(* Title: HOL/Library/FuncSet.thy
Author: Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
*)
section ‹Pi
and Function Sets
›
theory FuncSet
imports Main
abbrevs PiE =
"Pi\<^sub>E"
and PIE =
"\\<^sub>E"
begin
definition Pi ::
"'a set \ ('a \ 'b set) \ ('a \ 'b) set"
where "Pi A B = {f. \x. x \ A \ f x \ B x}"
definition extensional ::
"'a set \ ('a \ 'b) set"
where "extensional A = {f. \x. x \ A \ f x = undefined}"
definition "restrict" ::
"('a \ 'b) \ 'a set \ 'a \ 'b"
where "restrict f A = (\x. if x \ A then f x else undefined)"
abbreviation funcset ::
"'a set \ 'b set \ ('a \ 'b) set"
where "funcset A B \ Pi A (\_. B)"
open_bundle funcset_syntax
begin
notation funcset (
infixr ‹→› 60)
end
syntax
"_Pi" ::
"pttrn \ 'a set \ 'b set \ ('a \ 'b) set"
(
‹(
‹indent=3
notation=
‹binder Π
∈››Π _
∈_./ _)
› 10)
"_lam" ::
"pttrn \ 'a set \ ('a \ 'b) \ ('a \ 'b)"
(
‹(
‹indent=3
notation=
‹binder λ
∈››λ_
∈_./ _)
› [0, 0, 3] 3)
syntax_consts
"_Pi" ⇌ Pi
and
"_lam" ⇌ restrict
translations
"\ x\A. B" ⇌ "CONST Pi A (\x. B)"
"\x\A. f" ⇌ "CONST restrict (\x. f) A"
definition "compose" ::
"'a set \ ('b \ 'c) \ ('a \ 'b) \ ('a \ 'c)"
where "compose A g f = (\x\A. g (f x))"
subsection ‹Basic Properties of
🍋‹Pi
››
lemma Pi_I[intro!]:
"(\x. x \ A \ f x \ B x) \ f \ Pi A B"
by (simp add: Pi_def)
lemma Pi_I
'[simp]: "(\x. x \ A \ f x \ B x) \ f \ Pi A B"
by (simp add:Pi_def)
lemma funcsetI:
"(\x. x \ A \ f x \ B) \ f \ A \ B"
by (simp add: Pi_def)
lemma Pi_mem:
"f \ Pi A B \ x \ A \ f x \ B x"
by (simp add: Pi_def)
lemma Pi_iff:
"f \ Pi I X \ (\i\I. f i \ X i)"
unfolding Pi_def
by auto
lemma PiE [elim]:
"f \ Pi A B \ (f x \ B x \ Q) \ (x \ A \ Q) \ Q"
by (auto simp: Pi_def)
lemma Pi_cong:
"(\w. w \ A \ f w = g w) \ f \ Pi A B \ g \ Pi A B"
by (auto simp: Pi_def)
lemma funcset_id [simp]:
"(\x. x) \ A \ A"
by auto
lemma funcset_mem:
"f \ A \ B \ x \ A \ f x \ B"
by (simp add: Pi_def)
lemma funcset_image:
"f \ A \ B \ f ` A \ B"
by auto
lemma image_subset_iff_funcset:
"F ` A \ B \ F \ A \ B"
by auto
lemma funcset_to_empty_iff:
"A \ {} = (if A={} then UNIV else {})"
by auto
lemma Pi_eq_empty[simp]:
"(\ x \ A. B x) = {} \ (\x\A. B x = {})"
proof -
have "\x\A. B x = {}" if "\f. \y. y \ A \ f y \ B y"
using that [of
"\u. SOME y. y \ B u"] some_in_eq
by blast
then show ?thesis
by force
qed
lemma Pi_empty [simp]:
"Pi {} B = UNIV"
by (simp add: Pi_def)
lemma Pi_Int:
"Pi I E \ Pi I F = (\ i\I. E i \ F i)"
by auto
lemma Pi_UN:
fixes A ::
"nat \ 'i \ 'a set"
assumes "finite I"
and mono:
"\i n m. i \ I \ n \ m \ A n i \ A m i"
shows "(\n. Pi I (A n)) = (\ i\I. \n. A n i)"
proof (intro set_eqI iffI)
fix f
assume "f \ (\ i\I. \n. A n i)"
then have "\i\I. \n. f i \ A n i"
by auto
from bchoice[OF this]
obtain n
where n:
"f i \ A (n i) i" if "i \ I" for i
by auto
obtain k
where k:
"n i \ k" if "i \ I" for i
using ‹finite I
› finite_nat_set_iff_bounded_le[of
"n`I"]
by auto
have "f \ Pi I (A k)"
proof (intro Pi_I)
fix i
assume "i \ I"
from mono[OF this, of
"n i" k] k[OF this] n[OF this]
show "f i \ A k i" by auto
qed
then show "f \ (\n. Pi I (A n))"
by auto
qed auto
lemma Pi_UNIV [simp]:
"A \ UNIV = UNIV"
by (simp add: Pi_def)
text ‹Covariance of Pi-sets
in their second argument
›
lemma Pi_mono:
"(\x. x \ A \ B x \ C x) \ Pi A B \ Pi A C"
by auto
text ‹Contravariance of Pi-sets
in their first argument
›
lemma Pi_anti_mono:
"A' \ A \ Pi A B \ Pi A' B"
by auto
lemma prod_final:
assumes 1:
"fst \ f \ Pi A B"
and 2:
"snd \ f \ Pi A C"
shows "f \ (\ z \ A. B z \ C z)"
proof (rule Pi_I)
fix z
assume z:
"z \ A"
have "f z = (fst (f z), snd (f z))"
by simp
also have "\ \ B z \ C z"
by (metis SigmaI PiE o_apply 1 2 z)
finally show "f z \ B z \ C z" .
