(* Title: ZF/Zorn.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1994 University of Cambridge
*)
section\<open>Zorn's Lemma\<close>
theory Zorn imports OrderArith AC Inductive begin
text\<open>Based upon the unpublished article ``Towards the Mechanization of the
Proofs of Some Classical Theorems of Set Theory,'' by Abrial and Laffitte.\<close>
definition
Subset_rel :: "i=>i" where
"Subset_rel(A) == {z \ A*A . \x y. z= & x<=y & x\y}"
definition
chain :: "i=>i" where
"chain(A) == {F \ Pow(A). \X\F. \Y\F. X<=Y | Y<=X}"
definition
super :: "[i,i]=>i" where
"super(A,c) == {d \ chain(A). c<=d & c\d}"
definition
maxchain :: "i=>i" where
"maxchain(A) == {c \ chain(A). super(A,c)=0}"
definition
increasing :: "i=>i" where
"increasing(A) == {f \ Pow(A)->Pow(A). \x. x<=A \ x<=f`x}"
text\<open>Lemma for the inductive definition below\<close>
lemma Union_in_Pow: "Y \ Pow(Pow(A)) ==> \(Y) \ Pow(A)"
by blast
text\<open>We could make the inductive definition conditional on
\<^term>\<open>next \<in> increasing(S)\<close>
but instead we make this a side-condition of an introduction rule. Thus
the induction rule lets us assume that condition! Many inductive proofs
are therefore unconditional.\<close>
consts
"TFin" :: "[i,i]=>i"
inductive
domains "TFin(S,next)" \<subseteq> "Pow(S)"
intros
nextI: "[| x \ TFin(S,next); next \ increasing(S) |]
==> next`x \<in> TFin(S,next)"
Pow_UnionI: "Y \ Pow(TFin(S,next)) ==> \(Y) \ TFin(S,next)"
monos Pow_mono
con_defs increasing_def
type_intros CollectD1 [THEN apply_funtype] Union_in_Pow
subsection\<open>Mathematical Preamble\<close>
lemma Union_lemma0: "(\x\C. x<=A | B<=x) ==> \(C)<=A | B<=\(C)"
by blast
lemma Inter_lemma0:
"[| c \ C; \x\C. A<=x | x<=B |] ==> A \ \(C) | \(C) \ B"
by blast
subsection\<open>The Transfinite Construction\<close>
lemma increasingD1: "f \ increasing(A) ==> f \ Pow(A)->Pow(A)"
apply (unfold increasing_def)
apply (erule CollectD1)
done
lemma increasingD2: "[| f \ increasing(A); x<=A |] ==> x \ f`x"
by (unfold increasing_def, blast)
lemmas TFin_UnionI = PowI [THEN TFin.Pow_UnionI]
lemmas TFin_is_subset = TFin.dom_subset [THEN subsetD, THEN PowD]
text\<open>Structural induction on \<^term>\<open>TFin(S,next)\<close>\<close>
lemma TFin_induct:
"[| n \ TFin(S,next);
!!x. [| x \<in> TFin(S,next); P(x); next \<in> increasing(S) |] ==> P(next`x);
!!Y. [| Y \<subseteq> TFin(S,next); \<forall>y\<in>Y. P(y) |] ==> P(\<Union>(Y))
|] ==> P(n)"
by (erule TFin.induct, blast+)
subsection\<open>Some Properties of the Transfinite Construction\<close>
lemmas increasing_trans = subset_trans [OF _ increasingD2,
OF _ _ TFin_is_subset]
text\<open>Lemma 1 of section 3.1\<close>
lemma TFin_linear_lemma1:
"[| n \ TFin(S,next); m \ TFin(S,next);
\<forall>x \<in> TFin(S,next) . x<=m \<longrightarrow> x=m | next`x<=m |]
==> n<=m | next`m<=n"
apply (erule TFin_induct)
apply (erule_tac [2] Union_lemma0) (*or just Blast_tac*)
(*downgrade subsetI from intro! to intro*)
apply (blast dest: increasing_trans)
done
text\<open>Lemma 2 of section 3.2. Interesting in its own right!
