Theory Set_Interval

(*  Title:      HOL/Set_Interval.thy
    Author:     Tobias Nipkow, Clemens Ballarin, Jeremy Avigad

lessThan, greaterThan, atLeast, atMost and two-sided intervals

Modern convention: Ixy stands for an interval where x and y
describe the lower and upper bound and x,y : {c,o,i}
where c = closed, o = open, i = infinite.
Examples: Ico = {_ ..< _} and Ici = {_ ..}
*)

section Set intervals

theory Set_Interval
imports Divides
begin

(* Belongs in Finite_Set but 2 is not available there *)
lemma card_2_iff: "card S = 2  (x y. S = {x,y}  x  y)"
  by (auto simp: card_Suc_eq numeral_eq_Suc)

lemma card_2_iff': "card S = 2  (xS. yS. x  y  (zS. z = x  z = y))"
  by (auto simp: card_Suc_eq numeral_eq_Suc)

lemma card_3_iff: "card S = 3  (x y z. S = {x,y,z}  x  y  y  z  x  z)"
  by (fastforce simp: card_Suc_eq numeral_eq_Suc)

context ord
begin

definition
  lessThan    :: "'a => 'a set" ("(1{..<_})") where
  "{..<u} == {x. x < u}"

definition
  atMost      :: "'a => 'a set" ("(1{.._})") where
  "{..u} == {x. x  u}"

definition
  greaterThan :: "'a => 'a set" ("(1{_<..})") where
  "{l<..} == {x. l<x}"

definition
  atLeast     :: "'a => 'a set" ("(1{_..})") where
  "{l..} == {x. lx}"

definition
  greaterThanLessThan :: "'a => 'a => 'a set"  ("(1{_<..<_})") where
  "{l<..<u} == {l<..} Int {..<u}"

definition
  atLeastLessThan :: "'a => 'a => 'a set"      ("(1{_..<_})") where
  "{l..<u} == {l..} Int {..<u}"

definition
  greaterThanAtMost :: "'a => 'a => 'a set"    ("(1{_<.._})") where
  "{l<..u} == {l<..} Int {..u}"

definition
  atLeastAtMost :: "'a => 'a => 'a set"        ("(1{_.._})") where
  "{l..u} == {l..} Int {..u}"

end


textA note of warning when using term{..<n} on type typnat: it is equivalent to term{0::nat..<n} but some lemmas involving
term{m..<n} may not exist in term{..<n}-form as well.

syntax (ASCII)
  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3UN _<=_./ _)" [0, 0, 10] 10)
  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3UN _<_./ _)" [0, 0, 10] 10)
  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3INT _<=_./ _)" [0, 0, 10] 10)
  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3INT _<_./ _)" [0, 0, 10] 10)

syntax (latex output)
  "_UNION_le"   :: "'a  'a => 'b set => 'b set"       ("(3(‹unbreakable›_  _)/ _)" [0, 0, 10] 10)
  "_UNION_less" :: "'a  'a => 'b set => 'b set"       ("(3(‹unbreakable›_ < _)/ _)" [0, 0, 10] 10)
  "_INTER_le"   :: "'a  'a => 'b set => 'b set"       ("(3(‹unbreakable›_  _)/ _)" [0, 0, 10] 10)
  "_INTER_less" :: "'a  'a => 'b set => 'b set"       ("(3(‹unbreakable›_ < _)/ _)" [0, 0, 10] 10)

syntax
  "_UNION_le"   :: "'a => 'a => 'b set => 'b set"       ("(3__./ _)" [0, 0, 10] 10)
  "_UNION_less" :: "'a => 'a => 'b set => 'b set"       ("(3_<_./ _)" [0, 0, 10] 10)
  "_INTER_le"   :: "'a => 'a => 'b set => 'b set"       ("(3__./ _)" [0, 0, 10] 10)
  "_INTER_less" :: "'a => 'a => 'b set => 'b set"       ("(3_<_./ _)" [0, 0, 10] 10)

translations
  "in. A"  "i{..n}. A"
  "i<n. A"  "i{..<n}. A"
  "in. A"  "i{..n}. A"
  "i<n. A"  "i{..<n}. A"


subsection Various equivalences

lemma (in ord) lessThan_iff [iff]: "(i  lessThan k) = (i<k)"
by (simp add: lessThan_def)

lemma Compl_lessThan [simp]:
    "!!k:: 'a::linorder. -lessThan k = atLeast k"
  by (auto simp add: lessThan_def atLeast_def)

lemma single_Diff_lessThan [simp]: "!!k:: 'a::preorder. {k} - lessThan k = {k}"
  by auto

lemma (in ord) greaterThan_iff [iff]: "(i  greaterThan k) = (k<i)"
  by (simp add: greaterThan_def)

