(* Title: ZF/Zorn.thy ID: $Id: Zorn.thy,v 1.19 2007/10/07 19:19:33 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1994 University of Cambridge *) header{*Zorn's Lemma*} theory Zorn imports OrderArith AC Inductive begin text{*Based upon the unpublished article ``Towards the Mechanization of the Proofs of Some Classical Theorems of Set Theory,'' by Abrial and Laffitte.*} definition Subset_rel :: "i=>i" where "Subset_rel(A) == {z ∈ A*A . ∃x y. z=<x,y> & 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{*Lemma for the inductive definition below*} lemma Union_in_Pow: "Y ∈ Pow(Pow(A)) ==> Union(Y) ∈ Pow(A)" by blast text{*We could make the inductive definition conditional on @{term "next ∈ increasing(S)"} 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.*} consts "TFin" :: "[i,i]=>i" inductive domains "TFin(S,next)" <= "Pow(S)" intros nextI: "[| x ∈ TFin(S,next); next ∈ increasing(S) |] ==> next`x ∈ TFin(S,next)" Pow_UnionI: "Y ∈ Pow(TFin(S,next)) ==> Union(Y) ∈ TFin(S,next)" monos Pow_mono con_defs increasing_def type_intros CollectD1 [THEN apply_funtype] Union_in_Pow subsection{*Mathematical Preamble *} lemma Union_lemma0: "(∀x∈C. x<=A | B<=x) ==> Union(C)<=A | B<=Union(C)" by blast lemma Inter_lemma0: "[| c ∈ C; ∀x∈C. A<=x | x<=B |] ==> A <= Inter(C) | Inter(C) <= B" by blast subsection{*The Transfinite Construction *} 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, standard] lemmas TFin_is_subset = TFin.dom_subset [THEN subsetD, THEN PowD, standard] text{*Structural induction on @{term "TFin(S,next)"} *} lemma TFin_induct: "[| n ∈ TFin(S,next); !!x. [| x ∈ TFin(S,next); P(x); next ∈ increasing(S) |] ==> P(next`x); !!Y. [| Y <= TFin(S,next); ∀y∈Y. P(y) |] ==> P(Union(Y)) |] ==> P(n)" by (erule TFin.induct, blast+) subsection{*Some Properties of the Transfinite Construction *} lemmas increasing_trans = subset_trans [OF _ increasingD2, OF _ _ TFin_is_subset] text{*Lemma 1 of section 3.1*} lemma TFin_linear_lemma1: "[| n ∈ TFin(S,next); m ∈ TFin(S,next); ∀x ∈ TFin(S,next) . x<=m --> 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{*Lemma 2 of section 3.2. Interesting in its own right! Requires @{term "next ∈ increasing(S)"} in the second induction step.*} lemma TFin_linear_lemma2: "[| m ∈ TFin(S,next); next ∈ increasing(S) |] ==> ∀n ∈ TFin(S,next). n<=m --> n=m | next`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 (blast del: subsetI intro: increasing_trans subsetI, blast) txt{*second induction step*} 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{*a more convenient form for Lemma 2*} lemma TFin_subsetD: "[| n<=m; m ∈ TFin(S,next); n ∈ TFin(S,next); next ∈ increasing(S) |] ==> n=m | next`n <= m" by (blast dest: 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,next); n ∈ TFin(S,next); next ∈ increasing(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 increasingD2 [THEN subset_trans] TFin_is_subset) done text{*Lemma 3 of section 3.3*} 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{*Property 3.3 of section 3.3*} 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{*Hausdorff's Theorem: Every Set Contains a Maximal Chain*} text{*NOTE: We assume the partial ordering is @{text "⊆"}, the subset relation!