--- a/src/HOL/Set.thy Sun Oct 28 22:58:39 2001 +0100
+++ b/src/HOL/Set.thy Sun Oct 28 22:59:12 2001 +0100
@@ -4,70 +4,71 @@
Copyright 1993 University of Cambridge
*)
-Set = HOL +
+header {* Set theory for higher-order logic *}
+
+theory Set = HOL
+files ("subset.ML") ("equalities.ML") ("mono.ML"):
+
+text {* A set in HOL is simply a predicate. *}
-(** Core syntax **)
+subsection {* Basic syntax *}
global
-types
- 'a set
-
-arities
- set :: (term) term
-
-instance
- set :: (term) {ord, minus}
-
-syntax
- "op :" :: ['a, 'a set] => bool ("op :")
+typedecl 'a set
+arities set :: ("term") "term"
consts
- "{}" :: 'a set ("{}")
- UNIV :: 'a set
- insert :: ['a, 'a set] => 'a set
- Collect :: ('a => bool) => 'a set (*comprehension*)
- Int :: ['a set, 'a set] => 'a set (infixl 70)
- Un :: ['a set, 'a set] => 'a set (infixl 65)
- UNION, INTER :: ['a set, 'a => 'b set] => 'b set (*general*)
- Union, Inter :: (('a set) set) => 'a set (*of a set*)
- Pow :: 'a set => 'a set set (*powerset*)
- Ball, Bex :: ['a set, 'a => bool] => bool (*bounded quantifiers*)
- "image" :: ['a => 'b, 'a set] => ('b set) (infixr "`" 90)
- (*membership*)
- "op :" :: ['a, 'a set] => bool ("(_/ : _)" [50, 51] 50)
+ "{}" :: "'a set" ("{}")
+ UNIV :: "'a set"
+ insert :: "'a => 'a set => 'a set"
+ Collect :: "('a => bool) => 'a set" -- "comprehension"
+ Int :: "'a set => 'a set => 'a set" (infixl 70)
+ Un :: "'a set => 'a set => 'a set" (infixl 65)
+ UNION :: "'a set => ('a => 'b set) => 'b set" -- "general union"
+ INTER :: "'a set => ('a => 'b set) => 'b set" -- "general intersection"
+ Union :: "'a set set => 'a set" -- "union of a set"
+ Inter :: "'a set set => 'a set" -- "intersection of a set"
+ Pow :: "'a set => 'a set set" -- "powerset"
+ Ball :: "'a set => ('a => bool) => bool" -- "bounded universal quantifiers"
+ Bex :: "'a set => ('a => bool) => bool" -- "bounded existential quantifiers"
+ image :: "('a => 'b) => 'a set => 'b set" (infixr "`" 90)
+
+syntax
+ "op :" :: "'a => 'a set => bool" ("op :")
+consts
+ "op :" :: "'a => 'a set => bool" ("(_/ : _)" [50, 51] 50) -- "membership"
+
+local
+
+instance set :: ("term") ord ..
+instance set :: ("term") minus ..
-(** Additional concrete syntax **)
+subsection {* Additional concrete syntax *}
syntax
- range :: ('a => 'b) => 'b set (*of function*)
-
- (* Infix syntax for non-membership *)
+ range :: "('a => 'b) => 'b set" -- "of function"
- "op ~:" :: ['a, 'a set] => bool ("op ~:")
- "op ~:" :: ['a, 'a set] => bool ("(_/ ~: _)" [50, 51] 50)
-
+ "op ~:" :: "'a => 'a set => bool" ("op ~:") -- "non-membership"
+ "op ~:" :: "'a => 'a set => bool" ("(_/ ~: _)" [50, 51] 50)
- "@Finset" :: args => 'a set ("{(_)}")
- "@Coll" :: [pttrn, bool] => 'a set ("(1{_./ _})")
- "@SetCompr" :: ['a, idts, bool] => 'a set ("(1{_ |/_./ _})")
-
- (* Big Intersection / Union *)
+ "@Finset" :: "args => 'a set" ("{(_)}")
+ "@Coll" :: "pttrn => bool => 'a set" ("(1{_./ _})")
+ "@SetCompr" :: "'a => idts => bool => 'a set" ("(1{_ |/_./ _})")
- "@INTER1" :: [pttrns, 'b set] => 'b set ("(3INT _./ _)" 10)
- "@UNION1" :: [pttrns, 'b set] => 'b set ("(3UN _./ _)" 10)
- "@INTER" :: [pttrn, 'a set, 'b set] => 'b set ("(3INT _:_./ _)" 10)
- "@UNION" :: [pttrn, 'a set, 'b set] => 'b set ("(3UN _:_./ _)" 10)
+ "@INTER1" :: "pttrns => 'b set => 'b set" ("(3INT _./ _)" 10)
+ "@UNION1" :: "pttrns => 'b set => 'b set" ("(3UN _./ _)" 10)
+ "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3INT _:_./ _)" 10)
+ "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3UN _:_./ _)" 10)
- (* Bounded Quantifiers *)
- "_Ball" :: [pttrn, 'a set, bool] => bool ("(3ALL _:_./ _)" [0, 0, 10] 10)
- "_Bex" :: [pttrn, 'a set, bool] => bool ("(3EX _:_./ _)" [0, 0, 10] 10)
+ "_Ball" :: "pttrn => 'a set => bool => bool" ("(3ALL _:_./ _)" [0, 0, 10] 10)
+ "_Bex" :: "pttrn => 'a set => bool => bool" ("(3EX _:_./ _)" [0, 0, 10] 10)
syntax (HOL)
- "_Ball" :: [pttrn, 'a set, bool] => bool ("(3! _:_./ _)" [0, 0, 10] 10)
- "_Bex" :: [pttrn, 'a set, bool] => bool ("(3? _:_./ _)" [0, 0, 10] 10)
