--- a/src/CCL/Set.thy Tue Nov 11 13:50:56 2014 +0100
+++ b/src/CCL/Set.thy Tue Nov 11 15:55:31 2014 +0100
@@ -10,74 +10,74 @@
instance set :: ("term") "term" ..
consts
- Collect :: "['a => o] => 'a set" (*comprehension*)
- Compl :: "('a set) => 'a set" (*complement*)
- Int :: "['a set, 'a set] => 'a set" (infixl "Int" 70)
- Un :: "['a set, 'a set] => 'a set" (infixl "Un" 65)
- Union :: "(('a set)set) => 'a set" (*...of a set*)
- Inter :: "(('a set)set) => 'a set" (*...of a set*)
- UNION :: "['a set, 'a => 'b set] => 'b set" (*general*)
- INTER :: "['a set, 'a => 'b set] => 'b set" (*general*)
- Ball :: "['a set, 'a => o] => o" (*bounded quants*)
- Bex :: "['a set, 'a => o] => o" (*bounded quants*)
- mono :: "['a set => 'b set] => o" (*monotonicity*)
- mem :: "['a, 'a set] => o" (infixl ":" 50) (*membership*)
- subset :: "['a set, 'a set] => o" (infixl "<=" 50)
- singleton :: "'a => 'a set" ("{_}")
+ Collect :: "['a \<Rightarrow> o] \<Rightarrow> 'a set" (*comprehension*)
+ Compl :: "('a set) \<Rightarrow> 'a set" (*complement*)
+ Int :: "['a set, 'a set] \<Rightarrow> 'a set" (infixl "Int" 70)
+ Un :: "['a set, 'a set] \<Rightarrow> 'a set" (infixl "Un" 65)
+ Union :: "(('a set)set) \<Rightarrow> 'a set" (*...of a set*)
+ Inter :: "(('a set)set) \<Rightarrow> 'a set" (*...of a set*)
+ UNION :: "['a set, 'a \<Rightarrow> 'b set] \<Rightarrow> 'b set" (*general*)
+ INTER :: "['a set, 'a \<Rightarrow> 'b set] \<Rightarrow> 'b set" (*general*)
+ Ball :: "['a set, 'a \<Rightarrow> o] \<Rightarrow> o" (*bounded quants*)
+ Bex :: "['a set, 'a \<Rightarrow> o] \<Rightarrow> o" (*bounded quants*)
+ mono :: "['a set \<Rightarrow> 'b set] \<Rightarrow> o" (*monotonicity*)
+ mem :: "['a, 'a set] \<Rightarrow> o" (infixl ":" 50) (*membership*)
+ subset :: "['a set, 'a set] \<Rightarrow> o" (infixl "<=" 50)
+ singleton :: "'a \<Rightarrow> 'a set" ("{_}")
empty :: "'a set" ("{}")
syntax
- "_Coll" :: "[idt, o] => 'a set" ("(1{_./ _})") (*collection*)
+ "_Coll" :: "[idt, o] \<Rightarrow> 'a set" ("(1{_./ _})") (*collection*)
(* Big Intersection / Union *)
- "_INTER" :: "[idt, 'a set, 'b set] => 'b set" ("(INT _:_./ _)" [0, 0, 0] 10)
- "_UNION" :: "[idt, 'a set, 'b set] => 'b set" ("(UN _:_./ _)" [0, 0, 0] 10)
+ "_INTER" :: "[idt, 'a set, 'b set] \<Rightarrow> 'b set" ("(INT _:_./ _)" [0, 0, 0] 10)
+ "_UNION" :: "[idt, 'a set, 'b set] \<Rightarrow> 'b set" ("(UN _:_./ _)" [0, 0, 0] 10)
(* Bounded Quantifiers *)
- "_Ball" :: "[idt, 'a set, o] => o" ("(ALL _:_./ _)" [0, 0, 0] 10)
- "_Bex" :: "[idt, 'a set, o] => o" ("(EX _:_./ _)" [0, 0, 0] 10)
