (Field p2) f1 f2 ` (Field (r1 ^c r2)) = Field (p1 ^c p2)"
unfolding cexp_def Field_card_of using Func_map_surj[OF f1 f2 n, symmetric] .
have 00: "Field (p1 ^c p2) \<noteq> {}" unfolding cexp_def Field_card_of Func_is_emp
using False by simp
show ?thesis
using 0 card_of_ordLeq2[OF 00] unfolding cexp_def Field_card_of by blast
qed
lemma cexp_mono:
assumes 1: "p1 \<le>o r1" and 2: "p2 \<le>o r2"
and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
shows "p1 ^c p2 \<le>o r1 ^c r2"
by (metis (full_types) "1" "2" card cexp_mono' czeroE czeroI n)
lemma cexp_mono1:
assumes 1: "p1 \<le>o r1" and q: "Card_order q"
shows "p1 ^c q \<le>o r1 ^c q"
using ordLeq_refl[OF q] by (rule cexp_mono[OF 1]) (auto simp: q)
lemma cexp_mono2':
assumes 2: "p2 \<le>o r2" and q: "Card_order q"
and n: "Field p2 = {} \<Longrightarrow> Field r2 = {}"
shows "q ^c p2 \<le>o q ^c r2"
using ordLeq_refl[OF q] by (rule cexp_mono'[OF _ 2 n]) auto
lemma cexp_mono2:
assumes 2: "p2 \<le>o r2" and q: "Card_order q"
and n: "p2 =o czero \<Longrightarrow> r2 =o czero" and card: "Card_order p2"
shows "q ^c p2 \<le>o q ^c r2"
using ordLeq_refl[OF q] by (rule cexp_mono[OF _ 2 n card]) auto
lemma cexp_mono2_Cnotzero:
assumes "p2 \<le>o r2" "Card_order q" "Cnotzero p2"
shows "q ^c p2 \<le>o q ^c r2"
by (metis assms cexp_mono2' czeroI)
lemma cexp_cong:
assumes 1: "p1 =o r1" and 2: "p2 =o r2"
and Cr: "Card_order r2"
and Cp: "Card_order p2"
shows "p1 ^c p2 =o r1 ^c r2"
proof -
obtain f where "bij_betw f (Field p2) (Field r2)"
using 2 card_of_ordIso[of "Field p2" "Field r2"] card_of_cong by auto
hence 0: "Field p2 = {} \<longleftrightarrow> Field r2 = {}" unfolding bij_betw_def by auto
have r: "p2 =o czero \<Longrightarrow> r2 =o czero"
and p: "r2 =o czero \<Longrightarrow> p2 =o czero"
using 0 Cr Cp czeroE czeroI by auto
show ?thesis using 0 1 2 unfolding ordIso_iff_ordLeq
using r p cexp_mono[OF _ _ _ Cp] cexp_mono[OF _ _ _ Cr] by metis
qed
lemma cexp_cong1:
assumes 1: "p1 =o r1" and q: "Card_order q"
shows "p1 ^c q =o r1 ^c q"
by (rule cexp_cong[OF 1 _ q q]) (rule ordIso_refl[OF q])
lemma cexp_cong2:
assumes 2: "p2 =o r2" and q: "Card_order q" and p: "Card_order p2"
shows "q ^c p2 =o q ^c r2"
by (rule cexp_cong[OF _ 2]) (auto simp only: ordIso_refl Card_order_ordIso2[OF p 2] q p)
lemma cexp_cone:
assumes "Card_order r"
shows "r ^c cone =o r"
proof -
have "r ^c cone =o |Field r|"
unfolding cexp_def cone_def Field_card_of Func_empty
card_of_ordIso[symmetric] bij_betw_def Func_def inj_on_def image_def
by (rule exI[of _ "\<lambda>f. f ()"]) auto
also have "|Field r| =o r" by (rule card_of_Field_ordIso[OF assms])
finally show ?thesis .
qed
lemma cexp_cprod:
assumes r1: "Card_order r1"
shows "(r1 ^c r2) ^c r3 =o r1 ^c (r2 *c r3)" (is "?L =o ?R")
proof -
have "?L =o r1 ^c (r3 *c r2)"
unfolding cprod_def cexp_def Field_card_of
using card_of_Func_Times by(rule ordIso_symmetric)
also have "r1 ^c (r3 *c r2) =o ?R"
apply(rule cexp_cong2) using cprod_com r1 by (auto simp: Card_order_cprod)
finally show ?thesis .
qed
lemma cprod_infinite1': "\<lbrakk>Cinfinite r; Cnotzero p; p \<le>o r\<rbrakk> \<Longrightarrow> r *c p =o r"
unfolding cinfinite_def cprod_def
by (rule Card_order_Times_infinite[THEN conjunct1]) (blast intro: czeroI)+
lemma cexp_cprod_ordLeq:
assumes r1: "Card_order r1" and r2: "Cinfinite r2"
and r3: "Cnotzero r3" "r3 \<le>o r2"
shows "(r1 ^c r2) ^c r3 =o r1 ^c r2" (is "?L =o ?R")
proof-
have "?L =o r1 ^c (r2 *c r3)" using cexp_cprod[OF r1] .
