--- a/src/HOL/Library/Zorn.thy Wed Aug 31 15:46:36 2005 +0200
+++ b/src/HOL/Library/Zorn.thy Wed Aug 31 15:46:37 2005 +0200
@@ -42,18 +42,20 @@
subsection{*Mathematical Preamble*}
-lemma Union_lemma0: "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C)<=A | B \<subseteq> Union(C)"
-by blast
+lemma Union_lemma0:
+ "(\<forall>x \<in> C. x \<subseteq> A | B \<subseteq> x) ==> Union(C)<=A | B \<subseteq> Union(C)"
+ by blast
text{*This is theorem @{text increasingD2} of ZF/Zorn.thy*}
+
lemma Abrial_axiom1: "x \<subseteq> 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 swap, assumption)
-apply (rule someI2, blast+)
-done
+ apply (unfold succ_def)
+ apply (rule split_if [THEN iffD2])
+ apply (auto simp add: super_def maxchain_def psubset_def)
+ apply (rule swap, assumption)
+ apply (rule someI2, blast+)
+ done
lemmas TFin_UnionI = TFin.Pow_UnionI [OF PowI]
@@ -62,79 +64,77 @@
!!x. [| x \<in> TFin S; P(x) |] ==> P(succ S x);
!!Y. [| Y \<subseteq> TFin S; Ball Y P |] ==> P(Union Y) |]
==> P(n)"
-apply (erule TFin.induct, blast+)
-done
+ apply (erule TFin.induct)
+ apply blast+
+ done
lemma succ_trans: "x \<subseteq> y ==> x \<subseteq> succ S y"
-apply (erule subset_trans)
-apply (rule Abrial_axiom1)
-done
+ apply (erule subset_trans)
+ apply (rule Abrial_axiom1)
+ done
text{*Lemma 1 of section 3.1*}
lemma TFin_linear_lemma1:
"[| n \<in> TFin S; m \<in> TFin S;
\<forall>x \<in> TFin S. x \<subseteq> m --> x = m | succ S x \<subseteq> m
|] ==> n \<subseteq> m | succ S m \<subseteq> n"
-apply (erule TFin_induct)
-apply (erule_tac [2] Union_lemma0) (*or just blast*)
-apply (blast del: subsetI intro: succ_trans)
-done
+ 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 \<in> TFin S ==> \<forall>n \<in> TFin S. n \<subseteq> m --> n=m | succ S n \<subseteq> 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
+ 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 \<subseteq> m; m \<in> TFin S; n \<in> TFin S |] ==> n=m | succ S n \<subseteq> m"
-apply (rule TFin_linear_lemma2 [rule_format])
-apply (assumption+)
-done
+ 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 \<in> TFin S; n \<in> TFin S|] ==> n \<subseteq> m | m \<subseteq> 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
+ 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 \<in> TFin S; m \<in> TFin S; m = succ S m |] ==> n \<subseteq> m"
-apply (erule TFin_induct)
-apply (drule TFin_subsetD)
-apply (assumption+, force, blast)
-done
+ 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 \<in> TFin S ==> (m = succ S m) = (m = Union(TFin S))"
-apply (rule iffI)
-apply (rule Union_upper [THEN equalityI])
-apply (rule_tac [2] eq_succ_upper [THEN Union_least])
-apply (assumption+)
-apply (erule ssubst)
-apply (rule Abrial_axiom1 [THEN equalityI])
-apply (blast del: subsetI
- intro: subsetI TFin_UnionI TFin.succI)
-done
+ apply (rule iffI)
+ apply (rule Union_upper [THEN equalityI])
+ apply (rule_tac [2] eq_succ_upper [THEN Union_least])
+ apply (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.*}
@@ -142,60 +142,58 @@
the subset relation!*}
lemma empty_set_mem_chain: "({} :: 'a set set) \<in> chain S"
-by (unfold chain_def, auto)
+ by (unfold chain_def) auto
lemma super_subset_chain: "super S c \<subseteq> chain S"
-by (unfold super_def, fast)
+ by (unfold super_def) blast
lemma maxchain_subset_chain: "maxchain S \<subseteq> chain S"
-by (unfold maxchain_def, fast)
+ by (unfold maxchain_def) blast
