--- a/src/ZF/Zorn.thy Sun Mar 04 23:20:43 2012 +0100
+++ b/src/ZF/Zorn.thy Tue Mar 06 15:15:49 2012 +0000
@@ -28,11 +28,11 @@
definition
increasing :: "i=>i" where
- "increasing(A) == {f \<in> Pow(A)->Pow(A). \<forall>x. x<=A --> x<=f`x}"
+ "increasing(A) == {f \<in> Pow(A)->Pow(A). \<forall>x. x<=A \<longrightarrow> x<=f`x}"
text{*Lemma for the inductive definition below*}
-lemma Union_in_Pow: "Y \<in> Pow(Pow(A)) ==> Union(Y) \<in> Pow(A)"
+lemma Union_in_Pow: "Y \<in> Pow(Pow(A)) ==> \<Union>(Y) \<in> Pow(A)"
by blast
@@ -45,12 +45,12 @@
"TFin" :: "[i,i]=>i"
inductive
- domains "TFin(S,next)" <= "Pow(S)"
+ domains "TFin(S,next)" \<subseteq> "Pow(S)"
intros
nextI: "[| x \<in> TFin(S,next); next \<in> increasing(S) |]
==> next`x \<in> TFin(S,next)"
- Pow_UnionI: "Y \<in> Pow(TFin(S,next)) ==> Union(Y) \<in> TFin(S,next)"
+ Pow_UnionI: "Y \<in> Pow(TFin(S,next)) ==> \<Union>(Y) \<in> TFin(S,next)"
monos Pow_mono
con_defs increasing_def
@@ -59,11 +59,11 @@
subsection{*Mathematical Preamble *}
-lemma Union_lemma0: "(\<forall>x\<in>C. x<=A | B<=x) ==> Union(C)<=A | B<=Union(C)"
+lemma Union_lemma0: "(\<forall>x\<in>C. x<=A | B<=x) ==> \<Union>(C)<=A | B<=\<Union>(C)"
by blast
lemma Inter_lemma0:
- "[| c \<in> C; \<forall>x\<in>C. A<=x | x<=B |] ==> A <= Inter(C) | Inter(C) <= B"
+ "[| c \<in> C; \<forall>x\<in>C. A<=x | x<=B |] ==> A \<subseteq> \<Inter>(C) | \<Inter>(C) \<subseteq> B"
by blast
@@ -74,7 +74,7 @@
apply (erule CollectD1)
done
-lemma increasingD2: "[| f \<in> increasing(A); x<=A |] ==> x <= f`x"
+lemma increasingD2: "[| f \<in> increasing(A); x<=A |] ==> x \<subseteq> f`x"
by (unfold increasing_def, blast)
lemmas TFin_UnionI = PowI [THEN TFin.Pow_UnionI]
@@ -86,7 +86,7 @@
lemma TFin_induct:
"[| n \<in> TFin(S,next);
!!x. [| x \<in> TFin(S,next); P(x); next \<in> increasing(S) |] ==> P(next`x);
- !!Y. [| Y <= TFin(S,next); \<forall>y\<in>Y. P(y) |] ==> P(Union(Y))
+ !!Y. [| Y \<subseteq> TFin(S,next); \<forall>y\<in>Y. P(y) |] ==> P(\<Union>(Y))
|] ==> P(n)"
by (erule TFin.induct, blast+)
@@ -99,7 +99,7 @@
text{*Lemma 1 of section 3.1*}
lemma TFin_linear_lemma1:
"[| n \<in> TFin(S,next); m \<in> TFin(S,next);
- \<forall>x \<in> TFin(S,next) . x<=m --> x=m | next`x<=m |]
+ \<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*)
@@ -111,7 +111,7 @@
Requires @{term "next \<in> increasing(S)"} in the second induction step.*}
lemma TFin_linear_lemma2:
"[| m \<in> TFin(S,next); next \<in> increasing(S) |]
- ==> \<forall>n \<in> TFin(S,next). n<=m --> n=m | next`n <= m"
+ ==> \<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{*case split using @{text TFin_linear_lemma1}*}
@@ -136,13 +136,13 @@
text{*a more convenient form for Lemma 2*}
lemma TFin_subsetD:
"[| n<=m; m \<in> TFin(S,next); n \<in> TFin(S,next); next \<in> increasing(S) |]
- ==> n=m | next`n <= m"
+ ==> n=m | next`n \<subseteq> 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 \<in> TFin(S,next); n \<in> TFin(S,next); next \<in> increasing(S) |]
