(* Title: ZF/CardinalArith.thy ID: $Id: CardinalArith.thy,v 1.32 2007/10/07 19:19:31 wenzelm Exp $ 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 Un 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∈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 output) cadd (infixl "⊕" 65) and cmult (infixl "⊗" 70) notation (HTML output) cadd (infixl "⊕" 65) and cmult (infixl "⊗" 70) lemma Card_Union [simp,intro,TC]: "(ALL x:A. Card(x)) ==> Card(Union(A))" apply (rule CardI) apply (simp add: Card_is_Ord) apply (clarify dest!: ltD) apply (drule bspec, assumption) apply (frule lt_Card_imp_lesspoll, blast intro: ltI Card_is_Ord) apply (drule eqpoll_sym [THEN eqpoll_imp_lepoll]) apply (drule lesspoll_trans1, assumption) apply (subgoal_tac "B \<lesssim> \<Union>A") apply (drule lesspoll_trans1, assumption, blast) apply (blast intro: subset_imp_lepoll) done lemma Card_UN: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x∈A. K(x))" by (blast intro: Card_Union) lemma Card_OUN [simp,intro,TC]: "(!!x. x:A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))" by (simp add: OUnion_def Card_0) lemma n_lesspoll_nat: "n ∈ nat ==> n \<prec> nat" apply (unfold lesspoll_def) apply (rule conjI) apply (erule OrdmemD [THEN subset_imp_lepoll], rule Ord_nat) apply (rule notI) apply (erule eqpollE) apply (rule succ_lepoll_natE) apply (blast intro: nat_succI [THEN OrdmemD, THEN subset_imp_lepoll] lepoll_trans, assumption) done lemma in_Card_imp_lesspoll: "[| Card(K); b ∈ K |] ==> b \<prec> K" apply (unfold lesspoll_def) apply (simp add: Card_iff_initial) apply (fast intro!: le_imp_lepoll ltI leI) done lemma lesspoll_lemma: "[| ~ A \<prec> B; C \<prec> B |] ==> A - C ≠ 0" apply (unfold lesspoll_def) apply (fast dest!: Diff_eq_0_iff [THEN iffD1, THEN subset_imp_lepoll] intro!: eqpollI elim: notE elim!: eqpollE lepoll_trans) 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 ≈ B+A" apply (unfold eqpoll_def) apply (rule exI) apply (rule_tac c = "case(Inr,Inl)" and d = "case(Inr,Inl)" in lam_bijective) apply auto done lemma cadd_commute: "i |+| j = j |+| 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 ≈ A+(B+C)" apply (unfold eqpoll_def) apply (rule exI) apply (rule sum_assoc_bij) done (*Unconditional version requires AC*) lemma well_ord_cadd_assoc: "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> (i |+| j) |+| k = i |+| (j |+| k)" apply (unfold cadd_def) apply (rule cardinal_cong) apply (rule eqpoll_trans) apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) apply (blast intro: well_ord_radd ) apply (rule sum_assoc_eqpoll [THEN eqpoll_trans]) apply (rule eqpoll_sym) apply (rule sum_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) apply (blast intro: well_ord_radd ) done subsubsection{*0 is the identity for addition*} lemma sum_0_eqpoll: "0+A ≈ A" apply (unfold eqpoll_def) apply (rule exI) apply (rule bij_0_sum) done lemma cadd_0 [simp]: "Card(K) ==> 0 |+| 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" apply (unfold lepoll_def