also have "r1 ^c (r2 *c r3) =o ?R"
apply(rule cexp_cong2)
apply(rule cprod_infinite1'[OF r2 r3]) using r1 r2 by (fastforce simp: Card_order_cprod)+
finally show ?thesis .
qed
lemma Cnotzero_UNIV: "Cnotzero |UNIV|"
by (auto simp: card_of_Card_order card_of_ordIso_czero_iff_empty)
lemma ordLess_ctwo_cexp:
assumes "Card_order r"
shows "r <o ctwo ^c r"
proof -
have "r <o |Pow (Field r)|" using assms by (rule Card_order_Pow)
also have "|Pow (Field r)| =o ctwo ^c r"
unfolding ctwo_def cexp_def Field_card_of by (rule card_of_Pow_Func)
finally show ?thesis .
qed
lemma ordLeq_cexp1:
assumes "Cnotzero r" "Card_order q"
shows "q \<le>o q ^c r"
proof (cases "q =o (czero :: 'a rel)")
case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
next
case False
thus ?thesis
apply -
apply (rule ordIso_ordLeq_trans)
apply (rule ordIso_symmetric)
apply (rule cexp_cone)
apply (rule assms(2))
apply (rule cexp_mono2)
apply (rule cone_ordLeq_Cnotzero)
apply (rule assms(1))
apply (rule assms(2))
apply (rule notE)
apply (rule cone_not_czero)
apply assumption
apply (rule Card_order_cone)
done
qed
lemma ordLeq_cexp2:
assumes "ctwo \<le>o q" "Card_order r"
shows "r \<le>o q ^c r"
proof (cases "r =o (czero :: 'a rel)")
case True thus ?thesis by (simp only: card_of_empty cexp_def czero_def ordIso_ordLeq_trans)
next
case False thus ?thesis
apply -
apply (rule ordLess_imp_ordLeq)
apply (rule ordLess_ordLeq_trans)
apply (rule ordLess_ctwo_cexp)
apply (rule assms(2))
apply (rule cexp_mono1)
apply (rule assms(1))
apply (rule assms(2))
done
qed
lemma cinfinite_cexp: "\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> cinfinite (q ^c r)"
by (metis assms cinfinite_mono ordLeq_cexp2)
lemma Cinfinite_cexp:
"\<lbrakk>ctwo \<le>o q; Cinfinite r\<rbrakk> \<Longrightarrow> Cinfinite (q ^c r)"
by (simp add: cinfinite_cexp Card_order_cexp)
lemma ctwo_ordLess_natLeq: "ctwo <o natLeq"
unfolding ctwo_def using finite_UNIV natLeq_cinfinite natLeq_Card_order
by (intro Cfinite_ordLess_Cinfinite) (auto simp: cfinite_def card_of_Card_order)
lemma ctwo_ordLess_Cinfinite: "Cinfinite r \<Longrightarrow> ctwo <o r"
by (metis ctwo_ordLess_natLeq natLeq_ordLeq_cinfinite ordLess_ordLeq_trans)
lemma ctwo_ordLeq_Cinfinite:
assumes "Cinfinite r"
shows "ctwo \<le>o r"
by (rule ordLess_imp_ordLeq[OF ctwo_ordLess_Cinfinite[OF assms]])
lemma Un_Cinfinite_bound: "\<lbrakk>|A| \<le>o r; |B| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |A \<union> B| \<le>o r"
by (auto simp add: cinfinite_def card_of_Un_ordLeq_infinite_Field)
lemma UNION_Cinfinite_bound: "\<lbrakk>|I| \<le>o r; \<forall>i \<in> I. |A i| \<le>o r; Cinfinite r\<rbrakk> \<Longrightarrow> |\<Union>i \<in> I. A i| \<le>o r"
by (auto simp add: card_of_UNION_ordLeq_infinite_Field cinfinite_def)
lemma csum_cinfinite_bound:
assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
shows "p +c q \<le>o r"
proof -
from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
with assms show ?thesis unfolding cinfinite_def csum_def
by (blast intro: card_of_Plus_ordLeq_infinite_Field)
qed
lemma cprod_cinfinite_bound:
assumes "p \<le>o r" "q \<le>o r" "Card_order p" "Card_order q" "Cinfinite r"
shows "p *c q \<le>o r"
proof -
from assms(1-4) have "|Field p| \<le>o r" "|Field q| \<le>o r"
unfolding card_order_on_def using card_of_least ordLeq_transitive by blast+
with assms show ?thesis unfolding cinfinite_def cprod_def
by (blast intro: card_of_Times_ordLeq_infinite_Field)
qed
lemma cprod_csum_cexp:
"r1 *c r2 \<le>o (r1 +c r2) ^c ctwo"
unfolding cprod_def csum_def cexp_def ctwo_def Field_card_of
