(*  Title:      ZF/CardinalArith.thy

     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

     Copyright   1994  University of Cambridge

 *)

 header{*Cardinal Arithmetic Without the Axiom of Choice*}

 theory CardinalArith imports Cardinal OrderArith ArithSimp Finite begin

 definition

   InfCard       :: "i=>o"  where

     "InfCard(i) == Card(i) & nat \<le> i"

 definition

   cmult         :: "[i,i]=>i"       (infixl "|*|" 70)  where

     "i |*| j == |i*j|"

 definition

   cadd          :: "[i,i]=>i"       (infixl "|+|" 65)  where

     "i |+| j == |i+j|"

 definition

   csquare_rel   :: "i=>i"  where

     "csquare_rel(K) ==

           rvimage(K*K,

                   lam <x,y>:K*K. <x \<union> y, x, y>,

                   rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"

 definition

   jump_cardinal :: "i=>i"  where

     --{*This def is more complex than Kunen's but it more easily proved to

         be a cardinal*}

     "jump_cardinal(K) ==

          \<Union>X\<in>Pow(K). {z. r: Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"

 definition

   csucc         :: "i=>i"  where

     --{*needed because @{term "jump_cardinal(K)"} might not be the successor

         of @{term K}*}

     "csucc(K) == LEAST L. Card(L) & K<L"

 notation (xsymbols)

   cadd  (infixl "\<oplus>" 65) and

   cmult  (infixl "\<otimes>" 70)

 notation (HTML)

   cadd  (infixl "\<oplus>" 65) and

   cmult  (infixl "\<otimes>" 70)

 lemma Card_Union [simp,intro,TC]: 

   assumes A: "\<And>x. x\<in>A \<Longrightarrow> Card(x)" shows "Card(\<Union>(A))"

 proof (rule CardI)

   show "Ord(\<Union>A)" using A 

     by (simp add: Card_is_Ord)

 next

   fix j

   assume j: "j < \<Union>A"

   hence "\<exists>c\<in>A. j < c & Card(c)" using A

     by (auto simp add: lt_def intro: Card_is_Ord)

   then obtain c where c: "c\<in>A" "j < c" "Card(c)"

     by blast

   hence jls: "j \<prec> c" 

     by (simp add: lt_Card_imp_lesspoll) 

   { assume eqp: "j \<approx> \<Union>A"

     have  "c \<lesssim> \<Union>A" using c

       by (blast intro: subset_imp_lepoll)

     also have "... \<approx> j"  by (rule eqpoll_sym [OF eqp])

     also have "... \<prec> c"  by (rule jls)

     finally have "c \<prec> c" .

     hence False 

       by auto

   } thus "\<not> j \<approx> \<Union>A" by blast

 qed

 lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x\<in>A. K(x))"

   by blast

 lemma Card_OUN [simp,intro,TC]:

      "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))"

   by (auto simp add: OUnion_def Card_0)

 lemma in_Card_imp_lesspoll: "[| Card(K); b \<in> K |] ==> b \<prec> K"

 apply (unfold lesspoll_def)

 apply (simp add: Card_iff_initial)

 apply (fast intro!: le_imp_lepoll ltI leI)

 done

 subsection{*Cardinal addition*}

 text{*Note: Could omit proving the algebraic laws for cardinal addition and

 multiplication.  On finite cardinals these operations coincide with

 addition and multiplication of natural numbers; on infinite cardinals they

 coincide with union (maximum).  Either way we get most laws for free.*}

 subsubsection{*Cardinal addition is commutative*}

 lemma sum_commute_eqpoll: "A+B \<approx> B+A"

 proof (unfold eqpoll_def, rule exI)

   show "(\<lambda>z\<in>A+B. case(Inr,Inl,z)) \<in> bij(A+B, B+A)"

     by (auto intro: lam_bijective [where d = "case(Inr,Inl)"]) 

 qed

 lemma cadd_commute: "i \<oplus> j = j \<oplus> i"

 apply (unfold cadd_def)

 apply (rule sum_commute_eqpoll [THEN cardinal_cong])

 done

 subsubsection{*Cardinal addition is associative*}

 lemma sum_assoc_eqpoll: "(A+B)+C \<approx> A+(B+C)"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule sum_assoc_bij)

 done

 text{*Unconditional version requires AC*}

 lemma well_ord_cadd_assoc:

   assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"

   shows "(i \<oplus> j) \<oplus> k = i \<oplus> (j \<oplus> k)"

 proof (unfold cadd_def, rule cardinal_cong)

   have "|i + j| + k \<approx> (i + j) + k"

     by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j) 

   also have "...  \<approx> i + (j + k)"

     by (rule sum_assoc_eqpoll) 

   also have "...  \<approx> i + |j + k|"

     by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd j k eqpoll_sym) 

   finally show "|i + j| + k \<approx> i + |j + k|" .

 qed

 subsubsection{*0 is the identity for addition*}

 lemma sum_0_eqpoll: "0+A \<approx> A"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule bij_0_sum)

 done

 lemma cadd_0 [simp]: "Card(K) ==> 0 \<oplus> K = K"

 apply (unfold cadd_def)

 apply (simp add: sum_0_eqpoll [THEN cardinal_cong] Card_cardinal_eq)

 done

 subsubsection{*Addition by another cardinal*}

 lemma sum_lepoll_self: "A \<lesssim> A+B"

 proof (unfold lepoll_def, rule exI)

   show "(\<lambda>x\<in>A. Inl (x)) \<in> inj(A, A + B)"

     by (simp add: inj_def) 

 qed

 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)

 lemma cadd_le_self:

     "[| Card(K);  Ord(L) |] ==> K \<le> (K \<oplus> L)"

 apply (unfold cadd_def)

 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le],

        assumption)

 apply (rule_tac [2] sum_lepoll_self)

 apply (blast intro: well_ord_radd well_ord_Memrel Card_is_Ord)

 done

 subsubsection{*Monotonicity of addition*}

 lemma sum_lepoll_mono:

      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A + B \<lesssim> C + D"

 apply (unfold lepoll_def)

 apply (elim exE)

 apply (rule_tac x = "\<lambda>z\<in>A+B. case (%w. Inl(f`w), %y. Inr(fa`y), z)" in exI)

 apply (rule_tac d = "case (%w. Inl(converse(f) `w), %y. Inr(converse(fa) ` y))"

        in lam_injective)

 apply (typecheck add: inj_is_fun, auto)

 done

 lemma cadd_le_mono:

     "[| K' \<le> K;  L' \<le> L |] ==> (K' \<oplus> L') \<le> (K \<oplus> L)"

 apply (unfold cadd_def)

 apply (safe dest!: le_subset_iff [THEN iffD1])

 apply (rule well_ord_lepoll_imp_Card_le)

 apply (blast intro: well_ord_radd well_ord_Memrel)

 apply (blast intro: sum_lepoll_mono subset_imp_lepoll)

 done

 subsubsection{*Addition of finite cardinals is "ordinary" addition*}

 lemma sum_succ_eqpoll: "succ(A)+B \<approx> succ(A+B)"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule_tac c = "%z. if z=Inl (A) then A+B else z"

             and d = "%z. if z=A+B then Inl (A) else z" in lam_bijective)

    apply simp_all

 apply (blast dest: sym [THEN eq_imp_not_mem] elim: mem_irrefl)+

 done

 (*Pulling the  succ(...)  outside the |...| requires m, n: nat  *)

 (*Unconditional version requires AC*)

 lemma cadd_succ_lemma:

     "[| Ord(m);  Ord(n) |] ==> succ(m) \<oplus> n = |succ(m \<oplus> n)|"

 apply (unfold cadd_def)

 apply (rule sum_succ_eqpoll [THEN cardinal_cong, THEN trans])

 apply (rule succ_eqpoll_cong [THEN cardinal_cong])

 apply (rule well_ord_cardinal_eqpoll [THEN eqpoll_sym])

 apply (blast intro: well_ord_radd well_ord_Memrel)

 done

 lemma nat_cadd_eq_add: "[| m: nat;  n: nat |] ==> m \<oplus> n = m#+n"

 apply (induct_tac m)

 apply (simp add: nat_into_Card [THEN cadd_0])

 apply (simp add: cadd_succ_lemma nat_into_Card [THEN Card_cardinal_eq])

 done

 subsection{*Cardinal multiplication*}

 subsubsection{*Cardinal multiplication is commutative*}

 lemma prod_commute_eqpoll: "A*B \<approx> B*A"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule_tac c = "%<x,y>.<y,x>" and d = "%<x,y>.<y,x>" in lam_bijective,

        auto)

 done

 lemma cmult_commute: "i \<otimes> j = j \<otimes> i"

 apply (unfold cmult_def)

 apply (rule prod_commute_eqpoll [THEN cardinal_cong])

 done

 subsubsection{*Cardinal multiplication is associative*}

 lemma prod_assoc_eqpoll: "(A*B)*C \<approx> A*(B*C)"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule prod_assoc_bij)

 done

 text{*Unconditional version requires AC*}

 lemma well_ord_cmult_assoc:

   assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"

   shows "(i \<otimes> j) \<otimes> k = i \<otimes> (j \<otimes> k)"

 proof (unfold cmult_def, rule cardinal_cong)

   have "|i * j| * k \<approx> (i * j) * k"

     by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult i j) 

   also have "...  \<approx> i * (j * k)"

     by (rule prod_assoc_eqpoll) 

   also have "...  \<approx> i * |j * k|"

     by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult j k eqpoll_sym) 

   finally show "|i * j| * k \<approx> i * |j * k|" .

 qed

 subsubsection{*Cardinal multiplication distributes over addition*}

 lemma sum_prod_distrib_eqpoll: "(A+B)*C \<approx> (A*C)+(B*C)"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule sum_prod_distrib_bij)

 done

 lemma well_ord_cadd_cmult_distrib:

   assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"

   shows "(i \<oplus> j) \<otimes> k = (i \<otimes> k) \<oplus> (j \<otimes> k)"

 proof (unfold cadd_def cmult_def, rule cardinal_cong)

   have "|i + j| * k \<approx> (i + j) * k"

     by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j) 

   also have "...  \<approx> i * k + j * k"

     by (rule sum_prod_distrib_eqpoll) 

   also have "...  \<approx> |i * k| + |j * k|"

     by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll well_ord_rmult i j k eqpoll_sym) 

   finally show "|i + j| * k \<approx> |i * k| + |j * k|" .