lemma Compl_greaterThan [simp]:
    "!!k:: 'a::linorder. -greaterThan k = atMost k"
  by (auto simp add: greaterThan_def atMost_def)

lemma Compl_atMost [simp]: "!!k:: 'a::linorder. -atMost k = greaterThan k"
  by (metis Compl_greaterThan double_complement)

lemma (in ord) atLeast_iff [iff]: "(i  atLeast k) = (k<=i)"
  by (simp add: atLeast_def)

lemma Compl_atLeast [simp]: "!!k:: 'a::linorder. -atLeast k = lessThan k"
  by (auto simp add: lessThan_def atLeast_def)

lemma (in ord) atMost_iff [iff]: "(i  atMost k) = (i<=k)"
by (simp add: atMost_def)

lemma atMost_Int_atLeast: "!!n:: 'a::order. atMost n Int atLeast n = {n}"
by (blast intro: order_antisym)

lemma (in linorder) lessThan_Int_lessThan: "{ a <..}  { b <..} = { max a b <..}"
  by auto

lemma (in linorder) greaterThan_Int_greaterThan: "{..< a}  {..< b} = {..< min a b}"
  by auto

subsection Logical Equivalences for Set Inclusion and Equality

lemma atLeast_empty_triv [simp]: "{{}..} = UNIV"
  by auto

lemma atMost_UNIV_triv [simp]: "{..UNIV} = UNIV"
  by auto

lemma atLeast_subset_iff [iff]:
  "(atLeast x  atLeast y) = (y  (x::'a::preorder))"
  by (blast intro: order_trans)

lemma atLeast_eq_iff [iff]:
  "(atLeast x = atLeast y) = (x = (y::'a::order))"
  by (blast intro: order_antisym order_trans)

lemma greaterThan_subset_iff [iff]:
  "(greaterThan x  greaterThan y) = (y  (x::'a::linorder))"
  unfolding greaterThan_def by (auto simp: linorder_not_less [symmetric])

lemma greaterThan_eq_iff [iff]:
     "(greaterThan x = greaterThan y) = (x = (y::'a::linorder))"
  by (auto simp: elim!: equalityE)

lemma atMost_subset_iff [iff]: "(atMost x  atMost y) = (x  (y::'a::preorder))"
  by (blast intro: order_trans)

lemma atMost_eq_iff [iff]: "(atMost x = atMost y) = (x = (y::'a::order))"
  by (blast intro: order_antisym order_trans)

lemma lessThan_subset_iff [iff]:
     "(lessThan x  lessThan y) = (x  (y::'a::linorder))"
  unfolding lessThan_def by (auto simp: linorder_not_less [symmetric])

lemma lessThan_eq_iff [iff]:
     "(lessThan x = lessThan y) = (x = (y::'a::linorder))"
  by (auto simp: elim!: equalityE)

lemma lessThan_strict_subset_iff:
  fixes m n :: "'a::linorder"
  shows "{..<m} < {..<n}  m < n"
  by (metis leD lessThan_subset_iff linorder_linear not_less_iff_gr_or_eq psubset_eq)

lemma (in linorder) Ici_subset_Ioi_iff: "{a ..}  {b <..}  b < a"
  by auto

lemma (in linorder) Iic_subset_Iio_iff: "{.. a}  {..< b}  a < b"
  by auto

lemma (in preorder) Ioi_le_Ico: "{a <..}  {a ..}"
  by (auto intro: less_imp_le)

subsection Two-sided intervals

context ord
begin

lemma greaterThanLessThan_iff [simp]: "(i  {l<..<u}) = (l < i  i < u)"
  by (simp add: greaterThanLessThan_def)

lemma atLeastLessThan_iff [simp]: "(i  {l..<u}) = (l  i  i < u)"
  by (simp add: atLeastLessThan_def)

lemma greaterThanAtMost_iff [simp]: "(i  {l<..u}) = (l < i  i  u)"
  by (simp add: greaterThanAtMost_def)

lemma atLeastAtMost_iff [simp]: "(i  {l..u}) = (l  i  i  u)"
  by (simp add: atLeastAtMost_def)

text The above four lemmas could be declared as iffs. Unfortunately this
breaks many proofs. Since it only helps blast, it is better to leave them
alone.