qed
lemma Pi_split_domain[simp]:
"x \ Pi (I \ J) X \ x \ Pi I X \ x \ Pi J X"
by (auto simp: Pi_def)
lemma Pi_split_insert_domain[simp]:
"x \ Pi (insert i I) X \ x \ Pi I X \ x i \ X i"
by (auto simp: Pi_def)
lemma Pi_cancel_fupd_range[simp]:
"i \ I \ x \ Pi I (B(i := b)) \ x \ Pi I B"
by (auto simp: Pi_def)
lemma Pi_cancel_fupd[simp]:
"i \ I \ x(i := a) \ Pi I B \ x \ Pi I B"
by (auto simp: Pi_def)
lemma Pi_fupd_iff:
"i \ I \ f \ Pi I (B(i := A)) \ f \ Pi (I - {i}) B \ f i \ A"
using mk_disjoint_insert
by fastforce
lemma fst_Pi:
"fst \ A \ B \ A" and snd_Pi:
"snd \ A \ B \ B"
by auto
subsection ‹Composition
With a Restricted
Domain:
🍋‹compose››
lemma funcset_compose:
"f \ A \ B \ g \ B \ C \ compose A g f \ A \ C"
by (simp add: Pi_def compose_def restrict_def)
lemma compose_assoc:
assumes "f \ A \ B"
shows "compose A h (compose A g f) = compose A (compose B h g) f"
using assms
by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
lemma compose_eq:
"x \ A \ compose A g f x = g (f x)"
by (simp add: compose_def restrict_def)
lemma surj_compose:
"f ` A = B \ g ` B = C \ compose A g f ` A = C"
by (auto simp add: image_def compose_eq)
subsection ‹Bounded Abstraction:
🍋‹restrict››
lemma restrict_cong:
"I = J \ (\i. i \ J =simp=> f i = g i) \ restrict f I = restrict g J"
by (auto simp: restrict_def fun_eq_iff simp_implies_def)
lemma restrictI[intro!]:
"(\x. x \ A \ f x \ B x) \ (\x\A. f x) \ Pi A B"
by (simp add: Pi_def restrict_def)
lemma restrict_apply[simp]:
"(\y\A. f y) x = (if x \ A then f x else undefined)"
by (simp add: restrict_def)
lemma restrict_apply
': "x \ A \ (\y\A. f y) x = f x"
by simp
lemma restrict_ext:
"(\x. x \ A \ f x = g x) \ (\x\A. f x) = (\x\A. g x)"
by (simp add: fun_eq_iff Pi_def restrict_def)
lemma restrict_UNIV:
"restrict f UNIV = f"
by (simp add: restrict_def)
lemma inj_on_restrict_eq [simp]:
"inj_on (restrict f A) A \ inj_on f A"
by (simp add: inj_on_def restrict_def)
lemma inj_on_restrict_iff:
"A \ B \ inj_on (restrict f B) A \ inj_on f A"
by (metis inj_on_cong restrict_def subset_iff)
lemma Id_compose:
"f \ A \ B \ f \ extensional A \ compose A (\y\B. y) f = f"
by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
lemma compose_Id:
"g \ A \ B \ g \ extensional A \ compose A g (\x\A. x) = g"
by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
lemma image_restrict_eq [simp]:
"(restrict f A) ` A = f ` A"
by (auto simp add: restrict_def)
lemma restrict_restrict[simp]:
"restrict (restrict f A) B = restrict f (A \ B)"
unfolding restrict_def
by (simp add: fun_eq_iff)
lemma restrict_fupd[simp]:
"i \ I \ restrict (f (i := x)) I = restrict f I"
by (auto simp: restrict_def)
lemma restrict_upd[simp]:
"i \ I \ (restrict f I)(i := y) = restrict (f(i := y)) (insert i I)"
by (auto simp: fun_eq_iff)
lemma restrict_Pi_cancel:
"restrict x I \ Pi I A \ x \ Pi I A"
by (auto simp: restrict_def Pi_def)
lemma sum_restrict
' [simp]: "sum' (λi
∈I. g i) I = sum
' (\i. g i) I"
by (simp add: sum.G_def conj_commute cong: conj_cong)
lemma prod_restrict
' [simp]: "prod' (λi
∈I. g i) I = prod
' (\i. g i) I"
by (simp add: prod.G_def conj_commute cong: conj_cong)
subsection ‹Bijections Between Sets
›
text ‹The
definition of
🍋‹bij_betw
› is in ‹Fun.thy
›, but most of
the
theorems belong here, or need at least
🍋‹Hilbert_Choice
›.