+ "_Ball" :: "[idt, 'a set, o] \<Rightarrow> o" ("(ALL _:_./ _)" [0, 0, 0] 10)
+ "_Bex" :: "[idt, 'a set, o] \<Rightarrow> o" ("(EX _:_./ _)" [0, 0, 0] 10)
translations
- "{x. P}" == "CONST Collect(%x. P)"
- "INT x:A. B" == "CONST INTER(A, %x. B)"
- "UN x:A. B" == "CONST UNION(A, %x. B)"
- "ALL x:A. P" == "CONST Ball(A, %x. P)"
- "EX x:A. P" == "CONST Bex(A, %x. P)"
+ "{x. P}" == "CONST Collect(\<lambda>x. P)"
+ "INT x:A. B" == "CONST INTER(A, \<lambda>x. B)"
+ "UN x:A. B" == "CONST UNION(A, \<lambda>x. B)"
+ "ALL x:A. P" == "CONST Ball(A, \<lambda>x. P)"
+ "EX x:A. P" == "CONST Bex(A, \<lambda>x. P)"
axiomatization where
- mem_Collect_iff: "(a : {x. P(x)}) <-> P(a)" and
- set_extension: "A = B <-> (ALL x. x:A <-> x:B)"
+ mem_Collect_iff: "(a : {x. P(x)}) \<longleftrightarrow> P(a)" and
+ set_extension: "A = B \<longleftrightarrow> (ALL x. x:A \<longleftrightarrow> x:B)"
defs
- Ball_def: "Ball(A, P) == ALL x. x:A --> P(x)"
- Bex_def: "Bex(A, P) == EX x. x:A & P(x)"
- mono_def: "mono(f) == (ALL A B. A <= B --> f(A) <= f(B))"
+ Ball_def: "Ball(A, P) == ALL x. x:A \<longrightarrow> P(x)"
+ Bex_def: "Bex(A, P) == EX x. x:A \<and> P(x)"
+ mono_def: "mono(f) == (ALL A B. A <= B \<longrightarrow> f(A) <= f(B))"
subset_def: "A <= B == ALL x:A. x:B"
singleton_def: "{a} == {x. x=a}"
empty_def: "{} == {x. False}"
Un_def: "A Un B == {x. x:A | x:B}"
- Int_def: "A Int B == {x. x:A & x:B}"
- Compl_def: "Compl(A) == {x. ~x:A}"
+ Int_def: "A Int B == {x. x:A \<and> x:B}"
+ Compl_def: "Compl(A) == {x. \<not>x:A}"
INTER_def: "INTER(A, B) == {y. ALL x:A. y: B(x)}"
UNION_def: "UNION(A, B) == {y. EX x:A. y: B(x)}"
Inter_def: "Inter(S) == (INT x:S. x)"
Union_def: "Union(S) == (UN x:S. x)"
-lemma CollectI: "[| P(a) |] ==> a : {x. P(x)}"
+lemma CollectI: "P(a) \<Longrightarrow> a : {x. P(x)}"
apply (rule mem_Collect_iff [THEN iffD2])
apply assumption
done
-lemma CollectD: "[| a : {x. P(x)} |] ==> P(a)"
+lemma CollectD: "a : {x. P(x)} \<Longrightarrow> P(a)"
apply (erule mem_Collect_iff [THEN iffD1])
done
lemmas CollectE = CollectD [elim_format]
-lemma set_ext: "[| !!x. x:A <-> x:B |] ==> A = B"
+lemma set_ext: "(\<And>x. x:A \<longleftrightarrow> x:B) \<Longrightarrow> A = B"
apply (rule set_extension [THEN iffD2])
apply simp
done
@@ -85,80 +85,79 @@
subsection {* Bounded quantifiers *}
-lemma ballI: "[| !!x. x:A ==> P(x) |] ==> ALL x:A. P(x)"
+lemma ballI: "(\<And>x. x:A \<Longrightarrow> P(x)) \<Longrightarrow> ALL x:A. P(x)"
by (simp add: Ball_def)
-lemma bspec: "[| ALL x:A. P(x); x:A |] ==> P(x)"
+lemma bspec: "\<lbrakk>ALL x:A. P(x); x:A\<rbrakk> \<Longrightarrow> P(x)"
by (simp add: Ball_def)
