src/HOL/Product_Type.thy
 author nipkow Mon, 05 Jan 2004 00:46:06 +0100 changeset 14337 e13731554e50 parent 14208 144f45277d5a child 14359 3d9948163018 permissions -rw-r--r--
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
```
(*  Title:      HOL/Product_Type.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

theory Product_Type = Fun
files ("Tools/split_rule.ML"):

subsection {* Unit *}

typedef unit = "{True}"
proof
show "True : ?unit" by blast
qed

constdefs
Unity :: unit    ("'(')")
"() == Abs_unit True"

lemma unit_eq: "u = ()"
by (induct u) (simp add: unit_def Unity_def)

text {*
Simplification procedure for @{thm [source] unit_eq}.  Cannot use
this rule directly --- it loops!
*}

ML_setup {*
val unit_eq_proc =
let val unit_meta_eq = mk_meta_eq (thm "unit_eq") in
Simplifier.simproc (Theory.sign_of (the_context ())) "unit_eq" ["x::unit"]
(fn _ => fn _ => fn t => if HOLogic.is_unit t then None else Some unit_meta_eq)
end;

*}

lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()"
by simp

lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P"
by (rule triv_forall_equality)

lemma unit_induct [induct type: unit]: "P () ==> P x"
by simp

text {*
This rewrite counters the effect of @{text unit_eq_proc} on @{term
[source] "%u::unit. f u"}, replacing it by @{term [source]
f} rather than by @{term [source] "%u. f ()"}.
*}

lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f"
by (rule ext) simp

subsection {* Pairs *}

subsubsection {* Type definition *}

constdefs
Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
"Pair_Rep == (%a b. %x y. x=a & y=b)"

global

typedef (Prod)
('a, 'b) "*"    (infixr 20)
= "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}"
proof
fix a b show "Pair_Rep a b : ?Prod"
by blast
qed

syntax (xsymbols)
"*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
syntax (HTML output)
"*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)

local

subsubsection {* Abstract constants and syntax *}

global

consts
fst      :: "'a * 'b => 'a"
snd      :: "'a * 'b => 'b"
split    :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
curry    :: "['a * 'b => 'c, 'a, 'b] => 'c"
prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
Pair     :: "['a, 'b] => 'a * 'b"
Sigma    :: "['a set, 'a => 'b set] => ('a * 'b) set"

local

text {*
Patterns -- extends pre-defined type @{typ pttrn} used in
abstractions.
*}

nonterminals
tuple_args patterns

syntax
"_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
"_tuple_arg"  :: "'a => tuple_args"                   ("_")
"_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
"_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
""            :: "pttrn => patterns"                  ("_")
"_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
"@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10)
"@Times" ::"['a set,  'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80)

translations
"(x, y)"       == "Pair x y"
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))"
"%(x,y,zs).b"  == "split(%x (y,zs).b)"
"%(x,y).b"     == "split(%x y. b)"
"_abs (Pair x y) t" => "%(x,y).t"
(* The last rule accommodates tuples in `case C ... (x,y) ... => ...'
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *)

"SIGMA x:A. B" => "Sigma A (%x. B)"
"A <*> B"      => "Sigma A (_K B)"

syntax (xsymbols)
"@Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3\<Sigma> _\<in>_./ _)"   10)
"@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ \<times> _" [81, 80] 80)

print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *}

subsubsection {* Definitions *}

defs
Pair_def:     "Pair a b == Abs_Prod(Pair_Rep a b)"
fst_def:      "fst p == THE a. EX b. p = (a, b)"
snd_def:      "snd p == THE b. EX a. p = (a, b)"
split_def:    "split == (%c p. c (fst p) (snd p))"
curry_def:    "curry == (%c x y. c (x,y))"
prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))"
Sigma_def:    "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"

subsubsection {* Lemmas and proof tool setup *}

lemma ProdI: "Pair_Rep a b : Prod"
by (unfold Prod_def) blast

lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'"
apply (unfold Pair_Rep_def)
apply (drule fun_cong [THEN fun_cong], blast)
done

lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod"
apply (rule inj_on_inverseI)
apply (erule Abs_Prod_inverse)
done

lemma Pair_inject:
"(a, b) = (a', b') ==> (a = a' ==> b = b' ==> R) ==> R"
proof -
case rule_context [unfolded Pair_def]
show ?thesis
apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE])
apply (rule rule_context ProdI)+
.