›
lemma bij_betwI:
assumes "f \ A \ B"
and "g \ B \ A"
and g_f:
"\x. x\A \ g (f x) = x"
and f_g:
"\y. y\B \ f (g y) = y"
shows "bij_betw f A B"
unfolding bij_betw_def
proof
show "inj_on f A"
by (metis g_f inj_on_def)
have "f ` A \ B"
using ‹f
∈ A
→ B
› by auto
moreover
have "B \ f ` A"
by auto (metis Pi_mem
‹g
∈ B
→ A
› f_g image_iff)
ultimately show "f ` A = B"
by blast
qed
lemma bij_betw_imp_funcset:
"bij_betw f A B \ f \ A \ B"
by (auto simp add: bij_betw_def)
lemma inj_on_compose:
"bij_betw f A B \ inj_on g B \ inj_on (compose A g f) A"
by (auto simp add: bij_betw_def inj_on_def compose_eq)
lemma bij_betw_compose:
"bij_betw f A B \ bij_betw g B C \ bij_betw (compose A g f) A C"
by (simp add: bij_betw_def inj_on_compose surj_compose)
lemma bij_betw_restrict_eq [simp]:
"bij_betw (restrict f A) A B = bij_betw f A B"
by (simp add: bij_betw_def)
subsection ‹Extensionality
›
lemma extensional_empty[simp]:
"extensional {} = {\x. undefined}"
unfolding extensional_def
by auto
lemma extensional_arb:
"f \ extensional A \ x \ A \ f x = undefined"
by (simp add: extensional_def)
lemma restrict_extensional [simp]:
"restrict f A \ extensional A"
by (simp add: restrict_def extensional_def)
lemma compose_extensional [simp]:
"compose A f g \ extensional A"
by (simp add: compose_def)
lemma extensionalityI:
assumes "f \ extensional A"
and "g \ extensional A"
and "\x. x \ A \ f x = g x"
shows "f = g"
using assms
by (force simp add: fun_eq_iff extensional_def)
lemma extensional_restrict:
"f \ extensional A \ restrict f A = f"
by (rule extensionalityI[OF restrict_extensional]) auto
lemma extensional_subset:
"f \ extensional A \ A \ B \ f \ extensional B"
unfolding extensional_def
by auto
lemma inv_into_funcset:
"f ` A = B \ (\x\B. inv_into A f x) \ B \ A"
by (unfold inv_into_def) (fast intro: someI2)
lemma compose_inv_into_id:
"bij_betw f A B \ compose A (\y\B. inv_into A f y) f = (\x\A. x)"
by (smt (verit, best) bij_betwE bij_betw_inv_into_left compose_def restrict_apply
' restrict_ext)
lemma compose_id_inv_into:
"f ` A = B \ compose B f (\y\B. inv_into A f y) = (\x\B. x)"
by (smt (verit, best) compose_def f_inv_into_f restrict_apply
' restrict_ext)
lemma extensional_insert[intro, simp]:
assumes "a \ extensional (insert i I)"
shows "a(i := b) \ extensional (insert i I)"
using assms
unfolding extensional_def
by auto
lemma extensional_Int[simp]:
"extensional I \ extensional I' = extensional (I \ I')"
unfolding extensional_def
by auto
lemma extensional_UNIV[simp]:
"extensional UNIV = UNIV"
by (auto simp: extensional_def)
lemma restrict_extensional_sub[intro]:
"A \ B \ restrict f A \ extensional B"
unfolding restrict_def extensional_def
by auto
lemma extensional_insert_undefined[intro, simp]:
"a \ extensional (insert i I) \ a(i := undefined) \ extensional I"
unfolding extensional_def
by auto
lemma extensional_insert_cancel[intro, simp]:
"a \ extensional I \ a \ extensional (insert i I)"
unfolding extensional_def
by auto
subsection ‹Cardinality
›
lemma card_inj:
"f \ A \ B \ inj_on f A \ finite B \ card A \ card B"
by (rule card_inj_on_le) auto
lemma card_bij:
assumes "f \ A \ B" "inj_on f A"
and "g \ B \ A" "inj_on g B"
and "finite A" "finite B"
shows "card A = card B"
using assms