+ "_Ball" :: "pttrn => 'a set => bool => bool" ("(3! _:_./ _)" [0, 0, 10] 10)
+ "_Bex" :: "pttrn => 'a set => bool => bool" ("(3? _:_./ _)" [0, 0, 10] 10)
translations
"range f" == "f`UNIV"
@@ -85,120 +86,811 @@
"EX x:A. P" == "Bex A (%x. P)"
syntax ("" output)
- "_setle" :: ['a set, 'a set] => bool ("op <=")
- "_setle" :: ['a set, 'a set] => bool ("(_/ <= _)" [50, 51] 50)
- "_setless" :: ['a set, 'a set] => bool ("op <")
- "_setless" :: ['a set, 'a set] => bool ("(_/ < _)" [50, 51] 50)
+ "_setle" :: "'a set => 'a set => bool" ("op <=")
+ "_setle" :: "'a set => 'a set => bool" ("(_/ <= _)" [50, 51] 50)
+ "_setless" :: "'a set => 'a set => bool" ("op <")
+ "_setless" :: "'a set => 'a set => bool" ("(_/ < _)" [50, 51] 50)
syntax (symbols)
- "_setle" :: ['a set, 'a set] => bool ("op \\<subseteq>")
- "_setle" :: ['a set, 'a set] => bool ("(_/ \\<subseteq> _)" [50, 51] 50)
- "_setless" :: ['a set, 'a set] => bool ("op \\<subset>")
- "_setless" :: ['a set, 'a set] => bool ("(_/ \\<subset> _)" [50, 51] 50)
- "op Int" :: ['a set, 'a set] => 'a set (infixl "\\<inter>" 70)
- "op Un" :: ['a set, 'a set] => 'a set (infixl "\\<union>" 65)
- "op :" :: ['a, 'a set] => bool ("op \\<in>")
- "op :" :: ['a, 'a set] => bool ("(_/ \\<in> _)" [50, 51] 50)
- "op ~:" :: ['a, 'a set] => bool ("op \\<notin>")
- "op ~:" :: ['a, 'a set] => bool ("(_/ \\<notin> _)" [50, 51] 50)
- "@UNION1" :: [pttrns, 'b set] => 'b set ("(3\\<Union>_./ _)" 10)
- "@INTER1" :: [pttrns, 'b set] => 'b set ("(3\\<Inter>_./ _)" 10)
- "@UNION" :: [pttrn, 'a set, 'b set] => 'b set ("(3\\<Union>_\\<in>_./ _)" 10)
- "@INTER" :: [pttrn, 'a set, 'b set] => 'b set ("(3\\<Inter>_\\<in>_./ _)" 10)
- Union :: (('a set) set) => 'a set ("\\<Union>_" [90] 90)
- Inter :: (('a set) set) => 'a set ("\\<Inter>_" [90] 90)
- "_Ball" :: [pttrn, 'a set, bool] => bool ("(3\\<forall>_\\<in>_./ _)" [0, 0, 10] 10)
- "_Bex" :: [pttrn, 'a set, bool] => bool ("(3\\<exists>_\\<in>_./ _)" [0, 0, 10] 10)
+ "_setle" :: "'a set => 'a set => bool" ("op \<subseteq>")
+ "_setle" :: "'a set => 'a set => bool" ("(_/ \<subseteq> _)" [50, 51] 50)
+ "_setless" :: "'a set => 'a set => bool" ("op \<subset>")
+ "_setless" :: "'a set => 'a set => bool" ("(_/ \<subset> _)" [50, 51] 50)
+ "op Int" :: "'a set => 'a set => 'a set" (infixl "\<inter>" 70)
+ "op Un" :: "'a set => 'a set => 'a set" (infixl "\<union>" 65)
+ "op :" :: "'a => 'a set => bool" ("op \<in>")
+ "op :" :: "'a => 'a set => bool" ("(_/ \<in> _)" [50, 51] 50)
+ "op ~:" :: "'a => 'a set => bool" ("op \<notin>")
+ "op ~:" :: "'a => 'a set => bool" ("(_/ \<notin> _)" [50, 51] 50)
+ "@UNION1" :: "pttrns => 'b set => 'b set" ("(3\<Union>_./ _)" 10)
+ "@INTER1" :: "pttrns => 'b set => 'b set" ("(3\<Inter>_./ _)" 10)
+ "@UNION" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Union>_\<in>_./ _)" 10)
+ "@INTER" :: "pttrn => 'a set => 'b set => 'b set" ("(3\<Inter>_\<in>_./ _)" 10)
+ Union :: "'a set set => 'a set" ("\<Union>_" [90] 90)
+ Inter :: "'a set set => 'a set" ("\<Inter>_" [90] 90)
+ "_Ball" :: "pttrn => 'a set => bool => bool" ("(3\<forall>_\<in>_./ _)" [0, 0, 10] 10)
+ "_Bex" :: "pttrn => 'a set => bool => bool" ("(3\<exists>_\<in>_./ _)" [0, 0, 10] 10)
translations
- "op \\<subseteq>" => "op <= :: [_ set, _ set] => bool"
- "op \\<subset>" => "op < :: [_ set, _ set] => bool"
-
+ "op \<subseteq>" => "op <= :: _ set => _ set => bool"
+ "op \<subset>" => "op < :: _ set => _ set => bool"
-(** Rules and definitions **)
+typed_print_translation {*
+ let
+ fun le_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =
+ list_comb (Syntax.const "_setle", ts)
+ | le_tr' _ _ _ = raise Match;
+
+ fun less_tr' _ (Type ("fun", (Type ("set", _) :: _))) ts =
+ list_comb (Syntax.const "_setless", ts)
+ | less_tr' _ _ _ = raise Match;
+ in [("op <=", le_tr'), ("op <", less_tr')] end
+*}
-local
+text {*
+ \medskip Translate between @{text "{e | x1...xn. P}"} and @{text
+ "{u. EX x1..xn. u = e & P}"}; @{text "{y. EX x1..xn. y = e & P}"} is
+ only translated if @{text "[0..n] subset bvs(e)"}.
