(* Title: HOL/Product_Type.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
*)
section \<open>Cartesian products\<close>
theory Product_Type
imports Typedef Inductive Fun
keywords "inductive_set" "coinductive_set" :: thy_decl
begin
subsection \<open>@{typ bool} is a datatype\<close>
free_constructors (discs_sels) case_bool for True | False
by auto
text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
setup \<open>Sign.mandatory_path "old"\<close>
old_rep_datatype True False by (auto intro: bool_induct)
setup \<open>Sign.parent_path\<close>
text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
setup \<open>Sign.mandatory_path "bool"\<close>
lemmas induct = old.bool.induct
lemmas inducts = old.bool.inducts
lemmas rec = old.bool.rec
lemmas simps = bool.distinct bool.case bool.rec
setup \<open>Sign.parent_path\<close>
declare case_split [cases type: bool]
\<comment> \<open>prefer plain propositional version\<close>
lemma [code]: "HOL.equal False P \<longleftrightarrow> \<not> P"
and [code]: "HOL.equal True P \<longleftrightarrow> P"
and [code]: "HOL.equal P False \<longleftrightarrow> \<not> P"
and [code]: "HOL.equal P True \<longleftrightarrow> P"
and [code nbe]: "HOL.equal P P \<longleftrightarrow> True"
by (simp_all add: equal)
lemma If_case_cert:
assumes "CASE \<equiv> (\<lambda>b. If b f g)"
shows "(CASE True \<equiv> f) &&& (CASE False \<equiv> g)"
using assms by simp_all
setup \<open>Code.declare_case_global @{thm If_case_cert}\<close>
code_printing
constant "HOL.equal :: bool \<Rightarrow> bool \<Rightarrow> bool" \<rightharpoonup> (Haskell) infix 4 "=="
| class_instance "bool" :: "equal" \<rightharpoonup> (Haskell) -
subsection \<open>The \<open>unit\<close> type\<close>
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 \<open>
Simplification procedure for @{thm [source] unit_eq}. Cannot use
this rule directly --- it loops!
\<close>
simproc_setup unit_eq ("x::unit") = \<open>
fn _ => fn _ => fn ct =>
if HOLogic.is_unit (Thm.term_of ct) then NONE
else SOME (mk_meta_eq @{thm unit_eq})
\<close>
free_constructors case_unit for "()"
by auto
text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
setup \<open>Sign.mandatory_path "old"\<close>
old_rep_datatype "()" by simp
setup \<open>Sign.parent_path\<close>
text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
setup \<open>Sign.mandatory_path "unit"\<close>
lemmas induct = old.unit.induct
lemmas inducts = old.unit.inducts
lemmas rec = old.unit.rec
lemmas simps = unit.case unit.rec
setup \<open>Sign.parent_path\<close>
lemma unit_all_eq1: "(\<And>x::unit. PROP P x) \<equiv> PROP P ()"
by simp
lemma unit_all_eq2: "(\<And>x::unit. PROP P) \<equiv> PROP P"
by (rule triv_forall_equality)
text \<open>
This rewrite counters the effect of simproc \<open>unit_eq\<close> on @{term
[source] "\<lambda>u::unit. f u"}, replacing it by @{term [source]
f} rather than by @{term [source] "\<lambda>u. f ()"}.
\<close>
lemma unit_abs_eta_conv [simp]: "(\<lambda>u::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 \<Rightarrow> unit \<Rightarrow> bool"
where "(_::unit) \<le> _ \<longleftrightarrow> True"
lemma less_eq_unit [iff]: "u \<le> v" for u v :: unit
by (simp add: less_eq_unit_def)
definition less_unit :: "unit \<Rightarrow> unit \<Rightarrow> bool"
where "(_::unit) < _ \<longleftrightarrow> False"
lemma less_unit [iff]: "\<not> u < v" for u v :: unit
by (simp_all add: less_eq_unit_def less_unit_def)
definition bot_unit :: unit
where [code_unfold]: "\<bottom> = ()"
definition top_unit :: unit
where [code_unfold]: "\<top> = ()"
definition inf_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit"
where [simp]: "_ \<sqinter> _ = ()"
definition sup_unit :: "unit \<Rightarrow> unit \<Rightarrow> unit"
where [simp]: "_ \<squnion> _ = ()"
definition Inf_unit :: "unit set \<Rightarrow> unit"
where [simp]: "\<Sqinter>_ = ()"
definition Sup_unit :: "unit set \<Rightarrow> unit"
where [simp]: "\<Squnion>_ = ()"
definition uminus_unit :: "unit \<Rightarrow> 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 \<longleftrightarrow> True" for u v :: unit
unfolding equal unit_eq [of u] unit_eq [of v] by rule+
code_printing
type_constructor unit \<rightharpoonup>
(SML) "unit"
and (OCaml) "unit"
and (Haskell) "()"
and (Scala) "Unit"
| constant Unity \<rightharpoonup>
(SML) "()"
and (OCaml) "()"
and (Haskell) "()"
and (Scala) "()"