 qed

 subsubsection{*Multiplication by 0 yields 0*}

 lemma prod_0_eqpoll: "0*A \<approx> 0"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule lam_bijective, safe)

 done

 lemma cmult_0 [simp]: "0 \<otimes> i = 0"

 by (simp add: cmult_def prod_0_eqpoll [THEN cardinal_cong])

 subsubsection{*1 is the identity for multiplication*}

 lemma prod_singleton_eqpoll: "{x}*A \<approx> A"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule singleton_prod_bij [THEN bij_converse_bij])

 done

 lemma cmult_1 [simp]: "Card(K) ==> 1 \<otimes> K = K"

 apply (unfold cmult_def succ_def)

 apply (simp add: prod_singleton_eqpoll [THEN cardinal_cong] Card_cardinal_eq)

 done

 subsection{*Some inequalities for multiplication*}

 lemma prod_square_lepoll: "A \<lesssim> A*A"

 apply (unfold lepoll_def inj_def)

 apply (rule_tac x = "\<lambda>x\<in>A. <x,x>" in exI, simp)

 done

 (*Could probably weaken the premise to well_ord(K,r), or remove using AC*)

 lemma cmult_square_le: "Card(K) ==> K \<le> K \<otimes> K"

 apply (unfold cmult_def)

 apply (rule le_trans)

 apply (rule_tac [2] well_ord_lepoll_imp_Card_le)

 apply (rule_tac [3] prod_square_lepoll)

 apply (simp add: le_refl Card_is_Ord Card_cardinal_eq)

 apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)

 done

 subsubsection{*Multiplication by a non-zero cardinal*}

 lemma prod_lepoll_self: "b: B ==> A \<lesssim> A*B"

 apply (unfold lepoll_def inj_def)

 apply (rule_tac x = "\<lambda>x\<in>A. <x,b>" in exI, simp)

 done

 (*Could probably weaken the premises to well_ord(K,r), or removing using AC*)

 lemma cmult_le_self:

     "[| Card(K);  Ord(L);  0<L |] ==> K \<le> (K \<otimes> L)"

 apply (unfold cmult_def)

 apply (rule le_trans [OF Card_cardinal_le well_ord_lepoll_imp_Card_le])

   apply assumption

  apply (blast intro: well_ord_rmult well_ord_Memrel Card_is_Ord)

 apply (blast intro: prod_lepoll_self ltD)

 done

 subsubsection{*Monotonicity of multiplication*}

 lemma prod_lepoll_mono:

      "[| A \<lesssim> C;  B \<lesssim> D |] ==> A * B  \<lesssim>  C * D"

 apply (unfold lepoll_def)

 apply (elim exE)

 apply (rule_tac x = "lam <w,y>:A*B. <f`w, fa`y>" in exI)

 apply (rule_tac d = "%<w,y>. <converse (f) `w, converse (fa) `y>"

        in lam_injective)

 apply (typecheck add: inj_is_fun, auto)

 done

 lemma cmult_le_mono:

     "[| K' \<le> K;  L' \<le> L |] ==> (K' \<otimes> L') \<le> (K \<otimes> L)"

 apply (unfold cmult_def)

 apply (safe dest!: le_subset_iff [THEN iffD1])

 apply (rule well_ord_lepoll_imp_Card_le)

  apply (blast intro: well_ord_rmult well_ord_Memrel)

 apply (blast intro: prod_lepoll_mono subset_imp_lepoll)

 done

 subsection{*Multiplication of finite cardinals is "ordinary" multiplication*}

 lemma prod_succ_eqpoll: "succ(A)*B \<approx> B + A*B"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (rule_tac c = "%<x,y>. if x=A then Inl (y) else Inr (<x,y>)"

             and d = "case (%y. <A,y>, %z. z)" in lam_bijective)

 apply safe

 apply (simp_all add: succI2 if_type mem_imp_not_eq)

 done

 (*Unconditional version requires AC*)

 lemma cmult_succ_lemma:

     "[| Ord(m);  Ord(n) |] ==> succ(m) \<otimes> n = n \<oplus> (m \<otimes> n)"

 apply (unfold cmult_def cadd_def)

 apply (rule prod_succ_eqpoll [THEN cardinal_cong, THEN trans])

 apply (rule cardinal_cong [symmetric])

 apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll])

 apply (blast intro: well_ord_rmult well_ord_Memrel)

 done

 lemma nat_cmult_eq_mult: "[| m: nat;  n: nat |] ==> m \<otimes> n = m#*n"

 apply (induct_tac m)

 apply (simp_all add: cmult_succ_lemma nat_cadd_eq_add)

 done

 lemma cmult_2: "Card(n) ==> 2 \<otimes> n = n \<oplus> n"

 by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0])

 lemma sum_lepoll_prod: 

   assumes C: "2 \<lesssim> C" shows "B+B \<lesssim> C*B"

 proof -

   have "B+B \<lesssim> 2*B"

     by (simp add: sum_eq_2_times) 

   also have "... \<lesssim> C*B"

     by (blast intro: prod_lepoll_mono lepoll_refl C) 

   finally show "B+B \<lesssim> C*B" .

 qed

 lemma lepoll_imp_sum_lepoll_prod: "[| A \<lesssim> B; 2 \<lesssim> A |] ==> A+B \<lesssim> A*B"

 by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)

 subsection{*Infinite Cardinals are Limit Ordinals*}

 (*This proof is modelled upon one assuming nat<=A, with injection

   \<lambda>z\<in>cons(u,A). if z=u then 0 else if z \<in> nat then succ(z) else z

   and inverse %y. if y:nat then nat_case(u, %z. z, y) else y.  \

   If f: inj(nat,A) then range(f) behaves like the natural numbers.*)

 lemma nat_cons_lepoll: "nat \<lesssim> A ==> cons(u,A) \<lesssim> A"

 apply (unfold lepoll_def)

 apply (erule exE)

 apply (rule_tac x =

           "\<lambda>z\<in>cons (u,A).