+*}
+
+parse_translation {*
+ let
+ val ex_tr = snd (mk_binder_tr ("EX ", "Ex"));
-rules
+ fun nvars (Const ("_idts", _) $ _ $ idts) = nvars idts + 1
+ | nvars _ = 1;
+
+ fun setcompr_tr [e, idts, b] =
+ let
+ val eq = Syntax.const "op =" $ Bound (nvars idts) $ e;
+ val P = Syntax.const "op &" $ eq $ b;
+ val exP = ex_tr [idts, P];
+ in Syntax.const "Collect" $ Abs ("", dummyT, exP) end;
+
+ in [("@SetCompr", setcompr_tr)] end;
+*}
- (* Isomorphisms between Predicates and Sets *)
+print_translation {*
+ let
+ val ex_tr' = snd (mk_binder_tr' ("Ex", "DUMMY"));
+
+ fun setcompr_tr' [Abs (_, _, P)] =
+ let
+ fun check (Const ("Ex", _) $ Abs (_, _, P), n) = check (P, n + 1)
+ | check (Const ("op &", _) $ (Const ("op =", _) $ Bound m $ e) $ P, n) =
+ if n > 0 andalso m = n andalso not (loose_bvar1 (P, n)) andalso
+ ((0 upto (n - 1)) subset add_loose_bnos (e, 0, [])) then ()
+ else raise Match;
- mem_Collect_eq "(a : {x. P(x)}) = P(a)"
- Collect_mem_eq "{x. x:A} = A"
+ fun tr' (_ $ abs) =
+ let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr' [abs]
+ in Syntax.const "@SetCompr" $ e $ idts $ Q end;
+ in check (P, 0); tr' P end;
+ in [("Collect", setcompr_tr')] end;
+*}
+
+
+subsection {* Rules and definitions *}
+
+text {* Isomorphisms between predicates and sets. *}
+axioms
+ mem_Collect_eq [iff]: "(a : {x. P(x)}) = P(a)"
+ Collect_mem_eq [simp]: "{x. x:A} = A"
+
+defs
+ Ball_def: "Ball A P == ALL x. x:A --> P(x)"
+ Bex_def: "Bex A P == EX x. x:A & P(x)"
+
+defs (overloaded)
+ subset_def: "A <= B == ALL x:A. x:B"
+ psubset_def: "A < B == (A::'a set) <= B & ~ A=B"
+ Compl_def: "- A == {x. ~x:A}"
defs
- Ball_def "Ball A P == ! x. x:A --> P(x)"
- Bex_def "Bex A P == ? x. x:A & P(x)"
- subset_def "A <= B == ! x:A. x:B"
- psubset_def "A < B == (A::'a set) <= B & ~ A=B"
- Compl_def "- A == {x. ~x:A}"
- Un_def "A Un B == {x. x:A | x:B}"
- Int_def "A Int B == {x. x:A & x:B}"
- set_diff_def "A - B == {x. x:A & ~x:B}"
- INTER_def "INTER A B == {y. ! x:A. y: B(x)}"
- UNION_def "UNION A B == {y. ? x:A. y: B(x)}"
- Inter_def "Inter S == (INT x:S. x)"
- Union_def "Union S == (UN x:S. x)"
- Pow_def "Pow A == {B. B <= A}"
- empty_def "{} == {x. False}"
- UNIV_def "UNIV == {x. True}"
- insert_def "insert a B == {x. x=a} Un B"
- image_def "f`A == {y. ? x:A. y=f(x)}"
+ Un_def: "A Un B == {x. x:A | x:B}"
+ Int_def: "A Int B == {x. x:A & x:B}"
+ set_diff_def: "A - B == {x. x:A & ~x:B}"
+ 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)"
+ Pow_def: "Pow A == {B. B <= A}"
+ empty_def: "{} == {x. False}"
+ UNIV_def: "UNIV == {x. True}"
+ insert_def: "insert a B == {x. x=a} Un B"
+ image_def: "f`A == {y. EX x:A. y = f(x)}"
+
+
+subsection {* Lemmas and proof tool setup *}
+
+subsubsection {* Relating predicates and sets *}
+
+lemma CollectI [intro!]: "P(a) ==> a : {x. P(x)}"
+ by simp
+
+lemma CollectD: "a : {x. P(x)} ==> P(a)"
+ by simp
+
+lemma set_ext: "(!!x. (x:A) = (x:B)) ==> A = B"
+proof -
+ case rule_context
+ show ?thesis
+ apply (rule prems [THEN ext, THEN arg_cong, THEN box_equals])
+ apply (rule Collect_mem_eq)
+ apply (rule Collect_mem_eq)
+ done
+qed
+
+lemma Collect_cong: "(!!x. P x = Q x) ==> {x. P(x)} = {x. Q(x)}"
+ by simp
+
+lemmas CollectE [elim!] = CollectD [elim_format]
+
+
+subsubsection {* Bounded quantifiers *}
+
+lemma ballI [intro!]: "(!!x. x:A ==> P x) ==> ALL x:A. P x"
+ by (simp add: Ball_def)
+
+lemmas strip = impI allI ballI
+
+lemma bspec [dest?]: "ALL x:A. P x ==> x:A ==> P x"
+ by (simp add: Ball_def)
+
+lemma ballE [elim]: "ALL x:A. P x ==> (P x ==> Q) ==> (x ~: A ==> Q) ==> Q"
+ by (unfold Ball_def) blast
+
+text {*
+ \medskip This tactic takes assumptions @{prop "ALL x:A. P x"} and
+ @{prop "a:A"}; creates assumption @{prop "P a"}.