| class_instance unit :: equal \<rightharpoonup>
(Haskell) -
| constant "HOL.equal :: unit \<Rightarrow> unit \<Rightarrow> bool" \<rightharpoonup>
(Haskell) infix 4 "=="
code_reserved SML
unit
code_reserved OCaml
unit
code_reserved Scala
Unit
subsection \<open>The product type\<close>
subsubsection \<open>Type definition\<close>
definition Pair_Rep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
where "Pair_Rep a b = (\<lambda>x y. x = a \<and> y = b)"
definition "prod = {f. \<exists>a b. f = Pair_Rep (a::'a) (b::'b)}"
typedef ('a, 'b) prod ("(_ \<times>/ _)" [21, 20] 20) = "prod :: ('a \<Rightarrow> 'b \<Rightarrow> bool) set"
unfolding prod_def by auto
type_notation (ASCII)
prod (infixr "*" 20)
definition Pair :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<times> 'b"
where "Pair a b = Abs_prod (Pair_Rep a b)"
lemma prod_cases: "(\<And>a b. P (Pair a b)) \<Longrightarrow> 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 \<times> 'b"
show "(\<And>x1 x2. p = Pair x1 x2 \<Longrightarrow> P) \<Longrightarrow> 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 \<longleftrightarrow> a = c \<and> b = d"
by (auto simp add: Pair_Rep_def fun_eq_iff)
moreover have "Pair_Rep a b \<in> prod" and "Pair_Rep c d \<in> prod"
by (auto simp add: prod_def)
ultimately show "Pair a b = Pair c d \<longleftrightarrow> a = c \<and> b = d"
by (simp add: Pair_def Abs_prod_inject)
qed
text \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
setup \<open>Sign.mandatory_path "old"\<close>
old_rep_datatype Pair
by (erule prod_cases) (rule prod.inject)
setup \<open>Sign.parent_path\<close>
text \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
setup \<open>Sign.mandatory_path "prod"\<close>
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 \<open>Sign.parent_path\<close>
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 \<open>
@{thm [source] prod.split} could be declared as \<open>[split]\<close>
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.
\<close>
subsubsection \<open>Tuple syntax\<close>
text \<open>
Patterns -- extends pre-defined type @{typ pttrn} used in
abstractions.
\<close>
nonterminal tuple_args and patterns
syntax
"_tuple" :: "'a \<Rightarrow> tuple_args \<Rightarrow> 'a \<times> 'b" ("(1'(_,/ _'))")
"_tuple_arg" :: "'a \<Rightarrow> tuple_args" ("_")
"_tuple_args" :: "'a \<Rightarrow> tuple_args \<Rightarrow> tuple_args" ("_,/ _")
"_pattern" :: "pttrn \<Rightarrow> patterns \<Rightarrow> pttrn" ("'(_,/ _')")
"" :: "pttrn \<Rightarrow> patterns" ("_")
"_patterns" :: "pttrn \<Rightarrow> patterns \<Rightarrow> patterns" ("_,/ _")
"_unit" :: pttrn ("'(')")
translations
"(x, y)" \<rightleftharpoons> "CONST Pair x y"
"_pattern x y" \<rightleftharpoons> "CONST Pair x y"
"_patterns x y" \<rightleftharpoons> "CONST Pair x y"
"_tuple x (_tuple_args y z)" \<rightleftharpoons> "_tuple x (_tuple_arg (_tuple y z))"
"\<lambda>(x, y, zs). b" \<rightleftharpoons> "CONST case_prod (\<lambda>x (y, zs). b)"
"\<lambda>(x, y). b" \<rightleftharpoons> "CONST case_prod (\<lambda>x y. b)"
"_abs (CONST Pair x y) t" \<rightharpoonup> "\<lambda>(x, y). t"
\<comment> \<open>This rule accommodates tuples in \<open>case C \<dots> (x, y) \<dots> \<Rightarrow> \<dots>\<close>:
The \<open>(x, y)\<close> is parsed as \<open>Pair x y\<close> because it is \<open>logic\<close>,
not \<open>pttrn\<close>.\<close>
"\<lambda>(). b" \<rightleftharpoons> "CONST case_unit b"
"_abs (CONST Unity) t" \<rightharpoonup> "\<lambda>(). t"
text \<open>print @{term "case_prod f"} as @{term "\<lambda>(x, y). f x y"} and
@{term "case_prod (\<lambda>x. f x)"} as @{term "\<lambda>(x, y). f x y"}\<close>
typed_print_translation \<open>
let
fun case_prod_guess_names_tr' T [Abs (x, _, Abs _)] = raise Match
| case_prod_guess_names_tr' T [Abs (x, xT, t)] =
(case (head_of t) of
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] =
(case head_of t of
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
\<close>
text \<open>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.\<close>
print_translation \<open>
let
fun case_prod_tr' [Abs (x, T, t as (Abs abs))] =
(* case_prod (\<lambda>x y. t) \<Rightarrow> \<lambda>(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 (\<lambda>x. (case_prod (\<lambda>y z. t))) \<Rightarrow> \<lambda>(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 (\<lambda>x y z. t)) \<Rightarrow> \<lambda>((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 (\<lambda>pttrn z. t) \<Rightarrow> \<lambda>(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