-lemma ballE: "[| ALL x:A. P(x); P(x) ==> Q; ~ x:A ==> Q |] ==> Q"
+lemma ballE: "\<lbrakk>ALL x:A. P(x); P(x) \<Longrightarrow> Q; \<not> x:A \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
unfolding Ball_def by blast
-lemma bexI: "[| P(x); x:A |] ==> EX x:A. P(x)"
+lemma bexI: "\<lbrakk>P(x); x:A\<rbrakk> \<Longrightarrow> EX x:A. P(x)"
unfolding Bex_def by blast
-lemma bexCI: "[| EX x:A. ~P(x) ==> P(a); a:A |] ==> EX x:A. P(x)"
+lemma bexCI: "\<lbrakk>EX x:A. \<not>P(x) \<Longrightarrow> P(a); a:A\<rbrakk> \<Longrightarrow> EX x:A. P(x)"
unfolding Bex_def by blast
-lemma bexE: "[| EX x:A. P(x); !!x. [| x:A; P(x) |] ==> Q |] ==> Q"
+lemma bexE: "\<lbrakk>EX x:A. P(x); \<And>x. \<lbrakk>x:A; P(x)\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
unfolding Bex_def by blast
(*Trival rewrite rule; (! x:A.P)=P holds only if A is nonempty!*)
-lemma ball_rew: "(ALL x:A. True) <-> True"
+lemma ball_rew: "(ALL x:A. True) \<longleftrightarrow> True"
by (blast intro: ballI)
subsection {* Congruence rules *}
lemma ball_cong:
- "[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==>
- (ALL x:A. P(x)) <-> (ALL x:A'. P'(x))"
+ "\<lbrakk>A = A'; \<And>x. x:A' \<Longrightarrow> P(x) \<longleftrightarrow> P'(x)\<rbrakk> \<Longrightarrow>
+ (ALL x:A. P(x)) \<longleftrightarrow> (ALL x:A'. P'(x))"
by (blast intro: ballI elim: ballE)
lemma bex_cong:
- "[| A=A'; !!x. x:A' ==> P(x) <-> P'(x) |] ==>
- (EX x:A. P(x)) <-> (EX x:A'. P'(x))"
+ "\<lbrakk>A = A'; \<And>x. x:A' \<Longrightarrow> P(x) \<longleftrightarrow> P'(x)\<rbrakk> \<Longrightarrow>
+ (EX x:A. P(x)) \<longleftrightarrow> (EX x:A'. P'(x))"
by (blast intro: bexI elim: bexE)
subsection {* Rules for subsets *}
-lemma subsetI: "(!!x. x:A ==> x:B) ==> A <= B"
+lemma subsetI: "(\<And>x. x:A \<Longrightarrow> x:B) \<Longrightarrow> A <= B"
unfolding subset_def by (blast intro: ballI)
(*Rule in Modus Ponens style*)
-lemma subsetD: "[| A <= B; c:A |] ==> c:B"
+lemma subsetD: "\<lbrakk>A <= B; c:A\<rbrakk> \<Longrightarrow> c:B"
unfolding subset_def by (blast elim: ballE)
(*Classical elimination rule*)
-lemma subsetCE: "[| A <= B; ~(c:A) ==> P; c:B ==> P |] ==> P"
+lemma subsetCE: "\<lbrakk>A <= B; \<not>(c:A) \<Longrightarrow> P; c:B \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (blast dest: subsetD)
lemma subset_refl: "A <= A"
by (blast intro: subsetI)
-lemma subset_trans: "[| A<=B; B<=C |] ==> A<=C"
+lemma subset_trans: "\<lbrakk>A <= B; B <= C\<rbrakk> \<Longrightarrow> A <= C"
by (blast intro: subsetI dest: subsetD)
subsection {* Rules for equality *}
(*Anti-symmetry of the subset relation*)
-lemma subset_antisym: "[| A <= B; B <= A |] ==> A = B"