by (blast intro: card_inj order_antisym)
subsection ‹Extensional
Function Spaces
›
definition PiE ::
"'a set \ ('a \ 'b set) \ ('a \ 'b) set"
where "PiE S T = Pi S T \ extensional S"
abbreviation "Pi\<^sub>E A B \ PiE A B"
syntax
"_PiE" ::
"pttrn \ 'a set \ 'b set \ ('a \ 'b) set"
(
‹(
‹indent=3
notation=
‹binder Π
🚫E
∈››Π
🚫E _
∈_./ _)
› 10)
syntax_consts
"_PiE" ⇌ Pi
🚫E
translations
"\\<^sub>E x\A. B" ⇌ "CONST Pi\<^sub>E A (\x. B)"
abbreviation extensional_funcset ::
"'a set \ 'b set \ ('a \ 'b) set" (
infixr ‹→🚫E
› 60
)
where "A \\<^sub>E B \ (\\<^sub>E i\A. B)"
lemma extensional_funcset_def: "extensional_funcset S T = (S \ T) \ extensional S"
by (simp add: PiE_def)
lemma PiE_empty_domain[simp]: "Pi\<^sub>E {} T = {\x. undefined}"
unfolding PiE_def by simp
lemma PiE_UNIV_domain: "Pi\<^sub>E UNIV T = Pi UNIV T"
unfolding PiE_def by simp
lemma PiE_empty_range[simp]: "i \ I \ F i = {} \ (\\<^sub>E i\I. F i) = {}"
unfolding PiE_def by auto
lemma PiE_eq_empty_iff: "Pi\<^sub>E I F = {} \ (\i\I. F i = {})"
proof
assume "Pi\<^sub>E I F = {}"
show "\i\I. F i = {}"
proof (rule ccontr)
assume "\ ?thesis"
then have "\i. \y. (i \ I \ y \ F i) \ (i \ I \ y = undefined)"
by auto
from choice[OF this]
obtain f where " \x. (x \ I \ f x \ F x) \ (x \ I \ f x = undefined)" ..
then have "f \ Pi\<^sub>E I F"
by (auto simp: extensional_def PiE_def)
with ‹Pi🚫E I F = {}› show False
by auto
qed
qed (auto simp: PiE_def)
lemma PiE_arb: "f \ Pi\<^sub>E S T \ x \ S \ f x = undefined"
unfolding PiE_def by auto (auto dest!: extensional_arb)
lemma PiE_mem: "f \ Pi\<^sub>E S T \ x \ S \ f x \ T x"
unfolding PiE_def by auto
lemma PiE_fun_upd: "y \ T x \ f \ Pi\<^sub>E S T \ f(x := y) \ Pi\<^sub>E (insert x S) T"
unfolding PiE_def extensional_def by auto
lemma fun_upd_in_PiE: "x \ S \ f \ Pi\<^sub>E (insert x S) T \ f(x := undefined) \ Pi\<^sub>E S T"
unfolding PiE_def extensional_def by auto
lemma PiE_insert_eq: "Pi\<^sub>E (insert x S) T = (\(y, g). g(x := y)) ` (T x \ Pi\<^sub>E S T)"
proof -
have "f \ (\(y, g). g(x := y)) ` (T x \ Pi\<^sub>E S T)" if "f \ Pi\<^sub>E (insert x S) T" "x \ S" for f
using that
by (auto intro!: image_eqI[where x="(f x, f(x := undefined))"] intro: fun_upd_in_PiE PiE_mem)
moreover
have "f \ (\(y, g). g(x := y)) ` (T x \ Pi\<^sub>E S T)" if "f \ Pi\<^sub>E (insert x S) T" "x \ S" for f
using that
by (auto intro!: image_eqI[where x="(f x, f)"] intro: fun_upd_in_PiE PiE_mem simp: insert_absorb)
ultimately show ?thesis
by (auto intro: PiE_fun_upd)
qed
lemma PiE_Int: "Pi\<^sub>E I A \ Pi\<^sub>E I B = Pi\<^sub>E I (\x. A x \ B x)"
by (auto simp: PiE_def)
lemma PiE_cong: "(\i. i\I \ A i = B i) \ Pi\<^sub>E I A = Pi\<^sub>E I B"
unfolding PiE_def by (auto simp: Pi_cong)
lemma PiE_E [elim]:
assumes "f \ Pi\<^sub>E A B"
obtains "x \ A" and "f x \ B x"
| "x \ A" and "f x = undefined"
using assms by (auto simp: Pi_def PiE_def extensional_def)
lemma PiE_I[intro!]:
"(\x. x \ A \ f x \ B x) \ (\x. x \ A \ f x = undefined) \ f \ Pi\<^sub>E A B"
by (simp add: PiE_def extensional_def)
lemma PiE_mono: "(\x. x \ A \ B x \ C x) \ Pi\<^sub>E A B \ Pi\<^sub>E A C"
by auto
lemma PiE_iff: "f \ Pi\<^sub>E I X \ (\i\I. f i \ X i) \ f \ extensional I"
by (simp add: PiE_def Pi_iff)