\<close>
subsubsection \<open>Code generator setup\<close>
code_printing
type_constructor prod \<rightharpoonup>
(SML) infix 2 "*"
and (OCaml) infix 2 "*"
and (Haskell) "!((_),/ (_))"
and (Scala) "((_),/ (_))"
| constant Pair \<rightharpoonup>
(SML) "!((_),/ (_))"
and (OCaml) "!((_),/ (_))"
and (Haskell) "!((_),/ (_))"
and (Scala) "!((_),/ (_))"
| class_instance prod :: equal \<rightharpoonup>
(Haskell) -
| constant "HOL.equal :: 'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool" \<rightharpoonup>
(Haskell) infix 4 "=="
| constant fst \<rightharpoonup> (Haskell) "fst"
| constant snd \<rightharpoonup> (Haskell) "snd"
subsubsection \<open>Fundamental operations and properties\<close>
lemma Pair_inject: "(a, b) = (a', b') \<Longrightarrow> (a = a' \<Longrightarrow> b = b' \<Longrightarrow> R) \<Longrightarrow> R"
by simp
lemma surj_pair [simp]: "\<exists>x y. p = (x, y)"
by (cases p) simp
lemma fst_eqD: "fst (x, y) = a \<Longrightarrow> x = a"
by simp
lemma snd_eqD: "snd (x, y) = a \<Longrightarrow> y = a"
by simp
lemma case_prod_unfold [nitpick_unfold]: "case_prod = (\<lambda>c 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) \<Rightarrow> f c d) = f a b"
by (fact prod.case)
lemmas surjective_pairing = prod.collapse [symmetric]
lemma prod_eq_iff: "s = t \<longleftrightarrow> fst s = fst t \<and> snd s = snd t"
by (cases s, cases t) simp
lemma prod_eqI [intro?]: "fst p = fst q \<Longrightarrow> snd p = snd q \<Longrightarrow> p = q"
by (simp add: prod_eq_iff)
lemma case_prodI: "f a b \<Longrightarrow> case (a, b) of (c, d) \<Rightarrow> f c d"
by (rule prod.case [THEN iffD2])
lemma case_prodD: "(case (a, b) of (c, d) \<Rightarrow> f c d) \<Longrightarrow> 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: "(\<lambda>(x, y). f (x, y)) = f"
\<comment> \<open>Subsumes the old \<open>split_Pair\<close> when @{term f} is the identity function.\<close>
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]: "(\<lambda>(_,_). c) = (\<lambda>_. c)"
by(rule case_prod_eta)
*)
lemma case_prod_comp: "(case x of (a, b) \<Rightarrow> (f \<circ> 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))"
by (simp add: case_prod_unfold)
lemma cond_case_prod_eta: "(\<And>x y. f x y = g (x, y)) \<Longrightarrow> (\<lambda>(x, y). f x y) = g"
by (simp add: case_prod_eta)
lemma split_paired_all [no_atp]: "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
proof
fix a b
assume "\<And>x. PROP P x"
then show "PROP P (a, b)" .
next
fix x
assume "\<And>a b. PROP P (a, b)"
from \<open>PROP P (fst x, snd x)\<close> show "PROP P x" by simp
qed
text \<open>
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 \<open>\<And>a b. \<dots> = ?P(a, b)\<close>
which cannot be solved by reflexivity.
\<close>
lemmas split_tupled_all = split_paired_all unit_all_eq2
ML \<open>
(* 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}]
addsimprocs [@{simproc unit_eq}]);
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;
\<close>
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_all_tac", split_all_tac))\<close>
lemma split_paired_All [simp, no_atp]: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>a b. P (a, b))"
\<comment> \<open>\<open>[iff]\<close> is not a good idea because it makes \<open>blast\<close> loop\<close>
by fast
lemma split_paired_Ex [simp, no_atp]: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>a b. P (a, b))"
by fast
lemma split_paired_The [no_atp]: "(THE x. P x) = (THE (a, b). P (a, b))"
\<comment> \<open>Can't be added to simpset: loops!\<close>
by (simp add: case_prod_eta)
text \<open>
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 \<open>prod.case_eq_if\<close>.
\<close>
ML \<open>
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;
\<close>
simproc_setup case_prod_beta ("case_prod f z") =
\<open>fn _ => fn ctxt => fn ct => beta_proc ctxt (Thm.term_of ct)\<close>
simproc_setup case_prod_eta ("case_prod f") =
\<open>fn _ => fn ctxt => fn ct => eta_proc ctxt (Thm.term_of ct)\<close>
lemma case_prod_beta': "(\<lambda>(x,y). f x y) = (\<lambda>x. f (fst x) (snd x))"
by (auto simp: fun_eq_iff)
text \<open>
\<^medskip> @{const case_prod} used as a logical connective or set former.
