--- a/src/ZF/ZF.thy Fri Oct 10 12:12:35 2003 +0200
+++ b/src/ZF/ZF.thy Fri Oct 10 17:39:23 2003 +0200
@@ -172,14 +172,18 @@
"op *" :: "[i, i] => i" (infixr "\<times>" 80)
+finalconsts
+ 0 Pow Inf Union PrimReplace
+ "op :"
+
defs
(*don't try to use constdefs: the declaration order is tightly constrained*)
(* Bounded Quantifiers *)
- Ball_def: "Ball(A, P) == ALL x. x:A --> P(x)"
- Bex_def: "Bex(A, P) == EX x. x:A & P(x)"
+ Ball_def: "Ball(A, P) == \<forall>x. x\<in>A --> P(x)"
+ Bex_def: "Bex(A, P) == \<exists>x. x\<in>A & P(x)"
- subset_def: "A <= B == ALL x:A. x:B"
+ subset_def: "A <= B == \<forall>x\<in>A. x\<in>B"
local
@@ -191,18 +195,18 @@
uniqueness is derivable using extensionality. *)
extension: "A = B <-> A <= B & B <= A"
- Union_iff: "A : Union(C) <-> (EX B:C. A:B)"
- Pow_iff: "A : Pow(B) <-> A <= B"
+ Union_iff: "A \<in> Union(C) <-> (\<exists>B\<in>C. A\<in>B)"
+ Pow_iff: "A \<in> Pow(B) <-> A <= B"
(*We may name this set, though it is not uniquely defined.*)
- infinity: "0:Inf & (ALL y:Inf. succ(y): Inf)"
+ infinity: "0\<in>Inf & (\<forall>y\<in>Inf. succ(y): Inf)"
(*This formulation facilitates case analysis on A.*)
- foundation: "A=0 | (EX x:A. ALL y:x. y~:A)"
+ foundation: "A=0 | (\<exists>x\<in>A. \<forall>y\<in>x. y~:A)"
(*Schema axiom since predicate P is a higher-order variable*)
- replacement: "(ALL x:A. ALL y z. P(x,y) & P(x,z) --> y=z) ==>
- b : PrimReplace(A,P) <-> (EX x:A. P(x,b))"
+ replacement: "(\<forall>x\<in>A. \<forall>y z. P(x,y) & P(x,z) --> y=z) ==>
+ b \<in> PrimReplace(A,P) <-> (\<exists>x\<in>A. P(x,b))"
defs
@@ -214,61 +218,61 @@
(* Functional form of replacement -- analgous to ML's map functional *)
- RepFun_def: "RepFun(A,f) == {y . x:A, y=f(x)}"
+ RepFun_def: "RepFun(A,f) == {y . x\<in>A, y=f(x)}"
(* Separation and Pairing can be derived from the Replacement
and Powerset Axioms using the following definitions. *)
- Collect_def: "Collect(A,P) == {y . x:A, x=y & P(x)}"
+ Collect_def: "Collect(A,P) == {y . x\<in>A, x=y & P(x)}"
(*Unordered pairs (Upair) express binary union/intersection and cons;
set enumerations translate as {a,...,z} = cons(a,...,cons(z,0)...)*)
- Upair_def: "Upair(a,b) == {y. x:Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}"
+ Upair_def: "Upair(a,b) == {y. x\<in>Pow(Pow(0)), (x=0 & y=a) | (x=Pow(0) & y=b)}"
cons_def: "cons(a,A) == Upair(a,a) Un A"
succ_def: "succ(i) == cons(i, i)"
(* Difference, general intersection, binary union and small intersection *)
- Diff_def: "A - B == { x:A . ~(x:B) }"
- Inter_def: "Inter(A) == { x:Union(A) . ALL y:A. x:y}"
+ Diff_def: "A - B == { x\<in>A . ~(x\<in>B) }"
+ Inter_def: "Inter(A) == { x\<in>Union(A) . \<forall>y\<in>A. x\<in>y}"
Un_def: "A Un B == Union(Upair(A,B))"
Int_def: "A Int B == Inter(Upair(A,B))"
(* Definite descriptions -- via Replace over the set "1" *)