lemma greaterThanLessThan_eq: "{ a <..< b} = { a <..}  {..< b }"
  by auto

lemma (in order) atLeastLessThan_eq_atLeastAtMost_diff:
  "{a..<b} = {a..b} - {b}"
  by (auto simp add: atLeastLessThan_def atLeastAtMost_def)

lemma (in order) greaterThanAtMost_eq_atLeastAtMost_diff:
  "{a<..b} = {a..b} - {a}"
  by (auto simp add: greaterThanAtMost_def atLeastAtMost_def)

end

subsubsectionEmptyness, singletons, subset

context preorder
begin

lemma atLeastatMost_empty_iff[simp]:
  "{a..b} = {}  (¬ a  b)"
  by auto (blast intro: order_trans)

lemma atLeastatMost_empty_iff2[simp]:
  "{} = {a..b}  (¬ a  b)"
  by auto (blast intro: order_trans)

lemma atLeastLessThan_empty_iff[simp]:
  "{a..<b} = {}  (¬ a < b)"
  by auto (blast intro: le_less_trans)

lemma atLeastLessThan_empty_iff2[simp]:
  "{} = {a..<b}  (¬ a < b)"
  by auto (blast intro: le_less_trans)

lemma greaterThanAtMost_empty_iff[simp]: "{k<..l} = {}  ¬ k < l"
  by auto (blast intro: less_le_trans)

lemma greaterThanAtMost_empty_iff2[simp]: "{} = {k<..l}  ¬ k < l"
  by auto (blast intro: less_le_trans)

lemma atLeastatMost_subset_iff[simp]:
  "{a..b}  {c..d}  (¬ a  b)  c  a  b  d"
  unfolding atLeastAtMost_def atLeast_def atMost_def
  by (blast intro: order_trans)

lemma atLeastatMost_psubset_iff:
  "{a..b} < {c..d} 
   ((¬ a  b)  c  a  b  d  (c < a  b < d))  c  d"
  by(simp add: psubset_eq set_eq_iff less_le_not_le)(blast intro: order_trans)

lemma atLeastAtMost_subseteq_atLeastLessThan_iff:
  "{a..b}  {c ..< d}  (a  b  c  a  b < d)" 
  by auto (blast intro: local.order_trans local.le_less_trans elim: )+

lemma Icc_subset_Ici_iff[simp]:
  "{l..h}  {l'..} = (¬ lh  ll')"
  by(auto simp: subset_eq intro: order_trans)

lemma Icc_subset_Iic_iff[simp]:
  "{l..h}  {..h'} = (¬ lh  hh')"
  by(auto simp: subset_eq intro: order_trans)

lemma not_Ici_eq_empty[simp]: "{l..}  {}"
  by(auto simp: set_eq_iff)

lemma not_Iic_eq_empty[simp]: "{..h}  {}"
  by(auto simp: set_eq_iff)

lemmas not_empty_eq_Ici_eq_empty[simp] = not_Ici_eq_empty[symmetric]
lemmas not_empty_eq_Iic_eq_empty[simp] = not_Iic_eq_empty[symmetric]

end

context order
begin

lemma atLeastatMost_empty[simp]:
  "b < a  {a..b} = {}"
  by(auto simp: atLeastAtMost_def atLeast_def atMost_def)

lemma atLeastLessThan_empty[simp]:
  "b  a  {a..<b} = {}"
  by(auto simp: atLeastLessThan_def)

lemma greaterThanAtMost_empty[simp]: "l  k ==> {k<..l} = {}"
  by(auto simp:greaterThanAtMost_def greaterThan_def atMost_def)

lemma greaterThanLessThan_empty[simp]:"l  k ==> {k<..<l} = {}"
  by(auto simp:greaterThanLessThan_def greaterThan_def lessThan_def)

lemma atLeastAtMost_singleton [simp]: "{a..a} = {a}"
  by (auto simp add: atLeastAtMost_def atMost_def atLeast_def)

lemma atLeastAtMost_singleton': "a = b  {a .. b} = {a}" by simp

lemma Icc_eq_Icc[simp]:
  "{l..h} = {l'..h'} = (l=l'  h=h'  ¬ lh  ¬ l'h')"
  by (simp add: order_class.order.eq_iff) (auto intro: order_trans)

lemma (in linorder) Ico_eq_Ico:
  "{l..<h} = {l'..<h'} = (l=l'  h=h'  ¬ l<h  ¬ l'<h')"
  by (metis atLeastLessThan_empty_iff2 nle_le not_less ord.atLeastLessThan_iff)

lemma atLeastAtMost_singleton_iff[simp]:
  "{a .. b} = {c}  a = b  b = c"
proof
  assume "{a..b} = {c}"
  hence *: "¬ (¬ a  b)" unfolding atLeastatMost_empty_iff[symmetric] by simp
  with {a..b} = {c} have "c  a  b  c" by auto
  with * show "a = b  b = c" by auto
qed simp

end

context no_top
begin

(* also holds for no_bot but no_top should suffice *)
lemma not_UNIV_le_Icc[simp]: "¬ UNIV  {l..h}"
using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)

lemma not_UNIV_le_Iic[simp]: "¬ UNIV  {..h}"
using gt_ex[of h] by(auto simp: subset_eq less_le_not_le)

