# Theory Product_Type

theory Product_Type
imports Typedef Inductive
```(*  Title:      HOL/Product_Type.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section ‹Cartesian products›

theory Product_Type
imports Typedef Inductive Fun
keywords "inductive_set" "coinductive_set" :: thy_decl
begin

subsection ‹@{typ bool} is a datatype›

free_constructors (discs_sels) case_bool for True | False
by auto

text ‹Avoid name clashes by prefixing the output of ‹old_rep_datatype› with ‹old›.›

setup ‹Sign.mandatory_path "old"›

old_rep_datatype True False by (auto intro: bool_induct)

setup ‹Sign.parent_path›

text ‹But erase the prefix for properties that are not generated by ‹free_constructors›.›

setup ‹Sign.mandatory_path "bool"›

lemmas induct = old.bool.induct
lemmas inducts = old.bool.inducts
lemmas rec = old.bool.rec
lemmas simps = bool.distinct bool.case bool.rec

setup ‹Sign.parent_path›

declare case_split [cases type: bool]
― ‹prefer plain propositional version›

lemma [code]: "HOL.equal False P ⟷ ¬ P"
and [code]: "HOL.equal True P ⟷ P"
and [code]: "HOL.equal P False ⟷ ¬ P"
and [code]: "HOL.equal P True ⟷ P"
and [code nbe]: "HOL.equal P P ⟷ True"

lemma If_case_cert:
assumes "CASE ≡ (λb. If b f g)"
shows "(CASE True ≡ f) &&& (CASE False ≡ g)"
using assms by simp_all

setup ‹Code.declare_case_global @{thm If_case_cert}›

code_printing
constant "HOL.equal :: bool ⇒ bool ⇒ bool" ⇀ (Haskell) infix 4 "=="
| class_instance "bool" :: "equal" ⇀ (Haskell) -

subsection ‹The ‹unit› type›

typedef unit = "{True}"
by auto

definition Unity :: unit  ("'(')")
where "() = Abs_unit True"

lemma unit_eq [no_atp]: "u = ()"
by (induct u) (simp add: Unity_def)

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

simproc_setup unit_eq ("x::unit") = ‹
fn _ => fn _ => fn ct =>
if HOLogic.is_unit (Thm.term_of ct) then NONE
else SOME (mk_meta_eq @{thm unit_eq})
›

free_constructors case_unit for "()"
by auto

text ‹Avoid name clashes by prefixing the output of ‹old_rep_datatype› with ‹old›.›

setup ‹Sign.mandatory_path "old"›

old_rep_datatype "()" by simp

setup ‹Sign.parent_path›

text ‹But erase the prefix for properties that are not generated by ‹free_constructors›.›

setup ‹Sign.mandatory_path "unit"›

lemmas induct = old.unit.induct
lemmas inducts = old.unit.inducts
lemmas rec = old.unit.rec
lemmas simps = unit.case unit.rec

setup ‹Sign.parent_path›

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)

text ‹
This rewrite counters the effect of simproc ‹unit_eq› on @{term
[source] "λu::unit. f u"}, replacing it by @{term [source]
f} rather than by @{term [source] "λu. f ()"}.
›

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

lemma UNIV_unit: "UNIV = {()}"
by auto

instantiation unit :: default
begin

definition "default = ()"

instance ..

end

instantiation unit :: "{complete_boolean_algebra,complete_linorder,wellorder}"
begin

definition less_eq_unit :: "unit ⇒ unit ⇒ bool"
where "(_::unit) ≤ _ ⟷ True"

lemma less_eq_unit [iff]: "u ≤ v" for u v :: unit

definition less_unit :: "unit ⇒ unit ⇒ bool"
where "(_::unit) < _ ⟷ False"

lemma less_unit [iff]: "¬ u < v" for u v :: unit

definition bot_unit :: unit
where [code_unfold]: "⊥ = ()"

definition top_unit :: unit
where [code_unfold]: "⊤ = ()"

definition inf_unit :: "unit ⇒ unit ⇒ unit"
where [simp]: "_ ⊓ _ = ()"

definition sup_unit :: "unit ⇒ unit ⇒ unit"
where [simp]: "_ ⊔ _ = ()"

definition Inf_unit :: "unit set ⇒ unit"
where [simp]: "⨅_ = ()"

definition Sup_unit :: "unit set ⇒ unit"
where [simp]: "⨆_ = ()"

definition uminus_unit :: "unit ⇒ unit"
where [simp]: "- _ = ()"

declare less_eq_unit_def [abs_def, code_unfold]
less_unit_def [abs_def, code_unfold]
inf_unit_def [abs_def, code_unfold]
sup_unit_def [abs_def, code_unfold]
Inf_unit_def [abs_def, code_unfold]
Sup_unit_def [abs_def, code_unfold]
uminus_unit_def [abs_def, code_unfold]

instance
by intro_classes auto

end

lemma [code]: "HOL.equal u v ⟷ True" for u v :: unit
unfolding equal unit_eq [of u] unit_eq [of v] by rule+

code_printing
type_constructor unit ⇀
(SML) "unit"
and (OCaml) "unit"
and (Scala) "Unit"
| constant Unity ⇀
(SML) "()"
and (OCaml) "()"
and (Scala) "()"
| class_instance unit :: equal ⇀
| constant "HOL.equal :: unit ⇒ unit ⇒ bool" ⇀

code_reserved SML
unit

code_reserved OCaml
unit

code_reserved Scala
Unit

subsection ‹The product type›

subsubsection ‹Type definition›

definition Pair_Rep :: "'a ⇒ 'b ⇒ 'a ⇒ 'b ⇒ bool"
where "Pair_Rep a b = (λx y. x = a ∧ y = b)"