- the_def: "The(P) == Union({y . x:{0}, P(y)})"
+ the_def: "The(P) == Union({y . x \<in> {0}, P(y)})"
if_def: "if(P,a,b) == THE z. P & z=a | ~P & z=b"
(* this "symmetric" definition works better than {{a}, {a,b}} *)
Pair_def: "<a,b> == {{a,a}, {a,b}}"
- fst_def: "fst(p) == THE a. EX b. p=<a,b>"
- snd_def: "snd(p) == THE b. EX a. p=<a,b>"
+ fst_def: "fst(p) == THE a. \<exists>b. p=<a,b>"
+ snd_def: "snd(p) == THE b. \<exists>a. p=<a,b>"
split_def: "split(c) == %p. c(fst(p), snd(p))"
- Sigma_def: "Sigma(A,B) == UN x:A. UN y:B(x). {<x,y>}"
+ Sigma_def: "Sigma(A,B) == \<Union>x\<in>A. \<Union>y\<in>B(x). {<x,y>}"
(* Operations on relations *)
(*converse of relation r, inverse of function*)
- converse_def: "converse(r) == {z. w:r, EX x y. w=<x,y> & z=<y,x>}"
+ converse_def: "converse(r) == {z. w\<in>r, \<exists>x y. w=<x,y> & z=<y,x>}"
- domain_def: "domain(r) == {x. w:r, EX y. w=<x,y>}"
+ domain_def: "domain(r) == {x. w\<in>r, \<exists>y. w=<x,y>}"
range_def: "range(r) == domain(converse(r))"
field_def: "field(r) == domain(r) Un range(r)"
- relation_def: "relation(r) == ALL z:r. EX x y. z = <x,y>"
+ relation_def: "relation(r) == \<forall>z\<in>r. \<exists>x y. z = <x,y>"
function_def: "function(r) ==
- ALL x y. <x,y>:r --> (ALL y'. <x,y'>:r --> y=y')"
- image_def: "r `` A == {y : range(r) . EX x:A. <x,y> : r}"
+ \<forall>x y. <x,y>:r --> (\<forall>y'. <x,y'>:r --> y=y')"
+ image_def: "r `` A == {y : range(r) . \<exists>x\<in>A. <x,y> : r}"
vimage_def: "r -`` A == converse(r)``A"
(* Abstraction, application and Cartesian product of a family of sets *)
- lam_def: "Lambda(A,b) == {<x,b(x)> . x:A}"
+ lam_def: "Lambda(A,b) == {<x,b(x)> . x\<in>A}"
apply_def: "f`a == Union(f``{a})"
- Pi_def: "Pi(A,B) == {f: Pow(Sigma(A,B)). A<=domain(f) & function(f)}"
+ Pi_def: "Pi(A,B) == {f\<in>Pow(Sigma(A,B)). A<=domain(f) & function(f)}"
(* Restrict the relation r to the domain A *)
- restrict_def: "restrict(r,A) == {z : r. EX x:A. EX y. z = <x,y>}"
+ restrict_def: "restrict(r,A) == {z : r. \<exists>x\<in>A. \<exists>y. z = <x,y>}"
(* Pattern-matching and 'Dependent' type operators *)
@@ -280,63 +284,63 @@
subsection {* Substitution*}
(*Useful examples: singletonI RS subst_elem, subst_elem RSN (2,IntI) *)
-lemma subst_elem: "[| b:A; a=b |] ==> a:A"
+lemma subst_elem: "[| b\<in>A; a=b |] ==> a\<in>A"
by (erule ssubst, assumption)
subsection{*Bounded universal quantifier*}
-lemma ballI [intro!]: "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)"
+lemma ballI [intro!]: "[| !!x. x\<in>A ==> P(x) |] ==> \<forall>x\<in>A. P(x)"
by (simp add: Ball_def)
-lemma bspec [dest?]: "[| ALL x:A. P(x); x: A |] ==> P(x)"
+lemma bspec [dest?]: "[| \<forall>x\<in>A. P(x); x: A |] ==> P(x)"
by (simp add: Ball_def)
(*Instantiates x first: better for automatic theorem proving?*)