lemma not_Ici_le_Icc[simp]: "¬ {l..}  {l'..h'}"
using gt_ex[of h']
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)

lemma not_Ici_le_Iic[simp]: "¬ {l..}  {..h'}"
using gt_ex[of h']
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)

end

context no_bot
begin

lemma not_UNIV_le_Ici[simp]: "¬ UNIV  {l..}"
using lt_ex[of l] by(auto simp: subset_eq less_le_not_le)

lemma not_Iic_le_Icc[simp]: "¬ {..h}  {l'..h'}"
using lt_ex[of l']
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)

lemma not_Iic_le_Ici[simp]: "¬ {..h}  {l'..}"
using lt_ex[of l']
by(auto simp: subset_eq less_le)(blast dest:antisym_conv intro: order_trans)

end


context no_top
begin

(* also holds for no_bot but no_top should suffice *)
lemma not_UNIV_eq_Icc[simp]: "¬ UNIV = {l'..h'}"
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)

lemmas not_Icc_eq_UNIV[simp] = not_UNIV_eq_Icc[symmetric]

lemma not_UNIV_eq_Iic[simp]: "¬ UNIV = {..h'}"
using gt_ex[of h'] by(auto simp: set_eq_iff  less_le_not_le)

lemmas not_Iic_eq_UNIV[simp] = not_UNIV_eq_Iic[symmetric]

lemma not_Icc_eq_Ici[simp]: "¬ {l..h} = {l'..}"
unfolding atLeastAtMost_def using not_Ici_le_Iic[of l'] by blast

lemmas not_Ici_eq_Icc[simp] = not_Icc_eq_Ici[symmetric]

(* also holds for no_bot but no_top should suffice *)
lemma not_Iic_eq_Ici[simp]: "¬ {..h} = {l'..}"
using not_Ici_le_Iic[of l' h] by blast

lemmas not_Ici_eq_Iic[simp] = not_Iic_eq_Ici[symmetric]

end

context no_bot
begin

lemma not_UNIV_eq_Ici[simp]: "¬ UNIV = {l'..}"
using lt_ex[of l'] by(auto simp: set_eq_iff  less_le_not_le)

lemmas not_Ici_eq_UNIV[simp] = not_UNIV_eq_Ici[symmetric]

lemma not_Icc_eq_Iic[simp]: "¬ {l..h} = {..h'}"
unfolding atLeastAtMost_def using not_Iic_le_Ici[of h'] by blast

lemmas not_Iic_eq_Icc[simp] = not_Icc_eq_Iic[symmetric]

end


context dense_linorder
begin

lemma greaterThanLessThan_empty_iff[simp]:
  "{ a <..< b } = {}  b  a"
  using dense[of a b] by (cases "a < b") auto

lemma greaterThanLessThan_empty_iff2[simp]:
  "{} = { a <..< b }  b  a"
  using dense[of a b] by (cases "a < b") auto

lemma atLeastLessThan_subseteq_atLeastAtMost_iff:
  "{a ..< b}  { c .. d }  (a < b  c  a  b  d)"
  using dense[of "max a d" "b"]
  by (force simp: subset_eq Ball_def not_less[symmetric])

lemma greaterThanAtMost_subseteq_atLeastAtMost_iff:
  "{a <.. b}  { c .. d }  (a < b  c  a  b  d)"
  using dense[of "a" "min c b"]
  by (force simp: subset_eq Ball_def not_less[symmetric])

lemma greaterThanLessThan_subseteq_atLeastAtMost_iff:
  "{a <..< b}  { c .. d }  (a < b  c  a  b  d)"
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
  by (force simp: subset_eq Ball_def not_less[symmetric])

lemma greaterThanLessThan_subseteq_greaterThanLessThan:
  "{a <..< b}  {c <..< d}  (a < b  a  c  b  d)"
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
  by (force simp: subset_eq Ball_def not_less[symmetric])

lemma greaterThanAtMost_subseteq_atLeastLessThan_iff:
  "{a <.. b}  { c ..< d }  (a < b  c  a  b < d)"
  using dense[of "a" "min c b"]
  by (force simp: subset_eq Ball_def not_less[symmetric])

lemma greaterThanLessThan_subseteq_atLeastLessThan_iff:
  "{a <..< b}  { c ..< d }  (a < b  c  a  b  d)"
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
  by (force simp: subset_eq Ball_def not_less[symmetric])

lemma greaterThanLessThan_subseteq_greaterThanAtMost_iff:
  "{a <..< b}  { c <.. d }  (a < b  c  a  b  d)"
  using dense[of "a" "min c b"] dense[of "max a d" "b"]
  by (force simp: subset_eq Ball_def not_less[symmetric])