Requires \<^term>\<open>next \<in> increasing(S)\<close> in the second induction step.\<close>
lemma TFin_linear_lemma2:
"[| m \ TFin(S,next); next \ increasing(S) |]
==> \<forall>n \<in> TFin(S,next). n<=m \<longrightarrow> n=m | next`n \<subseteq> m"
apply (erule TFin_induct)
apply (rule impI [THEN ballI])
txt\<open>case split using \<open>TFin_linear_lemma1\<close>\<close>
apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
assumption+)
apply (blast del: subsetI
intro: increasing_trans subsetI, blast)
txt\<open>second induction step\<close>
apply (rule impI [THEN ballI])
apply (rule Union_lemma0 [THEN disjE])
apply (erule_tac [3] disjI2)
prefer 2 apply blast
apply (rule ballI)
apply (drule bspec, assumption)
apply (drule subsetD, assumption)
apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE],
assumption+, blast)
apply (erule increasingD2 [THEN subset_trans, THEN disjI1])
apply (blast dest: TFin_is_subset)+
done
text\<open>a more convenient form for Lemma 2\<close>
lemma TFin_subsetD:
"[| n<=m; m \ TFin(S,next); n \ TFin(S,next); next \ increasing(S) |]
==> n=m | next`n \<subseteq> m"
by (blast dest: TFin_linear_lemma2 [rule_format])
text\<open>Consequences from section 3.3 -- Property 3.2, the ordering is total\<close>
lemma TFin_subset_linear:
"[| m \ TFin(S,next); n \ TFin(S,next); next \ increasing(S) |]
==> n \<subseteq> m | m<=n"
apply (rule disjE)
apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
apply (assumption+, erule disjI2)
apply (blast del: subsetI
intro: subsetI increasingD2 [THEN subset_trans] TFin_is_subset)
done
text\<open>Lemma 3 of section 3.3\<close>
lemma equal_next_upper:
"[| n \ TFin(S,next); m \ TFin(S,next); m = next`m |] ==> n \ m"
apply (erule TFin_induct)
apply (drule TFin_subsetD)
apply (assumption+, force, blast)
done
text\<open>Property 3.3 of section 3.3\<close>
lemma equal_next_Union:
"[| m \ TFin(S,next); next \ increasing(S) |]
==> m = next`m <-> m = \<Union>(TFin(S,next))"
apply (rule iffI)
apply (rule Union_upper [THEN equalityI])
apply (rule_tac [2] equal_next_upper [THEN Union_least])
apply (assumption+)
apply (erule ssubst)
apply (rule increasingD2 [THEN equalityI], assumption)
apply (blast del: subsetI
intro: subsetI TFin_UnionI TFin.nextI TFin_is_subset)+
done
subsection\<open>Hausdorff's Theorem: Every Set Contains a Maximal Chain\<close>
text\<open>NOTE: We assume the partial ordering is \<open>\<subseteq>\<close>, the subset
relation!\<close>
text\<open>* Defining the "next" operation for Hausdorff's Theorem *\<close>
lemma chain_subset_Pow: "chain(A) \ Pow(A)"
apply (unfold chain_def)
apply (rule Collect_subset)
done
lemma super_subset_chain: "super(A,c) \ chain(A)"
apply (unfold super_def)
apply (rule Collect_subset)
done
lemma maxchain_subset_chain: "maxchain(A) \ chain(A)"
apply (unfold maxchain_def)
apply (rule Collect_subset)
done
lemma choice_super:
"[| ch \ (\X \ Pow(chain(S)) - {0}. X); X \ chain(S); X \ maxchain(S) |]
==> ch ` super(S,X) \<in> super(S,X)"
apply (erule apply_type)
apply (unfold super_def maxchain_def, blast)
done
lemma choice_not_equals:
"[| ch \ (\X \ Pow(chain(S)) - {0}. X); X \ chain(S); X \ maxchain(S) |]
==> ch ` super(S,X) \<noteq> X"
apply (rule notI)
apply (drule choice_super, assumption, assumption)
apply (simp add: super_def)
done
text\<open>This justifies Definition 4.4\<close>
lemma Hausdorff_next_exists:
"ch \ (\X \ Pow(chain(S))-{0}. X) ==>
\<exists>next \<in> increasing(S). \<forall>X \<in> Pow(S).
next`X = if(X \<in> chain(S)-maxchain(S), ch`super(S,X), X)"
apply (rule_tac x="\X\Pow(S).