lemma rev_ballE [elim]:
- "[| ALL x:A. P(x); x~:A ==> Q; P(x) ==> Q |] ==> Q"
+ "[| \<forall>x\<in>A. P(x); x~:A ==> Q; P(x) ==> Q |] ==> Q"
by (simp add: Ball_def, blast)
-lemma ballE: "[| ALL x:A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"
+lemma ballE: "[| \<forall>x\<in>A. P(x); P(x) ==> Q; x~:A ==> Q |] ==> Q"
by blast
(*Used in the datatype package*)
-lemma rev_bspec: "[| x: A; ALL x:A. P(x) |] ==> P(x)"
+lemma rev_bspec: "[| x: A; \<forall>x\<in>A. P(x) |] ==> P(x)"
by (simp add: Ball_def)
-(*Trival rewrite rule; (ALL x:A.P)<->P holds only if A is nonempty!*)
-lemma ball_triv [simp]: "(ALL x:A. P) <-> ((EX x. x:A) --> P)"
+(*Trival rewrite rule; (\<forall>x\<in>A.P)<->P holds only if A is nonempty!*)
+lemma ball_triv [simp]: "(\<forall>x\<in>A. P) <-> ((\<exists>x. x\<in>A) --> P)"
by (simp add: Ball_def)
(*Congruence rule for rewriting*)
lemma ball_cong [cong]:
- "[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==> (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))"
+ "[| A=A'; !!x. x\<in>A' ==> P(x) <-> P'(x) |] ==> (\<forall>x\<in>A. P(x)) <-> (\<forall>x\<in>A'. P'(x))"
by (simp add: Ball_def)
subsection{*Bounded existential quantifier*}
-lemma bexI [intro]: "[| P(x); x: A |] ==> EX x:A. P(x)"
+lemma bexI [intro]: "[| P(x); x: A |] ==> \<exists>x\<in>A. P(x)"
by (simp add: Bex_def, blast)
-(*The best argument order when there is only one x:A*)
-lemma rev_bexI: "[| x:A; P(x) |] ==> EX x:A. P(x)"
+(*The best argument order when there is only one x\<in>A*)
+lemma rev_bexI: "[| x\<in>A; P(x) |] ==> \<exists>x\<in>A. P(x)"
by blast
-(*Not of the general form for such rules; ~EX has become ALL~ *)
-lemma bexCI: "[| ALL x:A. ~P(x) ==> P(a); a: A |] ==> EX x:A. P(x)"
+(*Not of the general form for such rules; ~\<exists>has become ALL~ *)
+lemma bexCI: "[| \<forall>x\<in>A. ~P(x) ==> P(a); a: A |] ==> \<exists>x\<in>A. P(x)"
by blast
-lemma bexE [elim!]: "[| EX x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"
+lemma bexE [elim!]: "[| \<exists>x\<in>A. P(x); !!x. [| x\<in>A; P(x) |] ==> Q |] ==> Q"
by (simp add: Bex_def, blast)
-(*We do not even have (EX x:A. True) <-> True unless A is nonempty!!*)
-lemma bex_triv [simp]: "(EX x:A. P) <-> ((EX x. x:A) & P)"
+(*We do not even have (\<exists>x\<in>A. True) <-> True unless A is nonempty!!*)
+lemma bex_triv [simp]: "(\<exists>x\<in>A. P) <-> ((\<exists>x. x\<in>A) & P)"
by (simp add: Bex_def)
lemma bex_cong [cong]:
- "[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |]
- ==> (EX x:A. P(x)) <-> (EX x:A'. P'(x))"
+ "[| A=A'; !!x. x\<in>A' ==> P(x) <-> P'(x) |]
+ ==> (\<exists>x\<in>A. P(x)) <-> (\<exists>x\<in>A'. P'(x))"
by (simp add: Bex_def cong: conj_cong)
@@ -344,22 +348,22 @@
subsection{*Rules for subsets*}
lemma subsetI [intro!]:
- "(!!x. x:A ==> x:B) ==> A <= B"