definition "prod = {f. ∃a b. f = Pair_Rep (a::'a) (b::'b)}"

typedef ('a, 'b) prod ("(_ ×/ _)" [21, 20] 20) = "prod :: ('a ⇒ 'b ⇒ bool) set"
unfolding prod_def by auto

type_notation (ASCII)
prod  (infixr "*" 20)

definition Pair :: "'a ⇒ 'b ⇒ 'a × 'b"
where "Pair a b = Abs_prod (Pair_Rep a b)"

lemma prod_cases: "(⋀a b. P (Pair a b)) ⟹ P p"
by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)

free_constructors case_prod for Pair fst snd
proof -
fix P :: bool and p :: "'a × 'b"
show "(⋀x1 x2. p = Pair x1 x2 ⟹ P) ⟹ P"
by (cases p) (auto simp add: prod_def Pair_def Pair_Rep_def)
next
fix a c :: 'a and b d :: 'b
have "Pair_Rep a b = Pair_Rep c d ⟷ a = c ∧ b = d"
by (auto simp add: Pair_Rep_def fun_eq_iff)
moreover have "Pair_Rep a b ∈ prod" and "Pair_Rep c d ∈ prod"
ultimately show "Pair a b = Pair c d ⟷ a = c ∧ b = d"
qed

text ‹Avoid name clashes by prefixing the output of ‹old_rep_datatype› with ‹old›.›

setup ‹Sign.mandatory_path "old"›

old_rep_datatype Pair
by (erule prod_cases) (rule prod.inject)

setup ‹Sign.parent_path›

text ‹But erase the prefix for properties that are not generated by ‹free_constructors›.›

setup ‹Sign.mandatory_path "prod"›

declare old.prod.inject [iff del]

lemmas induct = old.prod.induct
lemmas inducts = old.prod.inducts
lemmas rec = old.prod.rec
lemmas simps = prod.inject prod.case prod.rec

setup ‹Sign.parent_path›

declare prod.case [nitpick_simp del]
declare old.prod.case_cong_weak [cong del]
declare prod.case_eq_if [mono]
declare prod.split [no_atp]
declare prod.split_asm [no_atp]

text ‹
@{thm [source] prod.split} could be declared as ‹[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.
›

subsubsection ‹Tuple syntax›

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

nonterminal tuple_args and 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"      ("_,/ _")
"_unit"       :: pttrn                                ("'(')")
translations
"(x, y)" ⇌ "CONST Pair x y"
"_pattern x y" ⇌ "CONST Pair x y"
"_patterns x y" ⇌ "CONST Pair x y"
"_tuple x (_tuple_args y z)" ⇌ "_tuple x (_tuple_arg (_tuple y z))"
"λ(x, y, zs). b" ⇌ "CONST case_prod (λx (y, zs). b)"
"λ(x, y). b" ⇌ "CONST case_prod (λx y. b)"
"_abs (CONST Pair x y) t" ⇀ "λ(x, y). t"
― ‹This rule accommodates tuples in ‹case C … (x, y) … ⇒ …›:
The ‹(x, y)› is parsed as ‹Pair x y› because it is ‹logic›,
not ‹pttrn›.›
"λ(). b" ⇌ "CONST case_unit b"
"_abs (CONST Unity) t" ⇀ "λ(). t"

text ‹print @{term "case_prod f"} as @{term "λ(x, y). f x y"} and
@{term "case_prod (λx. f x)"} as @{term "λ(x, y). f x y"}›

typed_print_translation ‹
let
fun case_prod_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
| case_prod_guess_names_tr' T [Abs (x, xT, t)] =
Const (@{const_syntax case_prod}, _) => raise Match
| _ =>
let
val (_ :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
val (y, t') = Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 1 t \$ Bound 0);
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, xT, t');
in
Syntax.const @{syntax_const "_abs"} \$
(Syntax.const @{syntax_const "_pattern"} \$ x' \$ y) \$ t''
end)
| case_prod_guess_names_tr' T [t] =
Const (@{const_syntax case_prod}, _) => raise Match
| _ =>
let
val (xT :: yT :: _) = binder_types (domain_type T) handle Bind => raise Match;
val (y, t') =
Syntax_Trans.atomic_abs_tr' ("y", yT, incr_boundvars 2 t \$ Bound 1 \$ Bound 0);
val (x', t'') = Syntax_Trans.atomic_abs_tr' ("x", xT, t');
in
Syntax.const @{syntax_const "_abs"} \$
(Syntax.const @{syntax_const "_pattern"} \$ x' \$ y) \$ t''
end)
| case_prod_guess_names_tr' _ _ = raise Match;
in [(@{const_syntax case_prod}, K case_prod_guess_names_tr')] end
›

text ‹Reconstruct pattern from (nested) @{const case_prod}s,
avoiding eta-contraction of body; required for enclosing "let",
if "let" does not avoid eta-contraction, which has been observed to occur.›

print_translation ‹
let
fun case_prod_tr' [Abs (x, T, t as (Abs abs))] =
(* case_prod (λx y. t) ⇒ λ(x, y) t *)
let
val (y, t') = Syntax_Trans.atomic_abs_tr' abs;
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
in
Syntax.const @{syntax_const "_abs"} \$
(Syntax.const @{syntax_const "_pattern"} \$ x' \$ y) \$ t''
end
| case_prod_tr' [Abs (x, T, (s as Const (@{const_syntax case_prod}, _) \$ t))] =
(* case_prod (λx. (case_prod (λy z. t))) ⇒ λ(x, y, z). t *)
let
val Const (@{syntax_const "_abs"}, _) \$
(Const (@{syntax_const "_pattern"}, _) \$ y \$ z) \$ t' =
case_prod_tr' [t];
val (x', t'') = Syntax_Trans.atomic_abs_tr' (x, T, t');
in
Syntax.const @{syntax_const "_abs"} \$