+*}
+
+ML {*
+ local val ballE = thm "ballE"
+ in fun ball_tac i = etac ballE i THEN contr_tac (i + 1) end;
+*}
+
+text {*
+ Gives better instantiation for bound:
+*}
+
+ML_setup {*
+ claset_ref() := claset() addbefore ("bspec", datac (thm "bspec") 1);
+*}
+
+lemma bexI [intro]: "P x ==> x:A ==> EX x:A. P x"
+ -- {* Normally the best argument order: @{prop "P x"} constrains the
+ choice of @{prop "x:A"}. *}
+ by (unfold Bex_def) blast
+
+lemma rev_bexI: "x:A ==> P x ==> EX x:A. P x"
+ -- {* The best argument order when there is only one @{prop "x:A"}. *}
+ by (unfold Bex_def) blast
+
+lemma bexCI: "(ALL x:A. ~P x ==> P a) ==> a:A ==> EX x:A. P x"
+ by (unfold Bex_def) blast
+
+lemma bexE [elim!]: "EX x:A. P x ==> (!!x. x:A ==> P x ==> Q) ==> Q"
+ by (unfold Bex_def) blast
+
+lemma ball_triv [simp]: "(ALL x:A. P) = ((EX x. x:A) --> P)"
+ -- {* Trival rewrite rule. *}
+ by (simp add: Ball_def)
+
+lemma bex_triv [simp]: "(EX x:A. P) = ((EX x. x:A) & P)"
+ -- {* Dual form for existentials. *}
+ by (simp add: Bex_def)
+
+lemma bex_triv_one_point1 [simp]: "(EX x:A. x = a) = (a:A)"
+ by blast
+
+lemma bex_triv_one_point2 [simp]: "(EX x:A. a = x) = (a:A)"
+ by blast
+
+lemma bex_one_point1 [simp]: "(EX x:A. x = a & P x) = (a:A & P a)"
+ by blast
+
+lemma bex_one_point2 [simp]: "(EX x:A. a = x & P x) = (a:A & P a)"
+ by blast
+
+lemma ball_one_point1 [simp]: "(ALL x:A. x = a --> P x) = (a:A --> P a)"
+ by blast
+
+lemma ball_one_point2 [simp]: "(ALL x:A. a = x --> P x) = (a:A --> P a)"
+ by blast
+
+ML_setup {*
+ let
+ val Ball_def = thm "Ball_def";
+ val Bex_def = thm "Bex_def";
+
+ val ex_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
+ ("EX x:A. P x & Q x", HOLogic.boolT);
+
+ val prove_bex_tac =
+ rewrite_goals_tac [Bex_def] THEN Quantifier1.prove_one_point_ex_tac;
+ val rearrange_bex = Quantifier1.rearrange_bex prove_bex_tac;
+
+ val all_pattern = Thm.read_cterm (Theory.sign_of (the_context ()))
+ ("ALL x:A. P x --> Q x", HOLogic.boolT);
+
+ val prove_ball_tac =
+ rewrite_goals_tac [Ball_def] THEN Quantifier1.prove_one_point_all_tac;
+ val rearrange_ball = Quantifier1.rearrange_ball prove_ball_tac;
+
+ val defBEX_regroup = mk_simproc "defined BEX" [ex_pattern] rearrange_bex;
+ val defBALL_regroup = mk_simproc "defined BALL" [all_pattern] rearrange_ball;
+ in
+ Addsimprocs [defBALL_regroup, defBEX_regroup]
+ end;
+*}
+
+
+subsubsection {* Congruence rules *}
+
+lemma ball_cong [cong]:
+ "A = B ==> (!!x. x:B ==> P x = Q x) ==>
+ (ALL x:A. P x) = (ALL x:B. Q x)"
+ by (simp add: Ball_def)
+
+lemma bex_cong [cong]:
+ "A = B ==> (!!x. x:B ==> P x = Q x) ==>
+ (EX x:A. P x) = (EX x:B. Q x)"
+ by (simp add: Bex_def cong: conj_cong)
-end
+subsubsection {* Subsets *}
+
+lemma subsetI [intro!]: "(!!x. x:A ==> x:B) ==> A <= B"
+ by (simp add: subset_def)
+
+text {*
+ \medskip Map the type @{text "'a set => anything"} to just @{typ
+ 'a}; for overloading constants whose first argument has type @{typ
+ "'a set"}.