+ "(!!x. x\<in>A ==> x\<in>B) ==> A <= B"
by (simp add: subset_def)
(*Rule in Modus Ponens style [was called subsetE] *)
-lemma subsetD [elim]: "[| A <= B; c:A |] ==> c:B"
+lemma subsetD [elim]: "[| A <= B; c\<in>A |] ==> c\<in>B"
apply (unfold subset_def)
apply (erule bspec, assumption)
done
(*Classical elimination rule*)
lemma subsetCE [elim]:
- "[| A <= B; c~:A ==> P; c:B ==> P |] ==> P"
+ "[| A <= B; c~:A ==> P; c\<in>B ==> P |] ==> P"
by (simp add: subset_def, blast)
(*Sometimes useful with premises in this order*)
-lemma rev_subsetD: "[| c:A; A<=B |] ==> c:B"
+lemma rev_subsetD: "[| c\<in>A; A<=B |] ==> c\<in>B"
by blast
lemma contra_subsetD: "[| A <= B; c ~: B |] ==> c ~: A"
@@ -376,7 +380,7 @@
(*Useful for proving A<=B by rewriting in some cases*)
lemma subset_iff:
- "A<=B <-> (ALL x. x:A --> x:B)"
+ "A<=B <-> (\<forall>x. x\<in>A --> x\<in>B)"
apply (unfold subset_def Ball_def)
apply (rule iff_refl)
done
@@ -389,7 +393,7 @@
by (rule extension [THEN iffD2], rule conjI)
-lemma equality_iffI: "(!!x. x:A <-> x:B) ==> A = B"
+lemma equality_iffI: "(!!x. x\<in>A <-> x\<in>B) ==> A = B"
by (rule equalityI, blast+)
lemmas equalityD1 = extension [THEN iffD1, THEN conjunct1, standard]
@@ -399,13 +403,13 @@
by (blast dest: equalityD1 equalityD2)
lemma equalityCE:
- "[| A = B; [| c:A; c:B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"
+ "[| A = B; [| c\<in>A; c\<in>B |] ==> P; [| c~:A; c~:B |] ==> P |] ==> P"
by (erule equalityE, blast)
(*Lemma for creating induction formulae -- for "pattern matching" on p
To make the induction hypotheses usable, apply "spec" or "bspec" to
put universal quantifiers over the free variables in p.
- Would it be better to do subgoal_tac "ALL z. p = f(z) --> R(z)" ??*)
+ Would it be better to do subgoal_tac "\<forall>z. p = f(z) --> R(z)" ??*)
lemma setup_induction: "[| p: A; !!z. z: A ==> p=z --> R |] ==> R"
by auto
@@ -414,7 +418,7 @@
subsection{*Rules for Replace -- the derived form of replacement*}
lemma Replace_iff:
- "b : {y. x:A, P(x,y)} <-> (EX x:A. P(x,b) & (ALL y. P(x,y) --> y=b))"
+ "b : {y. x\<in>A, P(x,y)} <-> (\<exists>x\<in>A. P(x,b) & (\<forall>y. P(x,y) --> y=b))"
apply (unfold Replace_def)
apply (rule replacement [THEN iff_trans], blast+)
done
@@ -422,25 +426,25 @@
(*Introduction; there must be a unique y such that P(x,y), namely y=b. *)
lemma ReplaceI [intro]:
"[| P(x,b); x: A; !!y. P(x,y) ==> y=b |] ==>
- b : {y. x:A, P(x,y)}"
+ b : {y. x\<in>A, P(x,y)}"
by (rule Replace_iff [THEN iffD2], blast)
(*Elimination; may asssume there is a unique y such that P(x,y), namely y=b. *)
lemma ReplaceE:
- "[| b : {y. x:A, P(x,y)};
- !!x. [| x: A; P(x,b); ALL y. P(x,y)-->y=b |] ==> R
+ "[| b : {y. x\<in>A, P(x,y)};
+ !!x. [| x: A; P(x,b); \<forall>y. P(x,y)-->y=b |] ==> R
|] ==> R"
by (rule Replace_iff [THEN iffD1, THEN bexE], simp+)