lemma restrict_PiE_iff: "restrict f I \ Pi\<^sub>E I X \ (\i \ I. f i \ X i)"
by (simp add: PiE_iff)
lemma ext_funcset_to_sing_iff [simp]: "A \\<^sub>E {a} = {\x\A. a}"
by (auto simp: PiE_def Pi_iff extensionalityI)
lemma PiE_restrict[simp]: "f \ Pi\<^sub>E A B \ restrict f A = f"
by (simp add: extensional_restrict PiE_def)
lemma restrict_PiE[simp]: "restrict f I \ Pi\<^sub>E I S \ f \ Pi I S"
by (auto simp: PiE_iff)
lemma PiE_eq_subset:
assumes ne: "\i. i \ I \ F i \ {}" "\i. i \ I \ F' i \ {}"
and eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
and "i \ I"
shows "F i \ F' i"
proof
fix x
assume "x \ F i"
with ne have "\j. \y. (j \ I \ y \ F j \ (i = j \ x = y)) \ (j \ I \ y = undefined)"
by auto
from choice[OF this] obtain f
where f: " \j. (j \ I \ f j \ F j \ (i = j \ x = f j)) \ (j \ I \ f j = undefined)" ..
then have "f \ Pi\<^sub>E I F"
by (auto simp: extensional_def PiE_def)
then have "f \ Pi\<^sub>E I F'"
using assms by simp
then show "x \ F' i"
using f ‹i ∈ I› by (auto simp: PiE_def)
qed
lemma PiE_eq_iff_not_empty:
assumes ne: "\i. i \ I \ F i \ {}" "\i. i \ I \ F' i \ {}"
shows "Pi\<^sub>E I F = Pi\<^sub>E I F' \ (\i\I. F i = F' i)"
proof (intro iffI ballI)
fix i
assume eq: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
assume i: "i \ I"
show "F i = F' i"
using PiE_eq_subset[of I F F', OF ne eq i]
using PiE_eq_subset[of I F' F, OF ne(2,1) eq[symmetric] i]
by auto
qed (auto simp: PiE_def)
lemma PiE_eq_iff: "Pi\<^sub>E I F = Pi\<^sub>E I F' \ (\i\I. F i = F' i) \ ((\i\I. F i = {}) \ (\i\I. F' i = {}))"
proof (intro iffI disjCI)
assume eq[simp]: "Pi\<^sub>E I F = Pi\<^sub>E I F'"
assume "\ ((\i\I. F i = {}) \ (\i\I. F' i = {}))"
then have "(\i\I. F i \ {}) \ (\i\I. F' i \ {})"
using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by auto
with PiE_eq_iff_not_empty[of I F F'] show "\i\I. F i = F' i"
by auto
next
assume "(\i\I. F i = F' i) \ (\i\I. F i = {}) \ (\i\I. F' i = {})"
then show "Pi\<^sub>E I F = Pi\<^sub>E I F'"
using PiE_eq_empty_iff[of I F] PiE_eq_empty_iff[of I F'] by (auto simp: PiE_def)
qed
lemma extensional_funcset_fun_upd_restricts_rangeI:
"\y \ S. f x \ f y \ f \ (insert x S) \\<^sub>E T \ f(x := undefined) \ S \\<^sub>E (T - {f x})"
unfolding extensional_funcset_def extensional_def
by (auto split: if_split_asm)
lemma extensional_funcset_fun_upd_extends_rangeI:
assumes "a \ T" "f \ S \\<^sub>E (T - {a})"
shows "f(x := a) \ insert x S \\<^sub>E T"
using assms unfolding extensional_funcset_def extensional_def by auto
lemma subset_PiE:
"PiE I S \ PiE I T \ PiE I S = {} \ (\i \ I. S i \ T i)" (is "?lhs \ _ \ ?rhs")
proof (cases "PiE I S = {}")
case False
moreover have "?lhs = ?rhs"
proof
assume L: ?lhs
have "\i. i\I \ S i \ {}"
using False PiE_eq_empty_iff by blast
with L show ?rhs
by (simp add: PiE_Int PiE_eq_iff inf.absorb_iff2)
qed auto
ultimately show ?thesis
by simp
qed simp
lemma PiE_eq: "PiE I S = PiE I T \ PiE I S = {} \ PiE I T = {} \ (\i \ I. S i = T i)"
by (auto simp: PiE_eq_iff PiE_eq_empty_iff)
lemma PiE_UNIV [simp]: "PiE UNIV (\i. UNIV) = UNIV"
by blast
lemma image_projection_PiE:
"(\f. f i) ` (PiE I S) = (if PiE I S = {} then {} else if i \ I then S i else {undefined})"
proof -
have "(\f. f i) ` Pi\<^sub>E I S = S i" if "i \ I" "f \ PiE I S" for f
proof -