qed

lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')"
by (blast elim!: Pair_inject)

lemma fst_conv [simp]: "fst (a, b) = a"
by (unfold fst_def) blast

lemma snd_conv [simp]: "snd (a, b) = b"
by (unfold snd_def) blast

lemma fst_eqD: "fst (x, y) = a ==> x = a"
by simp

lemma snd_eqD: "snd (x, y) = a ==> y = a"
by simp

lemma PairE_lemma: "EX x y. p = (x, y)"
apply (unfold Pair_def)
apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE])
apply (erule exE, erule exE, rule exI, rule exI)
apply (rule Rep_Prod_inverse [symmetric, THEN trans])
apply (erule arg_cong)
done

lemma PairE [cases type: *]: "(!!x y. p = (x, y) ==> Q) ==> Q"
by (insert PairE_lemma [of p]) blast

ML_setup {*
local val PairE = thm "PairE" in
fun pair_tac s =
EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac];
end;
*}

lemma surjective_pairing: "p = (fst p, snd p)"
-- {* Do not add as rewrite rule: invalidates some proofs in IMP *}
by (cases p) simp

declare surjective_pairing [symmetric, simp]

lemma surj_pair [simp]: "EX x y. z = (x, y)"
apply (rule exI)
apply (rule exI)
apply (rule surjective_pairing)
done

lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))"
proof
fix a b
assume "!!x. PROP P x"
thus "PROP P (a, b)" .
next
fix x
assume "!!a b. PROP P (a, b)"
hence "PROP P (fst x, snd x)" .
thus "PROP P x" by simp
qed

lemmas split_tupled_all = split_paired_all unit_all_eq2

text {*
The rule @{thm [source] split_paired_all} does not work with the
Simplifier because it also affects premises in congrence rules,
where this can lead to premises of the form @{text "!!a b. ... =
?P(a, b)"} which cannot be solved by reflexivity.
*}

ML_setup "
(* replace parameters of product type by individual component parameters *)
val safe_full_simp_tac = generic_simp_tac true (true, false, false);
local (* filtering with exists_paired_all is an essential optimization *)
fun exists_paired_all (Const (\"all\", _) \$ Abs (_, T, t)) =
can HOLogic.dest_prodT T orelse exists_paired_all t
| exists_paired_all (t \$ u) = exists_paired_all t orelse exists_paired_all u
| exists_paired_all (Abs (_, _, t)) = exists_paired_all t
| exists_paired_all _ = false;
val ss = HOL_basic_ss
addsimps [thm \"split_paired_all\", thm \"unit_all_eq2\", thm \"unit_abs_eta_conv\"]
in
val split_all_tac = SUBGOAL (fn (t, i) =>
if exists_paired_all t then safe_full_simp_tac ss i else no_tac);
val unsafe_split_all_tac = SUBGOAL (fn (t, i) =>
if exists_paired_all t then full_simp_tac ss i else no_tac);
fun split_all th =
if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th;
end;

claset_ref() := claset() addSbefore (\"split_all_tac\", split_all_tac);
"

lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))"
-- {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *}
by fast

lemma curry_split [simp]: "curry (split f) = f"

lemma split_curry [simp]: "split (curry f) = f"

lemma curryI [intro!]: "f (a,b) ==> curry f a b"

lemma curryD [dest!]: "curry f a b ==> f (a,b)"

lemma curryE: "[| curry f a b ; f (a,b) ==> Q |] ==> Q"

lemma curry_conv [simp]: "curry f a b = f (a,b)"

lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x"
by fast

lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))"
by fast

lemma split_conv [simp]: "split c (a, b) = c a b"

lemmas split = split_conv  -- {* for backwards compatibility *}

lemmas splitI = split_conv [THEN iffD2, standard]
lemmas splitD = split_conv [THEN iffD1, standard]

lemma split_Pair_apply: "split (%x y. f (x, y)) = f"
-- {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *}