\<^medskip> These rules are for use with \<open>blast\<close>; could instead
call \<open>simp\<close> using @{thm [source] prod.split} as rewrite.\<close>
lemma case_prodI2:
"\<And>p. (\<And>a b. p = (a, b) \<Longrightarrow> c a b) \<Longrightarrow> case p of (a, b) \<Rightarrow> c a b"
by (simp add: split_tupled_all)
lemma case_prodI2':
"\<And>p. (\<And>a b. (a, b) = p \<Longrightarrow> c a b x) \<Longrightarrow> (case p of (a, b) \<Rightarrow> c a b) x"
by (simp add: split_tupled_all)
lemma case_prodE [elim!]:
"(case p of (a, b) \<Rightarrow> c a b) \<Longrightarrow> (\<And>x y. p = (x, y) \<Longrightarrow> c x y \<Longrightarrow> Q) \<Longrightarrow> Q"
by (induct p) simp
lemma case_prodE' [elim!]:
"(case p of (a, b) \<Rightarrow> c a b) z \<Longrightarrow> (\<And>x y. p = (x, y) \<Longrightarrow> c x y z \<Longrightarrow> Q) \<Longrightarrow> Q"
by (induct p) simp
lemma case_prodE2:
assumes q: "Q (case z of (a, b) \<Rightarrow> P a b)"
and r: "\<And>x y. z = (x, y) \<Longrightarrow> Q (P x y) \<Longrightarrow> 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) \<Rightarrow> R c d) c \<Longrightarrow> R a b c"
by simp
lemma mem_case_prodI: "z \<in> c a b \<Longrightarrow> z \<in> (case (a, b) of (d, e) \<Rightarrow> c d e)"
by simp
lemma mem_case_prodI2 [intro!]:
"\<And>p. (\<And>a b. p = (a, b) \<Longrightarrow> z \<in> c a b) \<Longrightarrow> z \<in> (case p of (a, b) \<Rightarrow> c a b)"
by (simp only: split_tupled_all) simp
declare mem_case_prodI [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close>
declare case_prodI2' [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close>
declare case_prodI2 [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close>
declare case_prodI [intro!] \<comment> \<open>postponed to maintain traditional declaration order!\<close>
lemma mem_case_prodE [elim!]:
assumes "z \<in> case_prod c p"
obtains x y where "p = (x, y)" and "z \<in> c x y"
using assms by (rule case_prodE2)
ML \<open>
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;
\<close>
(* This prevents applications of splitE for already splitted arguments leading
to quite time-consuming computations (in particular for nested tuples) *)
setup \<open>map_theory_claset (fn ctxt => ctxt addSbefore ("split_conv_tac", split_conv_tac))\<close>
lemma split_eta_SetCompr [simp, no_atp]: "(\<lambda>u. \<exists>x y. u = (x, y) \<and> P (x, y)) = P"
by (rule ext) fast
lemma split_eta_SetCompr2 [simp, no_atp]: "(\<lambda>u. \<exists>x y. u = (x, y) \<and> P x y) = case_prod P"
by (rule ext) fast
lemma split_part [simp]: "(\<lambda>(a,b). P \<and> Q a b) = (\<lambda>ab. P \<and> case_prod Q ab)"
\<comment> \<open>Allows simplifications of nested splits in case of independent predicates.\<close>
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 \<Rightarrow> 'b \<Rightarrow> 'c"
and g :: "'d \<Rightarrow> 'a"
shows "(\<lambda>u. f (g (fst u)) (snd u)) = case_prod (\<lambda>x. f (g x))"
by (rule ext) auto
lemma pair_imageI [intro]: "(a, b) \<in> A \<Longrightarrow> f a b \<in> (\<lambda>(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' \<and> 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]:
"(\<And>a b c. P (a, b, c)) \<Longrightarrow> 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]:
"(\<And>a b c d. P (a, b, c, d)) \<Longrightarrow> 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]:
"(\<And>a b c d e. P (a, b, c, d, e)) \<Longrightarrow> 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]:
"(\<And>a b c d e f. P (a, b, c, d, e, f)) \<Longrightarrow> 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]:
"(\<And>a b c d e f g. P (a, b, c, d, e, f, g)) \<Longrightarrow> P x"
by (cases x) blast
definition internal_case_prod :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"
where "internal_case_prod \<equiv> 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 \<open>Derived operations\<close>
definition curry :: "('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"
where "curry = (\<lambda>c x y. c (x, y))"
lemma curry_conv [simp, code]: "curry f a b = f (a, b)"
by (simp add: curry_def)