- ==> n <= m | m<=n"
+ ==> n \<subseteq> m | m<=n"
apply (rule disjE)
apply (rule TFin_linear_lemma1 [OF _ _TFin_linear_lemma2])
apply (assumption+, erule disjI2)
@@ -153,7 +153,7 @@
text{*Lemma 3 of section 3.3*}
lemma equal_next_upper:
- "[| n \<in> TFin(S,next); m \<in> TFin(S,next); m = next`m |] ==> n <= m"
+ "[| n \<in> TFin(S,next); m \<in> TFin(S,next); m = next`m |] ==> n \<subseteq> m"
apply (erule TFin_induct)
apply (drule TFin_subsetD)
apply (assumption+, force, blast)
@@ -162,7 +162,7 @@
text{*Property 3.3 of section 3.3*}
lemma equal_next_Union:
"[| m \<in> TFin(S,next); next \<in> increasing(S) |]
- ==> m = next`m <-> m = Union(TFin(S,next))"
+ ==> 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])
@@ -181,17 +181,17 @@
text{** Defining the "next" operation for Hausdorff's Theorem **}
-lemma chain_subset_Pow: "chain(A) <= Pow(A)"
+lemma chain_subset_Pow: "chain(A) \<subseteq> Pow(A)"
apply (unfold chain_def)
apply (rule Collect_subset)
done
-lemma super_subset_chain: "super(A,c) <= chain(A)"
+lemma super_subset_chain: "super(A,c) \<subseteq> chain(A)"
apply (unfold super_def)
apply (rule Collect_subset)
done
-lemma maxchain_subset_chain: "maxchain(A) <= chain(A)"
+lemma maxchain_subset_chain: "maxchain(A) \<subseteq> chain(A)"
apply (unfold maxchain_def)
apply (rule Collect_subset)
done
@@ -255,12 +255,12 @@
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)) \<in> chain (S) ")
+apply (subgoal_tac "\<Union>(TFin (S,next)) \<in> 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_tac x = "\<Union>(TFin (S,next))" in exI)
apply (rule classical)
-apply (subgoal_tac "next ` Union (TFin (S,next)) = Union (TFin (S,next))")
+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
@@ -280,11 +280,11 @@
"[| c \<in> chain(A); z \<in> A; \<forall>x \<in> c. x<=z |] ==> cons(z,c) \<in> chain(A)"
by (unfold chain_def, blast)
-lemma Zorn: "\<forall>c \<in> chain(S). Union(c) \<in> S ==> \<exists>y \<in> S. \<forall>z \<in> S. y<=z --> y=z"
+lemma Zorn: "\<forall>c \<in> chain(S). \<Union>(c) \<in> S ==> \<exists>y \<in> S. \<forall>z \<in> S. y<=z \<longrightarrow> y=z"
apply (rule Hausdorff [THEN exE])
apply (simp add: maxchain_def)
apply (rename_tac c)
-apply (rule_tac x = "Union (c)" in bexI)
+apply (rule_tac x = "\<Union>(c)" in bexI)
prefer 2 apply blast
apply safe
apply (rename_tac z)
@@ -299,7 +299,7 @@
text {* Alternative version of Zorn's Lemma *}
theorem Zorn2:
- "\<forall>c \<in> chain(S). \<exists>y \<in> S. \<forall>x \<in> c. x <= y ==> \<exists>y \<in> S. \<forall>z \<in> S. y<=z --> y=z"
+ "\<forall>c \<in> chain(S). \<exists>y \<in> S. \<forall>x \<in> c. x \<subseteq> y ==> \<exists>y \<in> S. \<forall>z \<in> S. y<=z \<longrightarrow> y=z"
apply (cut_tac Hausdorff maxchain_subset_chain)
apply (erule exE)
apply (drule subsetD, assumption)
@@ -321,20 +321,20 @@
text{*Lemma 5*}
lemma TFin_well_lemma5:
- "[| n \<in> TFin(S,next); Z <= TFin(S,next); z:Z; ~ Inter(Z) \<in> Z |]
- ==> \<forall>m \<in> Z. n <= m"
+ "[| n \<in> TFin(S,next); Z \<subseteq> TFin(S,next); z:Z; ~ \<Inter>(Z) \<in> Z |]
+ ==> \<forall>m \<in> Z. n \<subseteq> 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 (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 \<in> Z |] ==> Inter(Z) \<in> Z"
+lemma well_ord_TFin_lemma: "[| Z \<subseteq> TFin(S,next); z \<in> Z |] ==> \<Inter>(Z) \<in> Z"
apply (rule classical)
-apply (subgoal_tac "Z = {Union (TFin (S,next))}")
+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])
@@ -350,7 +350,7 @@
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 (drule_tac x = "\<Inter>(Z) " in bspec, assumption)
apply blast
txt{*Now prove the linearity goal*}
apply (intro ballI)
@@ -394,25 +394,25 @@
"[| ch \<in> (\<Pi> X \<in> 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}))
+ ==> (\<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 \<notin> Union ({y \<in> TFin (S,next) . x \<notin> y}) ")
+apply (subgoal_tac "x \<notin> \<Union>({y \<in> TFin (S,next) . x \<notin> y}) ")
prefer 2 apply (blast elim: equalityE)
-apply (subgoal_tac "Union ({y \<in> TFin (S,next) . x \<notin> y}) \<noteq> S")
+apply (subgoal_tac "\<Union>({y \<in> TFin (S,next) . x \<notin> y}) \<noteq> S")
prefer 2 apply (blast elim: equalityE)
-txt{*For proving @{text "x \<in> next`Union(...)"}.
+txt{*For proving @{text "x \<in> next`\<Union>(...)"}.
Abrial and Laffitte's justification appears to be faulty.*}
-apply (subgoal_tac "~ next ` Union ({y \<in> TFin (S,next) . x \<notin> y})
- <= Union ({y \<in> TFin (S,next) . x \<notin> y}) ")
+apply (subgoal_tac "~ next ` Union({y \<in> TFin (S,next) . x \<notin> 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 \<in> next ` Union ({y \<in> TFin (S,next) . x \<notin> y}) ")
+apply (subgoal_tac "x \<in> next ` Union({y \<in> TFin (S,next) . x \<notin> y}) ")
prefer 2
apply (blast intro!: Collect_subset [THEN TFin_UnionI] TFin.nextI)
txt{*End of the lemmas!*}
@@ -436,7 +436,7 @@
definition Chain :: "i => i" where
- "Chain(r) = {A : Pow(field(r)). ALL a:A. ALL b:A. <a, b> : r | <b, a> : r}"
+ "Chain(r) = {A \<in> Pow(field(r)). \<forall>a\<in>A. \<forall>b\<in>A. <a, b> \<in> r | <b, a> \<in> r}"
lemma mono_Chain:
"r \<subseteq> s ==> Chain(r) \<subseteq> Chain(s)"
@@ -445,17 +445,17 @@
theorem Zorn_po:
assumes po: "Partial_order(r)"
- and u: "ALL C:Chain(r). EX u:field(r). ALL a:C. <a, u> : r"
- shows "EX m:field(r). ALL a:field(r). <m, a> : r --> a = m"
+ and u: "\<forall>C\<in>Chain(r). \<exists>u\<in>field(r). \<forall>a\<in>C. <a, u> \<in> r"
+ shows "\<exists>m\<in>field(r). \<forall>a\<in>field(r). <m, a> \<in> r \<longrightarrow> a = m"
proof -
have "Preorder(r)" using po by (simp add: partial_order_on_def)
--{* Mirror r in the set of subsets below (wrt r) elements of A (?). *}
- let ?B = "lam x:field(r). r -`` {x}" let ?S = "?B `` field(r)"