lemma mem_super_Ex: "c \<in> chain S - maxchain S ==> ? d. d \<in> super S c"
-by (unfold super_def maxchain_def, auto)
+ by (unfold super_def maxchain_def) auto
lemma select_super: "c \<in> chain S - maxchain S ==>
- (@c'. c': super S c): super S c"
-apply (erule mem_super_Ex [THEN exE])
-apply (rule someI2, auto)
-done
+ (\<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 \<in> chain S - maxchain S ==>
- (@c'. c': super S c) \<noteq> c"
-apply (rule notI)
-apply (drule select_super)
-apply (simp add: super_def psubset_def)
-done
+ (\<some>c'. c': super S c) \<noteq> c"
+ apply (rule notI)
+ apply (drule select_super)
+ apply (simp add: super_def psubset_def)
+ done
-lemma succI3: "c \<in> chain S - maxchain S ==> succ S c = (@c'. c': super S c)"
-apply (unfold succ_def)
-apply (fast intro!: if_not_P)
-done
+lemma succI3: "c \<in> 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 \<in> chain S - maxchain S ==> succ S c \<noteq> c"
-apply (frule succI3)
-apply (simp (no_asm_simp))
-apply (rule select_not_equals, assumption)
-done
+ apply (frule succI3)
+ apply (simp (no_asm_simp))
+ apply (rule select_not_equals, assumption)
+ done
lemma TFin_chain_lemma4: "c \<in> 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
+ 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: "\<exists>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
+ 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
@@ -204,61 +202,61 @@
lemma chain_extend:
"[| c \<in> chain S; z \<in> S;
\<forall>x \<in> c. x<=(z:: 'a set) |] ==> {z} Un c \<in> chain S"
-by (unfold chain_def, blast)
+ by (unfold chain_def) blast
lemma chain_Union_upper: "[| c \<in> chain S; x \<in> c |] ==> x \<subseteq> Union(c)"
-by (unfold chain_def, auto)
+ by (unfold chain_def) auto
lemma chain_ball_Union_upper: "c \<in> chain S ==> \<forall>x \<in> c. x \<subseteq> Union(c)"
-by (unfold chain_def, auto)
+ by (unfold chain_def) auto
lemma maxchain_Zorn:
"[| c \<in> maxchain S; u \<in> S; Union(c) \<subseteq> u |] ==> Union(c) = u"
-apply (rule ccontr)
-apply (simp add: maxchain_def)
-apply (erule conjE)
-apply (subgoal_tac " ({u} Un c) \<in> super S c")
-apply simp
-apply (unfold super_def psubset_def)
-apply (blast intro: chain_extend dest: chain_Union_upper)
-done
+ apply (rule ccontr)
+ apply (simp add: maxchain_def)
+ apply (erule conjE)
+ apply (subgoal_tac " ({u} Un c) \<in> super S c")
+ apply simp
+ apply (unfold super_def psubset_def)
+ apply (blast intro: chain_extend dest: chain_Union_upper)
+ done
theorem Zorn_Lemma:
- "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> 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
+ "\<forall>c \<in> chain S. Union(c): S ==> \<exists>y \<in> S. \<forall>z \<in> S. y \<subseteq> 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:
- "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
- ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> 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
+ "\<forall>c \<in> chain S. \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y
+ ==> \<exists>y \<in> S. \<forall>x \<in> S. (y :: 'a set) \<subseteq> 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 \<in> chain S; x \<in> c; y \<in> c |] ==> x \<subseteq> y | y \<subseteq> x"
-by (unfold chain_def, blast)
+ by (unfold chain_def) blast
lemma chainD2: "!!(c :: 'a set set). c \<in> chain S ==> c \<subseteq> S"
-by (unfold chain_def, blast)
+ by (unfold chain_def) blast
end