apply (rule ext)
apply (tactic {* pair_tac "x" 1 *}, simp)
done

lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))"
-- {* Can't be added to simpset: loops! *}

lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))"

lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)"
by (simp only: split_tupled_all, simp)

lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q"

lemma split_weak_cong: "p = q ==> split c p = split c q"
-- {* Prevents simplification of @{term c}: much faster *}
by (erule arg_cong)

lemma split_eta: "(%(x, y). f (x, y)) = f"
apply (rule ext)
apply (simp only: split_tupled_all)
apply (rule split_conv)
done

lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g"

text {*
Simplification procedure for @{thm [source] cond_split_eta}.  Using
@{thm [source] split_eta} as a rewrite rule is not general enough,
and using @{thm [source] cond_split_eta} directly would render some
existing proofs very inefficient; similarly for @{text
split_beta}. *}

ML_setup {*

local
val cond_split_eta = thm "cond_split_eta";
fun  Pair_pat k 0 (Bound m) = (m = k)
|    Pair_pat k i (Const ("Pair",  _) \$ Bound m \$ t) = i > 0 andalso
m = k+i andalso Pair_pat k (i-1) t
|    Pair_pat _ _ _ = false;
fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
|   no_args k i (t \$ u) = no_args k i t andalso no_args k i u
|   no_args k i (Bound m) = m < k orelse m > k+i
|   no_args _ _ _ = true;
fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then Some (i,t) else None
|   split_pat tp i (Const ("split", _) \$ Abs (_, _, t)) = split_pat tp (i+1) t
|   split_pat tp i _ = None;
fun metaeq sg lhs rhs = mk_meta_eq (Tactic.prove sg [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs)))
(K (simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1)));

fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
|   beta_term_pat k i (t \$ u) = Pair_pat k i (t \$ u) orelse
(beta_term_pat k i t andalso beta_term_pat k i u)
|   beta_term_pat k i t = no_args k i t;
fun  eta_term_pat k i (f \$ arg) = no_args k i f andalso Pair_pat k i arg
|    eta_term_pat _ _ _ = false;
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
|   subst arg k i (t \$ u) = if Pair_pat k i (t \$ u) then incr_boundvars k arg
else (subst arg k i t \$ subst arg k i u)
|   subst arg k i t = t;
fun beta_proc sg _ (s as Const ("split", _) \$ Abs (_, _, t) \$ arg) =
(case split_pat beta_term_pat 1 t of
Some (i,f) => Some (metaeq sg s (subst arg 0 i f))
| None => None)
|   beta_proc _ _ _ = None;
fun eta_proc sg _ (s as Const ("split", _) \$ Abs (_, _, t)) =
(case split_pat eta_term_pat 1 t of
Some (_,ft) => Some (metaeq sg s (let val (f \$ arg) = ft in f end))
| None => None)
|   eta_proc _ _ _ = None;
in
val split_beta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
"split_beta" ["split f z"] beta_proc;
val split_eta_proc = Simplifier.simproc (Theory.sign_of (the_context ()))
"split_eta" ["split f"] eta_proc;
end;

*}

lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)"
by (subst surjective_pairing, rule split_conv)

lemma split_split: "R (split c p) = (ALL x y. p = (x, y) --> R (c x y))"
-- {* For use with @{text split} and the Simplifier. *}
apply (subst surjective_pairing)
apply (subst split_conv, blast)
done

text {*
@{thm [source] split_split} could be declared as @{text "[split]"}
done after the Splitter has been speeded up significantly;
precompute the constants involved and don't do anything unless the
current goal contains one of those constants.
