--- a/src/ZF/Zorn.thy Tue Sep 03 18:49:10 2002 +0200
+++ b/src/ZF/Zorn.thy Tue Sep 03 18:49:30 2002 +0200
@@ -14,29 +14,29 @@
constdefs
Subset_rel :: "i=>i"
- "Subset_rel(A) == {z: A*A . EX x y. z=<x,y> & x<=y & x~=y}"
+ "Subset_rel(A) == {z \<in> A*A . \<exists>x y. z=<x,y> & x<=y & x\<noteq>y}"
chain :: "i=>i"
- "chain(A) == {F: Pow(A). ALL X:F. ALL Y:F. X<=Y | Y<=X}"
+ "chain(A) == {F \<in> Pow(A). \<forall>X\<in>F. \<forall>Y\<in>F. X<=Y | Y<=X}"
maxchain :: "i=>i"
- "maxchain(A) == {c: chain(A). super(A,c)=0}"
-
+ "maxchain(A) == {c \<in> chain(A). super(A,c)=0}"
+
super :: "[i,i]=>i"
- "super(A,c) == {d: chain(A). c<=d & c~=d}"
+ "super(A,c) == {d \<in> chain(A). c<=d & c\<noteq>d}"
constdefs
increasing :: "i=>i"
- "increasing(A) == {f: Pow(A)->Pow(A). ALL x. x<=A --> x<=f`x}"
+ "increasing(A) == {f \<in> Pow(A)->Pow(A). \<forall>x. x<=A --> x<=f`x}"
-(*Lemma for the inductive definition below*)
-lemma Union_in_Pow: "Y : Pow(Pow(A)) ==> Union(Y) : Pow(A)"
+text{*Lemma for the inductive definition below*}
+lemma Union_in_Pow: "Y \<in> Pow(Pow(A)) ==> Union(Y) \<in> Pow(A)"
by blast
-text{*We could make the inductive definition conditional on
+text{*We could make the inductive definition conditional on
@{term "next \<in> 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
@@ -47,10 +47,10 @@
inductive
domains "TFin(S,next)" <= "Pow(S)"
intros
- nextI: "[| x : TFin(S,next); next: increasing(S) |]
- ==> next`x : TFin(S,next)"
+ nextI: "[| x \<in> TFin(S,next); next \<in> increasing(S) |]
+ ==> next`x \<in> TFin(S,next)"
- Pow_UnionI: "Y : Pow(TFin(S,next)) ==> Union(Y) : TFin(S,next)"
+ Pow_UnionI: "Y \<in> Pow(TFin(S,next)) ==> Union(Y) \<in> TFin(S,next)"
monos Pow_mono
con_defs increasing_def
@@ -59,22 +59,22 @@
subsection{*Mathematical Preamble *}
-lemma Union_lemma0: "(ALL x: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:C; ALL x:C. A<=x | x<=B |] ==> A<=Inter(C) | Inter(C)<=B"
+ "[| c \<in> C; \<forall>x\<in>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)"
+lemma increasingD1: "f \<in> increasing(A) ==> f \<in> Pow(A)->Pow(A)"
apply (unfold increasing_def)
apply (erule CollectD1)
done
-lemma increasingD2: "[| f: increasing(A); x<=A |] ==> x <= f`x"
+lemma increasingD2: "[| f \<in> increasing(A); x<=A |] ==> x <= f`x"
by (unfold increasing_def, blast)
lemmas TFin_UnionI = PowI [THEN TFin.Pow_UnionI, standard]
@@ -82,25 +82,24 @@
lemmas TFin_is_subset = TFin.dom_subset [THEN subsetD, THEN PowD, standard]
-(** Structural induction on TFin(S,next) **)
-
+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); ALL y:Y. P(y) |] ==> P(Union(Y))
+ "[| 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))
|] ==> P(n)"
by (erule TFin.induct, blast+)
subsection{*Some Properties of the Transfinite Construction *}
-lemmas increasing_trans = subset_trans [OF _ increasingD2,
+lemmas increasing_trans = subset_trans [OF _ increasingD2,
OF _ _ TFin_is_subset]
-(*Lemma 1 of section 3.1*)
+text{*Lemma 1 of section 3.1*}
lemma TFin_linear_lemma1:
- "[| n: TFin(S,next); m: TFin(S,next);
- ALL x: TFin(S,next) . x<=m --> x=m | next`x<=m |]
+ "[| n \<in> TFin(S,next); m \<in> TFin(S,next);
+ \<forall>x \<in> 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*)
@@ -108,61 +107,62 @@
apply (blast dest: increasing_trans)
done
-(*Lemma 2 of section 3.2. Interesting in its own right!