+*}
+
+ML {*
+ fun overload_1st_set s = Blast.overloaded (s, HOLogic.dest_setT o domain_type);
+*}
+
+ML "
+ (* While (:) is not, its type must be kept
+ for overloading of = to work. *)
+ Blast.overloaded (\"op :\", domain_type);
+
+ overload_1st_set \"Ball\"; (*need UNION, INTER also?*)
+ overload_1st_set \"Bex\";
+
+ (*Image: retain the type of the set being expressed*)
+ Blast.overloaded (\"image\", domain_type);
+"
+
+lemma subsetD [elim]: "A <= B ==> c:A ==> c:B"
+ -- {* Rule in Modus Ponens style. *}
+ by (unfold subset_def) blast
+
+declare subsetD [intro?] -- FIXME
+
+lemma rev_subsetD: "c:A ==> A <= B ==> c:B"
+ -- {* The same, with reversed premises for use with @{text erule} --
+ cf @{text rev_mp}. *}
+ by (rule subsetD)
+
+declare rev_subsetD [intro?] -- FIXME
+
+text {*
+ \medskip Converts @{prop "A <= B"} to @{prop "x:A ==> x:B"}.
+*}
+
+ML {*
+ local val rev_subsetD = thm "rev_subsetD"
+ in fun impOfSubs th = th RSN (2, rev_subsetD) end;
+*}
+
+lemma subsetCE [elim]: "A <= B ==> (c~:A ==> P) ==> (c:B ==> P) ==> P"
+ -- {* Classical elimination rule. *}
+ by (unfold subset_def) blast
+
+text {*
+ \medskip Takes assumptions @{prop "A <= B"}; @{prop "c:A"} and
+ creates the assumption @{prop "c:B"}.
+*}
+
+ML {*
+ local val subsetCE = thm "subsetCE"
+ in fun set_mp_tac i = etac subsetCE i THEN mp_tac i end;
+*}
+
+lemma contra_subsetD: "A <= B ==> c ~: B ==> c ~: A"
+ by blast
+
+lemma subset_refl: "A <= (A::'a set)"
+ by fast
+
+lemma subset_trans: "A <= B ==> B <= C ==> A <= (C::'a set)"
+ by blast
-ML
+subsubsection {* Equality *}
+
+lemma subset_antisym [intro!]: "A <= B ==> B <= A ==> A = (B::'a set)"
+ -- {* Anti-symmetry of the subset relation. *}
+ by (rule set_ext) (blast intro: subsetD)
+
+lemmas equalityI = subset_antisym
+
+text {*
+ \medskip Equality rules from ZF set theory -- are they appropriate
+ here?
+*}
+
+lemma equalityD1: "A = B ==> A <= (B::'a set)"
+ by (simp add: subset_refl)
+
+lemma equalityD2: "A = B ==> B <= (A::'a set)"
+ by (simp add: subset_refl)
+
+text {*
+ \medskip Be careful when adding this to the claset as @{text
+ subset_empty} is in the simpset: @{prop "A = {}"} goes to @{prop "{}
+ <= A"} and @{prop "A <= {}"} and then back to @{prop "A = {}"}!
+*}
+
+lemma equalityE: "A = B ==> (A <= B ==> B <= (A::'a set) ==> P) ==> P"
+ by (simp add: subset_refl)
-local
+lemma equalityCE [elim]:
+ "A = B ==> (c:A ==> c:B ==> P) ==> (c~:A ==> c~:B ==> P) ==> P"
+ by blast
+
+text {*
+ \medskip Lemma for creating induction formulae -- for "pattern
+ matching" on @{text p}. To make the induction hypotheses usable,
+ apply @{text spec} or @{text bspec} to put universal quantifiers over the free
+ variables in @{text p}.
+*}
+
+lemma setup_induction: "p:A ==> (!!z. z:A ==> p = z --> R) ==> R"
+ by simp
-(* Set inclusion *)
+lemma eqset_imp_iff: "A = B ==> (x : A) = (x : B)"
+ by simp
+
+
+subsubsection {* The universal set -- UNIV *}
+
+lemma UNIV_I [simp]: "x : UNIV"
+ by (simp add: UNIV_def)
+
+declare UNIV_I [intro] -- {* unsafe makes it less likely to cause problems *}
+
+lemma UNIV_witness [intro?]: "EX x. x : UNIV"
+ by simp
+
+lemma subset_UNIV: "A <= UNIV"
+ by (rule subsetI) (rule UNIV_I)
-fun le_tr' _ (*op <=*) (Type ("fun", (Type ("set", _) :: _))) ts =
- list_comb (Syntax.const "_setle", ts)
- | le_tr' _ (*op <=*) _ _ = raise Match;
+text {*
+ \medskip Eta-contracting these two rules (to remove @{text P})
+ causes them to be ignored because of their interaction with
+ congruence rules.