(Syntax.const @{syntax_const "_pattern"} \$ x' \$
(Syntax.const @{syntax_const "_patterns"} \$ y \$ z)) \$ t''
end
| case_prod_tr' [Const (@{const_syntax case_prod}, _) \$ t] =
(* case_prod (case_prod (λx y z. t)) ⇒ λ((x, y), z). t *)
case_prod_tr' [(case_prod_tr' [t])]
(* inner case_prod_tr' creates next pattern *)
| case_prod_tr' [Const (@{syntax_const "_abs"}, _) \$ x_y \$ Abs abs] =
(* case_prod (λpttrn z. t) ⇒ λ(pttrn, z). t *)
let val (z, t) = Syntax_Trans.atomic_abs_tr' abs in
Syntax.const @{syntax_const "_abs"} \$
(Syntax.const @{syntax_const "_pattern"} \$ x_y \$ z) \$ t
end
| case_prod_tr' _ = raise Match;
in [(@{const_syntax case_prod}, K case_prod_tr')] end
›

subsubsection ‹Code generator setup›

code_printing
type_constructor prod ⇀
(SML) infix 2 "*"
and (OCaml) infix 2 "*"
and (Scala) "((_),/ (_))"
| constant Pair ⇀
(SML) "!((_),/ (_))"
and (OCaml) "!((_),/ (_))"
and (Scala) "!((_),/ (_))"
| class_instance  prod :: equal ⇀
| constant "HOL.equal :: 'a × 'b ⇒ 'a × 'b ⇒ bool" ⇀
| constant fst ⇀ (Haskell) "fst"
| constant snd ⇀ (Haskell) "snd"

subsubsection ‹Fundamental operations and properties›

lemma Pair_inject: "(a, b) = (a', b') ⟹ (a = a' ⟹ b = b' ⟹ R) ⟹ R"
by simp

lemma surj_pair [simp]: "∃x y. p = (x, y)"
by (cases p) simp

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

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

lemma case_prod_unfold [nitpick_unfold]: "case_prod = (λc p. c (fst p) (snd p))"
by (simp add: fun_eq_iff split: prod.split)

lemma case_prod_conv [simp, code]: "(case (a, b) of (c, d) ⇒ f c d) = f a b"
by (fact prod.case)

lemmas surjective_pairing = prod.collapse [symmetric]

lemma prod_eq_iff: "s = t ⟷ fst s = fst t ∧ snd s = snd t"
by (cases s, cases t) simp

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

lemma case_prodI: "f a b ⟹ case (a, b) of (c, d) ⇒ f c d"
by (rule prod.case [THEN iffD2])

lemma case_prodD: "(case (a, b) of (c, d) ⇒ f c d) ⟹ f a b"
by (rule prod.case [THEN iffD1])

lemma case_prod_Pair [simp]: "case_prod Pair = id"
by (simp add: fun_eq_iff split: prod.split)

lemma case_prod_eta: "(λ(x, y). f (x, y)) = f"
― ‹Subsumes the old ‹split_Pair› when @{term f} is the identity function.›
by (simp add: fun_eq_iff split: prod.split)

(* This looks like a sensible simp-rule but appears to do more harm than good:
lemma case_prod_const [simp]: "(λ(_,_). c) = (λ_. c)"
by(rule case_prod_eta)
*)

lemma case_prod_comp: "(case x of (a, b) ⇒ (f ∘ g) a b) = f (g (fst x)) (snd x)"
by (cases x) simp

lemma The_case_prod: "The (case_prod P) = (THE xy. P (fst xy) (snd xy))"

lemma cond_case_prod_eta: "(⋀x y. f x y = g (x, y)) ⟹ (λ(x, y). f x y) = g"

lemma split_paired_all [no_atp]: "(⋀x. PROP P x) ≡ (⋀a b. PROP P (a, b))"
proof
fix a b
assume "⋀x. PROP P x"
then show "PROP P (a, b)" .
next
fix x
assume "⋀a b. PROP P (a, b)"
from ‹PROP P (fst x, snd x)› show "PROP P x" by simp
qed

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 ‹⋀a b. … = ?P(a, b)›
which cannot be solved by reflexivity.
›

lemmas split_tupled_all = split_paired_all unit_all_eq2

ML ‹
(* replace parameters of product type by individual component parameters *)
local (* filtering with exists_paired_all is an essential optimization *)
fun exists_paired_all (Const (@{const_name Pure.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 =
simpset_of
(put_simpset HOL_basic_ss @{context}
addsimps [@{thm split_paired_all}, @{thm unit_all_eq2}, @{thm unit_abs_eta_conv}]
in
fun split_all_tac ctxt = SUBGOAL (fn (t, i) =>
if exists_paired_all t then safe_full_simp_tac (put_simpset ss ctxt) i else no_tac);

fun unsafe_split_all_tac ctxt = SUBGOAL (fn (t, i) =>
if exists_paired_all t then full_simp_tac (put_simpset ss ctxt) i else no_tac);

fun split_all ctxt th =
if exists_paired_all (Thm.prop_of th)
then full_simplify (put_simpset ss ctxt) th else th;
end;
›

setup ‹map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac))›

lemma split_paired_All [simp, no_atp]: "(∀x. P x) ⟷ (∀a b. P (a, b))"
― ‹‹[iff]› is not a good idea because it makes ‹blast› loop›
by fast

lemma split_paired_Ex [simp, no_atp]: "(∃x. P x) ⟷ (∃a b. P (a, b))"
by fast

lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
― ‹Can't be added to simpset: loops!›

text ‹
Simplification procedure for @{thm [source] cond_case_prod_eta}.  Using
@{thm [source] case_prod_eta} as a rewrite rule is not general enough,
and using @{thm [source] cond_case_prod_eta} directly would render some
existing proofs very inefficient; similarly for ‹prod.case_eq_if›.