- Requires next: increasing(S) in the second induction step. *)
+text{*Lemma 2 of section 3.2. Interesting in its own right!
+ Requires @{term "next \<in> increasing(S)"} in the second induction step.*}
lemma TFin_linear_lemma2:
- "[| m: TFin(S,next); next: increasing(S) |]
- ==> ALL n: TFin(S,next) . n<=m --> n=m | next`n<=m"
+ "[| m \<in> TFin(S,next); next \<in> increasing(S) |]
+ ==> \<forall>n \<in> TFin(S,next). n<=m --> n=m | next`n <= m"
apply (erule TFin_induct)
apply (rule impI [THEN ballI])
-(*case split using TFin_linear_lemma1*)
+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)
-(*second induction step*)
+ 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
+prefer 2 apply blast
apply (rule ballI)
-apply (drule bspec, assumption)
-apply (drule subsetD, assumption)
+apply (drule bspec, assumption)
+apply (drule subsetD, assumption)
apply (rule_tac n1 = "n" and m1 = "x" in TFin_linear_lemma1 [THEN disjE],
- assumption+, blast)
+ assumption+, blast)
apply (erule increasingD2 [THEN subset_trans, THEN disjI1])
apply (blast dest: TFin_is_subset)+
done
-(*a more convenient form for Lemma 2*)
+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])
+ "[| n<=m; m \<in> TFin(S,next); n \<in> TFin(S,next); next \<in> increasing(S) |]
+ ==> n=m | next`n <= m"
+by (blast dest: TFin_linear_lemma2 [rule_format])
-(*Consequences from section 3.3 -- Property 3.2, the ordering is total*)
+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)
+ "[| m \<in> TFin(S,next); n \<in> TFin(S,next); next \<in> 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
+apply (blast del: subsetI
intro: subsetI increasingD2 [THEN subset_trans] TFin_is_subset)
done
-(*Lemma 3 of section 3.3*)
+text{*Lemma 3 of section 3.3*}
lemma equal_next_upper:
- "[| n: TFin(S,next); m: TFin(S,next); m = next`m |] ==> n<=m"
+ "[| n \<in> TFin(S,next); m \<in> TFin(S,next); m = next`m |] ==> n <= m"
apply (erule TFin_induct)
apply (drule TFin_subsetD)
-apply (assumption+, force)
+apply (assumption+, force)
apply blast
done
-(*Property 3.3 of section 3.3*)
-lemma equal_next_Union: "[| m: TFin(S,next); next: increasing(S) |]
+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))"
apply (rule iffI)
apply (rule Union_upper [THEN equalityI])
@@ -180,7 +180,7 @@
text{*NOTE: We assume the partial ordering is @{text "\<subseteq>"}, the subset
relation!*}
-(** Defining the "next" operation for Hausdorff's Theorem **)
+text{** Defining the "next" operation for Hausdorff's Theorem **}
lemma chain_subset_Pow: "chain(A) <= Pow(A)"
apply (unfold chain_def)
@@ -197,38 +197,38 @@
apply (rule Collect_subset)
done
-lemma choice_super: "[| ch : (PROD X:Pow(chain(S)) - {0}. X);
- X : chain(S); X ~: maxchain(S) |]
- ==> ch ` super(S,X) : super(S,X)"
+lemma choice_super:
+ "[| ch \<in> (\<Pi>X \<in> Pow(chain(S)) - {0}. X); X \<in> chain(S); X \<notin> 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 : (PROD X:Pow(chain(S)) - {0}. X);
- X : chain(S); X ~: maxchain(S) |]
- ==> ch ` super(S,X) ~= X"
+ "[| ch \<in> (\<Pi>X \<in> Pow(chain(S)) - {0}. X); X \<in> chain(S); X \<notin> maxchain(S) |]
+ ==> ch ` super(S,X) \<noteq> X"
apply (rule notI)
apply (drule choice_super, assumption)
apply assumption
apply (simp add: super_def)
done
-(*This justifies Definition 4.4*)
+text{*This justifies Definition 4.4*}
lemma Hausdorff_next_exists:
- "ch: (PROD X: Pow(chain(S))-{0}. X) ==>
- EX next: increasing(S). ALL X: Pow(S).
