(* Title : HOL/Library/Zorn.thy ID : $Id: Zorn.thy,v 1.11 2007/07/11 09:28:14 berghofe Exp $ Author : Jacques D. Fleuriot Description : Zorn's Lemma -- see Larry Paulson's Zorn.thy in ZF *) header {* Zorn's Lemma *} theory Zorn imports Main begin text{* The lemma and section numbers refer to an unpublished article \cite{Abrial-Laffitte}. *} definition chain :: "'a set set => 'a set set set" where "chain S = {F. F ⊆ S & (∀x ∈ F. ∀y ∈ F. x ⊆ y | y ⊆ x)}" definition super :: "['a set set,'a set set] => 'a set set set" where "super S c = {d. d ∈ chain S & c ⊂ d}" definition maxchain :: "'a set set => 'a set set set" where "maxchain S = {c. c ∈ chain S & super S c = {}}" definition succ :: "['a set set,'a set set] => 'a set set" where "succ S c = (if c ∉ chain S | c ∈ maxchain S then c else SOME c'. c' ∈ super S c)" inductive_set TFin :: "'a set set => 'a set set set" for S :: "'a set set" where succI: "x ∈ TFin S ==> succ S x ∈ TFin S" | Pow_UnionI: "Y ∈ Pow(TFin S) ==> Union(Y) ∈ TFin S" monos Pow_mono subsection{*Mathematical Preamble*} lemma Union_lemma0: "(∀x ∈ C. x ⊆ A | B ⊆ x) ==> Union(C) ⊆ A | B ⊆ Union(C)" by blast text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*} lemma Abrial_axiom1: "x ⊆ succ S x" apply (unfold succ_def) apply (rule split_if [THEN iffD2]) apply (auto simp add: super_def maxchain_def psubset_def) apply (rule contrapos_np, assumption) apply (rule someI2, blast+) done lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI] lemma TFin_induct: "[| n ∈ TFin S; !!x. [| x ∈ TFin S; P(x) |] ==> P(succ S x); !!Y. [| Y ⊆ TFin S; Ball Y P |] ==> P(Union Y) |] ==> P(n)" apply (induct set: TFin) apply blast+ done lemma succ_trans: "x ⊆ y ==> x ⊆ succ S y" apply (erule subset_trans) apply (rule Abrial_axiom1) done text{*Lemma 1 of section 3.1*} lemma TFin_linear_lemma1: "[| n ∈ TFin S; m ∈ TFin S; ∀x ∈ TFin S. x ⊆ m --> x = m | succ S x ⊆ m |] ==> n ⊆ m | succ S m ⊆ n" apply (erule TFin_induct) apply (erule_tac [2] Union_lemma0) apply (blast del: subsetI intro: succ_trans) done text{* Lemma 2 of section 3.2 *} lemma TFin_linear_lemma2: "m ∈ TFin S ==> ∀n ∈ TFin S. n ⊆ m --> n=m | succ S n ⊆ m" apply (erule TFin_induct) apply (rule impI [THEN ballI]) txt{*case split using @{text TFin_linear_lemma1}*} apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE], assumption+) apply (drule_tac x = n in bspec, assumption) apply (blast del: subsetI intro: succ_trans, blast) txt{*second induction step*} apply (rule impI [THEN ballI]) apply (rule Union_lemma0 [THEN disjE]) apply (rule_tac [3] disjI2) prefer 2 apply blast apply (rule ballI) apply (rule_tac n1 = n and m1 = x in TFin_linear_lemma1 [THEN disjE], assumption+, auto) apply (blast intro!: Abrial_axiom1 [THEN subsetD]) done text{*Re-ordering the premises of Lemma 2*} lemma TFin_subsetD: "[| n ⊆ m; m ∈ TFin S; n ∈ TFin S |] ==> n=m | succ S n ⊆ m" by (rule TFin_linear_lemma2 [rule_format]) text{*Consequences from section 3.3 -- Property 3.2, the ordering is total*} lemma TFin_subset_linear: "[| m ∈ TFin S; n ∈ TFin S|] ==> n ⊆ 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 Abrial_axiom1 [THEN subset_trans]) done text{*Lemma 3 of section 3.3*} lemma eq_succ_upper: "[| n ∈ TFin S; m ∈ TFin S; m = succ S m |] ==> n ⊆ m" apply (erule TFin_induct) apply (drule TFin_subsetD) apply (assumption+, force, blast) done text{*Property 3.3 of section 3.3*} lemma equal_succ_Union: "m ∈ TFin S ==> (m = succ S m) = (m = Union(TFin S))" apply (rule iffI) apply (rule Union_upper [THEN equalityI]) apply assumption apply (rule eq_succ_upper [THEN Union_least], assumption+) apply (erule ssubst) apply (rule Abrial_axiom1 [THEN equalityI]) apply (blast del: subsetI intro: subsetI TFin_UnionI TFin.succI) done subsection{*Hausdorff's Theorem: Every Set Contains a Maximal Chain.*} text{*NB: We assume the partial ordering is @{text "⊆"}, the subset relation!*} lemma empty_set_mem_chain: "({} :: 'a set set) ∈ chain S" by (unfold chain_def) auto lemma super_subset_chain: "super S c ⊆ chain S" by (unfold super_def) blast lemma maxchain_subset_chain: "maxchain S ⊆ chain S" by (unfold maxchain_def) blast lemma mem_super_Ex: "c ∈ chain S - maxchain S ==> ? d. d ∈ super S c" by (unfold super_def maxchain_def) auto lemma select_super: "c ∈ chain S - maxchain S ==> (\<some>c'. c': super S c): super S c" apply (erule mem_super_Ex [THEN exE]) apply (rule someI2, auto) done lemma select_not_equals: "c ∈ chain S - maxchain S ==> (\<some>c'. c': super S c) ≠ c" apply (rule notI) apply (drule select_super) apply (simp add: super_def psubset_def) done lemma succI3: "c ∈ chain S - maxchain S ==> succ S c = (\<some>c'. c': super S c)" by (unfold succ_def) (blast intro!: if_not_P) lemma succ_not_equals: "c ∈ chain S - maxchain S ==> succ S c ≠ c" apply (frule succI3) apply (simp (no_asm_simp)) apply (rule select_not_equals, assumption) done lemma TFin_chain_lemma4: "c ∈ TFin S ==> (c :: 'a set set): chain S" apply (erule TFin_induct) apply (simp add: succ_def select_super [THEN super_subset_chain[THEN subsetD]]) apply (unfold chain_def) apply (rule CollectI, safe) apply (drule bspec, assumption) apply (rule_tac [2] m1 = Xa and n1 = X in TFin_subset_linear [THEN disjE], blast+) done theorem Hausdorff: "∃c. (c :: 'a set set): maxchain S" apply (rule_tac x = "Union (TFin S)" in exI) apply (rule classical) apply (subgoal_tac "succ S (Union (TFin S)) = Union (TFin S) ") prefer 2 apply (blast intro!: TFin_UnionI equal_succ_Union [THEN iffD2, symmetric]) apply (cut_tac subset_refl [THEN TFin_UnionI, THEN TFin_chain_lemma4]) apply (drule DiffI [THEN succ_not_equals], blast+) done subsection{*Zorn's Lemma: If All Chains Have Upper Bounds Then There Is a Maximal Element*} lemma chain_extend: "[| c ∈ chain S; z ∈ S; ∀x ∈ c. x ⊆ (z:: 'a set) |] ==> {z} Un c ∈ chain S" by (unfold chain_def) blast lemma chain_Union_upper: "[| c ∈ chain S; x ∈ c |] ==> x ⊆ Union(c)" by (unfold chain_def) auto lemma chain_ball_Union_upper: "c ∈ chain S ==> ∀x ∈ c. x ⊆ Union(c)" by (unfold chain_def) auto lemma maxchain_Zorn: "[| c ∈ maxchain S; u ∈ S; Union(c) ⊆ u |] ==> Union(c) = u" apply (rule ccontr) apply (simp add: maxchain_def) apply (erule conjE) apply (subgoal_tac "({u} Un c) ∈ super S c") apply simp apply (unfold super_def psubset_def) apply (blast intro: chain_extend dest: chain_Union_upper) done theorem Zorn_Lemma: "∀c ∈ chain S. Union(c): S ==> ∃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) apply (rule_tac x = "Union(c)" in bexI) apply (rule ballI, rule impI) apply (blast dest!: maxchain_Zorn, assumption) done subsection{*Alternative version of Zorn's Lemma*} lemma Zorn_Lemma2: "∀c ∈ chain S. ∃y ∈ S. ∀x ∈ c. x ⊆ y ==> ∃y ∈ S. ∀x ∈ S. (y :: 'a set) ⊆ x --> y = x" 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 ccontr) apply (frule_tac z = x in chain_extend) apply (assumption, blast) apply (unfold maxchain_def super_def psubset_def) apply (blast elim!: equalityCE) done text{*Various other lemmas*} lemma chainD: "[| c ∈ chain S; x ∈ c; y ∈ c |] ==> x ⊆ y | y ⊆ x" by (unfold chain_def) blast lemma chainD2: "!!(c :: 'a set set). c ∈ chain S ==> c ⊆ S" by (unfold chain_def) blast end
lemma Union_lemma0:
∀x∈C. x ⊆ A ∨ B ⊆ x ==> Union C ⊆ A ∨ B ⊆ Union C
lemma Abrial_axiom1:
x ⊆ succ S x
lemma TFin_UnionI:
Y ⊆ TFin S ==> Union Y ∈ TFin S
lemma TFin_induct:
[| n ∈ TFin S; !!x. [| x ∈ TFin S; P x |] ==> P (succ S x);
!!Y. [| Y ⊆ TFin S; Ball Y P |] ==> P (Union Y) |]
==> P n
lemma succ_trans:
x ⊆ y ==> x ⊆ succ S y
lemma TFin_linear_lemma1:
[| n ∈ TFin S; m ∈ TFin S; ∀x∈TFin S. x ⊆ m --> x = m ∨ succ S x ⊆ m |]
==> n ⊆ m ∨ succ S m ⊆ n
lemma TFin_linear_lemma2:
m ∈ TFin S ==> ∀n∈TFin S. n ⊆ m --> n = m ∨ succ S n ⊆ m
lemma TFin_subsetD:
[| n ⊆ m; m ∈ TFin S; n ∈ TFin S |] ==> n = m ∨ succ S n ⊆ m
lemma TFin_subset_linear:
[| m ∈ TFin S; n ∈ TFin S |] ==> n ⊆ m ∨ m ⊆ n
lemma eq_succ_upper:
[| n ∈ TFin S; m ∈ TFin S; m = succ S m |] ==> n ⊆ m
lemma equal_succ_Union:
m ∈ TFin S ==> (m = succ S m) = (m = Union (TFin S))
lemma empty_set_mem_chain:
{} ∈ chain S
lemma super_subset_chain:
super S c ⊆ chain S
lemma maxchain_subset_chain:
maxchain S ⊆ chain S
lemma mem_super_Ex:
c ∈ chain S - maxchain S ==> ∃d. d ∈ super S c
lemma select_super:
c ∈ chain S - maxchain S ==> (SOME c'. c' ∈ super S c) ∈ super S c
lemma select_not_equals:
c ∈ chain S - maxchain S ==> (SOME c'. c' ∈ super S c) ≠ c
lemma succI3:
c ∈ chain S - maxchain S ==> succ S c = (SOME c'. c' ∈ super S c)
lemma succ_not_equals:
c ∈ chain S - maxchain S ==> succ S c ≠ c
lemma TFin_chain_lemma4:
c ∈ TFin S ==> c ∈ chain S
theorem Hausdorff:
∃c. c ∈ maxchain S
lemma chain_extend:
[| c ∈ chain S; z ∈ S; ∀x∈c. x ⊆ z |] ==> {z} ∪ c ∈ chain S
lemma chain_Union_upper:
[| c ∈ chain S; x ∈ c |] ==> x ⊆ Union c
lemma chain_ball_Union_upper:
c ∈ chain S ==> ∀x∈c. x ⊆ Union c
lemma maxchain_Zorn:
[| c ∈ maxchain S; u ∈ S; Union c ⊆ u |] ==> Union c = u
theorem Zorn_Lemma:
∀c∈chain S. Union c ∈ S ==> ∃y∈S. ∀z∈S. y ⊆ z --> y = z
lemma Zorn_Lemma2:
∀c∈chain S. ∃y∈S. ∀x∈c. x ⊆ y ==> ∃y∈S. ∀x∈S. y ⊆ x --> y = x
lemma chainD:
[| c ∈ chain S; x ∈ c; y ∈ c |] ==> x ⊆ y ∨ y ⊆ x
lemma chainD2:
c ∈ chain S ==> c ⊆ S