+lemma subset_antisym: "\<lbrakk>A <= B; B <= A\<rbrakk> \<Longrightarrow> A = B"
by (blast intro: set_ext dest: subsetD)
lemmas equalityI = subset_antisym
(* Equality rules from ZF set theory -- are they appropriate here? *)
-lemma equalityD1: "A = B ==> A<=B"
- and equalityD2: "A = B ==> B<=A"
+lemma equalityD1: "A = B \<Longrightarrow> A<=B"
+ and equalityD2: "A = B \<Longrightarrow> B<=A"
by (simp_all add: subset_refl)
-lemma equalityE: "[| A = B; [| A<=B; B<=A |] ==> P |] ==> P"
+lemma equalityE: "\<lbrakk>A = B; \<lbrakk>A <= B; B <= A\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (simp add: subset_refl)
-lemma equalityCE:
- "[| A = B; [| c:A; c:B |] ==> P; [| ~ c:A; ~ c:B |] ==> P |] ==> P"
+lemma equalityCE: "\<lbrakk>A = B; \<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> P; \<lbrakk>\<not> c:A; \<not> c:B\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (blast elim: equalityE subsetCE)
lemma trivial_set: "{x. x:A} = A"
@@ -167,40 +166,40 @@
subsection {* Rules for binary union *}
-lemma UnI1: "c:A ==> c : A Un B"
- and UnI2: "c:B ==> c : A Un B"
+lemma UnI1: "c:A \<Longrightarrow> c : A Un B"
+ and UnI2: "c:B \<Longrightarrow> c : A Un B"
unfolding Un_def by (blast intro: CollectI)+
(*Classical introduction rule: no commitment to A vs B*)
-lemma UnCI: "(~c:B ==> c:A) ==> c : A Un B"
+lemma UnCI: "(\<not>c:B \<Longrightarrow> c:A) \<Longrightarrow> c : A Un B"
by (blast intro: UnI1 UnI2)
-lemma UnE: "[| c : A Un B; c:A ==> P; c:B ==> P |] ==> P"
+lemma UnE: "\<lbrakk>c : A Un B; c:A \<Longrightarrow> P; c:B \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
unfolding Un_def by (blast dest: CollectD)
subsection {* Rules for small intersection *}
-lemma IntI: "[| c:A; c:B |] ==> c : A Int B"
+lemma IntI: "\<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> c : A Int B"
unfolding Int_def by (blast intro: CollectI)
-lemma IntD1: "c : A Int B ==> c:A"
- and IntD2: "c : A Int B ==> c:B"
+lemma IntD1: "c : A Int B \<Longrightarrow> c:A"
+ and IntD2: "c : A Int B \<Longrightarrow> c:B"
unfolding Int_def by (blast dest: CollectD)+
-lemma IntE: "[| c : A Int B; [| c:A; c:B |] ==> P |] ==> P"
+lemma IntE: "\<lbrakk>c : A Int B; \<lbrakk>c:A; c:B\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (blast dest: IntD1 IntD2)
subsection {* Rules for set complement *}
-lemma ComplI: "[| c:A ==> False |] ==> c : Compl(A)"
+lemma ComplI: "(c:A \<Longrightarrow> False) \<Longrightarrow> c : Compl(A)"
unfolding Compl_def by (blast intro: CollectI)
(*This form, with negated conclusion, works well with the Classical prover.