end

context no_top
begin

lemma greaterThan_non_empty[simp]: "{x <..}  {}"
  using gt_ex[of x] by auto

end

context no_bot
begin

lemma lessThan_non_empty[simp]: "{..< x}  {}"
  using lt_ex[of x] by auto

end

lemma (in linorder) atLeastLessThan_subset_iff:
  "{a..<b}  {c..<d}  b  a  ca  bd"
proof (cases "a < b")
  case True
  assume assm: "{a..<b}  {c..<d}"
  then have 1: "c  a  a  d"
    using True by (auto simp add: subset_eq Ball_def)
  then have 2: "b  d"
    using assm by (auto simp add: subset_eq)
  from 1 2 show ?thesis
    by simp
qed (auto)

lemma atLeastLessThan_inj:
  fixes a b c d :: "'a::linorder"
  assumes eq: "{a ..< b} = {c ..< d}" and "a < b" "c < d"
  shows "a = c" "b = d"
using assms by (metis atLeastLessThan_subset_iff eq less_le_not_le antisym_conv2 subset_refl)+

lemma atLeastLessThan_eq_iff:
  fixes a b c d :: "'a::linorder"
  assumes "a < b" "c < d"
  shows "{a ..< b} = {c ..< d}  a = c  b = d"
  using atLeastLessThan_inj assms by auto

lemma (in linorder) Ioc_inj: 
  {a <.. b} = {c <.. d}  (b  a  d  c)  a = c  b = d (is ?P  ?Q)
proof
  assume ?Q
  then show ?P
    by auto
next
  assume ?P
  then have a < x  x  b  c < x  x  d for x
    by (simp add: set_eq_iff)
  from this [of a] this [of b] this [of c] this [of d] show ?Q
    by auto
qed

lemma (in order) Iio_Int_singleton: "{..<k}  {x} = (if x < k then {x} else {})"
  by auto

lemma (in linorder) Ioc_subset_iff: "{a<..b}  {c<..d}  (b  a  c  a  b  d)"
  by (auto simp: subset_eq Ball_def) (metis less_le not_less)

lemma (in order_bot) atLeast_eq_UNIV_iff: "{x..} = UNIV  x = bot"
by (auto simp: set_eq_iff intro: le_bot)

lemma (in order_top) atMost_eq_UNIV_iff: "{..x} = UNIV  x = top"
by (auto simp: set_eq_iff intro: top_le)

lemma (in bounded_lattice) atLeastAtMost_eq_UNIV_iff:
  "{x..y} = UNIV  (x = bot  y = top)"
by (auto simp: set_eq_iff intro: top_le le_bot)

lemma Iio_eq_empty_iff: "{..< n::'a::{linorder, order_bot}} = {}  n = bot"
  by (auto simp: set_eq_iff not_less le_bot)

lemma lessThan_empty_iff: "{..< n::nat} = {}  n = 0"
  by (simp add: Iio_eq_empty_iff bot_nat_def)

lemma mono_image_least:
  assumes f_mono: "mono f" and f_img: "f ` {m ..< n} = {m' ..< n'}" "m < n"
  shows "f m = m'"
proof -
  from f_img have "{m' ..< n'}  {}"
    by (metis atLeastLessThan_empty_iff image_is_empty)
  with f_img have "m'  f ` {m ..< n}" by auto
  then obtain k where "f k = m'" "m  k" by auto
  moreover have "m'  f m" using f_img by auto
  ultimately show "f m = m'"
    using f_mono by (auto elim: monoE[where x=m and y=k])
qed


subsection Infinite intervals

context dense_linorder
begin

lemma infinite_Ioo:
  assumes "a < b"
  shows "¬ finite {a<..<b}"
proof
  assume fin: "finite {a<..<b}"
  moreover have ne: "{a<..<b}  {}"
    using a < b by auto
  ultimately have "a < Max {a <..< b}" "Max {a <..< b} < b"
    using Max_in[of "{a <..< b}"] by auto
  then obtain x where "Max {a <..< b} < x" "x < b"
    using dense[of "Max {a<..<b}" b] by auto
  then have "x  {a <..< b}"
    using a < Max {a <..< b} by auto
  then have "x  Max {a <..< b}"
    using fin by auto
  with Max {a <..< b} < x show False by auto
qed

lemma infinite_Icc: "a < b  ¬ finite {a .. b}"
  using greaterThanLessThan_subseteq_atLeastAtMost_iff[of a b a b] infinite_Ioo[of a b]
  by (auto dest: finite_subset)

lemma infinite_Ico: "a < b  ¬ finite {a ..< b}"
  using greaterThanLessThan_subseteq_atLeastLessThan_iff[of a b a b] infinite_Ioo[of a b]
  by (auto dest: finite_subset)

lemma infinite_Ioc: "a < b  ¬ finite {a <.. b}"
  using greaterThanLessThan_subseteq_greaterThanAtMost_iff[of a b a b] infinite_Ioo[of a b]
  by (auto dest: finite_subset)