(*As above but without the (generally useless) 3rd assumption*)
lemma ReplaceE2 [elim!]:
- "[| b : {y. x:A, P(x,y)};
+ "[| b : {y. x\<in>A, P(x,y)};
!!x. [| x: A; P(x,b) |] ==> R
|] ==> R"
by (erule ReplaceE, blast)
lemma Replace_cong [cong]:
- "[| A=B; !!x y. x:B ==> P(x,y) <-> Q(x,y) |] ==>
+ "[| A=B; !!x y. x\<in>B ==> P(x,y) <-> Q(x,y) |] ==>
Replace(A,P) = Replace(B,Q)"
apply (rule equality_iffI)
apply (simp add: Replace_iff)
@@ -449,52 +453,52 @@
subsection{*Rules for RepFun*}
-lemma RepFunI: "a : A ==> f(a) : {f(x). x:A}"
+lemma RepFunI: "a \<in> A ==> f(a) : {f(x). x\<in>A}"
by (simp add: RepFun_def Replace_iff, blast)
(*Useful for coinduction proofs*)
-lemma RepFun_eqI [intro]: "[| b=f(a); a : A |] ==> b : {f(x). x:A}"
+lemma RepFun_eqI [intro]: "[| b=f(a); a \<in> A |] ==> b : {f(x). x\<in>A}"
apply (erule ssubst)
apply (erule RepFunI)
done
lemma RepFunE [elim!]:
- "[| b : {f(x). x:A};
- !!x.[| x:A; b=f(x) |] ==> P |] ==>
+ "[| b : {f(x). x\<in>A};
+ !!x.[| x\<in>A; b=f(x) |] ==> P |] ==>
P"
by (simp add: RepFun_def Replace_iff, blast)
lemma RepFun_cong [cong]:
- "[| A=B; !!x. x:B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
+ "[| A=B; !!x. x\<in>B ==> f(x)=g(x) |] ==> RepFun(A,f) = RepFun(B,g)"
by (simp add: RepFun_def)
-lemma RepFun_iff [simp]: "b : {f(x). x:A} <-> (EX x:A. b=f(x))"
+lemma RepFun_iff [simp]: "b : {f(x). x\<in>A} <-> (\<exists>x\<in>A. b=f(x))"
by (unfold Bex_def, blast)
-lemma triv_RepFun [simp]: "{x. x:A} = A"
+lemma triv_RepFun [simp]: "{x. x\<in>A} = A"
by blast
subsection{*Rules for Collect -- forming a subset by separation*}
(*Separation is derivable from Replacement*)
-lemma separation [simp]: "a : {x:A. P(x)} <-> a:A & P(a)"
+lemma separation [simp]: "a : {x\<in>A. P(x)} <-> a\<in>A & P(a)"
by (unfold Collect_def, blast)
-lemma CollectI [intro!]: "[| a:A; P(a) |] ==> a : {x:A. P(x)}"
+lemma CollectI [intro!]: "[| a\<in>A; P(a) |] ==> a : {x\<in>A. P(x)}"
by simp
-lemma CollectE [elim!]: "[| a : {x:A. P(x)}; [| a:A; P(a) |] ==> R |] ==> R"
+lemma CollectE [elim!]: "[| a : {x\<in>A. P(x)}; [| a\<in>A; P(a) |] ==> R |] ==> R"
by simp
-lemma CollectD1: "a : {x:A. P(x)} ==> a:A"
+lemma CollectD1: "a : {x\<in>A. P(x)} ==> a\<in>A"
by (erule CollectE, assumption)
-lemma CollectD2: "a : {x:A. P(x)} ==> P(a)"
+lemma CollectD2: "a : {x\<in>A. P(x)} ==> P(a)"
by (erule CollectE, assumption)
lemma Collect_cong [cong]:
- "[| A=B; !!x. x:B ==> P(x) <-> Q(x) |]
+ "[| A=B; !!x. x\<in>B ==> P(x) <-> Q(x) |]
==> Collect(A, %x. P(x)) = Collect(B, %x. Q(x))"
by (simp add: Collect_def)
@@ -507,31 +511,31 @@
lemma UnionI [intro]: "[| B: C; A: B |] ==> A: Union(C)"
by (simp, blast)
-lemma UnionE [elim!]: "[| A : Union(C); !!B.[| A: B; B: C |] ==> R |] ==> R"
+lemma UnionE [elim!]: "[| A \<in> Union(C); !!B.[| A: B; B: C |] ==> R |] ==> R"