+*}
+
+lemma ball_UNIV [simp]: "Ball UNIV P = All P"
+ by (simp add: Ball_def)
+
+lemma bex_UNIV [simp]: "Bex UNIV P = Ex P"
+ by (simp add: Bex_def)
+
+
+subsubsection {* The empty set *}
+
+lemma empty_iff [simp]: "(c : {}) = False"
+ by (simp add: empty_def)
+
+lemma emptyE [elim!]: "a : {} ==> P"
+ by simp
+
+lemma empty_subsetI [iff]: "{} <= A"
+ -- {* One effect is to delete the ASSUMPTION @{prop "{} <= A"} *}
+ by blast
+
+lemma equals0I: "(!!y. y:A ==> False) ==> A = {}"
+ by blast
-fun less_tr' _ (*op <*) (Type ("fun", (Type ("set", _) :: _))) ts =
- list_comb (Syntax.const "_setless", ts)
- | less_tr' _ (*op <*) _ _ = raise Match;
+lemma equals0D: "A={} ==> a ~: A"
+ -- {* Use for reasoning about disjointness: @{prop "A Int B = {}"} *}
+ by blast
+
+lemma ball_empty [simp]: "Ball {} P = True"
+ by (simp add: Ball_def)
+
+lemma bex_empty [simp]: "Bex {} P = False"
+ by (simp add: Bex_def)
+
+lemma UNIV_not_empty [iff]: "UNIV ~= {}"
+ by (blast elim: equalityE)
+
+
+section {* The Powerset operator -- Pow *}
+
+lemma Pow_iff [iff]: "(A : Pow B) = (A <= B)"
+ by (simp add: Pow_def)
+
+lemma PowI: "A <= B ==> A : Pow B"
+ by (simp add: Pow_def)
+
+lemma PowD: "A : Pow B ==> A <= B"
+ by (simp add: Pow_def)
+
+lemma Pow_bottom: "{}: Pow B"
+ by simp
+
+lemma Pow_top: "A : Pow A"
+ by (simp add: subset_refl)
-(* Translates between { e | x1..xn. P} and {u. ? x1..xn. u=e & P} *)
-(* {y. ? x1..xn. y = e & P} is only translated if [0..n] subset bvs(e) *)
+subsubsection {* Set complement *}
+
+lemma Compl_iff [simp]: "(c : -A) = (c~:A)"
+ by (unfold Compl_def) blast
+
+lemma ComplI [intro!]: "(c:A ==> False) ==> c : -A"
+ by (unfold Compl_def) blast
+
+text {*
+ \medskip 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 : -A ==> c~:A"
+ by (unfold Compl_def) blast
+
+lemmas ComplE [elim!] = ComplD [elim_format]
+
+
+subsubsection {* Binary union -- Un *}
-val ex_tr = snd(mk_binder_tr("EX ","Ex"));
+lemma Un_iff [simp]: "(c : A Un B) = (c:A | c:B)"
+ by (unfold Un_def) blast
+
+lemma UnI1 [elim?]: "c:A ==> c : A Un B"
+ by simp
+
+lemma UnI2 [elim?]: "c:B ==> c : A Un B"
+ by simp
-fun nvars(Const("_idts",_) $ _ $ idts) = nvars(idts)+1
- | nvars(_) = 1;
+text {*
+ \medskip Classical introduction rule: no commitment to @{prop A} vs
+ @{prop B}.
+*}
+
+lemma UnCI [intro!]: "(c~:B ==> c:A) ==> c : A Un B"
+ by auto
+
+lemma UnE [elim!]: "c : A Un B ==> (c:A ==> P) ==> (c:B ==> P) ==> P"
+ by (unfold Un_def) blast
+
+
+section {* Binary intersection -- Int *}
-fun setcompr_tr[e,idts,b] =
- let val eq = Syntax.const("op =") $ Bound(nvars(idts)) $ e
- val P = Syntax.const("op &") $ eq $ b
- val exP = ex_tr [idts,P]
- in Syntax.const("Collect") $ Abs("",dummyT,exP) end;
+lemma Int_iff [simp]: "(c : A Int B) = (c:A & c:B)"
+ by (unfold Int_def) blast
+
+lemma IntI [intro!]: "c:A ==> c:B ==> c : A Int B"
+ by simp
+
+lemma IntD1: "c : A Int B ==> c:A"
+ by simp
+
+lemma IntD2: "c : A Int B ==> c:B"
+ by simp
+
+lemma IntE [elim!]: "c : A Int B ==> (c:A ==> c:B ==> P) ==> P"
+ by simp
+
+
+section {* Set difference *}
+
+lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
+ by (unfold set_diff_def) blast
-val ex_tr' = snd(mk_binder_tr' ("Ex","DUMMY"));
+lemma DiffI [intro!]: "c : A ==> c ~: B ==> c : A - B"
+ by simp
+
+lemma DiffD1: "c : A - B ==> c : A"
+ by simp
+
+lemma DiffD2: "c : A - B ==> c : B ==> P"
+ by simp
+
+lemma DiffE [elim!]: "c : A - B ==> (c:A ==> c~:B ==> P) ==> P"
+ by simp
+
+
+subsubsection {* Augmenting a set -- insert *}
+
+lemma insert_iff [simp]: "(a : insert b A) = (a = b | a:A)"
+ by (unfold insert_def) blast
+
+lemma insertI1: "a : insert a B"
+ by simp
+
+lemma insertI2: "a : B ==> a : insert b B"
+ by simp
-fun setcompr_tr'[Abs(_,_,P)] =
- let fun ok(Const("Ex",_)$Abs(_,_,P),n) = ok(P,n+1)
- | ok(Const("op &",_) $ (Const("op =",_) $ Bound(m) $ e) $ P, n) =
- if n>0 andalso m=n andalso not(loose_bvar1(P,n)) andalso
- ((0 upto (n-1)) subset add_loose_bnos(e,0,[]))
- then () else raise Match
+lemma insertE [elim!]: "a : insert b A ==> (a = b ==> P) ==> (a:A ==> P) ==> P"
+ by (unfold insert_def) blast
+
+lemma insertCI [intro!]: "(a~:B ==> a = b) ==> a: insert b B"
+ -- {* Classical introduction rule. *}
+ by auto
+
+lemma subset_insert_iff: "(A <= insert x B) = (if x:A then A - {x} <= B else A <= B)"
+ by auto
+
+
+subsubsection {* Singletons, using insert *}
+
+lemma singletonI [intro!]: "a : {a}"
+ -- {* Redundant? But unlike @{text insertCI}, it proves the subgoal immediately! *}
+ by (rule insertI1)
+
+lemma singletonD: "b : {a} ==> b = a"
+ by blast
+
+lemmas singletonE [elim!] = singletonD [elim_format]
+
+lemma singleton_iff: "(b : {a}) = (b = a)"
+ by blast
+
+lemma singleton_inject [dest!]: "{a} = {b} ==> a = b"
+ by blast
+
+lemma singleton_insert_inj_eq [iff]: "({b} = insert a A) = (a = b & A <= {b})"
+ by blast
+
+lemma singleton_insert_inj_eq' [iff]: "(insert a A = {b}) = (a = b & A <= {b})"
+ by blast
+
+lemma subset_singletonD: "A <= {x} ==> A={} | A = {x}"
+ by fast
+
+lemma singleton_conv [simp]: "{x. x = a} = {a}"
+ by blast
+
+lemma singleton_conv2 [simp]: "{x. a = x} = {a}"
+ by blast
- fun tr'(_ $ abs) =
- let val _ $ idts $ (_ $ (_ $ _ $ e) $ Q) = ex_tr'[abs]
- in Syntax.const("@SetCompr") $ e $ idts $ Q end
- in ok(P,0); tr'(P) end;
+lemma diff_single_insert: "A - {x} <= B ==> x : A ==> A <= insert x B"
+ by blast
+
+
+subsubsection {* Unions of families *}
+
+text {*
+ @{term [source] "UN x:A. B x"} is @{term "Union (B`A)"}.