have "x \ S i \ \f\Pi\<^sub>E I S. x = f i" for x
using that
by (force intro: bexI [where x="\k. if k=i then x else f k"])
then show ?thesis
using that by force
qed
then show ?thesis
by (smt (verit) PiE_arb equals0I image_cong image_constant image_empty)
qed
lemma PiE_singleton:
assumes "f \ extensional A"
shows "PiE A (\x. {f x}) = {f}"
proof -
have "g = f" if "g \ PiE A (\x. {f x})" for g
proof -
from that have "g x = f x" for x
using assms by (cases "x \ A") (auto simp: extensional_def)
then show ?thesis by (simp add: fun_eq_iff)
qed
with assms show ?thesis
by (auto simp: extensional_def)
qed
lemma PiE_eq_singleton: "(\\<^sub>E i\I. S i) = {\i\I. f i} \ (\i\I. S i = {f i})"
by (metis (mono_tags, lifting) PiE_eq PiE_singleton insert_not_empty restrict_apply' restrict_extensional)
lemma PiE_over_singleton_iff: "(\\<^sub>E x\{a}. B x) = (\b \ B a. {\x \ {a}. b})"
proof -
have "\xa\B a. x = (\x\{a}. xa)" if "x a \ B a" and "x \ extensional {a}" for x
using that PiE_singleton by fastforce
then show ?thesis
by (auto simp: PiE_iff split: if_split_asm)
qed
lemma all_PiE_elements:
"(\z \ PiE I S. \i \ I. P i (z i)) \ PiE I S = {} \ (\i \ I. \x \ S i. P i x)"
(is "?lhs = ?rhs")
proof (cases "PiE I S = {}")
case False
then obtain f where f: "\i. i \ I \ f i \ S i"
by fastforce
show ?thesis
proof
assume L: ?lhs
have "P i x"
if "i \ I" "x \ S i" for i x
proof -
have "(\j \ I. if j=i then x else f j) \ PiE I S"
by (simp add: f that(2))
then have "P i ((\j \ I. if j=i then x else f j) i)"
using L that by blast
with that show ?thesis
by simp
qed
then show ?rhs
by (simp add: False)
qed fastforce
qed simp
lemma PiE_ext: "\x \ PiE k s; y \ PiE k s; \i. i \ k \ x i = y i\ \ x = y"
by (metis ext PiE_E)
subsubsection ‹Injective Extensional Function Spaces›
lemma extensional_funcset_fun_upd_inj_onI:
assumes "f \ S \\<^sub>E (T - {a})"
and "inj_on f S"
shows "inj_on (f(x := a)) S"
using assms
unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
lemma extensional_funcset_extend_domain_inj_on_eq:
assumes "x \ S"
shows "{f. f \ (insert x S) \\<^sub>E T \ inj_on f (insert x S)} =
(λ(y, g). g(x:=y)) ` {(y, g). y ∈ T ∧ g ∈ S →🚫E (T - {y}) ∧ inj_on g S}"
proof -
have False if "f \ S \\<^sub>E T - {a}" and "a = (if y = x then a else f y)" and "y \ S" for a f y
using assms that by (auto dest!: PiE_mem split: if_split_asm)
moreover
have "\b. b \ S \\<^sub>E T - {f x} \ inj_on b S \ f = b(x := f x)"
if "f \ insert x S \\<^sub>E T" and "inj_on f S" and "\xb\S. f x \ f xb" for f
using that
unfolding inj_on_def
by (smt (verit, ccfv_threshold) PiE_restrict fun_upd_apply fun_upd_triv insert_Diff insert_iff
restrict_PiE_iff restrict_upd)
ultimately show ?thesis
using assms
apply (auto simp: image_iff intro: extensional_funcset_fun_upd_inj_onI
extensional_funcset_fun_upd_extends_rangeI del: PiE_I PiE_E)
apply (smt (verit, best) PiE_cong PiE_mem inj_on_def insertCI)
apply blast
done
qed
lemma extensional_funcset_extend_domain_inj_onI:
assumes "x \ S"
shows "inj_on (\(y, g). g(x := y)) {(y, g). y \ T \ g \ S \\<^sub>E (T - {y}) \ inj_on g S}"
using assms
by (simp add: inj_on_def) (metis PiE_restrict fun_upd_apply restrict_fupd)
subsubsection ‹Misc properties of functions, composition and restriction from HOL Light›