- have "ALL C:chain(?S). EX U:?S. ALL A:C. A \<subseteq> U"
+ let ?B = "\<lambda>x\<in>field(r). r -`` {x}" let ?S = "?B `` field(r)"
+ have "\<forall>C\<in>chain(?S). \<exists>U\<in>?S. \<forall>A\<in>C. A \<subseteq> U"
proof (clarsimp simp: chain_def Subset_rel_def bex_image_simp)
fix C
- assume 1: "C \<subseteq> ?S" and 2: "ALL A:C. ALL B:C. A \<subseteq> B | B \<subseteq> A"
- let ?A = "{x : field(r). EX M:C. M = ?B`x}"
+ assume 1: "C \<subseteq> ?S" and 2: "\<forall>A\<in>C. \<forall>B\<in>C. A \<subseteq> B | B \<subseteq> A"
+ let ?A = "{x \<in> field(r). \<exists>M\<in>C. M = ?B`x}"
have "C = ?B `` ?A" using 1
apply (auto simp: image_def)
apply rule
@@ -472,46 +472,46 @@
apply (thin_tac "C \<subseteq> ?X")
apply (fast elim: lamE)
done
- have "?A : Chain(r)"
+ have "?A \<in> 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"
+ assume "a \<in> field(r)" "r -`` {a} \<in> C" "b \<in> field(r)" "r -`` {b} \<in> C"
hence "r -`` {a} \<subseteq> r -`` {b} | r -`` {b} \<subseteq> r -`` {a}" using 2 by auto
- then show "<a, b> : r | <b, a> : r"
- using `Preorder(r)` `a : field(r)` `b : field(r)`
+ then show "<a, b> \<in> r | <b, a> \<in> r"
+ using `Preorder(r)` `a \<in> field(r)` `b \<in> field(r)`
by (simp add: subset_vimage1_vimage1_iff)
qed
- then obtain u where uA: "u : field(r)" "ALL a:?A. <a, u> : r"
+ then obtain u where uA: "u \<in> field(r)" "\<forall>a\<in>?A. <a, u> \<in> r"
using u
apply auto
apply (drule bspec) apply assumption
apply auto
done
- have "ALL A:C. A \<subseteq> r -`` {u}"
+ have "\<forall>A\<in>C. A \<subseteq> 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}"
+ assume aB: "B \<in> C" "a \<in> B"
+ with 1 obtain x where "x \<in> field(r)" "B = r -`` {x}"
apply -
apply (drule subsetD) apply assumption
apply (erule imageE)
apply (erule lamE)
apply simp
done
- then show "<a, u> : r" using uA aB `Preorder(r)`
+ then show "<a, u> \<in> r" using uA aB `Preorder(r)`
by (auto simp: preorder_on_def refl_def) (blast dest: trans_onD)+
qed
- then show "EX U:field(r). ALL A:C. A \<subseteq> r -`` {U}"
- using `u : field(r)` ..
+ then show "\<exists>U\<in>field(r). \<forall>A\<in>C. A \<subseteq> r -`` {U}"
+ using `u \<in> field(r)` ..
qed
from Zorn2 [OF this]
- obtain m B where "m : field(r)" "B = r -`` {m}"
- "ALL x:field(r). B \<subseteq> r -`` {x} --> B = r -`` {x}"
+ obtain m B where "m \<in> field(r)" "B = r -`` {m}"
+ "\<forall>x\<in>field(r). B \<subseteq> r -`` {x} \<longrightarrow> B = r -`` {x}"
by (auto elim!: lamE simp: ball_image_simp)
- then have "ALL a:field(r). <m, a> : r --> a = m"
- using po `Preorder(r)` `m : field(r)`
+ then have "\<forall>a\<in>field(r). <m, a> \<in> r \<longrightarrow> a = m"
+ using po `Preorder(r)` `m \<in> field(r)`
by (auto simp: subset_vimage1_vimage1_iff Partial_order_eq_vimage1_vimage1_iff)
- then show ?thesis using `m : field(r)` by blast
+ then show ?thesis using `m \<in> field(r)` by blast
qed
end