if X \<in> chain(S) - maxchain(S) then ch ` super(S, X) else X"
in bexI)
apply force
apply (unfold increasing_def)
apply (rule CollectI)
apply (rule lam_type)
apply (simp (no_asm_simp))
apply (blast dest: super_subset_chain [THEN subsetD]
chain_subset_Pow [THEN subsetD] choice_super)
txt\<open>Now, verify that it increases\<close>
apply (simp (no_asm_simp) add: Pow_iff subset_refl)
apply safe
apply (drule choice_super)
apply (assumption+)
apply (simp add: super_def, blast)
done
text\<open>Lemma 4\<close>
lemma TFin_chain_lemma4:
"[| c \ TFin(S,next);
ch \<in> (\<Prod>X \<in> Pow(chain(S))-{0}. X);
next \<in> increasing(S);
\<forall>X \<in> Pow(S). next`X =
if(X \<in> chain(S)-maxchain(S), ch`super(S,X), X) |]
==> c \<in> chain(S)"
apply (erule TFin_induct)
apply (simp (no_asm_simp) add: chain_subset_Pow [THEN subsetD, THEN PowD]
choice_super [THEN super_subset_chain [THEN subsetD]])
apply (unfold chain_def)
apply (rule CollectI, blast, safe)
apply (rule_tac m1=B and n1=Ba in TFin_subset_linear [THEN disjE], fast+)
txt\<open>\<open>Blast_tac's\<close> slow\<close>
done
theorem Hausdorff: "\c. c \ maxchain(S)"
apply (rule AC_Pi_Pow [THEN exE])
apply (rule Hausdorff_next_exists [THEN bexE], assumption)
apply (rename_tac ch "next")
apply (subgoal_tac "\(TFin (S,next)) \ chain (S) ")
prefer 2
apply (blast intro!: TFin_chain_lemma4 subset_refl [THEN TFin_UnionI])
apply (rule_tac x = "\(TFin (S,next))" in exI)
apply (rule classical)
apply (subgoal_tac "next ` Union(TFin (S,next)) = \(TFin (S,next))")
apply (rule_tac [2] equal_next_Union [THEN iffD2, symmetric])
apply (rule_tac [2] subset_refl [THEN TFin_UnionI])
prefer 2 apply assumption
apply (rule_tac [2] refl)
apply (simp add: subset_refl [THEN TFin_UnionI,
THEN TFin.dom_subset [THEN subsetD, THEN PowD]])
apply (erule choice_not_equals [THEN notE])
apply (assumption+)
done
subsection\<open>Zorn's Lemma: If All Chains in S Have Upper Bounds In S,
then S contains a Maximal Element\<close>
text\<open>Used in the proof of Zorn's Lemma\<close>
lemma chain_extend:
"[| c \ chain(A); z \ A; \x \ c. x<=z |] ==> cons(z,c) \ chain(A)"
by (unfold chain_def, blast)
lemma Zorn: "\c \ chain(S). \(c) \ S ==> \y \ S. \z \ S. y<=z \ y=z"
apply (rule Hausdorff [THEN exE])
apply (simp add: maxchain_def)
apply (rename_tac c)
apply (rule_tac x = "\(c)" in bexI)
prefer 2 apply blast
apply safe
apply (rename_tac z)
apply (rule classical)
apply (subgoal_tac "cons (z,c) \ super (S,c) ")
apply (blast elim: equalityE)
apply (unfold super_def, safe)
apply (fast elim: chain_extend)
apply (fast elim: equalityE)
done
text \<open>Alternative version of Zorn's Lemma\<close>
theorem Zorn2:
"\c \ chain(S). \y \ S. \x \ c. x \ y ==> \y \ S. \z \ S. y<=z \ y=z"
apply (cut_tac Hausdorff maxchain_subset_chain)
apply (erule exE)
apply (drule subsetD, assumption)
apply (drule bspec, assumption, erule bexE)
apply (rule_tac x = y in bexI)
prefer 2 apply assumption
apply clarify
apply rule apply assumption
apply rule
apply (rule ccontr)
apply (frule_tac z=z in chain_extend)
apply (assumption, blast)
apply (unfold maxchain_def super_def)
apply (blast elim!: equalityCE)
done
subsection\<open>Zermelo's Theorem: Every Set can be Well-Ordered\<close>
