section‹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 ≤ 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 ∪ y, x, y>,
rmult(K,Memrel(K), K*K, rmult(K,Memrel(K), K,Memrel(K))))"
definition
jump_cardinal :: "i=>i" where
"jump_cardinal(K) ==
⋃X∈Pow(K). {z. r ∈ Pow(K*K), well_ord(X,r) & z = ordertype(X,r)}"
definition
csucc :: "i=>i" where
"csucc(K) == μ L. Card(L) & K<L"
lemma Card_Union [simp,intro,TC]:
assumes A: "⋀x. x∈A ⟹ Card(x)" shows "Card(⋃(A))"
proof (rule CardI)
show "Ord(⋃A)" using A
by (simp add: Card_is_Ord)
next
fix j
assume j: "j < ⋃A"
hence "∃c∈A. j < c & Card(c)" using A
by (auto simp add: lt_def intro: Card_is_Ord)
then obtain c where c: "c∈A" "j < c" "Card(c)"
by blast
hence jls: "j ≺ c"
by (simp add: lt_Card_imp_lesspoll)
{ assume eqp: "j ≈ ⋃A"
have "c ≲ ⋃A" using c
by (blast intro: subset_imp_lepoll)
also have "... ≈ j" by (rule eqpoll_sym [OF eqp])
also have "... ≺ c" by (rule jls)
finally have "c ≺ c" .
hence False
by auto
} thus "¬ j ≈ ⋃A" by blast
qed
lemma Card_UN: "(!!x. x ∈ A ==> Card(K(x))) ==> Card(⋃x∈A. K(x))"
by blast
lemma Card_OUN [simp,intro,TC]:
"(!!x. x ∈ A ==> Card(K(x))) ==> Card(⋃x<A. K(x))"
by (auto simp add: OUnion_def Card_0)
lemma in_Card_imp_lesspoll: "[| Card(K); b ∈ K |] ==> b ≺ 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 ≈ B+A"
proof (unfold eqpoll_def, rule exI)
show "(λz∈A+B. case(Inr,Inl,z)) ∈ bij(A+B, B+A)"
by (auto intro: lam_bijective [where d = "case(Inr,Inl)"])
qed
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
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 ⊕ j) ⊕ k = i ⊕ (j ⊕ k)"
proof (unfold cadd_def, rule cardinal_cong)
have "|i + j| + k ≈ (i + j) + k"
by (blast intro: sum_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j)
also have "... ≈ i + (j + k)"
by (rule sum_assoc_eqpoll)
also have "... ≈ 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 ≈ i + |j + k|" .
qed
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 ≲ A+B"
proof (unfold lepoll_def, rule exI)
show "(λx∈A. Inl (x)) ∈ inj(A, A + B)"
by (simp add: inj_def)
qed
lemma cadd_le_self:
assumes K: "Card(K)" and L: "Ord(L)" shows "K ≤ (K ⊕ L)"
proof (unfold cadd_def)
have "K ≤ |K|"
by (rule Card_cardinal_le [OF K])
moreover have "|K| ≤ |K + L|" using K L
by (blast intro: well_ord_lepoll_imp_Card_le sum_lepoll_self
well_ord_radd well_ord_Memrel Card_is_Ord)
ultimately show "K ≤ |K + L|"
by (blast intro: le_trans)
qed
subsubsection‹Monotonicity of addition›
lemma sum_lepoll_mono:
"[| A ≲ C; B ≲ D |] ==> A + B ≲ C + D"
apply (unfold lepoll_def)
apply (elim exE)
apply (rule_tac x = "λz∈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' ≤ K; L' ≤ L |] ==> (K' ⊕ L') ≤ (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
lemma cadd_succ_lemma:
assumes "Ord(m)" "Ord(n)" shows "succ(m) ⊕ n = |succ(m ⊕ n)|"
proof (unfold cadd_def)
have [intro]: "m + n ≈ |m + n|" using assms
by (blast intro: eqpoll_sym well_ord_cardinal_eqpoll well_ord_radd well_ord_Memrel)
have "|succ(m) + n| = |succ(m + n)|"
by (rule sum_succ_eqpoll [THEN cardinal_cong])
also have "... = |succ(|m + n|)|"
by (blast intro: succ_eqpoll_cong cardinal_cong)
finally show "|succ(m) + n| = |succ(|m + n|)|" .
qed
lemma nat_cadd_eq_add:
assumes m: "m ∈ nat" and [simp]: "n ∈ nat" shows"m ⊕ n = m #+ n"
using m
proof (induct m)
case 0 thus ?case by (simp add: nat_into_Card cadd_0)
next
case (succ m) thus ?case by (simp add: cadd_succ_lemma nat_into_Card Card_cardinal_eq)
qed
subsection‹Cardinal multiplication›
subsubsection‹Cardinal multiplication is commutative›
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
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 ⊗ j) ⊗ k = i ⊗ (j ⊗ k)"
proof (unfold cmult_def, rule cardinal_cong)
have "|i * j| * k ≈ (i * j) * k"
by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_rmult i j)
also have "... ≈ i * (j * k)"
by (rule prod_assoc_eqpoll)
also have "... ≈ 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 ≈ i * |j * k|" .