              if z=u then f`0

              else if z: range (f) then f`succ (converse (f) `z) else z"

        in exI)

 apply (rule_tac d =

           "%y. if y: range(f) then nat_case (u, %z. f`z, converse(f) `y)

                               else y"

        in lam_injective)

 apply (fast intro!: if_type apply_type intro: inj_is_fun inj_converse_fun)

 apply (simp add: inj_is_fun [THEN apply_rangeI]

                  inj_converse_fun [THEN apply_rangeI]

                  inj_converse_fun [THEN apply_funtype])

 done

 lemma nat_cons_eqpoll: "nat \<lesssim> A ==> cons(u,A) \<approx> A"

 apply (erule nat_cons_lepoll [THEN eqpollI])

 apply (rule subset_consI [THEN subset_imp_lepoll])

 done

 (*Specialized version required below*)

 lemma nat_succ_eqpoll: "nat \<subseteq> A ==> succ(A) \<approx> A"

 apply (unfold succ_def)

 apply (erule subset_imp_lepoll [THEN nat_cons_eqpoll])

 done

 lemma InfCard_nat: "InfCard(nat)"

 apply (unfold InfCard_def)

 apply (blast intro: Card_nat le_refl Card_is_Ord)

 done

 lemma InfCard_is_Card: "InfCard(K) ==> Card(K)"

 apply (unfold InfCard_def)

 apply (erule conjunct1)

 done

 lemma InfCard_Un:

     "[| InfCard(K);  Card(L) |] ==> InfCard(K \<union> L)"

 apply (unfold InfCard_def)

 apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans]  Card_is_Ord)

 done

 (*Kunen's Lemma 10.11*)

 lemma InfCard_is_Limit: "InfCard(K) ==> Limit(K)"

 apply (unfold InfCard_def)

 apply (erule conjE)

 apply (frule Card_is_Ord)

 apply (rule ltI [THEN non_succ_LimitI])

 apply (erule le_imp_subset [THEN subsetD])

 apply (safe dest!: Limit_nat [THEN Limit_le_succD])

 apply (unfold Card_def)

 apply (drule trans)

 apply (erule le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong])

 apply (erule Ord_cardinal_le [THEN lt_trans2, THEN lt_irrefl])

 apply (rule le_eqI, assumption)

 apply (rule Ord_cardinal)

 done

 (*** An infinite cardinal equals its square (Kunen, Thm 10.12, page 29) ***)

 (*A general fact about ordermap*)

 lemma ordermap_eqpoll_pred:

     "[| well_ord(A,r);  x:A |] ==> ordermap(A,r)`x \<approx> Order.pred(A,x,r)"

 apply (unfold eqpoll_def)

 apply (rule exI)

 apply (simp add: ordermap_eq_image well_ord_is_wf)

 apply (erule ordermap_bij [THEN bij_is_inj, THEN restrict_bij,

                            THEN bij_converse_bij])

 apply (rule pred_subset)

 done

 subsubsection{*Establishing the well-ordering*}

 lemma well_ord_csquare: 

   assumes K: "Ord(K)" shows "well_ord(K*K, csquare_rel(K))"

 proof (unfold csquare_rel_def, rule well_ord_rvimage)

   show "(\<lambda>\<langle>x,y\<rangle>\<in>K \<times> K. \<langle>x \<union> y, x, y\<rangle>) \<in> inj(K \<times> K, K \<times> K \<times> K)" using K

     by (force simp add: inj_def intro: lam_type Un_least_lt [THEN ltD] ltI)

 next

   show "well_ord(K \<times> K \<times> K, rmult(K, Memrel(K), K \<times> K, rmult(K, Memrel(K), K, Memrel(K))))"

     using K by (blast intro: well_ord_rmult well_ord_Memrel)

 qed

 subsubsection{*Characterising initial segments of the well-ordering*}

 lemma csquareD:

  "[| <<x,y>, <z,z>> \<in> csquare_rel(K);  x<K;  y<K;  z<K |] ==> x \<le> z & y \<le> z"

 apply (unfold csquare_rel_def)

 apply (erule rev_mp)

 apply (elim ltE)

 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)

 apply (safe elim!: mem_irrefl intro!: Un_upper1_le Un_upper2_le)

 apply (simp_all add: lt_def succI2)

 done

 lemma pred_csquare_subset:

     "z<K ==> Order.pred(K*K, <z,z>, csquare_rel(K)) \<subseteq> succ(z)*succ(z)"

 apply (unfold Order.pred_def)

 apply (safe del: SigmaI dest!: csquareD)

 apply (unfold lt_def, auto)

 done

 lemma csquare_ltI:

  "[| x<z;  y<z;  z<K |] ==>  <<x,y>, <z,z>> \<in> csquare_rel(K)"

 apply (unfold csquare_rel_def)

 apply (subgoal_tac "x<K & y<K")

  prefer 2 apply (blast intro: lt_trans)

 apply (elim ltE)

 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)

 done

 (*Part of the traditional proof.  UNUSED since it's harder to prove & apply *)

 lemma csquare_or_eqI:

  "[| x \<le> z;  y \<le> z;  z<K |] ==> <<x,y>, <z,z>> \<in> csquare_rel(K) | x=z & y=z"

 apply (unfold csquare_rel_def)

 apply (subgoal_tac "x<K & y<K")

  prefer 2 apply (blast intro: lt_trans1)

 apply (elim ltE)

 apply (simp add: rvimage_iff Un_absorb Un_least_mem_iff ltD)

 apply (elim succE)

 apply (simp_all add: subset_Un_iff [THEN iff_sym]

                      subset_Un_iff2 [THEN iff_sym] OrdmemD)

 done

 subsubsection{*The cardinality of initial segments*}

 lemma ordermap_z_lt:

       "[| Limit(K);  x<K;  y<K;  z=succ(x \<union> y) |] ==>

           ordermap(K*K, csquare_rel(K)) ` <x,y> <

           ordermap(K*K, csquare_rel(K)) ` <z,z>"

 apply (subgoal_tac "z<K & well_ord (K*K, csquare_rel (K))")

 prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ

                               Limit_is_Ord [THEN well_ord_csquare], clarify)

 apply (rule csquare_ltI [THEN ordermap_mono, THEN ltI])

 apply (erule_tac [4] well_ord_is_wf)

 apply (blast intro!: Un_upper1_le Un_upper2_le Ord_ordermap elim!: ltE)+

 done

 text{*Kunen: "each @{term"\<langle>x,y\<rangle> \<in> K \<times> K"} has no more than @{term"z \<times> z"} predecessors..." (page 29) *}

 lemma ordermap_csquare_le:

   assumes K: "Limit(K)" and x: "x<K" and y: " y<K" and z: "z=succ(x \<union> y)"

   shows "|ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle>| \<le> |succ(z)| \<otimes> |succ(z)|"

 proof (unfold cmult_def, rule well_ord_lepoll_imp_Card_le)

   show "well_ord(|succ(z)| \<times> |succ(z)|, 

                  rmult(|succ(z)|, Memrel(|succ(z)|), |succ(z)|, Memrel(|succ(z)|)))"

     by (blast intro: Ord_cardinal well_ord_Memrel well_ord_rmult) 

 next

   have zK: "z<K" using x y z K 

     by (blast intro: Un_least_lt Limit_has_succ)

   hence oz: "Ord(z)" by (elim ltE) 

   have "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> ordermap(K \<times> K, csquare_rel(K)) ` \<langle>z,z\<rangle>"

     by (blast intro: ordermap_z_lt leI le_imp_lepoll K x y z) 

   also have "... \<approx>  Order.pred(K \<times> K, \<langle>z,z\<rangle>, csquare_rel(K))"

     proof (rule ordermap_eqpoll_pred)

       show "well_ord(K \<times> K, csquare_rel(K))" using K 

         by (rule Limit_is_Ord [THEN well_ord_csquare])

     next

       show "\<langle>z, z\<rangle> \<in> K \<times> K" using zK

         by (blast intro: ltD)

     qed

   also have "...  \<lesssim> succ(z) \<times> succ(z)" using zK

     by (rule pred_csquare_subset [THEN subset_imp_lepoll])

   also have "... \<approx> |succ(z)| \<times> |succ(z)|" using oz

     by (blast intro: prod_eqpoll_cong Ord_succ Ord_cardinal_eqpoll eqpoll_sym) 

   finally show "ordermap(K \<times> K, csquare_rel(K)) ` \<langle>x,y\<rangle> \<lesssim> |succ(z)| \<times> |succ(z)|" .

 qed

 text{*Kunen: "... so the order type is @{text"\<le>"} K" *}

 lemma ordertype_csquare_le:

      "[| InfCard(K);  \<forall>y\<in>K. InfCard(y) \<longrightarrow> y \<otimes> y = y |]

       ==> ordertype(K*K, csquare_rel(K)) \<le> K"

 apply (frule InfCard_is_Card [THEN Card_is_Ord])

 apply (rule all_lt_imp_le, assumption)

 apply (erule well_ord_csquare [THEN Ord_ordertype])

 apply (rule Card_lt_imp_lt)

 apply (erule_tac [3] InfCard_is_Card)

 apply (erule_tac [2] ltE)

 apply (simp add: ordertype_unfold)

 apply (safe elim!: ltE)

 apply (subgoal_tac "Ord (xa) & Ord (ya)")

  prefer 2 apply (blast intro: Ord_in_Ord, clarify)

 (*??WHAT A MESS!*)

 apply (rule InfCard_is_Limit [THEN ordermap_csquare_le, THEN lt_trans1],

        (assumption | rule refl | erule ltI)+)

 apply (rule_tac i = "xa \<union> ya" and j = nat in Ord_linear2,

        simp_all add: Ord_Un Ord_nat)

 prefer 2 (*case @{term"nat \<le> (xa \<union> ya)"} *)

  apply (simp add: le_imp_subset [THEN nat_succ_eqpoll, THEN cardinal_cong]

                   le_succ_iff InfCard_def Card_cardinal Un_least_lt Ord_Un

                 ltI nat_le_cardinal Ord_cardinal_le [THEN lt_trans1, THEN ltD])