proof -
let ?f = "\<lambda>(a, b). %x. if x then Inl a else Inr b"
have "inj_on ?f (Field r1 \<times> Field r2)" (is "inj_on _ ?LHS")
by (auto simp: inj_on_def fun_eq_iff split: bool.split)
moreover
have "?f ` ?LHS \<subseteq> Func (UNIV :: bool set) (Field r1 <+> Field r2)" (is "_ \<subseteq> ?RHS")
by (auto simp: Func_def)
ultimately show "|?LHS| \<le>o |?RHS|" using card_of_ordLeq by blast
qed
lemma Cfinite_cprod_Cinfinite: "\<lbrakk>Cfinite r; Cinfinite s\<rbrakk> \<Longrightarrow> r *c s \<le>o s"
by (intro cprod_cinfinite_bound)
(auto intro: ordLeq_refl ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite])
lemma cprod_cexp: "(r *c s) ^c t =o r ^c t *c s ^c t"
unfolding cprod_def cexp_def Field_card_of by (rule Func_Times_Range)
lemma cprod_cexp_csum_cexp_Cinfinite:
assumes t: "Cinfinite t"
shows "(r *c s) ^c t \<le>o (r +c s) ^c t"
proof -
have "(r *c s) ^c t \<le>o ((r +c s) ^c ctwo) ^c t"
by (rule cexp_mono1[OF cprod_csum_cexp conjunct2[OF t]])
also have "((r +c s) ^c ctwo) ^c t =o (r +c s) ^c (ctwo *c t)"
by (rule cexp_cprod[OF Card_order_csum])
also have "(r +c s) ^c (ctwo *c t) =o (r +c s) ^c (t *c ctwo)"
by (rule cexp_cong2[OF cprod_com Card_order_csum Card_order_cprod])
also have "(r +c s) ^c (t *c ctwo) =o ((r +c s) ^c t) ^c ctwo"
by (rule ordIso_symmetric[OF cexp_cprod[OF Card_order_csum]])
also have "((r +c s) ^c t) ^c ctwo =o (r +c s) ^c t"
by (rule cexp_cprod_ordLeq[OF Card_order_csum t ctwo_Cnotzero ctwo_ordLeq_Cinfinite[OF t]])
finally show ?thesis .
qed
lemma Cfinite_cexp_Cinfinite:
assumes s: "Cfinite s" and t: "Cinfinite t"
shows "s ^c t \<le>o ctwo ^c t"
proof (cases "s \<le>o ctwo")
case True thus ?thesis using t by (blast intro: cexp_mono1)
next
case False
hence "ctwo \<le>o s" by (metis card_order_on_well_order_on ctwo_Cnotzero ordLeq_total s)
hence "Cnotzero s" by (metis Cnotzero_mono ctwo_Cnotzero s)
hence st: "Cnotzero (s *c t)" by (metis Cinfinite_cprod2 cinfinite_not_czero t)
have "s ^c t \<le>o (ctwo ^c s) ^c t"
using assms by (blast intro: cexp_mono1 ordLess_imp_ordLeq[OF ordLess_ctwo_cexp])
also have "(ctwo ^c s) ^c t =o ctwo ^c (s *c t)"
by (blast intro: Card_order_ctwo cexp_cprod)
also have "ctwo ^c (s *c t) \<le>o ctwo ^c t"
using assms st by (intro cexp_mono2_Cnotzero Cfinite_cprod_Cinfinite Card_order_ctwo)
finally show ?thesis .
qed
lemma csum_Cfinite_cexp_Cinfinite:
assumes r: "Card_order r" and s: "Cfinite s" and t: "Cinfinite t"
shows "(r +c s) ^c t \<le>o (r +c ctwo) ^c t"
proof (cases "Cinfinite r")
case True
hence "r +c s =o r" by (intro csum_absorb1 ordLess_imp_ordLeq[OF Cfinite_ordLess_Cinfinite] s)
hence "(r +c s) ^c t =o r ^c t" using t by (blast intro: cexp_cong1)
also have "r ^c t \<le>o (r +c ctwo) ^c t" using t by (blast intro: cexp_mono1 ordLeq_csum1 r)
finally show ?thesis .
next
case False
with r have "Cfinite r" unfolding cinfinite_def cfinite_def by auto
hence "Cfinite (r +c s)" by (intro Cfinite_csum s)
hence "(r +c s) ^c t \<le>o ctwo ^c t" by (intro Cfinite_cexp_Cinfinite t)
also have "ctwo ^c t \<le>o (r +c ctwo) ^c t" using t
by (blast intro: cexp_mono1 ordLeq_csum2 Card_order_ctwo)
finally show ?thesis .
qed
(* cardSuc *)
lemma Cinfinite_cardSuc: "Cinfinite r \<Longrightarrow> Cinfinite (cardSuc r)"
by (simp add: cinfinite_def cardSuc_Card_order cardSuc_finite)
lemma cardSuc_UNION_Cinfinite:
assumes "Cinfinite r" "relChain (cardSuc r) As" "B \<le> (UN i : Field (cardSuc r). As i)" "|B| <=o r"
shows "EX i : Field (cardSuc r). B \<le> As i"
using cardSuc_UNION assms unfolding cinfinite_def by blast
end