qed
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:
assumes i: "well_ord(i,ri)" and j: "well_ord(j,rj)" and k: "well_ord(k,rk)"
shows "(i ⊕ j) ⊗ k = (i ⊗ k) ⊕ (j ⊗ k)"
proof (unfold cadd_def cmult_def, rule cardinal_cong)
have "|i + j| * k ≈ (i + j) * k"
by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_refl well_ord_radd i j)
also have "... ≈ i * k + j * k"
by (rule sum_prod_distrib_eqpoll)
also have "... ≈ |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 ≈ |i * k| + |j * k|" .
qed
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 ≲ A*A"
apply (unfold lepoll_def inj_def)
apply (rule_tac x = "λx∈A. <x,x>" in exI, simp)
done
lemma cmult_square_le: "Card(K) ==> K ≤ 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 ≲ A*B"
apply (unfold lepoll_def inj_def)
apply (rule_tac x = "λx∈A. <x,b>" in exI, simp)
done
lemma cmult_le_self:
"[| Card(K); Ord(L); 0<L |] ==> K ≤ (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 ≲ C; B ≲ D |] ==> A * B ≲ 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' ≤ K; L' ≤ L |] ==> (K' ⊗ L') ≤ (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
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:
assumes C: "2 ≲ C" shows "B+B ≲ C*B"
proof -
have "B+B ≲ 2*B"
by (simp add: sum_eq_2_times)
also have "... ≲ C*B"
by (blast intro: prod_lepoll_mono lepoll_refl C)
finally show "B+B ≲ C*B" .
qed
lemma lepoll_imp_sum_lepoll_prod: "[| A ≲ B; 2 ≲ A |] ==> A+B ≲ A*B"
by (blast intro: sum_lepoll_mono sum_lepoll_prod lepoll_trans lepoll_refl)
subsection‹Infinite Cardinals are Limit Ordinals›
lemma nat_cons_lepoll: "nat ≲ A ==> cons(u,A) ≲ A"
apply (unfold lepoll_def)
apply (erule exE)
apply (rule_tac x =
"λz∈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 ≲ A ==> cons(u,A) ≈ A"
apply (erule nat_cons_lepoll [THEN eqpollI])
apply (rule subset_consI [THEN subset_imp_lepoll])
done
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 ∪ L)"
apply (unfold InfCard_def)
apply (simp add: Card_Un Un_upper1_le [THEN [2] le_trans] Card_is_Ord)
done
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
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 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 "(λ⟨x,y⟩∈K × K. ⟨x ∪ y, x, y⟩) ∈ inj(K × K, K × K × K)" using K
by (force simp add: inj_def intro: lam_type Un_least_lt [THEN ltD] ltI)
next
show "well_ord(K × K × K, rmult(K, Memrel(K), K × 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>> ∈ csquare_rel(K); x<K; y<K; z<K |] ==> x ≤ z & y ≤ 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 dest!: csquareD)
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
lemma csquare_or_eqI:
"[| x ≤ z; y ≤ 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 ∪ 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>‹⟨x,y⟩ ∈ K × K› has no more than \<^term>‹z × z› predecessors..." (page 29)›
lemma ordermap_csquare_le:
assumes K: "Limit(K)" and x: "x<K" and y: " y<K"
defines "z ≡ succ(x ∪ y)"
shows "|ordermap(K × K, csquare_rel(K)) ` ⟨x,y⟩| ≤ |succ(z)| ⊗ |succ(z)|"
proof (unfold cmult_def, rule well_ord_lepoll_imp_Card_le)
show "well_ord(|succ(z)| × |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 K z_def
by (blast intro: Un_least_lt Limit_has_succ)
hence oz: "Ord(z)" by (elim ltE)
have "ordermap(K × K, csquare_rel(K)) ` ⟨x,y⟩ ≲ ordermap(K × K, csquare_rel(K)) ` ⟨z,z⟩"
using z_def
by (blast intro: ordermap_z_lt leI le_imp_lepoll K x y)
also have "... ≈ Order.pred(K × K, ⟨z,z⟩, csquare_rel(K))"
proof (rule ordermap_eqpoll_pred)
show "well_ord(K × K, csquare_rel(K))" using K
by (rule Limit_is_Ord [THEN well_ord_csquare])
next
show "⟨z, z⟩ ∈ K × K" using zK
by (blast intro: ltD)
qed
also have "... ≲ succ(z) × succ(z)" using zK
by (rule pred_csquare_subset [THEN subset_imp_lepoll])
also have "... ≈ |succ(z)| × |succ(z)|" using oz
by (blast intro: prod_eqpoll_cong Ord_succ Ord_cardinal_eqpoll eqpoll_sym)
finally show "ordermap(K × K, csquare_rel(K)) ` ⟨x,y⟩ ≲ |succ(z)| × |succ(z)|" .