 (*the finite case: @{term"xa \<union> ya < nat"} *)

 apply (rule_tac j = nat in lt_trans2)

  apply (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type

                   nat_into_Card [THEN Card_cardinal_eq]  Ord_nat)

 apply (simp add: InfCard_def)

 done

 (*Main result: Kunen's Theorem 10.12*)

 lemma InfCard_csquare_eq: "InfCard(K) ==> K \<otimes> K = K"

 apply (frule InfCard_is_Card [THEN Card_is_Ord])

 apply (erule rev_mp)

 apply (erule_tac i=K in trans_induct)

 apply (rule impI)

 apply (rule le_anti_sym)

 apply (erule_tac [2] InfCard_is_Card [THEN cmult_square_le])

 apply (rule ordertype_csquare_le [THEN [2] le_trans])

 apply (simp add: cmult_def Ord_cardinal_le

                  well_ord_csquare [THEN Ord_ordertype]

                  well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll,

                                    THEN cardinal_cong], assumption+)

 done

 (*Corollary for arbitrary well-ordered sets (all sets, assuming AC)*)

 lemma well_ord_InfCard_square_eq:

      "[| well_ord(A,r);  InfCard(|A|) |] ==> A*A \<approx> A"

 apply (rule prod_eqpoll_cong [THEN eqpoll_trans])

 apply (erule well_ord_cardinal_eqpoll [THEN eqpoll_sym])+

 apply (rule well_ord_cardinal_eqE)

 apply (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel, assumption)

 apply (simp add: cmult_def [symmetric] InfCard_csquare_eq)

 done

 lemma InfCard_square_eqpoll: "InfCard(K) ==> K \<times> K \<approx> K"

 apply (rule well_ord_InfCard_square_eq)

  apply (erule InfCard_is_Card [THEN Card_is_Ord, THEN well_ord_Memrel])

 apply (simp add: InfCard_is_Card [THEN Card_cardinal_eq])

 done

 lemma Inf_Card_is_InfCard: "[| ~Finite(i); Card(i) |] ==> InfCard(i)"

 by (simp add: InfCard_def Card_is_Ord [THEN nat_le_infinite_Ord])

 subsubsection{*Toward's Kunen's Corollary 10.13 (1)*}

 lemma InfCard_le_cmult_eq: "[| InfCard(K);  L \<le> K;  0<L |] ==> K \<otimes> L = K"

 apply (rule le_anti_sym)

  prefer 2

  apply (erule ltE, blast intro: cmult_le_self InfCard_is_Card)

 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])

 apply (rule cmult_le_mono [THEN le_trans], assumption+)

 apply (simp add: InfCard_csquare_eq)

 done

 (*Corollary 10.13 (1), for cardinal multiplication*)

 lemma InfCard_cmult_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<otimes> L = K \<union> L"

 apply (rule_tac i = K and j = L in Ord_linear_le)

 apply (typecheck add: InfCard_is_Card Card_is_Ord)

 apply (rule cmult_commute [THEN ssubst])

 apply (rule Un_commute [THEN ssubst])

 apply (simp_all add: InfCard_is_Limit [THEN Limit_has_0] InfCard_le_cmult_eq

                      subset_Un_iff2 [THEN iffD1] le_imp_subset)

 done

 lemma InfCard_cdouble_eq: "InfCard(K) ==> K \<oplus> K = K"

 apply (simp add: cmult_2 [symmetric] InfCard_is_Card cmult_commute)

 apply (simp add: InfCard_le_cmult_eq InfCard_is_Limit Limit_has_0 Limit_has_succ)

 done

 (*Corollary 10.13 (1), for cardinal addition*)

 lemma InfCard_le_cadd_eq: "[| InfCard(K);  L \<le> K |] ==> K \<oplus> L = K"

 apply (rule le_anti_sym)

  prefer 2

  apply (erule ltE, blast intro: cadd_le_self InfCard_is_Card)

 apply (frule InfCard_is_Card [THEN Card_is_Ord, THEN le_refl])

 apply (rule cadd_le_mono [THEN le_trans], assumption+)

 apply (simp add: InfCard_cdouble_eq)

 done

 lemma InfCard_cadd_eq: "[| InfCard(K);  InfCard(L) |] ==> K \<oplus> L = K \<union> L"

 apply (rule_tac i = K and j = L in Ord_linear_le)

 apply (typecheck add: InfCard_is_Card Card_is_Ord)

 apply (rule cadd_commute [THEN ssubst])

 apply (rule Un_commute [THEN ssubst])

 apply (simp_all add: InfCard_le_cadd_eq subset_Un_iff2 [THEN iffD1] le_imp_subset)

 done

 (*The other part, Corollary 10.13 (2), refers to the cardinality of the set

   of all n-tuples of elements of K.  A better version for the Isabelle theory

   might be  InfCard(K) ==> |list(K)| = K.