›

ML ‹
local
val cond_case_prod_eta_ss =
simpset_of (put_simpset HOL_basic_ss @{context} addsimps @{thms cond_case_prod_eta});
fun Pair_pat k 0 (Bound m) = (m = k)
| Pair_pat k i (Const (@{const_name 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 (@{const_name case_prod}, _) \$ Abs (_, _, t)) = split_pat tp (i + 1) t
| split_pat tp i _ = NONE;
fun metaeq ctxt lhs rhs = mk_meta_eq (Goal.prove ctxt [] []
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs)))
(K (simp_tac (put_simpset cond_case_prod_eta_ss ctxt) 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;
in
fun beta_proc ctxt (s as Const (@{const_name case_prod}, _) \$ Abs (_, _, t) \$ arg) =
(case split_pat beta_term_pat 1 t of
SOME (i, f) => SOME (metaeq ctxt s (subst arg 0 i f))
| NONE => NONE)
| beta_proc _ _ = NONE;
fun eta_proc ctxt (s as Const (@{const_name case_prod}, _) \$ Abs (_, _, t)) =
(case split_pat eta_term_pat 1 t of
SOME (_, ft) => SOME (metaeq ctxt s (let val f \$ _ = ft in f end))
| NONE => NONE)
| eta_proc _ _ = NONE;
end;
›
simproc_setup case_prod_beta ("case_prod f z") =
‹fn _ => fn ctxt => fn ct => beta_proc ctxt (Thm.term_of ct)›
simproc_setup case_prod_eta ("case_prod f") =
‹fn _ => fn ctxt => fn ct => eta_proc ctxt (Thm.term_of ct)›

lemma case_prod_beta': "(λ(x,y). f x y) = (λx. f (fst x) (snd x))"
by (auto simp: fun_eq_iff)

text ‹
┉ @{const case_prod} used as a logical connective or set former.

┉ These rules are for use with ‹blast›; could instead
call ‹simp› using @{thm [source] prod.split} as rewrite.›

lemma case_prodI2:
"⋀p. (⋀a b. p = (a, b) ⟹ c a b) ⟹ case p of (a, b) ⇒ c a b"

lemma case_prodI2':
"⋀p. (⋀a b. (a, b) = p ⟹ c a b x) ⟹ (case p of (a, b) ⇒ c a b) x"

lemma case_prodE [elim!]:
"(case p of (a, b) ⇒ c a b) ⟹ (⋀x y. p = (x, y) ⟹ c x y ⟹ Q) ⟹ Q"
by (induct p) simp

lemma case_prodE' [elim!]:
"(case p of (a, b) ⇒ c a b) z ⟹ (⋀x y. p = (x, y) ⟹ c x y z ⟹ Q) ⟹ Q"
by (induct p) simp

lemma case_prodE2:
assumes q: "Q (case z of (a, b) ⇒ P a b)"
and r: "⋀x y. z = (x, y) ⟹ Q (P x y) ⟹ R"
shows R
proof (rule r)
show "z = (fst z, snd z)" by simp
then show "Q (P (fst z) (snd z))"
using q by (simp add: case_prod_unfold)
qed

lemma case_prodD': "(case (a, b) of (c, d) ⇒ R c d) c ⟹ R a b c"
by simp

lemma mem_case_prodI: "z ∈ c a b ⟹ z ∈ (case (a, b) of (d, e) ⇒ c d e)"
by simp

lemma mem_case_prodI2 [intro!]:
"⋀p. (⋀a b. p = (a, b) ⟹ z ∈ c a b) ⟹ z ∈ (case p of (a, b) ⇒ c a b)"
by (simp only: split_tupled_all) simp

declare mem_case_prodI [intro!] ― ‹postponed to maintain traditional declaration order!›
declare case_prodI2' [intro!] ― ‹postponed to maintain traditional declaration order!›
declare case_prodI2 [intro!] ― ‹postponed to maintain traditional declaration order!›
declare case_prodI [intro!] ― ‹postponed to maintain traditional declaration order!›

lemma mem_case_prodE [elim!]:
assumes "z ∈ case_prod c p"
obtains x y where "p = (x, y)" and "z ∈ c x y"
using assms by (rule case_prodE2)

ML ‹
local (* filtering with exists_p_split is an essential optimization *)
fun exists_p_split (Const (@{const_name case_prod},_) \$ _ \$ (Const (@{const_name 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;
in
fun split_conv_tac ctxt = SUBGOAL (fn (t, i) =>
if exists_p_split t
then safe_full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps @{thms case_prod_conv}) i
else no_tac);
end;
›

to quite time-consuming computations (in particular for nested tuples) *)
setup ‹map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac))›

lemma split_eta_SetCompr [simp, no_atp]: "(λu. ∃x y. u = (x, y) ∧ P (x, y)) = P"