lemma infinite_Ioo_iff [simp]: "infinite {a<..<b}  a < b"
  using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioo)

lemma infinite_Icc_iff [simp]: "infinite {a .. b}  a < b"
  using not_less_iff_gr_or_eq by (fastforce simp: infinite_Icc)

lemma infinite_Ico_iff [simp]: "infinite {a..<b}  a < b"
  using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ico)

lemma infinite_Ioc_iff [simp]: "infinite {a<..b}  a < b"
  using not_less_iff_gr_or_eq by (fastforce simp: infinite_Ioc)

end

lemma infinite_Iio: "¬ finite {..< a :: 'a :: {no_bot, linorder}}"
proof
  assume "finite {..< a}"
  then have *: "x. x < a  Min {..< a}  x"
    by auto
  obtain x where "x < a"
    using lt_ex by auto

  obtain y where "y < Min {..< a}"
    using lt_ex by auto
  also have "Min {..< a}  x"
    using x < a by fact
  also note x < a
  finally have "Min {..< a}  y"
    by fact
  with y < Min {..< a} show False by auto
qed

lemma infinite_Iic: "¬ finite {.. a :: 'a :: {no_bot, linorder}}"
  using infinite_Iio[of a] finite_subset[of "{..< a}" "{.. a}"]
  by (auto simp: subset_eq less_imp_le)

lemma infinite_Ioi: "¬ finite {a :: 'a :: {no_top, linorder} <..}"
proof
  assume "finite {a <..}"
  then have *: "x. a < x  x  Max {a <..}"
    by auto

  obtain y where "Max {a <..} < y"
    using gt_ex by auto

  obtain x where x: "a < x"
    using gt_ex by auto
  also from x have "x  Max {a <..}"
    by fact
  also note Max {a <..} < y
  finally have "y  Max { a <..}"
    by fact
  with Max {a <..} < y show False by auto
qed

lemma infinite_Ici: "¬ finite {a :: 'a :: {no_top, linorder} ..}"
  using infinite_Ioi[of a] finite_subset[of "{a <..}" "{a ..}"]
  by (auto simp: subset_eq less_imp_le)

subsubsection Intersection

context linorder
begin

lemma Int_atLeastAtMost[simp]: "{a..b} Int {c..d} = {max a c .. min b d}"
by auto

lemma Int_atLeastAtMostR1[simp]: "{..b} Int {c..d} = {c .. min b d}"
by auto

lemma Int_atLeastAtMostR2[simp]: "{a..} Int {c..d} = {max a c .. d}"
by auto

lemma Int_atLeastAtMostL1[simp]: "{a..b} Int {..d} = {a .. min b d}"
by auto

lemma Int_atLeastAtMostL2[simp]: "{a..b} Int {c..} = {max a c .. b}"
by auto

lemma Int_atLeastLessThan[simp]: "{a..<b} Int {c..<d} = {max a c ..< min b d}"
by auto

lemma Int_greaterThanAtMost[simp]: "{a<..b} Int {c<..d} = {max a c <.. min b d}"
by auto

lemma Int_greaterThanLessThan[simp]: "{a<..<b} Int {c<..<d} = {max a c <..< min b d}"
by auto

lemma Int_atMost[simp]: "{..a}  {..b} = {.. min a b}"
  by (auto simp: min_def)

lemma Ioc_disjoint: "{a<..b}  {c<..d} = {}  b  a  d  c  b  c  d  a"
  by auto

end

context complete_lattice
begin

lemma
  shows Sup_atLeast[simp]: "Sup {x ..} = top"
    and Sup_greaterThanAtLeast[simp]: "x < top  Sup {x <..} = top"
    and Sup_atMost[simp]: "Sup {.. y} = y"
    and Sup_atLeastAtMost[simp]: "x  y  Sup { x .. y} = y"
    and Sup_greaterThanAtMost[simp]: "x < y  Sup { x <.. y} = y"
  by (auto intro!: Sup_eqI)

lemma
  shows Inf_atMost[simp]: "Inf {.. x} = bot"
    and Inf_atMostLessThan[simp]: "top < x  Inf {..< x} = bot"
    and Inf_atLeast[simp]: "Inf {x ..} = x"
    and Inf_atLeastAtMost[simp]: "x  y  Inf { x .. y} = x"
    and Inf_atLeastLessThan[simp]: "x < y  Inf { x ..< y} = x"
  by (auto intro!: Inf_eqI)