lemma function_factors_left_gen:
"(\x y. P x \ P y \ g x = g y \ f x = f y) \ (\h. \x. P x \ f x = h(g x))"
(is "?lhs = ?rhs")
proof
assume L: ?lhs
then show ?rhs
apply (rule_tac x="f \ inv_into (Collect P) g" in exI)
unfolding o_def
by (metis (mono_tags, opaque_lifting) f_inv_into_f imageI inv_into_into mem_Collect_eq)
qed auto
lemma function_factors_left: "(\x y. (g x = g y) \ (f x = f y)) \ (\h. f = h \ g)"
using function_factors_left_gen [of "\x. True" g f] unfolding o_def by blast
lemma function_factors_right_gen: "(\x. P x \ (\y. g y = f x)) \ (\h. \x. P x \ f x = g(h x))"
by metis
lemma function_factors_right: "(\x. \y. g y = f x) \ (\h. f = g \ h)"
unfolding o_def by metis
lemma restrict_compose_right: "restrict (g \ restrict f S) S = restrict (g \ f) S"
by auto
lemma restrict_compose_left: "f ` S \ T \ restrict (restrict g T \ f) S = restrict (g \ f) S"
by fastforce
subsubsection ‹Cardinality›
lemma finite_PiE: "finite S \ (\i. i \ S \ finite (T i)) \ finite (\\<^sub>E i \ S. T i)"
by (induct S arbitrary: T rule: finite_induct) (simp_all add: PiE_insert_eq)
lemma inj_combinator: "x \ S \ inj_on (\(y, g). g(x := y)) (T x \ Pi\<^sub>E S T)"
proof (safe intro!: inj_onI ext)
fix f y g z
assume "x \ S"
assume fg: "f \ Pi\<^sub>E S T" "g \ Pi\<^sub>E S T"
assume "f(x := y) = g(x := z)"
then have *: "\i. (f(x := y)) i = (g(x := z)) i"
unfolding fun_eq_iff by auto
from this[of x] show "y = z" by simp
fix i from *[of i] ‹x ∉ S› fg show "f i = g i"
by (auto split: if_split_asm simp: PiE_def extensional_def)
qed
lemma card_PiE: "finite S \ card (\\<^sub>E i \ S. T i) = (\ i\S. card (T i))"
proof (induct rule: finite_induct)
case empty
then show ?case
by auto
next
case (insert x S)
then show ?case
by (simp add: PiE_insert_eq inj_combinator card_image card_cartesian_product)
qed
lemma card_funcsetE: "finite A \ card (A \\<^sub>E B) = card B ^ card A"
by (subst card_PiE) auto
lemma card_inj_on_subset_funcset:
assumes finB: "finite B"
and finC: "finite C"
and AB: "A \ B"
shows "card {f \ B \\<^sub>E C. inj_on f A} =
card C^(card B - card A) * prod ((-) (card C)) {0 ..< card A}"
proof -
define D where "D = B - A"
from AB have B: "B = A \ D" and disj: "A \ D = {}"
unfolding D_def by auto
have sub: "card B - card A = card D"
unfolding D_def using finB AB
by (metis card_Diff_subset finite_subset)
from finB B have "finite A" "finite D" by auto
then show ?thesis
unfolding sub unfolding B using disj
proof (induct A rule: finite_induct)
case empty
from card_funcsetE[OF this(1), of C] show ?case
by auto
next
case (insert a A)
have "{f. f \ insert a A \ D \\<^sub>E C \ inj_on f (insert a A)} =
{f(a := c) | f c. f ∈ A ∪ D →🚫E C ∧ inj_on f A ∧ c ∈ C - f ` A}"
(is "?l = ?r")
proof
show "?r \ ?l"
by (auto intro: inj_on_fun_updI split: if_splits)
have "f \ ?r" if f: "f \ ?l" for f
proof -
let ?g = "f(a := undefined)"
let ?h = "?g(a := f a)"
have mem: "f a \ C - ?g ` A" using insert(1,2,4,5) f by auto
from f have f: "f \ insert a A \ D \\<^sub>E C" "inj_on f (insert a A)" by auto
hence "?g \ A \ D \\<^sub>E C" "inj_on ?g A" using ‹a ∉ A› ‹insert a A ∩ D = {}›
by (auto split: if_splits simp: inj_on_def)
with mem have "?h \ ?r" by blast
also have "?h = f" by auto
finally show ?thesis .