by (rule ext) fast

lemma split_eta_SetCompr2 [simp, no_atp]: "(λu. ∃x y. u = (x, y) ∧ P x y) = case_prod P"
by (rule ext) fast

lemma split_part [simp]: "(λ(a,b). P ∧ Q a b) = (λab. P ∧ case_prod Q ab)"
― ‹Allows simplifications of nested splits in case of independent predicates.›
by (rule ext) blast

(* 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:
fixes f :: "'a ⇒ 'b ⇒ 'c"
and g :: "'d ⇒ 'a"
shows "(λu. f (g (fst u)) (snd u)) = case_prod (λx. f (g x))"
by (rule ext) auto

lemma pair_imageI [intro]: "(a, b) ∈ A ⟹ f a b ∈ (λ(a, b). f a b) ` A"
by (rule image_eqI [where x = "(a, b)"]) auto

lemma Collect_const_case_prod[simp]: "{(a,b). P} = (if P then UNIV else {})"
by 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 case_prod_beta: "case_prod f p = f (fst p) (snd p)"
by (fact prod.case_eq_if)

lemma prod_cases3 [cases type]:
obtains (fields) a b c where "y = (a, b, c)"
by (cases y, case_tac b) blast

lemma prod_induct3 [case_names fields, induct type]:
"(⋀a b c. P (a, b, c)) ⟹ P x"
by (cases x) blast

lemma prod_cases4 [cases type]:
obtains (fields) a b c d where "y = (a, b, c, d)"
by (cases y, case_tac c) blast

lemma prod_induct4 [case_names fields, induct type]:
"(⋀a b c d. P (a, b, c, d)) ⟹ P x"
by (cases x) blast

lemma prod_cases5 [cases type]:
obtains (fields) a b c d e where "y = (a, b, c, d, e)"
by (cases y, case_tac d) blast

lemma prod_induct5 [case_names fields, induct type]:
"(⋀a b c d e. P (a, b, c, d, e)) ⟹ P x"
by (cases x) blast

lemma prod_cases6 [cases type]:
obtains (fields) a b c d e f where "y = (a, b, c, d, e, f)"
by (cases y, case_tac e) blast

lemma prod_induct6 [case_names fields, induct type]:
"(⋀a b c d e f. P (a, b, c, d, e, f)) ⟹ P x"
by (cases x) blast

lemma prod_cases7 [cases type]:
obtains (fields) a b c d e f g where "y = (a, b, c, d, e, f, g)"
by (cases y, case_tac f) blast

lemma prod_induct7 [case_names fields, induct type]:
"(⋀a b c d e f g. P (a, b, c, d, e, f, g)) ⟹ P x"
by (cases x) blast

definition internal_case_prod :: "('a ⇒ 'b ⇒ 'c) ⇒ 'a × 'b ⇒ 'c"
where "internal_case_prod ≡ case_prod"

lemma internal_case_prod_conv: "internal_case_prod c (a, b) = c a b"
by (simp only: internal_case_prod_def case_prod_conv)

ML_file "Tools/split_rule.ML"

hide_const internal_case_prod

subsubsection ‹Derived operations›

definition curry :: "('a × 'b ⇒ 'c) ⇒ 'a ⇒ 'b ⇒ 'c"
where "curry = (λc x y. c (x, y))"

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

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_case_prod [simp]: "curry (case_prod f) = f"

lemma case_prod_curry [simp]: "case_prod (curry f) = f"

lemma curry_K: "curry (λx. c) = (λx y. c)"

text ‹The composition-uncurry combinator.›

notation fcomp (infixl "∘>" 60)

definition scomp :: "('a ⇒ 'b × 'c) ⇒ ('b ⇒ 'c ⇒ 'd) ⇒ 'a ⇒ 'd"  (infixl "∘→" 60)
where "f ∘→ g = (λx. case_prod g (f x))"

lemma scomp_unfold: "scomp = (λf g x. g (fst (f x)) (snd (f x)))"
by (simp add: fun_eq_iff scomp_def case_prod_unfold)

lemma scomp_apply [simp]: "(f ∘→ g) x = case_prod g (f x)"

lemma Pair_scomp: "Pair x ∘→ f = f x"

lemma scomp_Pair: "x ∘→ Pair = x"

lemma scomp_scomp: "(f ∘→ g) ∘→ h = f ∘→ (λx. g x ∘→ h)"

lemma scomp_fcomp: "(f ∘→ g) ∘> h = f ∘→ (λx. g x ∘> h)"
by (simp add: fun_eq_iff scomp_unfold fcomp_def)

lemma fcomp_scomp: "(f ∘> g) ∘→ h = f ∘> (g ∘→ h)"

code_printing
constant scomp ⇀ (Eval) infixl 3 "#->"

no_notation fcomp (infixl "∘>" 60)
no_notation scomp (infixl "∘→" 60)

text ‹
@{term map_prod} --- action of the product functor upon functions.