Negated assumptions behave like formulae on the right side of the notional
turnstile...*)
-lemma ComplD: "[| c : Compl(A) |] ==> ~c:A"
+lemma ComplD: "c : Compl(A) \<Longrightarrow> \<not>c:A"
unfolding Compl_def by (blast dest: CollectD)
lemmas ComplE = ComplD [elim_format]
@@ -211,13 +210,13 @@
lemma empty_eq: "{x. False} = {}"
by (simp add: empty_def)
-lemma emptyD: "a : {} ==> P"
+lemma emptyD: "a : {} \<Longrightarrow> P"
unfolding empty_def by (blast dest: CollectD)
lemmas emptyE = emptyD [elim_format]
lemma not_emptyD:
- assumes "~ A={}"
+ assumes "\<not> A={}"
shows "EX x. x:A"
proof -
have "\<not> (EX x. x:A) \<Longrightarrow> A = {}"
@@ -231,7 +230,7 @@
lemma singletonI: "a : {a}"
unfolding singleton_def by (blast intro: CollectI)
-lemma singletonD: "b : {a} ==> b=a"
+lemma singletonD: "b : {a} \<Longrightarrow> b=a"
unfolding singleton_def by (blast dest: CollectD)
lemmas singletonE = singletonD [elim_format]
@@ -240,58 +239,54 @@
subsection {* Unions of families *}
(*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: "\<lbrakk>a:A; b: B(a)\<rbrakk> \<Longrightarrow> b: (UN x:A. B(x))"
unfolding UNION_def by (blast intro: bexI CollectI)
-lemma UN_E: "[| b : (UN x:A. B(x)); !!x.[| x:A; b: B(x) |] ==> R |] ==> R"
+lemma UN_E: "\<lbrakk>b : (UN x:A. B(x)); \<And>x. \<lbrakk>x:A; b: B(x)\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
unfolding UNION_def by (blast dest: CollectD elim: bexE)
-lemma UN_cong:
- "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==>
- (UN x:A. C(x)) = (UN x:B. D(x))"
+lemma UN_cong: "\<lbrakk>A = B; \<And>x. x:B \<Longrightarrow> C(x) = D(x)\<rbrakk> \<Longrightarrow> (UN x:A. C(x)) = (UN x:B. D(x))"
by (simp add: UNION_def cong: bex_cong)
subsection {* Intersections of families *}
-lemma INT_I: "(!!x. x:A ==> b: B(x)) ==> b : (INT x:A. B(x))"
+lemma INT_I: "(\<And>x. x:A \<Longrightarrow> b: B(x)) \<Longrightarrow> b : (INT x:A. B(x))"
unfolding INTER_def by (blast intro: CollectI ballI)
-lemma INT_D: "[| b : (INT x:A. B(x)); a:A |] ==> b: B(a)"
+lemma INT_D: "\<lbrakk>b : (INT x:A. B(x)); a:A\<rbrakk> \<Longrightarrow> b: B(a)"
unfolding INTER_def by (blast dest: CollectD bspec)
(*"Classical" elimination rule -- does not require proving X:C *)
-lemma INT_E: "[| b : (INT x:A. B(x)); b: B(a) ==> R; ~ a:A ==> R |] ==> R"
+lemma INT_E: "\<lbrakk>b : (INT x:A. B(x)); b: B(a) \<Longrightarrow> R; \<not> a:A \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
unfolding INTER_def by (blast dest: CollectD bspec)
-lemma INT_cong:
- "[| A=B; !!x. x:B ==> C(x) = D(x) |] ==>
- (INT x:A. C(x)) = (INT x:B. D(x))"
+lemma INT_cong: "\<lbrakk>A = B; \<And>x. x:B \<Longrightarrow> C(x) = D(x)\<rbrakk> \<Longrightarrow> (INT x:A. C(x)) = (INT x:B. D(x))"
by (simp add: INTER_def cong: ball_cong)
subsection {* Rules for Unions *}
(*The order of the premises presupposes that C is rigid; A may be flexible*)
-lemma UnionI: "[| X:C; A:X |] ==> A : Union(C)"
+lemma UnionI: "\<lbrakk>X:C; A:X\<rbrakk> \<Longrightarrow> A : Union(C)"
unfolding Union_def by (blast intro: UN_I)
-lemma UnionE: "[| A : Union(C); !!X.[| A:X; X:C |] ==> R |] ==> R"