qed
text‹Kunen: "... so the order type is ‹≤› K"›
lemma ordertype_csquare_le:
assumes IK: "InfCard(K)" and eq: "⋀y. y∈K ⟹ InfCard(y) ⟹ y ⊗ y = y"
shows "ordertype(K*K, csquare_rel(K)) ≤ K"
proof -
have CK: "Card(K)" using IK by (rule InfCard_is_Card)
hence OK: "Ord(K)" by (rule Card_is_Ord)
moreover have "Ord(ordertype(K × K, csquare_rel(K)))" using OK
by (rule well_ord_csquare [THEN Ord_ordertype])
ultimately show ?thesis
proof (rule all_lt_imp_le)
fix i
assume i: "i < ordertype(K × K, csquare_rel(K))"
hence Oi: "Ord(i)" by (elim ltE)
obtain x y where x: "x ∈ K" and y: "y ∈ K"
and ieq: "i = ordermap(K × K, csquare_rel(K)) ` ⟨x,y⟩"
using i by (auto simp add: ordertype_unfold elim: ltE)
hence xy: "Ord(x)" "Ord(y)" "x < K" "y < K" using OK
by (blast intro: Ord_in_Ord ltI)+
hence ou: "Ord(x ∪ y)"
by (simp add: Ord_Un)
show "i < K"
proof (rule Card_lt_imp_lt [OF _ Oi CK])
have "|i| ≤ |succ(succ(x ∪ y))| ⊗ |succ(succ(x ∪ y))|" using IK xy
by (auto simp add: ieq intro: InfCard_is_Limit [THEN ordermap_csquare_le])
moreover have "|succ(succ(x ∪ y))| ⊗ |succ(succ(x ∪ y))| < K"
proof (cases rule: Ord_linear2 [OF ou Ord_nat])
assume "x ∪ y < nat"
hence "|succ(succ(x ∪ y))| ⊗ |succ(succ(x ∪ y))| ∈ nat"
by (simp add: lt_def nat_cmult_eq_mult nat_succI mult_type
nat_into_Card [THEN Card_cardinal_eq] Ord_nat)
also have "... ⊆ K" using IK
by (simp add: InfCard_def le_imp_subset)
finally show "|succ(succ(x ∪ y))| ⊗ |succ(succ(x ∪ y))| < K"
by (simp add: ltI OK)
next
assume natxy: "nat ≤ x ∪ y"
hence seq: "|succ(succ(x ∪ y))| = |x ∪ y|" using xy
by (simp add: le_imp_subset nat_succ_eqpoll [THEN cardinal_cong] le_succ_iff)
also have "... < K" using xy
by (simp add: Un_least_lt Ord_cardinal_le [THEN lt_trans1])
finally have "|succ(succ(x ∪ y))| < K" .
moreover have "InfCard(|succ(succ(x ∪ y))|)" using xy natxy
by (simp add: seq InfCard_def Card_cardinal nat_le_cardinal)
ultimately show ?thesis by (simp add: eq ltD)
qed
ultimately show "|i| < K" by (blast intro: lt_trans1)
qed
qed
qed
lemma InfCard_csquare_eq:
assumes IK: "InfCard(K)" shows "InfCard(K) ==> K ⊗ K = K"
proof -
have OK: "Ord(K)" using IK by (simp add: Card_is_Ord InfCard_is_Card)
show "InfCard(K) ==> K ⊗ K = K" using OK
proof (induct rule: trans_induct)
case (step i)
show "i ⊗ i = i"
proof (rule le_anti_sym)
have "|i × i| = |ordertype(i × i, csquare_rel(i))|"
by (rule cardinal_cong,
simp add: step.hyps well_ord_csquare [THEN ordermap_bij, THEN bij_imp_eqpoll])
hence "i ⊗ i ≤ ordertype(i × i, csquare_rel(i))"
by (simp add: step.hyps cmult_def Ord_cardinal_le well_ord_csquare [THEN Ord_ordertype])
moreover
have "ordertype(i × i, csquare_rel(i)) ≤ i" using step
by (simp add: ordertype_csquare_le)
ultimately show "i ⊗ i ≤ i" by (rule le_trans)
next
show "i ≤ i ⊗ i" using step
by (blast intro: cmult_square_le InfCard_is_Card)
qed
qed
qed
lemma well_ord_InfCard_square_eq:
assumes r: "well_ord(A,r)" and I: "InfCard(|A|)" shows "A × A ≈ A"
proof -
have "A × A ≈ |A| × |A|"
by (blast intro: prod_eqpoll_cong well_ord_cardinal_eqpoll eqpoll_sym r)
also have "... ≈ A"
proof (rule well_ord_cardinal_eqE [OF _ r])
show "well_ord(|A| × |A|, rmult(|A|, Memrel(|A|), |A|, Memrel(|A|)))"
by (blast intro: Ord_cardinal well_ord_rmult well_ord_Memrel r)
next
show "||A| × |A|| = |A|" using InfCard_csquare_eq I
by (simp add: cmult_def)
qed
finally show ?thesis .