lemma curryI [intro!]: "f (a, b) \<Longrightarrow> curry f a b"
by (simp add: curry_def)
lemma curryD [dest!]: "curry f a b \<Longrightarrow> f (a, b)"
by (simp add: curry_def)
lemma curryE: "curry f a b \<Longrightarrow> (f (a, b) \<Longrightarrow> Q) \<Longrightarrow> Q"
by (simp add: curry_def)
lemma curry_case_prod [simp]: "curry (case_prod f) = f"
by (simp add: curry_def case_prod_unfold)
lemma case_prod_curry [simp]: "case_prod (curry f) = f"
by (simp add: curry_def case_prod_unfold)
lemma curry_K: "curry (\<lambda>x. c) = (\<lambda>x y. c)"
by (simp add: fun_eq_iff)
text \<open>The composition-uncurry combinator.\<close>
notation fcomp (infixl "\<circ>>" 60)
definition scomp :: "('a \<Rightarrow> 'b \<times> 'c) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd" (infixl "\<circ>\<rightarrow>" 60)
where "f \<circ>\<rightarrow> g = (\<lambda>x. case_prod g (f x))"
lemma scomp_unfold: "scomp = (\<lambda>f 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 \<circ>\<rightarrow> g) x = case_prod g (f x)"
by (simp add: scomp_unfold case_prod_unfold)
lemma Pair_scomp: "Pair x \<circ>\<rightarrow> f = f x"
by (simp add: fun_eq_iff)
lemma scomp_Pair: "x \<circ>\<rightarrow> Pair = x"
by (simp add: fun_eq_iff)
lemma scomp_scomp: "(f \<circ>\<rightarrow> g) \<circ>\<rightarrow> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>\<rightarrow> h)"
by (simp add: fun_eq_iff scomp_unfold)
lemma scomp_fcomp: "(f \<circ>\<rightarrow> g) \<circ>> h = f \<circ>\<rightarrow> (\<lambda>x. g x \<circ>> h)"
by (simp add: fun_eq_iff scomp_unfold fcomp_def)
lemma fcomp_scomp: "(f \<circ>> g) \<circ>\<rightarrow> h = f \<circ>> (g \<circ>\<rightarrow> h)"
by (simp add: fun_eq_iff scomp_unfold)
code_printing
constant scomp \<rightharpoonup> (Eval) infixl 3 "#->"
no_notation fcomp (infixl "\<circ>>" 60)
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
text \<open>
@{term map_prod} --- action of the product functor upon functions.
\<close>
definition map_prod :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'd"
where "map_prod f g = (\<lambda>(x, y). (f x, g y))"
lemma map_prod_simp [simp, code]: "map_prod f g (a, b) = (f a, g b)"
by (simp add: map_prod_def)
functor map_prod: map_prod
by (auto simp add: split_paired_all)
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 \<circ> map_prod f g = f \<circ> fst"
by (rule ext) simp_all
lemma snd_comp_map_prod [simp]: "snd \<circ> map_prod f g = g \<circ> snd"
by (rule ext) simp_all
lemma map_prod_compose: "map_prod (f1 \<circ> f2) (g1 \<circ> g2) = (map_prod f1 g1 \<circ> map_prod f2 g2)"
by (rule ext) (simp add: map_prod.compositionality comp_def)
lemma map_prod_ident [simp]: "map_prod (\<lambda>x. x) (\<lambda>y. y) = (\<lambda>z. z)"
by (rule ext) (simp add: map_prod.identity)
lemma map_prod_imageI [intro]: "(a, b) \<in> R \<Longrightarrow> (f a, g b) \<in> map_prod f g ` R"
by (rule image_eqI) simp_all
lemma prod_fun_imageE [elim!]:
assumes major: "c \<in> map_prod f g ` R"
and cases: "\<And>x y. c = (f x, g y) \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> P"
shows P
apply (rule major [THEN imageE])
apply (case_tac x)
apply (rule cases)
apply simp_all
done
definition apfst :: "('a \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c \<times> 'b"
where "apfst f = map_prod f id"
definition apsnd :: "('b \<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'a \<times> 'c"
where "apsnd f = map_prod id f"
lemma apfst_conv [simp, code]: "apfst f (x, y) = (f x, y)"
by (simp add: apfst_def)
lemma apsnd_conv [simp, code]: "apsnd f (x, y) = (x, f y)"
by (simp add: apsnd_def)
lemma fst_apfst [simp]: "fst (apfst f x) = f (fst x)"
by (cases x) simp
lemma fst_comp_apfst [simp]: "fst \<circ> apfst f = f \<circ> fst"
by (simp add: fun_eq_iff)
lemma fst_apsnd [simp]: "fst (apsnd f x) = fst x"
by (cases x) simp
lemma fst_comp_apsnd [simp]: "fst \<circ> apsnd f = fst"
by (simp add: fun_eq_iff)
lemma snd_apfst [simp]: "snd (apfst f x) = snd x"
by (cases x) simp
lemma snd_comp_apfst [simp]: "snd \<circ> apfst f = snd"
by (simp add: fun_eq_iff)
lemma snd_apsnd [simp]: "snd (apsnd f x) = f (snd x)"
by (cases x) simp