+*}
+
+lemma UN_iff [simp]: "(b: (UN x:A. B x)) = (EX x:A. b: B x)"
+ by (unfold UNION_def) blast
+
+lemma UN_I [intro]: "a:A ==> b: B a ==> b: (UN x:A. B x)"
+ -- {* The order of the premises presupposes that @{term A} is rigid;
+ @{term b} may be flexible. *}
+ by auto
+
+lemma UN_E [elim!]: "b : (UN x:A. B x) ==> (!!x. x:A ==> b: B x ==> R) ==> R"
+ by (unfold UNION_def) blast
-in
+lemma UN_cong [cong]:
+ "A = B ==> (!!x. x:B ==> C x = D x) ==> (UN x:A. C x) = (UN x:B. D x)"
+ by (simp add: UNION_def)
+
+
+subsubsection {* Intersections of families *}
+
+text {* @{term [source] "INT x:A. B x"} is @{term "Inter (B`A)"}. *}
+
+lemma INT_iff [simp]: "(b: (INT x:A. B x)) = (ALL x:A. b: B x)"
+ by (unfold INTER_def) blast
-val parse_translation = [("@SetCompr", setcompr_tr)];
-val print_translation = [("Collect", setcompr_tr')];
-val typed_print_translation = [("op <=", le_tr'), ("op <", less_tr')];
+lemma INT_I [intro!]: "(!!x. x:A ==> b: B x) ==> b : (INT x:A. B x)"
+ by (unfold INTER_def) blast
+
+lemma INT_D [elim]: "b : (INT x:A. B x) ==> a:A ==> b: B a"
+ by auto
+
+lemma INT_E [elim]: "b : (INT x:A. B x) ==> (b: B a ==> R) ==> (a~:A ==> R) ==> R"
+ -- {* "Classical" elimination -- by the Excluded Middle on @{prop "a:A"}. *}
+ by (unfold INTER_def) blast
+
+lemma INT_cong [cong]:
+ "A = B ==> (!!x. x:B ==> C x = D x) ==> (INT x:A. C x) = (INT x:B. D x)"
+ by (simp add: INTER_def)
-end;
+subsubsection {* Union *}
+
+lemma Union_iff [simp]: "(A : Union C) = (EX X:C. A:X)"
+ by (unfold Union_def) blast
+
+lemma UnionI [intro]: "X:C ==> A:X ==> A : Union C"
+ -- {* The order of the premises presupposes that @{term C} is rigid;
+ @{term A} may be flexible. *}
+ by auto
+
+lemma UnionE [elim!]: "A : Union C ==> (!!X. A:X ==> X:C ==> R) ==> R"
+ by (unfold Union_def) blast
+
+
+subsubsection {* Inter *}
+
+lemma Inter_iff [simp]: "(A : Inter C) = (ALL X:C. A:X)"
+ by (unfold Inter_def) blast
+
+lemma InterI [intro!]: "(!!X. X:C ==> A:X) ==> A : Inter C"
+ by (simp add: Inter_def)
+
+text {*
+ \medskip A ``destruct'' rule -- every @{term X} in @{term C}
+ contains @{term A} as an element, but @{prop "A:X"} can hold when
+ @{prop "X:C"} does not! This rule is analogous to @{text spec}.
+*}
+
+lemma InterD [elim]: "A : Inter C ==> X:C ==> A:X"
+ by auto
+
+lemma InterE [elim]: "A : Inter C ==> (X~:C ==> R) ==> (A:X ==> R) ==> R"
+ -- {* ``Classical'' elimination rule -- does not require proving
+ @{prop "X:C"}. *}
+ by (unfold Inter_def) blast
+
+text {*
+ \medskip Image of a set under a function. Frequently @{term b} does
+ not have the syntactic form of @{term "f x"}.