text\<open>Lemma 5\<close>
lemma TFin_well_lemma5:
"[| n \ TFin(S,next); Z \ TFin(S,next); z:Z; ~ \(Z) \ Z |]
==> \<forall>m \<in> Z. n \<subseteq> m"
apply (erule TFin_induct)
prefer 2 apply blast txt\<open>second induction step is easy\<close>
apply (rule ballI)
apply (rule bspec [THEN TFin_subsetD, THEN disjE], auto)
apply (subgoal_tac "m = \(Z) ")
apply blast+
done
text\<open>Well-ordering of \<^term>\<open>TFin(S,next)\<close>\<close>
lemma well_ord_TFin_lemma: "[| Z \ TFin(S,next); z \ Z |] ==> \(Z) \ Z"
apply (rule classical)
apply (subgoal_tac "Z = {\(TFin (S,next))}")
apply (simp (no_asm_simp) add: Inter_singleton)
apply (erule equal_singleton)
apply (rule Union_upper [THEN equalityI])
apply (rule_tac [2] subset_refl [THEN TFin_UnionI, THEN TFin_well_lemma5, THEN bspec], blast+)
done
text\<open>This theorem just packages the previous result\<close>
lemma well_ord_TFin:
"next \ increasing(S)
==> well_ord(TFin(S,next), Subset_rel(TFin(S,next)))"
apply (rule well_ordI)
apply (unfold Subset_rel_def linear_def)
txt\<open>Prove the well-foundedness goal\<close>
apply (rule wf_onI)
apply (frule well_ord_TFin_lemma, assumption)
apply (drule_tac x = "\(Z) " in bspec, assumption)
apply blast
txt\<open>Now prove the linearity goal\<close>
apply (intro ballI)
apply (case_tac "x=y")
apply blast
txt\<open>The \<^term>\<open>x\<noteq>y\<close> case remains\<close>
apply (rule_tac n1=x and m1=y in TFin_subset_linear [THEN disjE],
assumption+, blast+)
done
text\<open>* Defining the "next" operation for Zermelo's Theorem *\<close>
lemma choice_Diff:
"[| ch \ (\X \ Pow(S) - {0}. X); X \ S; X\S |] ==> ch ` (S-X) \ S-X"
apply (erule apply_type)
apply (blast elim!: equalityE)
done
text\<open>This justifies Definition 6.1\<close>
lemma Zermelo_next_exists:
"ch \ (\X \ Pow(S)-{0}. X) ==>
\<exists>next \<in> increasing(S). \<forall>X \<in> Pow(S).
next`X = (if X=S then S else cons(ch`(S-X), X))"
apply (rule_tac x="\X\Pow(S). if X=S then S else cons(ch`(S-X), X)"
in bexI)
apply force
apply (unfold increasing_def)
apply (rule CollectI)
apply (rule lam_type)
txt\<open>Type checking is surprisingly hard!\<close>
apply (simp (no_asm_simp) add: Pow_iff cons_subset_iff subset_refl)
apply (blast intro!: choice_Diff [THEN DiffD1])
txt\<open>Verify that it increases\<close>
apply (intro allI impI)
apply (simp add: Pow_iff subset_consI subset_refl)
done
text\<open>The construction of the injection\<close>
lemma choice_imp_injection:
"[| ch \ (\X \ Pow(S)-{0}. X);
next \<in> increasing(S);
\<forall>X \<in> Pow(S). next`X = if(X=S, S, cons(ch`(S-X), X)) |]
==> (\<lambda> x \<in> S. \<Union>({y \<in> TFin(S,next). x \<notin> y}))
\<in> inj(S, TFin(S,next) - {S})"
apply (rule_tac d = "%y. ch` (S-y) " in lam_injective)
apply (rule DiffI)
apply (rule Collect_subset [THEN TFin_UnionI])
apply (blast intro!: Collect_subset [THEN TFin_UnionI] elim: equalityE)
apply (subgoal_tac "x \ \({y \ TFin (S,next) . x \ y}) ")
prefer 2 apply (blast elim: equalityE)
apply (subgoal_tac "\({y \ TFin (S,next) . x \ y}) \ S")
prefer 2 apply (blast elim: equalityE)
txt\<open>For proving \<open>x \<in> next`\<Union>(...)\<close>.