qed
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: "[| Card(i); ~ Finite(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 ≤ 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
lemma InfCard_cmult_eq: "[| InfCard(K); InfCard(L) |] ==> K ⊗ L = K ∪ 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
lemma InfCard_le_cadd_eq: "[| InfCard(K); L ≤ 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 ∪ 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
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
lemma jump_cardinal_iff:
"i ∈ jump_cardinal(K) ⟷
(∃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
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
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
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) ≤ 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| ≤ 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' ≤ 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 Finite_imp_cardinal_cons [simp]:
assumes FA: "Finite(A)" and a: "a∉A" shows "|cons(a,A)| = succ(|A|)"
proof -
{ fix X
have "Finite(X) ==> a ∉ X ⟹ cons(a,X) ≲ X ⟹ False"
proof (induct X rule: Finite_induct)
case 0 thus False by (simp add: lepoll_0_iff)
next
case (cons x Y)
hence "cons(x, cons(a, Y)) ≲ cons(x, Y)" by (simp add: cons_commute)
hence "cons(a, Y) ≲ Y" using cons by (blast dest: cons_lepoll_consD)
thus False using cons by auto
qed
}
hence [simp]: "~ cons(a,A) ≲ A" using a FA by auto
have [simp]: "|A| ≈ A" using Finite_imp_well_ord [OF FA]
by (blast intro: well_ord_cardinal_eqpoll)
have "(μ i. i ≈ cons(a, A)) = succ(|A|)"
proof (rule Least_equality [OF _ _ notI])
show "succ(|A|) ≈ cons(a, A)"
by (simp add: succ_def cons_eqpoll_cong mem_not_refl a)
next
show "Ord(succ(|A|))" by simp
next
fix i
assume i: "i ≤ |A|" "i ≈ cons(a, A)"
have "cons(a, A) ≈ i" by (rule eqpoll_sym) (rule i)
also have "... ≲ |A|" by (rule le_imp_lepoll) (rule i)
also have "... ≈ A" by simp
finally have "cons(a, A) ≲ A" .
thus False by simp
qed
thus ?thesis by (simp add: cardinal_def)
qed
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"
proof (induct rule: Finite_induct)
case 0 thus ?case by (simp add: cardinal_0)
next
case (cons x A) thus ?case by (simp add: Finite_imp_cardinal_cons)
qed
lemma card_Un_Int:
"[|Finite(A); Finite(B)|] ==> |A| #+ |B| = |A ∪ B| #+ |A ∩ 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 ∩ B = 0|] ==> |A ∪ B| = |A| #+ |B|"
by (simp add: Finite_Un card_Un_Int)
lemma card_partition:
assumes FC: "Finite(C)"
shows
"Finite (⋃ C) ⟹
(∀c∈C. |c| = k) ⟹
(∀c1 ∈ C. ∀c2 ∈ C. c1 ≠ c2 ⟶ c1 ∩ c2 = 0) ⟹
k #* |C| = |⋃ C|"
using FC
proof (induct rule: Finite_induct)
case 0 thus ?case by simp
next
case (cons x B)
hence "x ∩ ⋃B = 0" by auto
thus ?case using cons
by (auto simp add: card_Un_disjoint)
qed
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:
assumes m: "m ∈ nat" and n: "n ∈ nat" shows "m + n ≈ m #+ n"
proof -
have "m + n ≈ |m+n|" using m n
by (blast intro: nat_implies_well_ord well_ord_radd well_ord_cardinal_eqpoll eqpoll_sym)
also have "... = m #+ n" using m n
by (simp add: nat_cadd_eq_add [symmetric] cadd_def)
finally show ?thesis .
qed
lemma Ord_subset_natD [rule_format]: "Ord(i) ==> i ⊆ nat ⟹ i ∈ nat | i=nat"
proof (induct i rule: trans_induct3)
case 0 thus ?case by auto
next
case (succ i) thus ?case by auto
next
case (limit l) thus ?case
by (blast dest: nat_le_Limit le_imp_subset)
qed
lemma Ord_nat_subset_into_Card: "[| Ord(i); i ⊆ nat |] ==> Card(i)"
by (blast dest: Ord_subset_natD intro: Card_nat nat_into_Card)
end