lemma snd_comp_apsnd [simp]: "snd \<circ> apsnd f = f \<circ> snd"
by (simp add: fun_eq_iff)
lemma apfst_compose: "apfst f (apfst g x) = apfst (f \<circ> g) x"
by (cases x) simp
lemma apsnd_compose: "apsnd f (apsnd g x) = apsnd (f \<circ> 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"
by (simp add: fun_eq_iff)
lemma apsnd_id [simp]: "apsnd id = id"
by (simp add: fun_eq_iff)
lemma apfst_eq_conv [simp]: "apfst f x = apfst g x \<longleftrightarrow> f (fst x) = g (fst x)"
by (cases x) simp
lemma apsnd_eq_conv [simp]: "apsnd f x = apsnd g x \<longleftrightarrow> 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 \<open>Local_Theory.map_background_naming (Name_Space.mandatory_path "prod")\<close>
definition swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a"
where "swap p = (snd p, fst p)"
end
lemma swap_simp [simp]: "prod.swap (x, y) = (y, x)"
by (simp add: prod.swap_def)
lemma swap_swap [simp]: "prod.swap (prod.swap p) = p"
by (cases p) simp
lemma swap_comp_swap [simp]: "prod.swap \<circ> prod.swap = id"
by (simp add: fun_eq_iff)
lemma pair_in_swap_image [simp]: "(y, x) \<in> prod.swap ` A \<longleftrightarrow> (x, y) \<in> 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 (\<lambda>(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"
by (simp add: bij_def)
lemma case_swap [simp]: "(case prod.swap p of (y, x) \<Rightarrow> f x y) = (case p of (x, y) \<Rightarrow> 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 \<open>Disjoint union of a family of sets -- Sigma.\<close>
definition Sigma :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<times> 'b) set"
where "Sigma A B \<equiv> \<Union>x\<in>A. \<Union>y\<in>B x. {Pair x y}"
abbreviation Times :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" (infixr "\<times>" 80)
where "A \<times> B \<equiv> Sigma A (\<lambda>_. B)"
hide_const (open) Times
bundle no_Set_Product_syntax begin
no_notation Product_Type.Times (infixr "\<times>" 80)
end
bundle Set_Product_syntax begin
notation Product_Type.Times (infixr "\<times>" 80)
end
syntax
"_Sigma" :: "pttrn \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set" ("(3SIGMA _:_./ _)" [0, 0, 10] 10)
translations
"SIGMA x:A. B" \<rightleftharpoons> "CONST Sigma A (\<lambda>x. B)"
lemma SigmaI [intro!]: "a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> (a, b) \<in> Sigma A B"
unfolding Sigma_def by blast
lemma SigmaE [elim!]: "c \<in> Sigma A B \<Longrightarrow> (\<And>x y. x \<in> A \<Longrightarrow> y \<in> B x \<Longrightarrow> c = (x, y) \<Longrightarrow> P) \<Longrightarrow> P"
\<comment> \<open>The general elimination rule.\<close>
unfolding Sigma_def by blast
text \<open>
Elimination of @{term "(a, b) \<in> A \<times> B"} -- introduces no
eigenvariables.
\<close>
lemma SigmaD1: "(a, b) \<in> Sigma A B \<Longrightarrow> a \<in> A"
by blast
lemma SigmaD2: "(a, b) \<in> Sigma A B \<Longrightarrow> b \<in> B a"
by blast
lemma SigmaE2: "(a, b) \<in> Sigma A B \<Longrightarrow> (a \<in> A \<Longrightarrow> b \<in> B a \<Longrightarrow> P) \<Longrightarrow> P"
by blast
lemma Sigma_cong: "A = B \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> C x = D x) \<Longrightarrow> (SIGMA x:A. C x) = (SIGMA x:B. D x)"
by auto
lemma Sigma_mono: "A \<subseteq> C \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> D x) \<Longrightarrow> Sigma A B \<subseteq> Sigma C D"
by blast
lemma Sigma_empty1 [simp]: "Sigma {} B = {}"
by blast
lemma Sigma_empty2 [simp]: "A \<times> {} = {}"
by blast
lemma UNIV_Times_UNIV [simp]: "UNIV \<times> UNIV = UNIV"
by auto
lemma Compl_Times_UNIV1 [simp]: "- (UNIV \<times> A) = UNIV \<times> (-A)"
by auto
lemma Compl_Times_UNIV2 [simp]: "- (A \<times> UNIV) = (-A) \<times> UNIV"
by auto
lemma mem_Sigma_iff [iff]: "(a, b) \<in> Sigma A B \<longleftrightarrow> a \<in> A \<and> b \<in> B a"
by blast
lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
by (induct x) simp
lemma Sigma_empty_iff: "(SIGMA i:I. X i) = {} \<longleftrightarrow> (\<forall>i\<in>I. X i = {})"
by auto
lemma Times_subset_cancel2: "x \<in> C \<Longrightarrow> A \<times> C \<subseteq> B \<times> C \<longleftrightarrow> A \<subseteq> B"
by blast