+*}
+
+lemma image_eqI [simp, intro]: "b = f x ==> x:A ==> b : f`A"
+ by (unfold image_def) blast
+
+lemma imageI: "x : A ==> f x : f ` A"
+ by (rule image_eqI) (rule refl)
+
+lemma rev_image_eqI: "x:A ==> b = f x ==> b : f`A"
+ -- {* This version's more effective when we already have the
+ required @{term x}. *}
+ by (unfold image_def) blast
+
+lemma imageE [elim!]:
+ "b : (%x. f x)`A ==> (!!x. b = f x ==> x:A ==> P) ==> P"
+ -- {* The eta-expansion gives variable-name preservation. *}
+ by (unfold image_def) blast
+
+lemma image_Un: "f`(A Un B) = f`A Un f`B"
+ by blast
+
+lemma image_iff: "(z : f`A) = (EX x:A. z = f x)"
+ by blast
+
+lemma image_subset_iff: "(f`A <= B) = (ALL x:A. f x: B)"
+ -- {* This rewrite rule would confuse users if made default. *}
+ by blast
+
+lemma subset_image_iff: "(B <= f ` A) = (EX AA. AA <= A & B = f ` AA)"
+ apply safe
+ prefer 2 apply fast
+ apply (rule_tac x = "{a. a : A & f a : B}" in exI)
+ apply fast
+ done
+
+lemma image_subsetI: "(!!x. x:A ==> f x : B) ==> f`A <= B"
+ -- {* Replaces the three steps @{text subsetI}, @{text imageE},
+ @{text hypsubst}, but breaks too many existing proofs. *}
+ by blast
+
+text {*
+ \medskip Range of a function -- just a translation for image!
+*}
+
+lemma range_eqI: "b = f x ==> b : range f"
+ by simp
+
+lemma rangeI: "f x : range f"
+ by simp
+
+lemma rangeE [elim?]: "b : range (%x. f x) ==> (!!x. b = f x ==> P) ==> P"
+ by blast
+
+
+subsubsection {* Set reasoning tools *}
+
+text {*
+ Rewrite rules for boolean case-splitting: faster than @{text
+ "split_if [split]"}.
+*}
+
+lemma split_if_eq1: "((if Q then x else y) = b) = ((Q --> x = b) & (~ Q --> y = b))"
+ by (rule split_if)
+
+lemma split_if_eq2: "(a = (if Q then x else y)) = ((Q --> a = x) & (~ Q --> a = y))"
+ by (rule split_if)
+
+text {*
+ Split ifs on either side of the membership relation. Not for @{text
+ "[simp]"} -- can cause goals to blow up!
+*}
+
+lemma split_if_mem1: "((if Q then x else y) : b) = ((Q --> x : b) & (~ Q --> y : b))"
+ by (rule split_if)
+
+lemma split_if_mem2: "(a : (if Q then x else y)) = ((Q --> a : x) & (~ Q --> a : y))"
+ by (rule split_if)
+
+lemmas split_ifs = if_bool_eq_conj split_if_eq1 split_if_eq2 split_if_mem1 split_if_mem2
+
+lemmas mem_simps =
+ insert_iff empty_iff Un_iff Int_iff Compl_iff Diff_iff
+ mem_Collect_eq UN_iff Union_iff INT_iff Inter_iff
+ -- {* Each of these has ALREADY been added @{text "[simp]"} above. *}
+
+(*Would like to add these, but the existing code only searches for the
+ outer-level constant, which in this case is just "op :"; we instead need
+ to use term-nets to associate patterns with rules. Also, if a rule fails to
+ apply, then the formula should be kept.
+ [("uminus", Compl_iff RS iffD1), ("op -", [Diff_iff RS iffD1]),
+ ("op Int", [IntD1,IntD2]),
+ ("Collect", [CollectD]), ("Inter", [InterD]), ("INTER", [INT_D])]
+ *)
+
+ML_setup {*
+ val mksimps_pairs = [("Ball", [thm "bspec"])] @ mksimps_pairs;
+ simpset_ref() := simpset() setmksimps (mksimps mksimps_pairs);
+*}
+
+declare subset_UNIV [simp] subset_refl [simp]
+
+
+subsubsection {* The ``proper subset'' relation *}
+
+lemma psubsetI [intro!]: "(A::'a set) <= B ==> A ~= B ==> A < B"
+ by (unfold psubset_def) blast
+
+lemma psubset_insert_iff:
+ "(A < insert x B) = (if x:B then A < B else if x:A then A - {x} < B else A <= B)"
+ apply (simp add: psubset_def subset_insert_iff)
+ apply blast
+ done
+
+lemma psubset_eq: "((A::'a set) < B) = (A <= B & A ~= B)"
+ by (simp only: psubset_def)
+
+lemma psubset_imp_subset: "(A::'a set) < B ==> A <= B"
+ by (simp add: psubset_eq)
+
+lemma psubset_subset_trans: "(A::'a set) < B ==> B <= C ==> A < C"
+ by (auto simp add: psubset_eq)
+
+lemma subset_psubset_trans: "(A::'a set) <= B ==> B < C ==> A < C"
+ by (auto simp add: psubset_eq)
+
+lemma psubset_imp_ex_mem: "A < B ==> EX b. b : (B - A)"
+ by (unfold psubset_def) blast
+
+lemma atomize_ball:
+ "(!!x. x:A ==> P x) == Trueprop (ALL x:A. P x)"
+ by (simp only: Ball_def atomize_all atomize_imp)
+
+declare atomize_ball [symmetric, rulify]
+
+
+subsection {* Further set-theory lemmas *}
+
+use "subset.ML"
+use "equalities.ML"
+use "mono.ML"
+
+end