 *)

 subsection{*For Every Cardinal Number There Exists A Greater One*}

 text{*This result is Kunen's Theorem 10.16, which would be trivial using AC*}

 lemma Ord_jump_cardinal: "Ord(jump_cardinal(K))"

 apply (unfold jump_cardinal_def)

 apply (rule Ord_is_Transset [THEN [2] OrdI])

  prefer 2 apply (blast intro!: Ord_ordertype)

 apply (unfold Transset_def)

 apply (safe del: subsetI)

 apply (simp add: ordertype_pred_unfold, safe)

 apply (rule UN_I)

 apply (rule_tac [2] ReplaceI)

    prefer 4 apply (blast intro: well_ord_subset elim!: predE)+

 done

 (*Allows selective unfolding.  Less work than deriving intro/elim rules*)

 lemma jump_cardinal_iff:

      "i \<in> jump_cardinal(K) \<longleftrightarrow>

       (\<exists>r X. r \<subseteq> K*K & X \<subseteq> K & well_ord(X,r) & i = ordertype(X,r))"

 apply (unfold jump_cardinal_def)

 apply (blast del: subsetI)

 done

 (*The easy part of Theorem 10.16: jump_cardinal(K) exceeds K*)

 lemma K_lt_jump_cardinal: "Ord(K) ==> K < jump_cardinal(K)"

 apply (rule Ord_jump_cardinal [THEN [2] ltI])

 apply (rule jump_cardinal_iff [THEN iffD2])

 apply (rule_tac x="Memrel(K)" in exI)

 apply (rule_tac x=K in exI)

 apply (simp add: ordertype_Memrel well_ord_Memrel)

 apply (simp add: Memrel_def subset_iff)

 done

 (*The proof by contradiction: the bijection f yields a wellordering of X

   whose ordertype is jump_cardinal(K).  *)

 lemma Card_jump_cardinal_lemma:

      "[| well_ord(X,r);  r \<subseteq> K * K;  X \<subseteq> K;

          f \<in> bij(ordertype(X,r), jump_cardinal(K)) |]

       ==> jump_cardinal(K) \<in> jump_cardinal(K)"

 apply (subgoal_tac "f O ordermap (X,r) \<in> bij (X, jump_cardinal (K))")

  prefer 2 apply (blast intro: comp_bij ordermap_bij)

 apply (rule jump_cardinal_iff [THEN iffD2])

 apply (intro exI conjI)

 apply (rule subset_trans [OF rvimage_type Sigma_mono], assumption+)

 apply (erule bij_is_inj [THEN well_ord_rvimage])

 apply (rule Ord_jump_cardinal [THEN well_ord_Memrel])

 apply (simp add: well_ord_Memrel [THEN [2] bij_ordertype_vimage]

                  ordertype_Memrel Ord_jump_cardinal)

 done

 (*The hard part of Theorem 10.16: jump_cardinal(K) is itself a cardinal*)

 lemma Card_jump_cardinal: "Card(jump_cardinal(K))"

 apply (rule Ord_jump_cardinal [THEN CardI])

 apply (unfold eqpoll_def)

 apply (safe dest!: ltD jump_cardinal_iff [THEN iffD1])

 apply (blast intro: Card_jump_cardinal_lemma [THEN mem_irrefl])

 done

 subsection{*Basic Properties of Successor Cardinals*}

 lemma csucc_basic: "Ord(K) ==> Card(csucc(K)) & K < csucc(K)"

 apply (unfold csucc_def)

 apply (rule LeastI)

 apply (blast intro: Card_jump_cardinal K_lt_jump_cardinal Ord_jump_cardinal)+

 done

 lemmas Card_csucc = csucc_basic [THEN conjunct1]

 lemmas lt_csucc = csucc_basic [THEN conjunct2]

 lemma Ord_0_lt_csucc: "Ord(K) ==> 0 < csucc(K)"

 by (blast intro: Ord_0_le lt_csucc lt_trans1)

 lemma csucc_le: "[| Card(L);  K<L |] ==> csucc(K) \<le> L"

 apply (unfold csucc_def)

 apply (rule Least_le)

 apply (blast intro: Card_is_Ord)+

 done

 lemma lt_csucc_iff: "[| Ord(i); Card(K) |] ==> i < csucc(K) \<longleftrightarrow> |i| \<le> K"

 apply (rule iffI)

 apply (rule_tac [2] Card_lt_imp_lt)

 apply (erule_tac [2] lt_trans1)

 apply (simp_all add: lt_csucc Card_csucc Card_is_Ord)

 apply (rule notI [THEN not_lt_imp_le])

 apply (rule Card_cardinal [THEN csucc_le, THEN lt_trans1, THEN lt_irrefl], assumption)

 apply (rule Ord_cardinal_le [THEN lt_trans1])

 apply (simp_all add: Ord_cardinal Card_is_Ord)

 done

 lemma Card_lt_csucc_iff:

      "[| Card(K'); Card(K) |] ==> K' < csucc(K) \<longleftrightarrow> K' \<le> K"

 by (simp add: lt_csucc_iff Card_cardinal_eq Card_is_Ord)

 lemma InfCard_csucc: "InfCard(K) ==> InfCard(csucc(K))"

 by (simp add: InfCard_def Card_csucc Card_is_Ord

               lt_csucc [THEN leI, THEN [2] le_trans])

 subsubsection{*Removing elements from a finite set decreases its cardinality*}

 lemma Fin_imp_not_cons_lepoll: "A: Fin(U) ==> x\<notin>A \<longrightarrow> ~ cons(x,A) \<lesssim> A"

 apply (erule Fin_induct)

 apply (simp add: lepoll_0_iff)

 apply (subgoal_tac "cons (x,cons (xa,y)) = cons (xa,cons (x,y))")

 apply simp

 apply (blast dest!: cons_lepoll_consD, blast)

 done

 lemma Finite_imp_cardinal_cons [simp]:

      "[| Finite(A);  a\<notin>A |] ==> |cons(a,A)| = succ(|A|)"

 apply (unfold cardinal_def)

 apply (rule Least_equality)

 apply (fold cardinal_def)

 apply (simp add: succ_def)

 apply (blast intro: cons_eqpoll_cong well_ord_cardinal_eqpoll

              elim!: mem_irrefl  dest!: Finite_imp_well_ord)

 apply (blast intro: Card_cardinal Card_is_Ord)

 apply (rule notI)

 apply (rule Finite_into_Fin [THEN Fin_imp_not_cons_lepoll, THEN mp, THEN notE],

        assumption, assumption)

 apply (erule eqpoll_sym [THEN eqpoll_imp_lepoll, THEN lepoll_trans])

 apply (erule le_imp_lepoll [THEN lepoll_trans])

 apply (blast intro: well_ord_cardinal_eqpoll [THEN eqpoll_imp_lepoll]

              dest!: Finite_imp_well_ord)

 done

 lemma Finite_imp_succ_cardinal_Diff:

      "[| Finite(A);  a:A |] ==> succ(|A-{a}|) = |A|"

 apply (rule_tac b = A in cons_Diff [THEN subst], assumption)

 apply (simp add: Finite_imp_cardinal_cons Diff_subset [THEN subset_Finite])

 apply (simp add: cons_Diff)

 done

 lemma Finite_imp_cardinal_Diff: "[| Finite(A);  a:A |] ==> |A-{a}| < |A|"

 apply (rule succ_leE)

 apply (simp add: Finite_imp_succ_cardinal_Diff)

 done

 lemma Finite_cardinal_in_nat [simp]: "Finite(A) ==> |A| \<in> nat"

 apply (erule Finite_induct)

 apply (auto simp add: cardinal_0 Finite_imp_cardinal_cons)

 done

 lemma card_Un_Int:

      "[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A \<union> B| #+ |A \<inter> B|"

 apply (erule Finite_induct, simp)

 apply (simp add: Finite_Int cons_absorb Un_cons Int_cons_left)

 done

 lemma card_Un_disjoint:

      "[|Finite(A); Finite(B); A \<inter> B = 0|] ==> |A \<union> B| = |A| #+ |B|"

 by (simp add: Finite_Un card_Un_Int)

 lemma card_partition [rule_format]:

      "Finite(C) ==>

         Finite (\<Union> C) \<longrightarrow>

         (\<forall>c\<in>C. |c| = k) \<longrightarrow>

         (\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = 0) \<longrightarrow>

         k #* |C| = |\<Union> C|"

 apply (erule Finite_induct, auto)

 apply (subgoal_tac " x \<inter> \<Union>B = 0")

 apply (auto simp add: card_Un_disjoint Finite_Union

        subset_Finite [of _ "\<Union> (cons(x,F))"])

 done

 subsubsection{*Theorems by Krzysztof Grabczewski, proofs by lcp*}

 lemmas nat_implies_well_ord = nat_into_Ord [THEN well_ord_Memrel]

 lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n \<approx> m #+ n"

 apply (rule eqpoll_trans)

 apply (rule well_ord_radd [THEN well_ord_cardinal_eqpoll, THEN eqpoll_sym])

 apply (erule nat_implies_well_ord)+

 apply (simp add: nat_cadd_eq_add [symmetric] cadd_def eqpoll_refl)

 done

 lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i \<subseteq> nat \<longrightarrow> i \<in> nat | i=nat"

 apply (erule trans_induct3, auto)

 apply (blast dest!: nat_le_Limit [THEN le_imp_subset])

 done

 lemma Ord_nat_subset_into_Card: "[| Ord(i); i \<subseteq> nat |] ==> Card(i)"

 by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)

 lemma Finite_Diff_sing_eq_diff_1: "[| Finite(A); x:A |] ==> |A-{x}| = |A| #- 1"

 apply (rule succ_inject)

 apply (rule_tac b = "|A|" in trans)

  apply (simp add: Finite_imp_succ_cardinal_Diff)

 apply (subgoal_tac "1 \<lesssim> A")

  prefer 2 apply (blast intro: not_0_is_lepoll_1)

 apply (frule Finite_imp_well_ord, clarify)

 apply (drule well_ord_lepoll_imp_Card_le)

  apply (auto simp add: cardinal_1)

 apply (rule trans)

  apply (rule_tac [2] diff_succ)

   apply (auto simp add: Finite_cardinal_in_nat)

 done

 lemma cardinal_lt_imp_Diff_not_0 [rule_format]:

      "Finite(B) ==> \<forall>A. |B|<|A| \<longrightarrow> A - B \<noteq> 0"

 apply (erule Finite_induct, auto)

 apply (case_tac "Finite (A)")

  apply (subgoal_tac [2] "Finite (cons (x, B))")

   apply (drule_tac [2] B = "cons (x, B) " in Diff_Finite)

    apply (auto simp add: Finite_0 Finite_cons)

 apply (subgoal_tac "|B|<|A|")

  prefer 2 apply (blast intro: lt_trans Ord_cardinal)

 apply (case_tac "x:A")

  apply (subgoal_tac [2] "A - cons (x, B) = A - B")

   apply auto

 apply (subgoal_tac "|A| \<le> |cons (x, B) |")

  prefer 2

  apply (blast dest: Finite_cons [THEN Finite_imp_well_ord]

               intro: well_ord_lepoll_imp_Card_le subset_imp_lepoll)

 apply (auto simp add: Finite_imp_cardinal_cons)

 apply (auto dest!: Finite_cardinal_in_nat simp add: le_iff)

 apply (blast intro: lt_trans)

 done

 end