lemma Times_eq_cancel2: "x \<in> C \<Longrightarrow> A \<times> C = B \<times> C \<longleftrightarrow> A = B"
by (blast elim: equalityE)
lemma Collect_case_prod_Sigma: "{(x, y). P x \<and> Q x y} = (SIGMA x:Collect P. Collect (Q x))"
by blast
lemma Collect_case_prod [simp]: "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q "
by (fact Collect_case_prod_Sigma)
lemma Collect_case_prodD: "x \<in> Collect (case_prod A) \<Longrightarrow> A (fst x) (snd x)"
by auto
lemma Collect_case_prod_mono: "A \<le> B \<Longrightarrow> Collect (case_prod A) \<subseteq> Collect (case_prod B)"
by auto (auto elim!: le_funE)
lemma Collect_split_mono_strong:
"X = fst ` A \<Longrightarrow> Y = snd ` A \<Longrightarrow> \<forall>a\<in>X. \<forall>b \<in> Y. P a b \<longrightarrow> Q a b
\<Longrightarrow> A \<subseteq> Collect (case_prod P) \<Longrightarrow> A \<subseteq> Collect (case_prod Q)"
by fastforce
lemma UN_Times_distrib: "(\<Union>(a, b)\<in>A \<times> B. E a \<times> F b) = UNION A E \<times> UNION B F"
\<comment> \<open>Suggested by Pierre Chartier\<close>
by blast
lemma split_paired_Ball_Sigma [simp, no_atp]: "(\<forall>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B x. P (x, y))"
by blast
lemma split_paired_Bex_Sigma [simp, no_atp]: "(\<exists>z\<in>Sigma A B. P z) \<longleftrightarrow> (\<exists>x\<in>A. \<exists>y\<in>B x. P (x, y))"
by blast
lemma Sigma_Un_distrib1: "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
by blast
lemma Sigma_Un_distrib2: "(SIGMA i:I. A i \<union> B i) = Sigma I A \<union> Sigma I B"
by blast
lemma Sigma_Int_distrib1: "Sigma (I \<inter> J) C = Sigma I C \<inter> Sigma J C"
by blast
lemma Sigma_Int_distrib2: "(SIGMA i:I. A i \<inter> B i) = Sigma I A \<inter> 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 (\<Union>X) B = (\<Union>A\<in>X. Sigma A B)"
by blast
lemma Pair_vimage_Sigma: "Pair x -` Sigma A f = (if x \<in> A then f x else {})"
by auto
text \<open>
Non-dependent versions are needed to avoid the need for higher-order
matching, especially when the rules are re-oriented.
\<close>
lemma Times_Un_distrib1: "(A \<union> B) \<times> C = A \<times> C \<union> B \<times> C "
by (fact Sigma_Un_distrib1)
lemma Times_Int_distrib1: "(A \<inter> B) \<times> C = A \<times> C \<inter> B \<times> C "
by (fact Sigma_Int_distrib1)
lemma Times_Diff_distrib1: "(A - B) \<times> C = A \<times> C - B \<times> C "
by (fact Sigma_Diff_distrib1)
lemma Times_empty [simp]: "A \<times> B = {} \<longleftrightarrow> A = {} \<or> B = {}"
by auto
lemma times_eq_iff: "A \<times> B = C \<times> D \<longleftrightarrow> A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {})"
by auto
lemma fst_image_times [simp]: "fst ` (A \<times> B) = (if B = {} then {} else A)"
by force
lemma snd_image_times [simp]: "snd ` (A \<times> B) = (if A = {} then {} else B)"
by force
lemma fst_image_Sigma: "fst ` (Sigma A B) = {x \<in> A. B(x) \<noteq> {}}"
by force
lemma snd_image_Sigma: "snd ` (Sigma A B) = (\<Union> x \<in> A. B x)"
by force
lemma vimage_fst: "fst -` A = A \<times> UNIV"
by auto
lemma vimage_snd: "snd -` A = UNIV \<times> A"
by auto
lemma insert_times_insert [simp]:
"insert a A \<times> insert b B = insert (a,b) (A \<times> insert b B \<union> insert a A \<times> B)"
by blast
lemma vimage_Times: "f -` (A \<times> B) = (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B"
proof (rule set_eqI)
show "x \<in> f -` (A \<times> B) \<longleftrightarrow> x \<in> (fst \<circ> f) -` A \<inter> (snd \<circ> f) -` B" for x
by (cases "f x") (auto split: prod.split)
qed
lemma times_Int_times: "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
by auto
lemma product_swap: "prod.swap ` (A \<times> B) = B \<times> A"
by (auto simp add: set_eq_iff)
lemma swap_product: "(\<lambda>(i, j). (j, i)) ` (A \<times> B) = B \<times> A"
by (auto simp add: set_eq_iff)
lemma image_split_eq_Sigma: "(\<lambda>x. (f x, g x)) ` A = Sigma (f ` A) (\<lambda>x. g ` (f -` {x} \<inter> A))"
proof (safe intro!: imageI)
fix a b
assume *: "a \<in> A" "b \<in> A" and eq: "f a = f b"
show "(f b, g a) \<in> (\<lambda>x. (f x, g x)) ` A"
using * eq[symmetric] by auto
qed simp_all