*}

lemma split_split_asm: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))"
by (subst split_split, simp)

text {*
\medskip @{term split} used as a logical connective or set former.

\medskip These rules are for use with @{text blast}; could instead
call @{text simp} using @{thm [source] split} as rewrite. *}

lemma splitI2: "!!p. [| !!a b. p = (a, b) ==> c a b |] ==> split c p"
apply (simp only: split_tupled_all)
apply (simp (no_asm_simp))
done

lemma splitI2': "!!p. [| !!a b. (a, b) = p ==> c a b x |] ==> split c p x"
apply (simp only: split_tupled_all)
apply (simp (no_asm_simp))
done

lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q"
by (induct p) (auto simp add: split_def)

lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q"
by (induct p) (auto simp add: split_def)

lemma splitE2:
"[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R |] ==> R"
proof -
assume q: "Q (split P z)"
assume r: "!!x y. [|z = (x, y); Q (P x y)|] ==> R"
show R
apply (rule r surjective_pairing)+
apply (rule split_beta [THEN subst], rule q)
done
qed

lemma splitD': "split R (a,b) c ==> R a b c"
by simp

lemma mem_splitI: "z: c a b ==> z: split c (a, b)"
by simp

lemma mem_splitI2: "!!p. [| !!a b. p = (a, b) ==> z: c a b |] ==> z: split c p"
by (simp only: split_tupled_all, simp)

lemma mem_splitE: "[| z: split c p; !!x y. [| p = (x,y); z: c x y |] ==> Q |] ==> Q"
proof -
case rule_context [unfolded split_def]
show ?thesis by (rule rule_context surjective_pairing)+
qed

declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!]
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!]

ML_setup "
local (* filtering with exists_p_split is an essential optimization *)
fun exists_p_split (Const (\"split\",_) \$ _ \$ (Const (\"Pair\",_)\$_\$_)) = true
| exists_p_split (t \$ u) = exists_p_split t orelse exists_p_split u
| exists_p_split (Abs (_, _, t)) = exists_p_split t
| exists_p_split _ = false;
val ss = HOL_basic_ss addsimps [thm \"split_conv\"];
in
val split_conv_tac = SUBGOAL (fn (t, i) =>
if exists_p_split t then safe_full_simp_tac ss i else no_tac);
end;
to quite time-consuming computations (in particular for nested tuples) *)
claset_ref() := claset() addSbefore (\"split_conv_tac\", split_conv_tac);
"

lemma split_eta_SetCompr [simp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P"
by (rule ext, fast)

lemma split_eta_SetCompr2 [simp]: "(%u. EX x y. u = (x, y) & P x y) = split P"
by (rule ext, fast)

lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)"
-- {* Allows simplifications of nested splits in case of independent predicates. *}
apply (rule ext, blast)
done

(* Do NOT make this a simp rule as it
a) only helps in special situations
b) can lead to nontermination in the presence of split_def
*)
lemma split_comp_eq:
"(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))"
by (rule ext, auto)

lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)"
by blast

(*
the following  would be slightly more general,
but cannot be used as rewrite rule:
### Cannot add premise as rewrite rule because it contains (type) unknowns:
### ?y = .x
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)"
by (rtac some_equality 1)
by ( Simp_tac 1)
by (split_all_tac 1)
by (Asm_full_simp_tac 1)
qed "The_split_eq";
*)

lemma injective_fst_snd: "!!x y. [|fst x = fst y; snd x = snd y|] ==> x = y"
by auto

text {*
\bigskip @{term prod_fun} --- action of the product functor upon
functions.