inj_def) apply (rule_tac x = "lam x:A. Inl (x) " in exI) apply simp done (*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 |+| 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 = "lam z: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' |+| L') le (K |+| 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 ≈ 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) |+| n = |succ(m |+| 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 |+| 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*} (*Easier to prove the two directions separately*) lemma prod_commute_eqpoll: "A*B ≈ 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 |*| j = j |*| 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 ≈ A*(B*C)" apply (unfold eqpoll_def) apply (rule exI) apply (rule prod_assoc_bij) done (*Unconditional version requires AC*) lemma well_ord_cmult_assoc: "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> (i |*| j) |*| k = i |*| (j |*| k)" apply (unfold cmult_def) apply (rule cardinal_cong) apply (rule eqpoll_trans) apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) apply (blast intro: well_ord_rmult) apply (rule prod_assoc_eqpoll [THEN eqpoll_trans]) apply (rule eqpoll_sym) apply (rule prod_eqpoll_cong [OF eqpoll_refl well_ord_cardinal_eqpoll]) apply (blast intro: well_ord_rmult) done subsubsection{*Cardinal multiplication distributes over addition*} lemma sum_prod_distrib_eqpoll: "(A+B)*C ≈ (A*C)+(B*C)" apply (unfold eqpoll_def) apply (rule exI) apply (rule sum_prod_distrib_bij) done lemma well_ord_cadd_cmult_distrib: "[| well_ord(i,ri); well_ord(j,rj); well_ord(k,rk) |] ==> (i |+| j) |*| k = (i |*| k) |+| (j |*| k)" apply (unfold cadd_def cmult_def) apply (rule cardinal_cong) apply (rule eqpoll_trans) apply (rule prod_eqpoll_cong [OF well_ord_cardinal_eqpoll eqpoll_refl]) apply (blast intro: well_ord_radd) apply (rule sum_prod_distrib_eqpoll [THEN eqpoll_trans]) apply (rule eqpoll_sym) apply (rule sum_eqpoll_cong [OF well_ord_cardinal_eqpoll well_ord_cardinal_eqpoll]) apply (blast intro: well_ord_rmult)+ done subsubsection{*Multiplication by 0 yields 0*} lemma prod_0_eqpoll: "0*A ≈ 0" apply (unfold eqpoll_def) apply (rule exI) apply (rule lam_bijective, safe) done lemma cmult_0 [simp]: "0 |*| 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 ≈ 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 |*| 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 = "lam x: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 |*| 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 = "lam x: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 |*| 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' |*| L') le (K |*| 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 ≈ 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) |*| n = n |+| (m |*| 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 |*| 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 |*| n = n |+| n" by (simp add: cmult_succ_lemma Card_is_Ord cadd_commute [of _ 