lemma subset_fst_snd: "A \<subseteq> (fst ` A \<times> snd ` A)"
by force
lemma inj_on_apfst [simp]: "inj_on (apfst f) (A \<times> UNIV) \<longleftrightarrow> inj_on f A"
by (auto simp add: inj_on_def)
lemma inj_apfst [simp]: "inj (apfst f) \<longleftrightarrow> inj f"
using inj_on_apfst[of f UNIV] by simp
lemma inj_on_apsnd [simp]: "inj_on (apsnd f) (UNIV \<times> A) \<longleftrightarrow> inj_on f A"
by (auto simp add: inj_on_def)
lemma inj_apsnd [simp]: "inj (apsnd f) \<longleftrightarrow> inj f"
using inj_on_apsnd[of f UNIV] by simp
context
begin
qualified definition product :: "'a set \<Rightarrow> 'b set \<Rightarrow> ('a \<times> 'b) set"
where [code_abbrev]: "product A B = A \<times> B"
lemma member_product: "x \<in> Product_Type.product A B \<longleftrightarrow> x \<in> A \<times> B"
by (simp add: product_def)
end
text \<open>The following @{const map_prod} lemmas are due to Joachim Breitner:\<close>
lemma map_prod_inj_on:
assumes "inj_on f A"
and "inj_on g B"
shows "inj_on (map_prod f g) (A \<times> B)"
proof (rule inj_onI)
fix x :: "'a \<times> 'c"
fix y :: "'a \<times> 'c"
assume "x \<in> A \<times> B"
then have "fst x \<in> A" and "snd x \<in> B" by auto
assume "y \<in> A \<times> B"
then have "fst y \<in> A" and "snd y \<in> 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 \<open>inj_on f A\<close> and \<open>fst x \<in> A\<close> and \<open>fst y \<in> A\<close> have "fst x = fst y"
by (auto dest: inj_onD)
moreover from \<open>map_prod f g x = map_prod f g y\<close>
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 \<open>inj_on g B\<close> and \<open>snd x \<in> B\<close> and \<open>snd y \<in> B\<close> 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 \<Rightarrow> 'b"
and g :: "'c \<Rightarrow> 'd"
assumes "surj f" and "surj g"
shows "surj (map_prod f g)"
unfolding surj_def
proof
fix y :: "'b \<times> 'd"
from \<open>surj f\<close> obtain a where "fst y = f a"
by (auto elim: surjE)
moreover
from \<open>surj g\<close> 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 "\<exists>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 \<times> B) = A' \<times> B'"
unfolding image_def
proof (rule set_eqI, rule iffI)
fix x :: "'a \<times> 'c"
assume "x \<in> {y::'a \<times> 'c. \<exists>x::'b \<times> 'd\<in>A \<times> B. y = map_prod f g x}"
then obtain y where "y \<in> A \<times> B" and "x = map_prod f g y"
by blast
from \<open>image f A = A'\<close> and \<open>y \<in> A \<times> B\<close> have "f (fst y) \<in> A'"
by auto
moreover from \<open>image g B = B'\<close> and \<open>y \<in> A \<times> B\<close> have "g (snd y) \<in> B'"
by auto
ultimately have "(f (fst y), g (snd y)) \<in> (A' \<times> B')"
by auto
with \<open>x = map_prod f g y\<close> show "x \<in> A' \<times> B'"
by (cases y) auto
next
fix x :: "'a \<times> 'c"
assume "x \<in> A' \<times> B'"
then have "fst x \<in> A'" and "snd x \<in> B'"
by auto
from \<open>image f A = A'\<close> and \<open>fst x \<in> A'\<close> have "fst x \<in> image f A"
by auto
then obtain a where "a \<in> A" and "fst x = f a"
by (rule imageE)
moreover from \<open>image g B = B'\<close> and \<open>snd x \<in> B'\<close> obtain b where "b \<in> B" and "snd x = g b"
by auto
ultimately have "(fst x, snd x) = map_prod f g (a, b)"
by auto
moreover from \<open>a \<in> A\<close> and \<open>b \<in> B\<close> have "(a , b) \<in> A \<times> B"
by auto
ultimately have "\<exists>y \<in> A \<times> B. x = map_prod f g y"
by auto
then show "x \<in> {x. \<exists>y \<in> A \<times> B. x = map_prod f g y}"
by auto
qed
subsection \<open>Simproc for rewriting a set comprehension into a pointfree expression\<close>
ML_file "Tools/set_comprehension_pointfree.ML"
setup \<open>
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}])
\<close>
subsection \<open>Inductively defined sets\<close>
(* simplify {(x1, ..., xn). (x1, ..., xn) : S} to S *)
simproc_setup Collect_mem ("Collect t") = \<open>
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)
\<close>
ML_file "Tools/inductive_set.ML"
subsection \<open>Legacy theorem bindings and duplicates\<close>
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