- next`X = if(X: chain(S)-maxchain(S), ch`super(S,X), X)"
-apply (rule_tac x="\<lambda>X\<in>Pow(S).
- if X \<in> chain(S) - maxchain(S) then ch ` super(S, X) else X"
+ "ch \<in> (\<Pi>X \<in> 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="\<lambda>X\<in>Pow(S).
+ if X \<in> chain(S) - maxchain(S) then ch ` super(S, X) else X"
in bexI)
-apply force
+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)
-(*Now, verify that it increases*)
+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)
@@ -236,28 +236,28 @@
apply (simp add: super_def, blast)
done
-(*Lemma 4*)
+text{*Lemma 4*}
lemma TFin_chain_lemma4:
- "[| c: TFin(S,next);
- ch: (PROD X: Pow(chain(S))-{0}. X);
- next: increasing(S);
- ALL X: Pow(S). next`X =
- if(X: chain(S)-maxchain(S), ch`super(S,X), X) |]
- ==> c: chain(S)"
+ "[| c \<in> TFin(S,next);
+ ch \<in> (\<Pi>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]
+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+)
- (*Blast_tac's slow*)
+apply (rule_tac m1=B and n1=Ba in TFin_subset_linear [THEN disjE], fast+)
+ txt{*@{text "Blast_tac's"} slow*}
done
-theorem Hausdorff: "EX c. c : maxchain(S)"
+theorem Hausdorff: "\<exists>c. c \<in> 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) ")
+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)
@@ -267,24 +267,22 @@
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,
+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*}
+subsection{*Zorn's Lemma: If All Chains in S Have Upper Bounds In S,
+ then S contains a Maximal Element*}
-text{*If all chains in S have upper bounds in S,
- then S contains a maximal element *}
-
-(*Used in the proof of Zorn's Lemma*)
-lemma chain_extend:
- "[| c: chain(A); z: A; ALL x:c. x<=z |] ==> cons(z,c) : chain(A)"
+text{*Used in the proof of Zorn's Lemma*}
+lemma chain_extend:
+ "[| 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: "ALL c: chain(S). Union(c) : S ==> EX y:S. ALL z: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 --> y=z"
apply (rule Hausdorff [THEN exE])
apply (simp add: maxchain_def)
apply (rename_tac c)
@@ -293,7 +291,7 @@
apply safe
apply (rename_tac z)
apply (rule classical)
-apply (subgoal_tac "cons (z,c) : super (S,c) ")
+apply (subgoal_tac "cons (z,c) \<in> super (S,c) ")
apply (blast elim: equalityE)
apply (unfold super_def, safe)
apply (fast elim: chain_extend)
@@ -303,20 +301,20 @@
subsection{*Zermelo's Theorem: Every Set can be Well-Ordered*}
-(*Lemma 5*)
+text{*Lemma 5*}
lemma TFin_well_lemma5:
- "[| n: TFin(S,next); Z <= TFin(S,next); z:Z; ~ Inter(Z) : Z |]
- ==> ALL m:Z. n<=m"
+ "[| n \<in> TFin(S,next); Z <= TFin(S,next); z:Z; ~ Inter(Z) \<in> Z |]
+ ==> \<forall>m \<in> Z. n <= m"
apply (erule TFin_induct)
-prefer 2 apply blast (*second induction step is easy*)
+prefer 2 apply blast txt{*second induction step is easy*}
apply (rule ballI)
-apply (rule bspec [THEN TFin_subsetD, THEN disjE], auto)
+apply (rule bspec [THEN TFin_subsetD, THEN disjE], auto)
apply (subgoal_tac "m = Inter (Z) ")
apply blast+
done
-(*Well-ordering of TFin(S,next)*)
-lemma well_ord_TFin_lemma: "[| Z <= TFin(S,next); z:Z |] ==> Inter(Z) : Z"
+text{*Well-ordering of @{term "TFin(S,next)"} *}
+lemma well_ord_TFin_lemma: "[| Z <= TFin(S,next); z \<in> Z |] ==> Inter(Z) \<in> Z"