+lemma UnionE: "\<lbrakk>A : Union(C); \<And>X. \<lbrakk> A:X; X:C\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
unfolding Union_def by (blast elim: UN_E)
subsection {* Rules for Inter *}
-lemma InterI: "[| !!X. X:C ==> A:X |] ==> A : Inter(C)"
+lemma InterI: "(\<And>X. X:C \<Longrightarrow> A:X) \<Longrightarrow> A : Inter(C)"
unfolding Inter_def by (blast intro: INT_I)
(*A "destruct" rule -- every X in C contains A as an element, but
A:X can hold when X:C does not! This rule is analogous to "spec". *)
-lemma InterD: "[| A : Inter(C); X:C |] ==> A:X"
+lemma InterD: "\<lbrakk>A : Inter(C); X:C\<rbrakk> \<Longrightarrow> A:X"
unfolding Inter_def by (blast dest: INT_D)
(*"Classical" elimination rule -- does not require proving X:C *)
-lemma InterE: "[| A : Inter(C); A:X ==> R; ~ X:C ==> R |] ==> R"
+lemma InterE: "\<lbrakk>A : Inter(C); A:X \<Longrightarrow> R; \<not> X:C \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
unfolding Inter_def by (blast elim: INT_E)
@@ -299,19 +294,19 @@
subsection {* Big Union -- least upper bound of a set *}
-lemma Union_upper: "B:A ==> B <= Union(A)"
+lemma Union_upper: "B:A \<Longrightarrow> B <= Union(A)"
by (blast intro: subsetI UnionI)
-lemma Union_least: "[| !!X. X:A ==> X<=C |] ==> Union(A) <= C"
+lemma Union_least: "(\<And>X. X:A \<Longrightarrow> X<=C) \<Longrightarrow> Union(A) <= C"
by (blast intro: subsetI dest: subsetD elim: UnionE)
subsection {* Big Intersection -- greatest lower bound of a set *}
-lemma Inter_lower: "B:A ==> Inter(A) <= B"
+lemma Inter_lower: "B:A \<Longrightarrow> Inter(A) <= B"
by (blast intro: subsetI dest: InterD)
-lemma Inter_greatest: "[| !!X. X:A ==> C<=X |] ==> C <= Inter(A)"
+lemma Inter_greatest: "(\<And>X. X:A \<Longrightarrow> C<=X) \<Longrightarrow> C <= Inter(A)"
by (blast intro: subsetI InterI dest: subsetD)
@@ -323,7 +318,7 @@
lemma Un_upper2: "B <= A Un B"
by (blast intro: subsetI UnI2)
-lemma Un_least: "[| A<=C; B<=C |] ==> A Un B <= C"
+lemma Un_least: "\<lbrakk>A<=C; B<=C\<rbrakk> \<Longrightarrow> A Un B <= C"
by (blast intro: subsetI elim: UnE dest: subsetD)
@@ -335,22 +330,22 @@
lemma Int_lower2: "A Int B <= B"
by (blast intro: subsetI elim: IntE)
-lemma Int_greatest: "[| C<=A; C<=B |] ==> C <= A Int B"
+lemma Int_greatest: "\<lbrakk>C<=A; C<=B\<rbrakk> \<Longrightarrow> C <= A Int B"
by (blast intro: subsetI IntI dest: subsetD)
subsection {* Monotonicity *}
-lemma monoI: "[| !!A B. A <= B ==> f(A) <= f(B) |] ==> mono(f)"
+lemma monoI: "(\<And>A B. A <= B \<Longrightarrow> f(A) <= f(B)) \<Longrightarrow> mono(f)"
unfolding mono_def by blast
-lemma monoD: "[| mono(f); A <= B |] ==> f(A) <= f(B)"
+lemma monoD: "\<lbrakk>mono(f); A <= B\<rbrakk> \<Longrightarrow> f(A) <= f(B)"
unfolding mono_def by blast
-lemma mono_Un: "mono(f) ==> f(A) Un f(B) <= f(A Un B)"
+lemma mono_Un: "mono(f) \<Longrightarrow> f(A) Un f(B) <= f(A Un B)"
by (blast intro: Un_least dest: monoD intro: Un_upper1 Un_upper2)
-lemma mono_Int: "mono(f) ==> f(A Int B) <= f(A) Int f(B)"
+lemma mono_Int: "mono(f) \<Longrightarrow> f(A Int B) <= f(A) Int f(B)"
by (blast intro: Int_greatest dest: monoD intro: Int_lower1 Int_lower2)