0]) lemma sum_lepoll_prod: "2 \<lesssim> C ==> B+B \<lesssim> C*B" apply (rule lepoll_trans) apply (rule sum_eq_2_times [THEN equalityD1, THEN subset_imp_lepoll]) apply (erule prod_lepoll_mono) apply (rule lepoll_refl) done 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 lam z:cons(u,A). if z=u then 0 else if z : 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 = "lam z: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) ≈ 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 <= A ==> succ(A) ≈ 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 Un 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 ≈ 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 csquare_lam_inj: "Ord(K) ==> (lam <x,y>:K*K. <x Un y, x, y>) : inj(K*K, K*K*K)" apply (unfold inj_def) apply (force intro: lam_type Un_least_lt [THEN ltD] ltI) done lemma well_ord_csquare: "Ord(K) ==> well_ord(K*K, csquare_rel(K))" apply (unfold csquare_rel_def) apply (rule csquare_lam_inj [THEN well_ord_rvimage], assumption) apply (blast intro: well_ord_rmult well_ord_Memrel) done subsubsection{*Characterising initial segments of the well-ordering*} lemma csquareD: "[| <<x,y>, <z,z>> : 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)) <= succ(z)*succ(z)" apply (unfold Order.pred_def) apply (safe del: SigmaI succCI) apply (erule csquareD [THEN conjE]) apply (unfold lt_def, auto) done lemma csquare_ltI: "[| x<z; y<z; z<K |] ==> <<x,y>, <z,z>> : 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>> : 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 Un 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 (*Kunen: "each <x,y>: K*K has no more than z*z predecessors..." (page 29) *) lemma ordermap_csquare_le: "[| Limit(K); x<K; y<K; z=succ(x Un y) |] ==> | ordermap(K*K, csquare_rel(K)) ` <x,y> | le |succ(z)| |*| |succ(z)|" apply (unfold cmult_def) apply (rule well_ord_rmult [THEN well_ord_lepoll_imp_Card_le]) apply (rule Ord_cardinal [THEN well_ord_Memrel])+ apply (subgoal_tac "z<K") prefer 2 apply (blast intro!: Un_least_lt Limit_has_succ) apply (rule ordermap_z_lt [THEN leI, THEN le_imp_lepoll, THEN lepoll_trans], assumption+) apply (rule ordermap_eqpoll_pred [THEN eqpoll_imp_lepoll, THEN lepoll_trans]) apply (erule Limit_is_Ord [THEN well_ord_csquare]) apply (blast intro: ltD) apply (rule pred_csquare_subset [THEN subset_imp_lepoll, THEN lepoll_trans], assumption) apply (elim ltE) apply (rule prod_eqpoll_cong [THEN eqpoll_sym, THEN eqpoll_imp_lepoll]) apply (erule Ord_succ [THEN Ord_cardinal_eqpoll])+ done (*Kunen: "... so the order type <= K" *) lemma ordertype_csquare_le: "[| InfCard(K); ALL y:K. InfCard(y) --> y |*| 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 Un ya" and j = nat in Ord_linear2, simp_all add: Ord_Un Ord_nat) prefer 2 (*case nat le (xa Un 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: xa Un 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 |*| 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 ≈ 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 × K ≈ 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 |*| 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 |*| L = K Un 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 |+| 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 |+| 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 |+| L = K Un 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 : jump_cardinal(K) <-> (EX r X. r <= K*K & X <= 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 <= K * K; X <= K; f : bij(ordertype(X,r), jump_cardinal(K)) |] ==> jump_cardinal(K) : jump_cardinal(K)" apply (subgoal_tac "f O ordermap (X,r) : 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, standard] lemmas lt_csucc = csucc_basic [THEN conjunct2, standard] 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) <-> |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) <-> 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~:A --> ~ 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~: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| : 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 Un B| #+ |A Int 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 Int B = 0|] ==> |A Un B| = |A| #+ |B|" by (simp add: Finite_Un card_Un_Int) lemma card_partition [rule_format]: "Finite(C) ==> Finite (\<Union> C) --> (∀c∈C. |c| = k) --> (∀c1 ∈ C. ∀c2 ∈ C. c1 ≠ c2 --> c1 ∩ c2 = 0) --> k #* |C| = |\<Union> C|" apply (erule Finite_induct, auto) apply (subgoal_tac " x ∩ \<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, standard] lemma nat_sum_eqpoll_sum: "[| m:nat; n:nat |] ==> m + n ≈ 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 <= nat --> i : 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 <= 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) ==> ALL A. |B|<|A| --> A - B ~= 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 ML{* val InfCard_def = thm "InfCard_def" val cmult_def = thm "cmult_def" val cadd_def = thm "cadd_def" val jump_cardinal_def = thm "jump_cardinal_def" val csucc_def = thm "csucc_def" val sum_commute_eqpoll = thm "sum_commute_eqpoll"; val cadd_commute = thm "cadd_commute"; val sum_assoc_eqpoll = thm "sum_assoc_eqpoll"; val well_ord_cadd_assoc = thm "well_ord_cadd_assoc"; val sum_0_eqpoll = thm "sum_0_eqpoll"; val cadd_0 = thm "cadd_0"; val sum_lepoll_self = thm "sum_lepoll_self"; val cadd_le_self = thm "cadd_le_self"; val sum_lepoll_mono = thm "sum_lepoll_mono"; val cadd_le_mono = thm "cadd_le_mono"; val eq_imp_not_mem = thm "eq_imp_not_mem"; val sum_succ_eqpoll = thm "sum_succ_eqpoll"; val nat_cadd_eq_add = thm "nat_cadd_eq_add"; val prod_commute_eqpoll = thm "prod_commute_eqpoll"; val cmult_commute = thm "cmult_commute"; val prod_assoc_eqpoll = thm "prod_assoc_eqpoll"; val well_ord_cmult_assoc = thm "well_ord_cmult_assoc"; val sum_prod_distrib_eqpoll = thm "sum_prod_distrib_eqpoll"; val well_ord_cadd_cmult_distrib = thm "well_ord_cadd_cmult_distrib"; val prod_0_eqpoll = thm "prod_0_eqpoll"; val cmult_0 = thm "cmult_0"; val prod_singleton_eqpoll = thm "prod_singleton_eqpoll"; val cmult_1 = thm "cmult_1"; val prod_lepoll_self = thm "prod_lepoll_self"; val cmult_le_self = thm "cmult_le_self"; val prod_lepoll_mono = thm "prod_lepoll_mono"; val cmult_le_mono = thm "cmult_le_mono"; val prod_succ_eqpoll = thm "prod_succ_eqpoll"; val nat_cmult_eq_mult = thm "nat_cmult_eq_mult"; val cmult_2 = thm "cmult_2"; val sum_lepoll_prod = thm "sum_lepoll_prod"; val lepoll_imp_sum_lepoll_prod = thm "lepoll_imp_sum_lepoll_prod"; val nat_cons_lepoll = thm "nat_cons_lepoll"; val nat_cons_eqpoll = thm "nat_cons_eqpoll"; val nat_succ_eqpoll = thm "nat_succ_eqpoll"; val InfCard_nat = thm "InfCard_nat"; val InfCard_is_Card = thm "InfCard_is_Card"; val InfCard_Un = thm "InfCard_Un"; val InfCard_is_Limit = thm "InfCard_is_Limit"; val ordermap_eqpoll_pred = thm "ordermap_eqpoll_pred"; val ordermap_z_lt = thm "ordermap_z_lt"; val InfCard_le_cmult_eq = thm "InfCard_le_cmult_eq"; val InfCard_cmult_eq = thm "InfCard_cmult_eq"; val InfCard_cdouble_eq = thm "InfCard_cdouble_eq"; val InfCard_le_cadd_eq = thm "InfCard_le_cadd_eq"; val InfCard_cadd_eq = thm "InfCard_cadd_eq"; val Ord_jump_cardinal = thm "Ord_jump_cardinal"; val jump_cardinal_iff = thm "jump_cardinal_iff"; val K_lt_jump_cardinal = thm "K_lt_jump_cardinal"; val Card_jump_cardinal = thm "Card_jump_cardinal"; val csucc_basic = thm "csucc_basic"; val Card_csucc = thm "Card_csucc"; val lt_csucc = thm "lt_csucc"; val Ord_0_lt_csucc = thm "Ord_0_lt_csucc"; val csucc_le = thm "csucc_le"; val lt_csucc_iff = thm "lt_csucc_iff"; val Card_lt_csucc_iff = thm "Card_lt_csucc_iff"; val InfCard_csucc = thm "InfCard_csucc"; val Finite_into_Fin = thm "Finite_into_Fin"; val Fin_into_Finite = thm "Fin_into_Finite"; val Finite_Fin_iff = thm "Finite_Fin_iff"; val Finite_Un = thm "Finite_Un"; val Finite_Union = thm "Finite_Union"; val Finite_induct = thm "Finite_induct"; val Fin_imp_not_cons_lepoll = thm "Fin_imp_not_cons_lepoll"; val Finite_imp_cardinal_cons = thm "Finite_imp_cardinal_cons"; val Finite_imp_succ_cardinal_Diff = thm "Finite_imp_succ_cardinal_Diff"; val Finite_imp_cardinal_Diff = thm "Finite_imp_cardinal_Diff"; val nat_implies_well_ord = thm "nat_implies_well_ord"; val nat_sum_eqpoll_sum = thm "nat_sum_eqpoll_sum"; val Diff_sing_Finite = thm "Diff_sing_Finite"; val Diff_Finite = thm "Diff_Finite"; val Ord_subset_natD = thm "Ord_subset_natD"; val Ord_nat_subset_into_Card = thm "Ord_nat_subset_into_Card"; val Finite_cardinal_in_nat = thm "Finite_cardinal_in_nat"; val Finite_Diff_sing_eq_diff_1 = thm "Finite_Diff_sing_eq_diff_1"; val cardinal_lt_imp_Diff_not_0 = thm "cardinal_lt_imp_Diff_not_0"; *} end
lemma Card_Union:
∀x∈A. Card(x) ==> Card(\<Union>A)
lemma Card_UN:
(!!x. x ∈ A ==> Card(K(x))) ==> Card(\<Union>x∈A. K(x))
lemma Card_OUN:
(!!x. x ∈ A ==> Card(K(x))) ==> Card(\<Union>x<A. K(x))
lemma n_lesspoll_nat:
n ∈ nat ==> n lesspoll nat
lemma in_Card_imp_lesspoll:
[| Card(K); b ∈ K |] ==> b lesspoll K
lemma lesspoll_lemma:
[| ¬ A lesspoll B; C lesspoll B |] ==> A - C ≠ 0
lemma sum_commute_eqpoll:
A + B ≈ B + A
lemma cadd_commute:
i ⊕ j = j ⊕ i
lemma sum_assoc_eqpoll:
(A + B) + C ≈ A + B + C
lemma well_ord_cadd_assoc:
[| well_ord(i, ri); well_ord(j, rj); well_ord(k, rk) |]
==> i ⊕ j ⊕ k = i ⊕ (j ⊕ k)
lemma sum_0_eqpoll:
0 + A ≈ A
lemma cadd_0:
Card(K) ==> 0 ⊕ K = K
lemma sum_lepoll_self:
A lepoll A + B
lemma cadd_le_self:
[| Card(K); Ord(L) |] ==> K ≤ K ⊕ L
lemma sum_lepoll_mono:
[| A lepoll C; B lepoll D |] ==> A + B lepoll C + D
lemma cadd_le_mono:
[| K' ≤ K; L' ≤ L |] ==> K' ⊕ L' ≤ K ⊕ L
lemma sum_succ_eqpoll:
succ(A) + B ≈ succ(A + B)
lemma cadd_succ_lemma:
[| Ord(m); Ord(n) |] ==> succ(m) ⊕ n = |succ(m ⊕ n)|
lemma nat_cadd_eq_add:
[| m ∈ nat; n ∈ nat |] ==> m ⊕ n = m #+ n
lemma prod_commute_eqpoll:
A × B ≈ B × A
lemma cmult_commute:
i ⊗ j = j ⊗ i
lemma prod_assoc_eqpoll:
(A × B) × C ≈ A × B × C
lemma well_ord_cmult_assoc:
[| well_ord(i, ri); well_ord(j, rj); well_ord(k, rk) |]
==> i ⊗ j ⊗ k = i ⊗ (j ⊗ k)
lemma sum_prod_distrib_eqpoll:
(A + B) × C ≈ A × C + B × C
lemma well_ord_cadd_cmult_distrib:
[| well_ord(i, ri); well_ord(j, rj); well_ord(k, rk) |]
==> (i ⊕ j) ⊗ k = i ⊗ k ⊕ j ⊗ k
lemma prod_0_eqpoll:
0 × A ≈ 0
lemma cmult_0:
0 ⊗ i = 0
lemma prod_singleton_eqpoll:
{x} × A ≈ A
lemma cmult_1:
Card(K) ==> 1 ⊗ K = K
lemma prod_square_lepoll:
A lepoll A × A
lemma cmult_square_le:
Card(K) ==> K ≤ K ⊗ K
lemma prod_lepoll_self:
b ∈ B ==> A lepoll A × B
lemma cmult_le_self:
[| Card(K); Ord(L); 0 < L |] ==> K ≤ K ⊗ L
lemma prod_lepoll_mono:
[| A lepoll C; B lepoll D |] ==> A × B lepoll C × D
lemma cmult_le_mono:
[| K' ≤ K; L' ≤ L |] ==> K' ⊗ L' ≤ K ⊗ L
lemma prod_succ_eqpoll:
succ(A) × B ≈ B + A × B
lemma cmult_succ_lemma:
[| Ord(m); Ord(n) |] ==> succ(m) ⊗ n = n ⊕ m ⊗ n
lemma nat_cmult_eq_mult:
[| m ∈ nat; n ∈ nat |] ==> m ⊗ n = m #× n
lemma cmult_2:
Card(n) ==> 2 ⊗ n = n ⊕ n
lemma sum_lepoll_prod:
2 lepoll C ==> B + B lepoll C × B
lemma lepoll_imp_sum_lepoll_prod:
[| A lepoll B; 2 lepoll A |] ==> A + B lepoll A × B
lemma nat_cons_lepoll:
nat lepoll A ==> cons(u, A) lepoll A
lemma nat_cons_eqpoll:
nat lepoll A ==> cons(u, A) ≈ A
lemma nat_succ_eqpoll:
nat ⊆ A ==> succ(A) ≈ A
lemma InfCard_nat:
InfCard(nat)
lemma InfCard_is_Card:
InfCard(K) ==> Card(K)
lemma InfCard_Un:
[| InfCard(K); Card(L) |] ==> InfCard(K ∪ L)
lemma InfCard_is_Limit:
InfCard(K) ==> Limit(K)
lemma ordermap_eqpoll_pred:
[| well_ord(A, r); x ∈ A |] ==> ordermap(A, r) ` x ≈ Order.pred(A, x, r)
lemma csquare_lam_inj:
Ord(K) ==> (λ〈x,y〉∈K × K. 〈x ∪ y, x, y〉) ∈ inj(K × K, K × K × K)
lemma well_ord_csquare:
Ord(K) ==> well_ord(K × K, csquare_rel(K))
lemma csquareD:
[| 〈〈x, y〉, z, z〉 ∈ csquare_rel(K); x < K; y < K; z < K |] ==> x ≤ z ∧ y ≤ z
lemma pred_csquare_subset:
z < K ==> Order.pred(K × K, 〈z, z〉, csquare_rel(K)) ⊆ succ(z) × succ(z)
lemma csquare_ltI:
[| x < z; y < z; z < K |] ==> 〈〈x, y〉, z, z〉 ∈ csquare_rel(K)
lemma csquare_or_eqI:
[| x ≤ z; y ≤ z; z < K |] ==> 〈〈x, y〉, z, z〉 ∈ csquare_rel(K) ∨ x = z ∧ y = z