by (simp, blast)
subsection{*Rules for Unions of families*}
-(* UN x:A. B(x) abbreviates Union({B(x). x:A}) *)
+(* \<Union>x\<in>A. B(x) abbreviates Union({B(x). x\<in>A}) *)
-lemma UN_iff [simp]: "b : (UN x:A. B(x)) <-> (EX x:A. b : B(x))"
+lemma UN_iff [simp]: "b : (\<Union>x\<in>A. B(x)) <-> (\<exists>x\<in>A. b \<in> B(x))"
by (simp add: Bex_def, blast)
(*The order of the premises presupposes that A is rigid; b may be flexible*)
-lemma UN_I: "[| a: A; b: B(a) |] ==> b: (UN x:A. B(x))"
+lemma UN_I: "[| a: A; b: B(a) |] ==> b: (\<Union>x\<in>A. B(x))"
by (simp, blast)
lemma UN_E [elim!]:
- "[| b : (UN x:A. B(x)); !!x.[| x: A; b: B(x) |] ==> R |] ==> R"
+ "[| b : (\<Union>x\<in>A. B(x)); !!x.[| x: A; b: B(x) |] ==> R |] ==> R"
by blast
lemma UN_cong:
- "[| A=B; !!x. x:B ==> C(x)=D(x) |] ==> (UN x:A. C(x)) = (UN x:B. D(x))"
+ "[| A=B; !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Union>x\<in>A. C(x)) = (\<Union>x\<in>B. D(x))"
by simp
-(*No "Addcongs [UN_cong]" because UN is a combination of constants*)
+(*No "Addcongs [UN_cong]" because \<Union>is a combination of constants*)
(* UN_E appears before UnionE so that it is tried first, to avoid expensive
calls to hyp_subst_tac. Cannot include UN_I as it is unsafe: would enlarge
@@ -540,7 +544,7 @@
subsection{*Rules for the empty set*}
-(*The set {x:0.False} is empty; by foundation it equals 0
+(*The set {x\<in>0. False} is empty; by foundation it equals 0
See Suppes, page 21.*)
lemma not_mem_empty [simp]: "a ~: 0"
apply (cut_tac foundation)
@@ -553,7 +557,7 @@
lemma empty_subsetI [simp]: "0 <= A"
by blast
-lemma equals0I: "[| !!y. y:A ==> False |] ==> A=0"
+lemma equals0I: "[| !!y. y\<in>A ==> False |] ==> A=0"
by blast
lemma equals0D [dest]: "A=0 ==> a ~: A"
@@ -561,75 +565,75 @@
declare sym [THEN equals0D, dest]
-lemma not_emptyI: "a:A ==> A ~= 0"
+lemma not_emptyI: "a\<in>A ==> A ~= 0"
by blast
-lemma not_emptyE: "[| A ~= 0; !!x. x:A ==> R |] ==> R"
+lemma not_emptyE: "[| A ~= 0; !!x. x\<in>A ==> R |] ==> R"
by blast
subsection{*Rules for Inter*}
(*Not obviously useful for proving InterI, InterD, InterE*)
-lemma Inter_iff: "A : Inter(C) <-> (ALL x:C. A: x) & C\<noteq>0"
+lemma Inter_iff: "A \<in> Inter(C) <-> (\<forall>x\<in>C. A: x) & C\<noteq>0"
by (simp add: Inter_def Ball_def, blast)
(* Intersection is well-behaved only if the family is non-empty! *)
lemma InterI [intro!]:
- "[| !!x. x: C ==> A: x; C\<noteq>0 |] ==> A : Inter(C)"
+ "[| !!x. x: C ==> A: x; C\<noteq>0 |] ==> A \<in> Inter(C)"
by (simp add: Inter_iff)
(*A "destruct" rule -- every B in C contains A as an element, but
- A:B can hold when B:C does not! This rule is analogous to "spec". *)
-lemma InterD [elim]: "[| A : Inter(C); B : C |] ==> A : B"
+ A\<in>B can hold when B\<in>C does not! This rule is analogous to "spec". *)