*} text{** Defining the "next" operation for Hausdorff's Theorem **} 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 ∈ (Π X ∈ Pow(chain(S)) - {0}. X); X ∈ chain(S); X ∉ maxchain(S) |] ==> ch ` super(S,X) ∈ super(S,X)" apply (erule apply_type) apply (unfold super_def maxchain_def, blast) done lemma choice_not_equals: "[| ch ∈ (Π X ∈ Pow(chain(S)) - {0}. X); X ∈ chain(S); X ∉ maxchain(S) |] ==> ch ` super(S,X) ≠ X" apply (rule notI) apply (drule choice_super, assumption, assumption) apply (simp add: super_def) done text{*This justifies Definition 4.4*} lemma Hausdorff_next_exists: "ch ∈ (Π X ∈ Pow(chain(S))-{0}. X) ==> ∃next ∈ increasing(S). ∀X ∈ Pow(S). next`X = if(X ∈ chain(S)-maxchain(S), ch`super(S,X), X)" apply (rule_tac x="λX∈Pow(S). if X ∈ 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{*Now, verify that it increases*} 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{*Lemma 4*} lemma TFin_chain_lemma4: "[| c ∈ TFin(S,next); ch ∈ (Π X ∈ Pow(chain(S))-{0}. X); next ∈ increasing(S); ∀X ∈ Pow(S). next`X = if(X ∈ chain(S)-maxchain(S), ch`super(S,X), X) |] ==> c ∈ 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{*@{text "Blast_tac's"} slow*} 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 "Union (TFin (S,next)) ∈ chain (S) ") prefer 2 apply (blast intro!: TFin_chain_lemma4 subset_refl [THEN TFin_UnionI]) apply (rule_tac x = "Union (TFin (S,next))" in exI) apply (rule classical) apply (subgoal_tac "next ` Union (TFin (S,next)) = Union (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{*Zorn's Lemma: If All Chains in S Have Upper Bounds In S, then S contains a Maximal Element*} text{*Used in the proof of Zorn's Lemma*} 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). Union(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 = "Union (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 subsection{*Zermelo's Theorem: Every Set can be Well-Ordered*} text{*Lemma 5*} lemma TFin_well_lemma5: "[| n ∈ TFin(S,next); Z <= TFin(S,next); z:Z; ~ Inter(Z) ∈ Z |] ==> ∀m ∈ Z. n <= m" apply (erule TFin_induct) prefer 2 apply blast txt{*second induction step is easy*} apply (rule ballI) apply (rule bspec [THEN TFin_subsetD, THEN disjE], auto) apply (subgoal_tac "m = Inter (Z) ") apply blast+ done text{*Well-ordering of @{term "TFin(S,next)"} *} lemma well_ord_TFin_lemma: "[| Z <= TFin(S,next); z ∈ Z |] ==> Inter(Z) ∈ Z" apply (rule classical) apply (subgoal_tac "Z = {Union (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{*This theorem just packages the previous result*} 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{*Prove the well-foundedness goal*} apply (rule wf_onI) apply (frule well_ord_TFin_lemma, assumption) apply (drule_tac x = "Inter (Z) " in bspec, assumption) apply blast txt{*Now prove the linearity goal*} apply (intro ballI) apply (case_tac "x=y") apply blast txt{*The @{term "x≠y"} case remains*} apply (rule_tac n1=x and m1=y in TFin_subset_linear [THEN disjE], assumption+, blast+) done text{** Defining the "next" operation for Zermelo's Theorem **} lemma choice_Diff: "[| ch ∈ (Π X ∈ Pow(S) - {0}. X); X ⊆ S; X≠S |] ==> ch ` (S-X) ∈ S-X" apply (erule apply_type) apply (blast elim!: equalityE) done text{*This justifies Definition 6.1*} lemma Zermelo_next_exists: "ch ∈ (Π X ∈ Pow(S)-{0}. X) ==> ∃next ∈ increasing(S). ∀X ∈ Pow(S). next`X = (if X=S then S else cons(ch`(S-X), X))" apply (rule_tac x="λX∈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{*Type checking is surprisingly hard!*} apply (simp (no_asm_simp) add: Pow_iff cons_subset_iff subset_refl) apply (blast intro!: choice_Diff [THEN DiffD1]) txt{*Verify that it increases*} apply (intro allI impI) apply (simp add: Pow_iff subset_consI subset_refl) done text{*The construction of the injection*} lemma choice_imp_injection: "[| ch ∈ (Π X ∈ Pow(S)-{0}. X); next ∈ increasing(S); ∀X ∈ Pow(S). next`X = if(X=S, S, cons(ch`(S-X), X)) |] ==> (λ x ∈ S. Union({y ∈ TFin(S,next). x ∉ y})) ∈ 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 ∉ Union ({y ∈ TFin (S,next) . x ∉ y}) ") prefer 2 apply (blast elim: equalityE) apply (subgoal_tac "Union ({y ∈ TFin (S,next) . x ∉ y}) ≠ S") prefer 2 apply (blast elim: equalityE) txt{*For proving @{text "x ∈ next`Union(...)"}. Abrial and Laffitte's justification appears to be faulty.*} apply (subgoal_tac "~ next ` Union ({y ∈ TFin (S,next) . x ∉ y}) <= Union ({y ∈ TFin (S,next) . x ∉ 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{*End of the lemmas!*} apply (simp add: Collect_subset [THEN TFin_UnionI, THEN TFin_is_subset]) done text{*The wellordering theorem*} 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 end