qed
then show "?l \ ?r" by auto
qed
also have "\ = (\ (f, c). f (a := c)) `
(Sigma {f . f ∈ A ∪ D →🚫E C ∧ inj_on f A} (λ f. C - f ` A))"
by auto
also have "card (...) = card (Sigma {f . f \ A \ D \\<^sub>E C \ inj_on f A} (\ f. C - f ` A))"
proof (rule card_image, intro inj_onI, clarsimp, goal_cases)
case (1 f c g d)
let ?f = "f(a := c, a := undefined)"
let ?g = "g(a := d, a := undefined)"
from 1 have id: "f(a := c) = g(a := d)"
by auto
from fun_upd_eqD[OF id]
have cd: "c = d"
by auto
from id have "?f = ?g"
by auto
also have "?f = f" using `f ∈ A ∪ D →🚫E C` insert(1,2,4,5)
by (intro ext, auto)
also have "?g = g" using `g ∈ A ∪ D →🚫E C` insert(1,2,4,5)
by (intro ext, auto)
finally show "f = g \ c = d"
using cd by auto
qed
also have "\ = (\f\{f \ A \ D \\<^sub>E C. inj_on f A}. card (C - f ` A))"
by (rule card_SigmaI, rule finite_subset[of _ "A \ D \\<^sub>E C"],
insert ‹finite C› ‹finite D› ‹finite A›, auto intro!: finite_PiE)
also have "\ = (\f\{f \ A \ D \\<^sub>E C. inj_on f A}. card C - card A)"
by (rule sum.cong[OF refl], subst card_Diff_subset, insert ‹finite A›, auto simp: card_image)
also have "\ = (card C - card A) * card {f \ A \ D \\<^sub>E C. inj_on f A}"
by simp
also have "\ = card C ^ card D * ((card C - card A) * prod ((-) (card C)) {0..
using insert by (auto simp: ac_simps)
also have "(card C - card A) * prod ((-) (card C)) {0..
prod ((-) (card C)) {0..<Suc (card A)}" by simp
also have "Suc (card A) = card (insert a A)" using insert by auto
finally show ?case .
qed
qed
subsection ‹The pigeonhole principle›
text ‹
An alternative formulation of this is that for a function mapping a finite set ‹A› of
cardinality ‹m› to a finite set ‹B› of cardinality ‹n›, there exists an element ‹y ∈ B› that
is hit at least $\lceil \frac{m}{n}\rceil$ times. However, since we do not have real numbers
or rounding yet, we state it in the following equivalent form:
›
lemma pigeonhole_card:
assumes "f \ A \ B" "finite A" "finite B" "B \ {}"
shows "\y\B. card (f -` {y} \ A) * card B \ card A"
proof -
from assms have "card B > 0"
by auto
define M where "M = Max ((\y. card (f -` {y} \ A)) ` B)"
have "A = (\y\B. f -` {y} \ A)"
using assms by auto
also have "card \ = (\i\B. card (f -` {i} \ A))"
using assms by (subst card_UN_disjoint) auto
also have "\ \ (\i\B. M)"
unfolding M_def using assms by (intro sum_mono Max.coboundedI) auto
also have "\ = card B * M"
by simp
finally have *: "M * card B \ card A"
by (simp add: mult_ac)
from assms have "M \ (\y. card (f -` {y} \ A)) ` B"
unfolding M_def by (intro Max_in) auto
with * show ?thesis
by blast
qed
subsection ‹Products of sums›
lemma prod_sum_PiE:
fixes f :: "'a \ 'b \ 'c :: comm_semiring_1"
assumes finite: "finite A" and finite: "\x. x \ A \ finite (B x)"
shows "(\x\A. \y\B x. f x y) = (\g\PiE A B. \x\A. f x (g x))"
using assms
proof (induction A rule: finite_induct)
case empty
thus ?case by auto
next
case (insert x A)
have "(\g\Pi\<^sub>E (insert x A) B. \x\insert x A. f x (g x)) =
(∑g∈Pi🚫E (insert x A) B. f x (g x) * (∏x'\A. f x' (g x')))"
using insert by simp
also have "(\g. \x'\A. f x' (g x')) = (\g. \x'\A. f x' (if x' = x then undefined else g x'))"
using insert by (intro ext prod.cong) auto
also have "(\g\Pi\<^sub>E (insert x A) B. f x (g x) * \ g) =
(∑(y,g)∈B x × Pi🚫E A B. f x y * (∏x'\A. f x' (g x')))"
using insert.prems insert.hyps
by (intro sum.reindex_bij_witness[of _ "\(y,g). g(x := y)" "\g. (g x, g(x := undefined))"])
(auto simp: PiE_def extensional_def)
also have "\ = (\y\B x. \g\Pi\<^sub>E A B. f x y * (\x'\A. f x' (g x')))"
by (subst sum.cartesian_product) auto
also have "\ = (\y\B x. f x y) * (\g\Pi\<^sub>E A B. \x'\A. f x' (g x'))"
using insert by (subst sum.swap) (simp add: sum_distrib_left sum_distrib_right)
also have "(\g\Pi\<^sub>E A B. \x'\A. f x' (g x')) = (\x\A. \y\B x. f x y)"
using insert.prems by (intro insert.IH [symmetric]) auto
also have "(\y\B x. f x y) * \ = (\x\insert x A. \y\B x. f x y)"
using insert.hyps by simp
finally show ?case ..
qed
end