end

lemma
  fixes x y :: "'a :: {complete_lattice, dense_linorder}"
  shows Sup_lessThan[simp]: "Sup {..< y} = y"
    and Sup_atLeastLessThan[simp]: "x < y  Sup { x ..< y} = y"
    and Sup_greaterThanLessThan[simp]: "x < y  Sup { x <..< y} = y"
    and Inf_greaterThan[simp]: "Inf {x <..} = x"
    and Inf_greaterThanAtMost[simp]: "x < y  Inf { x <.. y} = x"
    and Inf_greaterThanLessThan[simp]: "x < y  Inf { x <..< y} = x"
  by (auto intro!: Inf_eqI Sup_eqI intro: dense_le dense_le_bounded dense_ge dense_ge_bounded)

subsection Intervals of natural numbers

subsubsection The Constant termlessThan

lemma lessThan_0 [simp]: "lessThan (0::nat) = {}"
by (simp add: lessThan_def)

lemma lessThan_Suc: "lessThan (Suc k) = insert k (lessThan k)"
by (simp add: lessThan_def less_Suc_eq, blast)

text The following proof is convenient in induction proofs where
new elements get indices at the beginning. So it is used to transform
term{..<Suc n} to term0::nat and term{..< n}.

lemma zero_notin_Suc_image [simp]: "0  Suc ` A"
  by auto

lemma lessThan_Suc_eq_insert_0: "{..<Suc n} = insert 0 (Suc ` {..<n})"
  by (auto simp: image_iff less_Suc_eq_0_disj)

lemma lessThan_Suc_atMost: "lessThan (Suc k) = atMost k"
by (simp add: lessThan_def atMost_def less_Suc_eq_le)

lemma atMost_Suc_eq_insert_0: "{.. Suc n} = insert 0 (Suc ` {.. n})"
  unfolding lessThan_Suc_atMost[symmetric] lessThan_Suc_eq_insert_0[of "Suc n"] ..

lemma UN_lessThan_UNIV: "(m::nat. lessThan m) = UNIV"
by blast

subsubsection The Constant termgreaterThan

lemma greaterThan_0: "greaterThan 0 = range Suc"
  unfolding greaterThan_def
  by (blast dest: gr0_conv_Suc [THEN iffD1])

lemma greaterThan_Suc: "greaterThan (Suc k) = greaterThan k - {Suc k}"
  unfolding greaterThan_def
  by (auto elim: linorder_neqE)

lemma INT_greaterThan_UNIV: "(m::nat. greaterThan m) = {}"
  by blast

subsubsection The Constant termatLeast

lemma atLeast_0 [simp]: "atLeast (0::nat) = UNIV"
by (unfold atLeast_def UNIV_def, simp)

lemma atLeast_Suc: "atLeast (Suc k) = atLeast k - {k}"
  unfolding atLeast_def by (auto simp: order_le_less Suc_le_eq)

lemma atLeast_Suc_greaterThan: "atLeast (Suc k) = greaterThan k"
  by (auto simp add: greaterThan_def atLeast_def less_Suc_eq_le)

lemma UN_atLeast_UNIV: "(m::nat. atLeast m) = UNIV"
  by blast

subsubsection The Constant termatMost

lemma atMost_0 [simp]: "atMost (0::nat) = {0}"
  by (simp add: atMost_def)

lemma atMost_Suc: "atMost (Suc k) = insert (Suc k) (atMost k)"
  unfolding atMost_def by (auto simp add: less_Suc_eq order_le_less)

lemma UN_atMost_UNIV: "(m::nat. atMost m) = UNIV"
  by blast

subsubsection The Constant termatLeastLessThan

textThe orientation of the following 2 rules is tricky. The lhs is
defined in terms of the rhs.  Hence the chosen orientation makes sense
in this theory --- the reverse orientation complicates proofs (eg
nontermination). But outside, when the definition of the lhs is rarely
used, the opposite orientation seems preferable because it reduces a
specific concept to a more general one.

lemma atLeast0LessThan [code_abbrev]: "{0::nat..<n} = {..<n}"
  by(simp add:lessThan_def atLeastLessThan_def)

lemma atLeast0AtMost [code_abbrev]: "{0..n::nat} = {..n}"
  by(simp add:atMost_def atLeastAtMost_def)

lemma lessThan_atLeast0: "{..<n} = {0::nat..<n}"
  by (simp add: atLeast0LessThan)

lemma atMost_atLeast0: "{..n} = {0::nat..n}"
  by (simp add: atLeast0AtMost)

lemma atLeastLessThan0: "{m..<0::nat} = {}"
  by (simp add: atLeastLessThan_def)

lemma atLeast0_lessThan_Suc: "{0..<Suc n} = insert n {0..<n}"
  by (simp add: atLeast0LessThan lessThan_Suc)

lemma atLeast0_lessThan_Suc_eq_insert_0: "{0..<Suc n} = insert 0 (Suc ` {0..<n})"
  by (simp add: atLeast0LessThan lessThan_Suc_eq_insert_0)