*}

lemma prod_fun [simp]: "prod_fun f g (a, b) = (f a, g b)"

lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)"
apply (rule ext)
apply (tactic {* pair_tac "x" 1 *}, simp)
done

lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)"
apply (rule ext)
apply (tactic {* pair_tac "z" 1 *}, simp)
done

lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r"
apply (rule image_eqI)
apply (rule prod_fun [symmetric], assumption)
done

lemma prod_fun_imageE [elim!]:
"[| c: (prod_fun f g)`r;  !!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P
|] ==> P"
proof -
case rule_context
assume major: "c: (prod_fun f g)`r"
show ?thesis
apply (rule major [THEN imageE])
apply (rule_tac p = x in PairE)
apply (rule rule_context)
prefer 2
apply blast
apply (blast intro: prod_fun)
done
qed

constdefs
upd_fst :: "('a => 'c) => 'a * 'b => 'c * 'b"
"upd_fst f == prod_fun f id"

upd_snd :: "('b => 'c) => 'a * 'b => 'a * 'c"
"upd_snd f == prod_fun id f"

lemma upd_fst_conv [simp]: "upd_fst f (x,y) = (f x,y)"

lemma upd_snd_conv [simp]: "upd_snd f (x,y) = (x,f y)"

text {*
\bigskip Disjoint union of a family of sets -- Sigma.
*}

lemma SigmaI [intro!]: "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B"
by (unfold Sigma_def) blast

lemma SigmaE:
"[| c: Sigma A B;
!!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P
|] ==> P"
-- {* The general elimination rule. *}
by (unfold Sigma_def) blast

text {*
Elimination of @{term "(a, b) : A \<times> B"} -- introduces no
eigenvariables.
*}

lemma SigmaD1: "(a, b) : Sigma A B ==> a : A"
by (erule SigmaE, blast)

lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a"
by (erule SigmaE, blast)

lemma SigmaE2:
"[| (a, b) : Sigma A B;
[| a:A;  b:B(a) |] ==> P
|] ==> P"

declare SigmaE [elim!] SigmaE2 [elim!]

lemma Sigma_mono: "[| A <= C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D"
by blast

lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
by blast

lemma Sigma_empty2 [simp]: "A <*> {} = {}"
by blast

lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV"
by auto

lemma Compl_Times_UNIV1 [simp]: "- (UNIV <*> A) = UNIV <*> (-A)"
by auto

lemma Compl_Times_UNIV2 [simp]: "- (A <*> UNIV) = (-A) <*> UNIV"
by auto

lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))"
by blast

lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)"
by blast

lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)"
by (blast elim: equalityE)

lemma SetCompr_Sigma_eq:
"Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))"
by blast

text {*
\bigskip Complex rules for Sigma.
*}

lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q"
by blast

lemma UN_Times_distrib:
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)"
-- {* Suggested by Pierre Chartier *}
by blast

lemma split_paired_Ball_Sigma [simp]:
"(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))"
by blast

lemma split_paired_Bex_Sigma [simp]:
"(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))"
by blast

lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))"
by blast

lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))"
by blast

lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))"
by blast

lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))"
by blast

lemma Sigma_Diff_distrib1: "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))"
by blast

lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))"
by blast

lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)"
by blast

text {*
Non-dependent versions are needed to avoid the need for higher-order
matching, especially when the rules are re-oriented.
*}

lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)"
by blast

lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)"
by blast

lemma Times_Diff_distrib1: "(A - B) <*> C = (A <*> C) - (B <*> C)"
by blast

lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A"
apply (rule_tac x = "(a, b)" in image_eqI)
apply auto
done

text {*
Setup of internal @{text split_rule}.
*}

constdefs
internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c"
"internal_split == split"

lemma internal_split_conv: "internal_split c (a, b) = c a b"
by (simp only: internal_split_def split_conv)

hide const internal_split

use "Tools/split_rule.ML"
setup SplitRule.setup

end
```