Abrial and Laffitte's justification appears to be faulty.\
apply (subgoal_tac "~ next ` Union({y \ TFin (S,next) . x \ y})
\<subseteq> \<Union>({y \<in> TFin (S,next) . x \<notin> y}) ")
prefer 2
apply (simp del: Union_iff
add: Collect_subset [THEN TFin_UnionI, THEN TFin_is_subset]
Pow_iff cons_subset_iff subset_refl choice_Diff [THEN DiffD2])
apply (subgoal_tac "x \ next ` Union({y \ TFin (S,next) . x \ y}) ")
prefer 2
apply (blast intro!: Collect_subset [THEN TFin_UnionI] TFin.nextI)
txt\<open>End of the lemmas!\<close>
apply (simp add: Collect_subset [THEN TFin_UnionI, THEN TFin_is_subset])
done
text\<open>The wellordering theorem\<close>
theorem AC_well_ord: "\r. well_ord(S,r)"
apply (rule AC_Pi_Pow [THEN exE])
apply (rule Zermelo_next_exists [THEN bexE], assumption)
apply (rule exI)
apply (rule well_ord_rvimage)
apply (erule_tac [2] well_ord_TFin)
apply (rule choice_imp_injection [THEN inj_weaken_type], blast+)
done
subsection \<open>Zorn's Lemma for Partial Orders\<close>
text \<open>Reimported from HOL by Clemens Ballarin.\<close>
definition Chain :: "i => i" where
"Chain(r) = {A \ Pow(field(r)). \a\A. \b\A. \ r | \ r}"
lemma mono_Chain:
"r \ s ==> Chain(r) \ Chain(s)"
unfolding Chain_def
by blast
theorem Zorn_po:
assumes po: "Partial_order(r)"
and u: "\C\Chain(r). \u\field(r). \a\C. \ r"
shows "\m\field(r). \a\field(r). \ r \ a = m"
proof -
have "Preorder(r)" using po by (simp add: partial_order_on_def)
\<comment> \<open>Mirror r in the set of subsets below (wrt r) elements of A (?).\<close>
let ?B = "\x\field(r). r -`` {x}" let ?S = "?B `` field(r)"
have "\C\chain(?S). \U\?S. \A\C. A \ U"
proof (clarsimp simp: chain_def Subset_rel_def bex_image_simp)
fix C
assume 1: "C \ ?S" and 2: "\A\C. \B\C. A \ B | B \ A"
let ?A = "{x \ field(r). \M\C. M = ?B`x}"
have "C = ?B `` ?A" using 1
apply (auto simp: image_def)
apply rule
apply rule
apply (drule subsetD) apply assumption
apply (erule CollectE)
apply rule apply assumption
apply (erule bexE)
apply rule prefer 2 apply assumption
apply rule
apply (erule lamE) apply simp
apply assumption
apply (thin_tac "C \ X" for X)
apply (fast elim: lamE)
done
have "?A \ Chain(r)"
proof (simp add: Chain_def subsetI, intro conjI ballI impI)
fix a b
assume "a \ field(r)" "r -`` {a} \ C" "b \ field(r)" "r -`` {b} \ C"
hence "r -`` {a} \ r -`` {b} | r -`` {b} \ r -`` {a}" using 2 by auto
then show " \ r | \ r"
using \<open>Preorder(r)\<close> \<open>a \<in> field(r)\<close> \<open>b \<in> field(r)\<close>
by (simp add: subset_vimage1_vimage1_iff)
qed
then obtain u where uA: "u \ field(r)" "\a\?A. \ r"
using u
apply auto
apply (drule bspec) apply assumption
apply auto
done
have "\A\C. A \ r -`` {u}"
proof (auto intro!: vimageI)
fix a B
assume aB: "B \ C" "a \ B"
with 1 obtain x where "x \ field(r)" "B = r -`` {x}"
apply -
apply (drule subsetD) apply assumption
apply (erule imageE)
apply (erule lamE)
apply simp
done
then show " \ r" using uA aB \Preorder(r)\
by (auto simp: preorder_on_def refl_def) (blast dest: trans_onD)+
qed
then show "\U\field(r). \A\C. A \ r -`` {U}"
using \<open>u \<in> field(r)\<close> ..
qed
from Zorn2 [OF this]
obtain m B where "m \ field(r)" "B = r -`` {m}"
"\x\field(r). B \ r -`` {x} \ B = r -`` {x}"
by (auto elim!: lamE simp: ball_image_simp)
then have "\a\field(r). \ r \ a = m"
using po \<open>Preorder(r)\<close> \<open>m \<in> field(r)\<close>
by (auto simp: subset_vimage1_vimage1_iff Partial_order_eq_vimage1_vimage1_iff)
then show ?thesis using \<open>m \<in> field(r)\<close> by blast
qed
end
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