apply (rule classical)
apply (subgoal_tac "Z = {Union (TFin (S,next))}")
apply (simp (no_asm_simp) add: Inter_singleton)
@@ -325,26 +323,27 @@
apply (rule_tac [2] subset_refl [THEN TFin_UnionI, THEN TFin_well_lemma5, THEN bspec], blast+)
done
-(*This theorem just packages the previous result*)
+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)))"
+ "next \<in> increasing(S)
+ ==> well_ord(TFin(S,next), Subset_rel(TFin(S,next)))"
apply (rule well_ordI)
apply (unfold Subset_rel_def linear_def)
-(*Prove the well-foundedness goal*)
+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
-(*Now prove the linearity goal*)
+txt{*Now prove the linearity goal*}
apply (intro ballI)
apply (case_tac "x=y")
apply blast
-(*The x~=y case remains*)
+txt{*The @{term "x\<noteq>y"} case remains*}
apply (rule_tac n1=x and m1=y in TFin_subset_linear [THEN disjE],
assumption+, blast+)
done
-(** Defining the "next" operation for Zermelo's Theorem **)
+text{** Defining the "next" operation for Zermelo's Theorem **}
lemma choice_Diff:
"[| ch \<in> (\<Pi>X \<in> Pow(S) - {0}. X); X \<subseteq> S; X\<noteq>S |] ==> ch ` (S-X) \<in> S-X"
@@ -352,57 +351,58 @@
apply (blast elim!: equalityE)
done
-(*This justifies Definition 6.1*)
+text{*This justifies Definition 6.1*}
lemma Zermelo_next_exists:
- "ch: (PROD X: Pow(S)-{0}. X) ==>
- EX next: increasing(S). ALL X: Pow(S).
+ "ch \<in> (\<Pi>X \<in> 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="\<lambda>X\<in>Pow(S). if X=S then S else cons(ch`(S-X), X)"
in bexI)
-apply force
+apply force
apply (unfold increasing_def)
apply (rule CollectI)
apply (rule lam_type)
-(*Type checking is surprisingly hard!*)
+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])
-(*Verify that it increases*)
-apply (intro allI impI)
+txt{*Verify that it increases*}
+apply (intro allI impI)
apply (simp add: Pow_iff subset_consI subset_refl)
done
-(*The construction of the injection*)
+text{*The construction of the injection*}
lemma choice_imp_injection:
- "[| ch: (PROD X: Pow(S)-{0}. X);
- next: increasing(S);
- ALL X: Pow(S). next`X = if(X=S, S, cons(ch`(S-X), X)) |]
- ==> (lam x:S. Union({y: TFin(S,next). x~: y}))
- : inj(S, TFin(S,next) - {S})"
+ "[| 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}))
+ \<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 ~: Union ({y: TFin (S,next) . x~: 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: TFin (S,next) . x~: y}) ~= S")
+apply (subgoal_tac "Union ({y \<in> TFin (S,next) . x \<notin> y}) \<noteq> S")
prefer 2 apply (blast elim: equalityE)
-(*For proving x : next`Union(...)
- Abrial & 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)
-(*End of the lemmas!*)
+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}) ")
+ 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}) ")
+ 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
-(*The wellordering theorem*)
-theorem AC_well_ord: "EX r. well_ord(S,r)"
+text{*The wellordering theorem*}
+theorem AC_well_ord: "\<exists>r. well_ord(S,r)"
apply (rule AC_Pi_Pow [THEN exE])
apply (rule Zermelo_next_exists [THEN bexE], assumption)
apply (rule exI)
@@ -410,5 +410,5 @@
apply (erule_tac [2] well_ord_TFin)
apply (rule choice_imp_injection [THEN inj_weaken_type], blast+)
done
-
+
end