@@ -362,12 +357,12 @@
and [elim] = ballE InterD InterE INT_D INT_E subsetD subsetCE
lemma mem_rews:
- "(a : A Un B) <-> (a:A | a:B)"
- "(a : A Int B) <-> (a:A & a:B)"
- "(a : Compl(B)) <-> (~a:B)"
- "(a : {b}) <-> (a=b)"
- "(a : {}) <-> False"
- "(a : {x. P(x)}) <-> P(a)"
+ "(a : A Un B) \<longleftrightarrow> (a:A | a:B)"
+ "(a : A Int B) \<longleftrightarrow> (a:A \<and> a:B)"
+ "(a : Compl(B)) \<longleftrightarrow> (\<not>a:B)"
+ "(a : {b}) \<longleftrightarrow> (a=b)"
+ "(a : {}) \<longleftrightarrow> False"
+ "(a : {x. P(x)}) \<longleftrightarrow> P(a)"
by blast+
lemmas [simp] = trivial_set empty_eq mem_rews
@@ -390,7 +385,7 @@
lemma Int_Un_distrib: "(A Un B) Int C = (A Int C) Un (B Int C)"
by (blast intro: equalityI)
-lemma subset_Int_eq: "(A<=B) <-> (A Int B = A)"
+lemma subset_Int_eq: "(A<=B) \<longleftrightarrow> (A Int B = A)"
by (blast intro: equalityI elim: equalityE)
@@ -412,7 +407,7 @@
"(A Int B) Un (B Int C) Un (C Int A) = (A Un B) Int (B Un C) Int (C Un A)"
by (blast intro: equalityI)
-lemma subset_Un_eq: "(A<=B) <-> (A Un B = B)"
+lemma subset_Un_eq: "(A<=B) \<longleftrightarrow> (A Un B = B)"
by (blast intro: equalityI elim: equalityE)
@@ -440,7 +435,7 @@
by (blast intro: equalityI)
(*Halmos, Naive Set Theory, page 16.*)
-lemma Un_Int_assoc_eq: "((A Int B) Un C = A Int (B Un C)) <-> (C<=A)"
+lemma Un_Int_assoc_eq: "((A Int B) Un C = A Int (B Un C)) \<longleftrightarrow> (C<=A)"
by (blast intro: equalityI elim: equalityE)
@@ -450,7 +445,7 @@
by (blast intro: equalityI)
lemma Union_disjoint:
- "(Union(C) Int A = {x. False}) <-> (ALL B:C. B Int A = {x. False})"
+ "(Union(C) Int A = {x. False}) \<longleftrightarrow> (ALL B:C. B Int A = {x. False})"
by (blast intro: equalityI elim: equalityE)
lemma Inter_Un_distrib: "Inter(A Un B) = Inter(A) Int Inter(B)"
@@ -475,29 +470,25 @@
section {* Monotonicity of various operations *}
-lemma Union_mono: "A<=B ==> Union(A) <= Union(B)"
+lemma Union_mono: "A<=B \<Longrightarrow> Union(A) <= Union(B)"
by blast
-lemma Inter_anti_mono: "[| B<=A |] ==> Inter(A) <= Inter(B)"
+lemma Inter_anti_mono: "B <= A \<Longrightarrow> Inter(A) <= Inter(B)"
by blast
-lemma UN_mono:
- "[| A<=B; !!x. x:A ==> f(x)<=g(x) |] ==>
- (UN x:A. f(x)) <= (UN x:B. g(x))"
+lemma UN_mono: "\<lbrakk>A <= B; \<And>x. x:A \<Longrightarrow> f(x)<=g(x)\<rbrakk> \<Longrightarrow> (UN x:A. f(x)) <= (UN x:B. g(x))"
by blast
-lemma INT_anti_mono:
- "[| B<=A; !!x. x:A ==> f(x)<=g(x) |] ==>
- (INT x:A. f(x)) <= (INT x:A. g(x))"
+lemma INT_anti_mono: "\<lbrakk>B <= A; \<And>x. x:A \<Longrightarrow> f(x) <= g(x)\<rbrakk> \<Longrightarrow> (INT x:A. f(x)) <= (INT x:A. g(x))"
by blast
-lemma Un_mono: "[| A<=C; B<=D |] ==> A Un B <= C Un D"
+lemma Un_mono: "\<lbrakk>A <= C; B <= D\<rbrakk> \<Longrightarrow> A Un B <= C Un D"
by blast
-lemma Int_mono: "[| A<=C; B<=D |] ==> A Int B <= C Int D"
+lemma Int_mono: "\<lbrakk>A <= C; B <= D\<rbrakk> \<Longrightarrow> A Int B <= C Int D"
by blast
-lemma Compl_anti_mono: "[| A<=B |] ==> Compl(B) <= Compl(A)"
+lemma Compl_anti_mono: "A <= B \<Longrightarrow> Compl(B) <= Compl(A)"
by blast
end