›

definition map_prod :: "('a ⇒ 'c) ⇒ ('b ⇒ 'd) ⇒ 'a × 'b ⇒ 'c × 'd"
where "map_prod f g = (λ(x, y). (f x, g y))"

lemma map_prod_simp [simp, code]: "map_prod f g (a, b) = (f a, g b)"

functor map_prod: map_prod

lemma fst_map_prod [simp]: "fst (map_prod f g x) = f (fst x)"
by (cases x) simp_all

lemma snd_map_prod [simp]: "snd (map_prod f g x) = g (snd x)"
by (cases x) simp_all

lemma fst_comp_map_prod [simp]: "fst ∘ map_prod f g = f ∘ fst"
by (rule ext) simp_all

lemma snd_comp_map_prod [simp]: "snd ∘ map_prod f g = g ∘ snd"
by (rule ext) simp_all

lemma map_prod_compose: "map_prod (f1 ∘ f2) (g1 ∘ g2) = (map_prod f1 g1 ∘ map_prod f2 g2)"
by (rule ext) (simp add: map_prod.compositionality comp_def)

lemma map_prod_ident [simp]: "map_prod (λx. x) (λy. y) = (λz. z)"
by (rule ext) (simp add: map_prod.identity)

lemma map_prod_imageI [intro]: "(a, b) ∈ R ⟹ (f a, g b) ∈ map_prod f g ` R"
by (rule image_eqI) simp_all

lemma prod_fun_imageE [elim!]:
assumes major: "c ∈ map_prod f g ` R"
and cases: "⋀x y. c = (f x, g y) ⟹ (x, y) ∈ R ⟹ P"
shows P
apply (rule major [THEN imageE])
apply (case_tac x)
apply (rule cases)
apply simp_all
done

definition apfst :: "('a ⇒ 'c) ⇒ 'a × 'b ⇒ 'c × 'b"
where "apfst f = map_prod f id"

definition apsnd :: "('b ⇒ 'c) ⇒ 'a × 'b ⇒ 'a × 'c"
where "apsnd f = map_prod id f"

lemma apfst_conv [simp, code]: "apfst f (x, y) = (f x, y)"

lemma apsnd_conv [simp, code]: "apsnd f (x, y) = (x, f y)"

lemma fst_apfst [simp]: "fst (apfst f x) = f (fst x)"
by (cases x) simp

lemma fst_comp_apfst [simp]: "fst ∘ apfst f = f ∘ fst"

lemma fst_apsnd [simp]: "fst (apsnd f x) = fst x"
by (cases x) simp

lemma fst_comp_apsnd [simp]: "fst ∘ apsnd f = fst"

lemma snd_apfst [simp]: "snd (apfst f x) = snd x"
by (cases x) simp

lemma snd_comp_apfst [simp]: "snd ∘ apfst f = snd"

lemma snd_apsnd [simp]: "snd (apsnd f x) = f (snd x)"
by (cases x) simp

lemma snd_comp_apsnd [simp]: "snd ∘ apsnd f = f ∘ snd"

lemma apfst_compose: "apfst f (apfst g x) = apfst (f ∘ g) x"
by (cases x) simp

lemma apsnd_compose: "apsnd f (apsnd g x) = apsnd (f ∘ g) x"
by (cases x) simp

lemma apfst_apsnd [simp]: "apfst f (apsnd g x) = (f (fst x), g (snd x))"
by (cases x) simp

lemma apsnd_apfst [simp]: "apsnd f (apfst g x) = (g (fst x), f (snd x))"
by (cases x) simp

lemma apfst_id [simp]: "apfst id = id"

lemma apsnd_id [simp]: "apsnd id = id"

lemma apfst_eq_conv [simp]: "apfst f x = apfst g x ⟷ f (fst x) = g (fst x)"
by (cases x) simp

lemma apsnd_eq_conv [simp]: "apsnd f x = apsnd g x ⟷ f (snd x) = g (snd x)"
by (cases x) simp

lemma apsnd_apfst_commute: "apsnd f (apfst g p) = apfst g (apsnd f p)"
by simp

context
begin

local_setup ‹Local_Theory.map_background_naming (Name_Space.mandatory_path "prod")›

definition swap :: "'a × 'b ⇒ 'b × 'a"
where "swap p = (snd p, fst p)"

end

lemma swap_simp [simp]: "prod.swap (x, y) = (y, x)"

lemma swap_swap [simp]: "prod.swap (prod.swap p) = p"
by (cases p) simp

lemma swap_comp_swap [simp]: "prod.swap ∘ prod.swap = id"

lemma pair_in_swap_image [simp]: "(y, x) ∈ prod.swap ` A ⟷ (x, y) ∈ A"
by (auto intro!: image_eqI)

lemma inj_swap [simp]: "inj_on prod.swap A"
by (rule inj_onI) auto

lemma swap_inj_on: "inj_on (λ(i, j). (j, i)) A"
by (rule inj_onI) auto

lemma surj_swap [simp]: "surj prod.swap"
by (rule surjI [of _ prod.swap]) simp

lemma bij_swap [simp]: "bij prod.swap"

lemma case_swap [simp]: "(case prod.swap p of (y, x) ⇒ f x y) = (case p of (x, y) ⇒ f x y)"
by (cases p) simp

lemma fst_swap [simp]: "fst (prod.swap x) = snd x"
by (cases x) simp

lemma snd_swap [simp]: "snd (prod.swap x) = fst x"
by (cases x) simp

text ‹Disjoint union of a family of sets -- Sigma.›

definition Sigma :: "'a set ⇒ ('a ⇒ 'b set) ⇒ ('a × 'b) set"
where "Sigma A B ≡ ⋃x∈A. ⋃y∈B x. {Pair x y}"

abbreviation Times :: "'a set ⇒ 'b set ⇒ ('a × 'b) set"  (infixr "×" 80)
where "A × B ≡ Sigma A (λ_. B)"

hide_const (open) Times

bundle no_Set_Product_syntax begin
no_notation Product_Type.Times (infixr "×" 80)
end
bundle Set_Product_syntax begin
notation Product_Type.Times (infixr "×" 80)
end

syntax
"_Sigma" :: "pttrn ⇒ 'a set ⇒ 'b set ⇒ ('a × 'b) set"  ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
translations
"SIGMA x:A. B" ⇌ "CONST Sigma A (λx. B)"

lemma SigmaI [intro!]: "a ∈ A ⟹ b ∈ B a ⟹ (a, b) ∈ Sigma A B"
unfolding Sigma_def by blast

lemma SigmaE [elim!]: "c ∈ Sigma A B ⟹ (⋀x y. x ∈ A ⟹ y ∈ B x ⟹ c = (x, y) ⟹ P) ⟹ P"
― ‹The general elimination rule.›
unfolding Sigma_def by blast

text ‹
Elimination of @{term "(a, b) ∈ A × B"} -- introduces no
eigenvariables.