lemma Union_in_Pow:
Y ∈ Pow(Pow(A)) ==> \<Union>Y ∈ Pow(A)
lemma Union_lemma0:
∀x∈C. x ⊆ A ∨ B ⊆ x ==> \<Union>C ⊆ A ∨ B ⊆ \<Union>C
lemma Inter_lemma0:
[| c ∈ C; ∀x∈C. A ⊆ x ∨ x ⊆ B |] ==> A ⊆ \<Inter>C ∨ \<Inter>C ⊆ B
lemma increasingD1:
f ∈ increasing(A) ==> f ∈ Pow(A) -> Pow(A)
lemma increasingD2:
[| f ∈ increasing(A); x ⊆ A |] ==> x ⊆ f ` x
lemma TFin_UnionI:
Y ⊆ TFin(S, next) ==> \<Union>Y ∈ TFin(S, next)
lemma TFin_is_subset:
A ∈ TFin(B, next) ==> A ⊆ B
lemma TFin_induct:
[| n ∈ TFin(S, next);
!!x. [| x ∈ TFin(S, next); P(x); next ∈ increasing(S) |] ==> P(next ` x);
!!Y. [| Y ⊆ TFin(S, next); ∀y∈Y. P(y) |] ==> P(\<Union>Y) |]
==> P(n)
lemma increasing_trans:
[| A ⊆ B; f1 ∈ increasing(A1); B ∈ TFin(A1, next3) |] ==> A ⊆ f1 ` B
lemma TFin_linear_lemma1:
[| n ∈ TFin(S, next); m ∈ TFin(S, next);
∀x∈TFin(S, next). x ⊆ m --> x = m ∨ next ` x ⊆ m |]
==> n ⊆ m ∨ next ` m ⊆ n
lemma TFin_linear_lemma2:
[| m ∈ TFin(S, next); next ∈ increasing(S) |]
==> ∀n∈TFin(S, next). n ⊆ m --> n = m ∨ next ` n ⊆ m
lemma TFin_subsetD:
[| n ⊆ m; m ∈ TFin(S, next); n ∈ TFin(S, next); next ∈ increasing(S) |]
==> n = m ∨ next ` n ⊆ m
lemma TFin_subset_linear:
[| m ∈ TFin(S, next); n ∈ TFin(S, next); next ∈ increasing(S) |]
==> n ⊆ m ∨ m ⊆ n
lemma equal_next_upper:
[| n ∈ TFin(S, next); m ∈ TFin(S, next); m = next ` m |] ==> n ⊆ m
lemma equal_next_Union:
[| m ∈ TFin(S, next); next ∈ increasing(S) |]
==> m = next ` m <-> m = \<Union>TFin(S, next)
lemma chain_subset_Pow:
chain(A) ⊆ Pow(A)
lemma super_subset_chain:
super(A, c) ⊆ chain(A)
lemma maxchain_subset_chain:
maxchain(A) ⊆ chain(A)
lemma choice_super:
[| ch ∈ (ΠX∈Pow(chain(S)) - {0}. X); X ∈ chain(S); X ∉ maxchain(S) |]
==> ch ` super(S, X) ∈ super(S, X)
lemma choice_not_equals:
[| ch ∈ (ΠX∈Pow(chain(S)) - {0}. X); X ∈ chain(S); X ∉ maxchain(S) |]
==> ch ` super(S, X) ≠ X
lemma Hausdorff_next_exists:
ch ∈ (ΠX∈Pow(chain(S)) - {0}. X)
==> ∃next∈increasing(S).
∀X∈Pow(S).
next ` X =
(if X ∈ chain(S) - maxchain(S) then ch ` super(S, X) else X)
lemma TFin_chain_lemma4:
[| c ∈ TFin(S, next); ch ∈ (ΠX∈Pow(chain(S)) - {0}. X); next ∈ increasing(S);
∀X∈Pow(S).
next ` X = (if X ∈ chain(S) - maxchain(S) then ch ` super(S, X) else X) |]
==> c ∈ chain(S)
theorem Hausdorff:
∃c. c ∈ maxchain(S)
lemma chain_extend:
[| c ∈ chain(A); z ∈ A; ∀x∈c. x ⊆ z |] ==> cons(z, c) ∈ chain(A)
lemma Zorn:
∀c∈chain(S). \<Union>c ∈ S ==> ∃y∈S. ∀z∈S. y ⊆ z --> y = z
lemma TFin_well_lemma5:
[| n ∈ TFin(S, next); Z ⊆ TFin(S, next); z ∈ Z; \<Inter>Z ∉ Z |] ==> ∀m∈Z. n ⊆ m
lemma well_ord_TFin_lemma:
[| Z ⊆ TFin(S, next); z ∈ Z |] ==> \<Inter>Z ∈ Z
lemma well_ord_TFin:
next ∈ increasing(S) ==> well_ord(TFin(S, next), Subset_rel(TFin(S, next)))
lemma choice_Diff:
[| ch ∈ (ΠX∈Pow(S) - {0}. X); X ⊆ S; X ≠ S |] ==> ch ` (S - X) ∈ S - X
lemma Zermelo_next_exists:
ch ∈ (ΠX∈Pow(S) - {0}. X)
==> ∃next∈increasing(S).
∀X∈Pow(S). next ` X = (if X = S then S else cons(ch ` (S - X), X))
lemma choice_imp_injection:
[| ch ∈ (ΠX∈Pow(S) - {0}. X); next ∈ increasing(S);
∀X∈Pow(S). next ` X = (if X = S then S else cons(ch ` (S - X), X)) |]
==> (λx∈S. \<Union>{y ∈ TFin(S, next) . x ∉ y}) ∈ inj(S, TFin(S, next) - {S})
theorem AC_well_ord:
∃r. well_ord(S, r)