lemma ordermap_z_lt:
[| Limit(K); x < K; y < K; z = succ(x ∪ y) |]
==> ordermap(K × K, csquare_rel(K)) ` 〈x, y〉 <
ordermap(K × K, csquare_rel(K)) ` 〈z, z〉
lemma ordermap_csquare_le:
[| Limit(K); x < K; y < K; z = succ(x ∪ y) |]
==> |ordermap(K × K, csquare_rel(K)) ` 〈x, y〉| ≤ |succ(z)| ⊗ |succ(z)|
lemma ordertype_csquare_le:
[| InfCard(K); ∀y∈K. InfCard(y) --> y ⊗ y = y |]
==> ordertype(K × K, csquare_rel(K)) ≤ K
lemma InfCard_csquare_eq:
InfCard(K) ==> K ⊗ K = K
lemma well_ord_InfCard_square_eq:
[| well_ord(A, r); InfCard(|A|) |] ==> A × A ≈ A
lemma InfCard_square_eqpoll:
InfCard(K) ==> K × K ≈ K
lemma Inf_Card_is_InfCard:
[| ¬ Finite(i); Card(i) |] ==> InfCard(i)
lemma InfCard_le_cmult_eq:
[| InfCard(K); L ≤ K; 0 < L |] ==> K ⊗ L = K
lemma InfCard_cmult_eq:
[| InfCard(K); InfCard(L) |] ==> K ⊗ L = K ∪ L
lemma InfCard_cdouble_eq:
InfCard(K) ==> K ⊕ K = K
lemma InfCard_le_cadd_eq:
[| InfCard(K); L ≤ K |] ==> K ⊕ L = K
lemma InfCard_cadd_eq:
[| InfCard(K); InfCard(L) |] ==> K ⊕ L = K ∪ L
lemma Ord_jump_cardinal:
Ord(jump_cardinal(K))
lemma jump_cardinal_iff:
i ∈ jump_cardinal(K) <->
(∃r X. r ⊆ K × K ∧ X ⊆ K ∧ well_ord(X, r) ∧ i = ordertype(X, r))
lemma K_lt_jump_cardinal:
Ord(K) ==> K < jump_cardinal(K)
lemma Card_jump_cardinal_lemma:
[| well_ord(X, r); r ⊆ K × K; X ⊆ K;
f ∈ bij(ordertype(X, r), jump_cardinal(K)) |]
==> jump_cardinal(K) ∈ jump_cardinal(K)
lemma Card_jump_cardinal:
Card(jump_cardinal(K))
lemma csucc_basic:
Ord(K) ==> Card(csucc(K)) ∧ K < csucc(K)
lemma Card_csucc:
Ord(K) ==> Card(csucc(K))
lemma lt_csucc:
Ord(K) ==> K < csucc(K)
lemma Ord_0_lt_csucc:
Ord(K) ==> 0 < csucc(K)
lemma csucc_le:
[| Card(L); K < L |] ==> csucc(K) ≤ L
lemma lt_csucc_iff:
[| Ord(i); Card(K) |] ==> i < csucc(K) <-> |i| ≤ K
lemma Card_lt_csucc_iff:
[| Card(K'); Card(K) |] ==> K' < csucc(K) <-> K' ≤ K
lemma InfCard_csucc:
InfCard(K) ==> InfCard(csucc(K))
lemma Fin_imp_not_cons_lepoll:
A ∈ Fin(U) ==> x ∉ A --> ¬ cons(x, A) lepoll A
lemma Finite_imp_cardinal_cons:
[| Finite(A); a ∉ A |] ==> |cons(a, A)| = succ(|A|)
lemma Finite_imp_succ_cardinal_Diff:
[| Finite(A); a ∈ A |] ==> succ(|A - {a}|) = |A|
lemma Finite_imp_cardinal_Diff:
[| Finite(A); a ∈ A |] ==> |A - {a}| < |A|
lemma Finite_cardinal_in_nat:
Finite(A) ==> |A| ∈ nat
lemma card_Un_Int:
[| Finite(A); Finite(B) |] ==> |A| #+ |B| = |A ∪ B| #+ |A ∩ B|
lemma card_Un_disjoint:
[| Finite(A); Finite(B); A ∩ B = 0 |] ==> |A ∪ B| = |A| #+ |B|
lemma card_partition:
[| Finite(C); Finite(\<Union>C); !!c. c ∈ C ==> |c| = k;
!!c1 c2. [| c1 ∈ C; c2 ∈ C; c1 ≠ c2 |] ==> c1 ∩ c2 = 0 |]
==> k #× |C| = |\<Union>C|
lemma nat_implies_well_ord:
i ∈ nat ==> well_ord(i, Memrel(i))
lemma nat_sum_eqpoll_sum:
[| m ∈ nat; n ∈ nat |] ==> m + n ≈ m #+ n
lemma Ord_subset_natD:
[| Ord(i); i ⊆ nat |] ==> i ∈ nat ∨ i = nat
lemma Ord_nat_subset_into_Card:
[| Ord(i); i ⊆ nat |] ==> Card(i)
lemma Finite_Diff_sing_eq_diff_1:
[| Finite(A); x ∈ A |] ==> |A - {x}| = |A| #- 1
lemma cardinal_lt_imp_Diff_not_0:
[| Finite(B); |B| < |A| |] ==> A - B ≠ 0