›

lemma SigmaD1: "(a, b) ∈ Sigma A B ⟹ a ∈ A"
by blast

lemma SigmaD2: "(a, b) ∈ Sigma A B ⟹ b ∈ B a"
by blast

lemma SigmaE2: "(a, b) ∈ Sigma A B ⟹ (a ∈ A ⟹ b ∈ B a ⟹ P) ⟹ P"
by blast

lemma Sigma_cong: "A = B ⟹ (⋀x. x ∈ B ⟹ C x = D x) ⟹ (SIGMA x:A. C x) = (SIGMA x:B. D x)"
by auto

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 mem_Times_iff: "x ∈ A × B ⟷ fst x ∈ A ∧ snd x ∈ B"
by (induct x) simp

lemma Sigma_empty_iff: "(SIGMA i:I. X i) = {} ⟷ (∀i∈I. X i = {})"
by auto

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 Collect_case_prod_Sigma: "{(x, y). P x ∧ Q x y} = (SIGMA x:Collect P. Collect (Q x))"
by blast

lemma Collect_case_prod [simp]: "{(a, b). P a ∧ Q b} = Collect P × Collect Q "
by (fact Collect_case_prod_Sigma)

lemma Collect_case_prodD: "x ∈ Collect (case_prod A) ⟹ A (fst x) (snd x)"
by auto

lemma Collect_case_prod_mono: "A ≤ B ⟹ Collect (case_prod A) ⊆ Collect (case_prod B)"
by auto (auto elim!: le_funE)

lemma Collect_split_mono_strong:
"X = fst ` A ⟹ Y = snd ` A ⟹ ∀a∈X. ∀b ∈ Y. P a b ⟶ Q a b
⟹ A ⊆ Collect (case_prod P) ⟹ A ⊆ Collect (case_prod Q)"
by fastforce

lemma UN_Times_distrib: "(⋃(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, no_atp]: "(∀z∈Sigma A B. P z) ⟷ (∀x∈A. ∀y∈B x. P (x, y))"
by blast

lemma split_paired_Bex_Sigma [simp, no_atp]: "(∃z∈Sigma A B. P z) ⟷ (∃x∈A. ∃y∈B x. P (x, y))"
by blast

lemma Sigma_Un_distrib1: "Sigma (I ∪ J) C = Sigma I C ∪ Sigma J C"
by blast

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

lemma Sigma_Int_distrib1: "Sigma (I ∩ J) C = Sigma I C ∩ Sigma J C"
by blast

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

lemma Sigma_Diff_distrib1: "Sigma (I - J) C = Sigma I C - Sigma J C"
by blast

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

lemma Sigma_Union: "Sigma (⋃X) B = (⋃A∈X. Sigma A B)"
by blast

lemma Pair_vimage_Sigma: "Pair x -` Sigma A f = (if x ∈ A then f x else {})"
by auto

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 ∪ B) × C = A × C ∪ B × C "
by (fact Sigma_Un_distrib1)

lemma Times_Int_distrib1: "(A ∩ B) × C = A × C ∩ B × C "
by (fact Sigma_Int_distrib1)

lemma Times_Diff_distrib1: "(A - B) × C = A × C - B × C "
by (fact Sigma_Diff_distrib1)

lemma Times_empty [simp]: "A × B = {} ⟷ A = {} ∨ B = {}"
by auto

lemma times_eq_iff: "A × B = C × D ⟷ A = C ∧ B = D ∨ (A = {} ∨ B = {}) ∧ (C = {} ∨ D = {})"
by auto

lemma fst_image_times [simp]: "fst ` (A × B) = (if B = {} then {} else A)"
by force

lemma snd_image_times [simp]: "snd ` (A × B) = (if A = {} then {} else B)"
by force

lemma fst_image_Sigma: "fst ` (Sigma A B) = {x ∈ A. B(x) ≠ {}}"
by force

lemma snd_image_Sigma: "snd ` (Sigma A B) = (⋃ x ∈ A. B x)"
by force

lemma vimage_fst: "fst -` A = A × UNIV"
by auto

lemma vimage_snd: "snd -` A = UNIV × A"
by auto

lemma insert_times_insert [simp]:
"insert a A × insert b B = insert (a,b) (A × insert b B ∪ insert a A × B)"
by blast

lemma vimage_Times: "f -` (A × B) = (fst ∘ f) -` A ∩ (snd ∘ f) -` B"
proof (rule set_eqI)
show "x ∈ f -` (A × B) ⟷ x ∈ (fst ∘ f) -` A ∩ (snd ∘ f) -` B" for x
by (cases "f x") (auto split: prod.split)
qed

lemma times_Int_times: "A × B ∩ C × D = (A ∩ C) × (B ∩ D)"
by auto

lemma product_swap: "prod.swap ` (A × B) = B × A"

lemma swap_product: "(λ(i, j). (j, i)) ` (A × B) = B × A"

lemma image_split_eq_Sigma: "(λx. (f x, g x)) ` A = Sigma (f ` A) (λx. g ` (f -` {x} ∩ A))"
proof (safe intro!: imageI)
fix a b
assume *: "a ∈ A" "b ∈ A" and eq: "f a = f b"
show "(f b, g a) ∈ (λx. (f x, g x)) ` A"
using * eq[symmetric] by auto
qed simp_all

lemma subset_fst_snd: "A ⊆ (fst ` A × snd ` A)"
by force

lemma inj_on_apfst [simp]: "inj_on (apfst f) (A × UNIV) ⟷ inj_on f A"

lemma inj_apfst [simp]: "inj (apfst f) ⟷ inj f"
using inj_on_apfst[of f UNIV] by simp

lemma inj_on_apsnd [simp]: "inj_on (apsnd f) (UNIV × A) ⟷ inj_on f A"

lemma inj_apsnd [simp]: "inj (apsnd f) ⟷ inj f"
using inj_on_apsnd[of f UNIV] by simp

context
begin

qualified definition product :: "'a set ⇒ 'b set ⇒ ('a × 'b) set"
where [code_abbrev]: "product A B = A × B"

lemma member_product: "x ∈ Product_Type.product A B ⟷ x ∈ A × B"

end

text ‹The following @{const map_prod} lemmas are due to Joachim Breitner:›