+lemma InterD [elim]: "[| A \<in> Inter(C); B \<in> C |] ==> A \<in> B"
by (unfold Inter_def, blast)
-(*"Classical" elimination rule -- does not require exhibiting B:C *)
+(*"Classical" elimination rule -- does not require exhibiting B\<in>C *)
lemma InterE [elim]:
- "[| A : Inter(C); B~:C ==> R; A:B ==> R |] ==> R"
+ "[| A \<in> Inter(C); B~:C ==> R; A\<in>B ==> R |] ==> R"
by (simp add: Inter_def, blast)
subsection{*Rules for Intersections of families*}
-(* INT x:A. B(x) abbreviates Inter({B(x). x:A}) *)
+(* \<Inter>x\<in>A. B(x) abbreviates Inter({B(x). x\<in>A}) *)
-lemma INT_iff: "b : (INT x:A. B(x)) <-> (ALL x:A. b : B(x)) & A\<noteq>0"
+lemma INT_iff: "b : (\<Inter>x\<in>A. B(x)) <-> (\<forall>x\<in>A. b \<in> B(x)) & A\<noteq>0"
by (force simp add: Inter_def)
-lemma INT_I: "[| !!x. x: A ==> b: B(x); A\<noteq>0 |] ==> b: (INT x:A. B(x))"
+lemma INT_I: "[| !!x. x: A ==> b: B(x); A\<noteq>0 |] ==> b: (\<Inter>x\<in>A. B(x))"
by blast
-lemma INT_E: "[| b : (INT x:A. B(x)); a: A |] ==> b : B(a)"
+lemma INT_E: "[| b : (\<Inter>x\<in>A. B(x)); a: A |] ==> b \<in> B(a)"
by blast
lemma INT_cong:
- "[| A=B; !!x. x:B ==> C(x)=D(x) |] ==> (INT x:A. C(x)) = (INT x:B. D(x))"
+ "[| A=B; !!x. x\<in>B ==> C(x)=D(x) |] ==> (\<Inter>x\<in>A. C(x)) = (\<Inter>x\<in>B. D(x))"
by simp
-(*No "Addcongs [INT_cong]" because INT is a combination of constants*)
+(*No "Addcongs [INT_cong]" because \<Inter>is a combination of constants*)
subsection{*Rules for Powersets*}
-lemma PowI: "A <= B ==> A : Pow(B)"
+lemma PowI: "A <= B ==> A \<in> Pow(B)"
by (erule Pow_iff [THEN iffD2])
-lemma PowD: "A : Pow(B) ==> A<=B"
+lemma PowD: "A \<in> Pow(B) ==> A<=B"
by (erule Pow_iff [THEN iffD1])
declare Pow_iff [iff]
-lemmas Pow_bottom = empty_subsetI [THEN PowI] (* 0 : Pow(B) *)
-lemmas Pow_top = subset_refl [THEN PowI] (* A : Pow(A) *)
+lemmas Pow_bottom = empty_subsetI [THEN PowI] (* 0 \<in> Pow(B) *)
+lemmas Pow_top = subset_refl [THEN PowI] (* A \<in> Pow(A) *)
subsection{*Cantor's Theorem: There is no surjection from a set to its powerset.*}
(*The search is undirected. Allowing redundant introduction rules may
make it diverge. Variable b represents ANY map, such as
- (lam x:A.b(x)): A->Pow(A). *)
-lemma cantor: "EX S: Pow(A). ALL x:A. b(x) ~= S"
+ (lam x\<in>A.b(x)): A->Pow(A). *)
+lemma cantor: "\<exists>S \<in> Pow(A). \<forall>x\<in>A. b(x) ~= S"
by (best elim!: equalityCE del: ReplaceI RepFun_eqI)
ML
@@ -722,10 +726,10 @@
(*Functions for ML scripts*)
ML
{*
-(*Converts A<=B to x:A ==> x:B*)
+(*Converts A<=B to x\<in>A ==> x\<in>B*)
fun impOfSubs th = th RSN (2, rev_subsetD);
-(*Takes assumptions ALL x:A.P(x) and a:A; creates assumption P(a)*)
+(*Takes assumptions \<forall>x\<in>A.P(x) and a\<in>A; creates assumption P(a)*)
val ball_tac = dtac bspec THEN' assume_tac
*}