added Induct/Binary_Trees.thy, Induct/Tree_Forest (converted from
former ex/TF.ML ex/TF.thy ex/Term.ML ex/Term.thy);
(* Title: HOL/Set.thy
ID: $Id$
Author: Tobias Nipkow and Markus Wenzel, TU Muenchen
License: GPL (GNU GENERAL PUBLIC LICENSE)
*)
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. *}
subsection {* Basic syntax *}
global
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 :: "'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 ..
subsection {* Additional concrete syntax *}
syntax
range :: "('a => 'b) => 'b set" -- "of function"
"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{_ |/_./ _})")
"@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)
"_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)
translations
"range f" == "f`UNIV"
"x ~: y" == "~ (x : y)"
"{x, xs}" == "insert x {xs}"
"{x}" == "insert x {}"
"{x. P}" == "Collect (%x. P)"
"UN x y. B" == "UN x. UN y. B"
"UN x. B" == "UNION UNIV (%x. B)"
"INT x y. B" == "INT x. INT y. B"
"INT x. B" == "INTER UNIV (%x. B)"
"UN x:A. B" == "UNION A (%x. B)"
"INT x:A. B" == "INTER A (%x. B)"
"ALL x:A. P" == "Ball A (%x. P)"
"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)
syntax (xsymbols)
"_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"
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
*}
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"));
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;
*}
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;
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
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)
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
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)
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
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)
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
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)
subsubsection {* 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)
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 *}
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
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
subsubsection {* Binary intersection -- Int *}
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
subsubsection {* Set difference *}
lemma Diff_iff [simp]: "(c : A - B) = (c:A & c~:B)"
by (unfold set_diff_def) blast
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
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
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
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
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)
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"
lemma Least_mono:
"mono (f::'a::order => 'b::order) ==> EX x:S. ALL y:S. x <= y
==> (LEAST y. y : f ` S) = f (LEAST x. x : S)"
-- {* Courtesy of Stephan Merz *}
apply clarify
apply (erule_tac P = "%x. x : S" in LeastI2)
apply fast
apply (rule LeastI2)
apply (auto elim: monoD intro!: order_antisym)
done
subsection {* Transitivity rules for calculational reasoning *}
lemma forw_subst: "a = b ==> P b ==> P a"
by (rule ssubst)
lemma back_subst: "P a ==> a = b ==> P b"
by (rule subst)
lemma set_rev_mp: "x:A ==> A <= B ==> x:B"
by (rule subsetD)
lemma set_mp: "A <= B ==> x:A ==> x:B"
by (rule subsetD)
lemma order_neq_le_trans: "a ~= b ==> (a::'a::order) <= b ==> a < b"
by (simp add: order_less_le)
lemma order_le_neq_trans: "(a::'a::order) <= b ==> a ~= b ==> a < b"
by (simp add: order_less_le)
lemma order_less_asym': "(a::'a::order) < b ==> b < a ==> P"
by (rule order_less_asym)
lemma ord_le_eq_trans: "a <= b ==> b = c ==> a <= c"
by (rule subst)
lemma ord_eq_le_trans: "a = b ==> b <= c ==> a <= c"
by (rule ssubst)
lemma ord_less_eq_trans: "a < b ==> b = c ==> a < c"
by (rule subst)
lemma ord_eq_less_trans: "a = b ==> b < c ==> a < c"
by (rule ssubst)
lemma order_less_subst2: "(a::'a::order) < b ==> f b < (c::'c::order) ==>
(!!x y. x < y ==> f x < f y) ==> f a < c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b < c"
finally (order_less_trans) show ?thesis .
qed
lemma order_less_subst1: "(a::'a::order) < f b ==> (b::'b::order) < c ==>
(!!x y. x < y ==> f x < f y) ==> a < f c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a < f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (order_less_trans) show ?thesis .
qed
lemma order_le_less_subst2: "(a::'a::order) <= b ==> f b < (c::'c::order) ==>
(!!x y. x <= y ==> f x <= f y) ==> f a < c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b < c"
finally (order_le_less_trans) show ?thesis .
qed
lemma order_le_less_subst1: "(a::'a::order) <= f b ==> (b::'b::order) < c ==>
(!!x y. x < y ==> f x < f y) ==> a < f c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a <= f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (order_le_less_trans) show ?thesis .
qed
lemma order_less_le_subst2: "(a::'a::order) < b ==> f b <= (c::'c::order) ==>
(!!x y. x < y ==> f x < f y) ==> f a < c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b <= c"
finally (order_less_le_trans) show ?thesis .
qed
lemma order_less_le_subst1: "(a::'a::order) < f b ==> (b::'b::order) <= c ==>
(!!x y. x <= y ==> f x <= f y) ==> a < f c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a < f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (order_less_le_trans) show ?thesis .
qed
lemma order_subst1: "(a::'a::order) <= f b ==> (b::'b::order) <= c ==>
(!!x y. x <= y ==> f x <= f y) ==> a <= f c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a <= f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (order_trans) show ?thesis .
qed
lemma order_subst2: "(a::'a::order) <= b ==> f b <= (c::'c::order) ==>
(!!x y. x <= y ==> f x <= f y) ==> f a <= c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b <= c"
finally (order_trans) show ?thesis .
qed
lemma ord_le_eq_subst: "a <= b ==> f b = c ==>
(!!x y. x <= y ==> f x <= f y) ==> f a <= c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b = c"
finally (ord_le_eq_trans) show ?thesis .
qed
lemma ord_eq_le_subst: "a = f b ==> b <= c ==>
(!!x y. x <= y ==> f x <= f y) ==> a <= f c"
proof -
assume r: "!!x y. x <= y ==> f x <= f y"
assume "a = f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (ord_eq_le_trans) show ?thesis .
qed
lemma ord_less_eq_subst: "a < b ==> f b = c ==>
(!!x y. x < y ==> f x < f y) ==> f a < c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b = c"
finally (ord_less_eq_trans) show ?thesis .
qed
lemma ord_eq_less_subst: "a = f b ==> b < c ==>
(!!x y. x < y ==> f x < f y) ==> a < f c"
proof -
assume r: "!!x y. x < y ==> f x < f y"
assume "a = f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (ord_eq_less_trans) show ?thesis .
qed
text {*
Note that this list of rules is in reverse order of priorities.
*}
lemmas basic_trans_rules [trans] =
order_less_subst2
order_less_subst1
order_le_less_subst2
order_le_less_subst1
order_less_le_subst2
order_less_le_subst1
order_subst2
order_subst1
ord_le_eq_subst
ord_eq_le_subst
ord_less_eq_subst
ord_eq_less_subst
forw_subst
back_subst
rev_mp
mp
set_rev_mp
set_mp
order_neq_le_trans
order_le_neq_trans
order_less_trans
order_less_asym'
order_le_less_trans
order_less_le_trans
order_trans
order_antisym
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans
trans
end