lemma map_prod_inj_on:
assumes "inj_on f A"
and "inj_on g B"
shows "inj_on (map_prod f g) (A × B)"
proof (rule inj_onI)
fix x :: "'a × 'c"
fix y :: "'a × 'c"
assume "x ∈ A × B"
then have "fst x ∈ A" and "snd x ∈ B" by auto
assume "y ∈ A × B"
then have "fst y ∈ A" and "snd y ∈ B" by auto
assume "map_prod f g x = map_prod f g y"
then have "fst (map_prod f g x) = fst (map_prod f g y)" by auto
then have "f (fst x) = f (fst y)" by (cases x, cases y) auto
with ‹inj_on f A› and ‹fst x ∈ A› and ‹fst y ∈ A› have "fst x = fst y"
by (auto dest: inj_onD)
moreover from ‹map_prod f g x = map_prod f g y›
have "snd (map_prod f g x) = snd (map_prod f g y)" by auto
then have "g (snd x) = g (snd y)" by (cases x, cases y) auto
with ‹inj_on g B› and ‹snd x ∈ B› and ‹snd y ∈ B› have "snd x = snd y"
by (auto dest: inj_onD)
ultimately show "x = y" by (rule prod_eqI)
qed

lemma map_prod_surj:
fixes f :: "'a ⇒ 'b"
and g :: "'c ⇒ 'd"
assumes "surj f" and "surj g"
shows "surj (map_prod f g)"
unfolding surj_def
proof
fix y :: "'b × 'd"
from ‹surj f› obtain a where "fst y = f a"
by (auto elim: surjE)
moreover
from ‹surj g› obtain b where "snd y = g b"
by (auto elim: surjE)
ultimately have "(fst y, snd y) = map_prod f g (a,b)"
by auto
then show "∃x. y = map_prod f g x"
by auto
qed

lemma map_prod_surj_on:
assumes "f ` A = A'" and "g ` B = B'"
shows "map_prod f g ` (A × B) = A' × B'"
unfolding image_def
proof (rule set_eqI, rule iffI)
fix x :: "'a × 'c"
assume "x ∈ {y::'a × 'c. ∃x::'b × 'd∈A × B. y = map_prod f g x}"
then obtain y where "y ∈ A × B" and "x = map_prod f g y"
by blast
from ‹image f A = A'› and ‹y ∈ A × B› have "f (fst y) ∈ A'"
by auto
moreover from ‹image g B = B'› and ‹y ∈ A × B› have "g (snd y) ∈ B'"
by auto
ultimately have "(f (fst y), g (snd y)) ∈ (A' × B')"
by auto
with ‹x = map_prod f g y› show "x ∈ A' × B'"
by (cases y) auto
next
fix x :: "'a × 'c"
assume "x ∈ A' × B'"
then have "fst x ∈ A'" and "snd x ∈ B'"
by auto
from ‹image f A = A'› and ‹fst x ∈ A'› have "fst x ∈ image f A"
by auto
then obtain a where "a ∈ A" and "fst x = f a"
by (rule imageE)
moreover from ‹image g B = B'› and ‹snd x ∈ B'› obtain b where "b ∈ B" and "snd x = g b"
by auto
ultimately have "(fst x, snd x) = map_prod f g (a, b)"
by auto
moreover from ‹a ∈ A› and  ‹b ∈ B› have "(a , b) ∈ A × B"
by auto
ultimately have "∃y ∈ A × B. x = map_prod f g y"
by auto
then show "x ∈ {x. ∃y ∈ A × B. x = map_prod f g y}"
by auto
qed

subsection ‹Simproc for rewriting a set comprehension into a pointfree expression›

ML_file "Tools/set_comprehension_pointfree.ML"

setup ‹
Code_Preproc.map_pre (fn ctxt => ctxt addsimprocs
[Simplifier.make_simproc @{context} "set comprehension"
{lhss = [@{term "Collect P"}],
proc = K Set_Comprehension_Pointfree.code_simproc}])
›

subsection ‹Inductively defined sets›

(* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *)
simproc_setup Collect_mem ("Collect t") = ‹
fn _ => fn ctxt => fn ct =>
(case Thm.term_of ct of
S as Const (@{const_name Collect}, Type (@{type_name fun}, [_, T])) \$ t =>
let val (u, _, ps) = HOLogic.strip_ptupleabs t in
(case u of
(c as Const (@{const_name Set.member}, _)) \$ q \$ S' =>
(case try (HOLogic.strip_ptuple ps) q of
NONE => NONE
| SOME ts =>
if not (Term.is_open S') andalso
ts = map Bound (length ps downto 0)
then
let val simp =
full_simp_tac (put_simpset HOL_basic_ss ctxt
addsimps [@{thm split_paired_all}, @{thm case_prod_conv}]) 1
in
SOME (Goal.prove ctxt [] []
(Const (@{const_name Pure.eq}, T --> T --> propT) \$ S \$ S')
(K (EVERY
[resolve_tac ctxt [eq_reflection] 1,
resolve_tac ctxt @{thms subset_antisym} 1,
resolve_tac ctxt @{thms subsetI} 1,
dresolve_tac ctxt @{thms CollectD} 1, simp,
resolve_tac ctxt @{thms subsetI} 1,
resolve_tac ctxt @{thms CollectI} 1, simp])))
end
else NONE)
| _ => NONE)
end
| _ => NONE)
›

ML_file "Tools/inductive_set.ML"

subsection ‹Legacy theorem bindings and duplicates›

lemmas fst_conv = prod.sel(1)
lemmas snd_conv = prod.sel(2)
lemmas split_def = case_prod_unfold
lemmas split_beta' = case_prod_beta'
lemmas split_beta = prod.case_eq_if
lemmas split_conv = case_prod_conv
lemmas split = case_prod_conv

hide_const (open) prod

end
```