subsubsection The Constant termatLeastAtMost

lemma Icc_eq_insert_lb_nat: "m  n  {m..n} = insert m {Suc m..n}"
by auto

lemma atLeast0_atMost_Suc:
  "{0..Suc n} = insert (Suc n) {0..n}"
  by (simp add: atLeast0AtMost atMost_Suc)

lemma atLeast0_atMost_Suc_eq_insert_0:
  "{0..Suc n} = insert 0 (Suc ` {0..n})"
  by (simp add: atLeast0AtMost atMost_Suc_eq_insert_0)


subsubsection Intervals of nats with termSuc

textNot a simprule because the RHS is too messy.
lemma atLeastLessThanSuc:
    "{m..<Suc n} = (if m  n then insert n {m..<n} else {})"
by (auto simp add: atLeastLessThan_def)

lemma atLeastLessThan_singleton [simp]: "{m..<Suc m} = {m}"
by (auto simp add: atLeastLessThan_def)

lemma atLeastLessThanSuc_atLeastAtMost: "{l..<Suc u} = {l..u}"
  by (simp add: lessThan_Suc_atMost atLeastAtMost_def atLeastLessThan_def)

lemma atLeastSucAtMost_greaterThanAtMost: "{Suc l..u} = {l<..u}"
  by (simp add: atLeast_Suc_greaterThan atLeastAtMost_def
      greaterThanAtMost_def)

lemma atLeastSucLessThan_greaterThanLessThan: "{Suc l..<u} = {l<..<u}"
  by (simp add: atLeast_Suc_greaterThan atLeastLessThan_def
    greaterThanLessThan_def)

lemma atLeastAtMostSuc_conv: "m  Suc n  {m..Suc n} = insert (Suc n) {m..n}"
  by auto

lemma atLeastAtMost_insertL: "m  n  insert m {Suc m..n} = {m ..n}"
  by auto

text The analogous result is useful on typint:
(* here, because we don't have an own int section *)
lemma atLeastAtMostPlus1_int_conv:
  "m  1+n  {m..1+n} = insert (1+n) {m..n::int}"
  by (auto intro: set_eqI)

lemma atLeastLessThan_add_Un: "i  j  {i..<j+k} = {i..<j}  {j..<j+k::nat}"
  by (induct k) (simp_all add: atLeastLessThanSuc)


subsubsection Intervals and numerals

lemma lessThan_nat_numeral:  ― ‹Evaluation for specific numerals
  "lessThan (numeral k :: nat) = insert (pred_numeral k) (lessThan (pred_numeral k))"
  by (simp add: numeral_eq_Suc lessThan_Suc)

lemma atMost_nat_numeral:  ― ‹Evaluation for specific numerals
  "atMost (numeral k :: nat) = insert (numeral k) (atMost (pred_numeral k))"
  by (simp add: numeral_eq_Suc atMost_Suc)

lemma atLeastLessThan_nat_numeral:  ― ‹Evaluation for specific numerals
  "atLeastLessThan m (numeral k :: nat) =
     (if m  (pred_numeral k) then insert (pred_numeral k) (atLeastLessThan m (pred_numeral k))
                 else {})"
  by (simp add: numeral_eq_Suc atLeastLessThanSuc)


subsubsection Image

context linordered_semidom
begin

lemma image_add_atLeast[simp]: "plus k ` {i..} = {k + i..}"
proof -
  have "n = k + (n - k)" if "i + k  n" for n
  proof -
    have "n = (n - (k + i)) + (k + i)" using that
      by (metis add_commute le_add_diff_inverse)
    then show "n = k + (n - k)"
      by (metis local.add_diff_cancel_left' add_assoc add_commute)
  qed
  then show ?thesis
    by (fastforce simp: add_le_imp_le_diff add.commute)
qed

lemma image_add_atLeastAtMost [simp]:
  "plus k ` {i..j} = {i + k..j + k}" (is "?A = ?B")
proof
  show "?A  ?B"
    by (auto simp add: ac_simps)
next
  show "?B  ?A"
  proof
    fix n
    assume "n  ?B"
    then have "i  n - k"
      by (simp add: add_le_imp_le_diff)
    have "n = n - k + k"
    proof -
      from n  ?B have "n = n - (i + k) + (i + k)"
        by simp
      also have " = n - k - i + i + k"
        by (simp add: algebra_simps)
      also have " = n - k + k"
        using i  n - k by simp
      finally show ?thesis .
    qed
    moreover have "n - k  {i..j}"
      using n  ?B
      by (auto simp: add_le_imp_le_diff add_le_add_imp_diff_le)
    ultimately show "n  ?A"
      by (simp add: ac_simps) 
  qed
qed

lemma image_add_atLeastAtMost' [simp]:
  "(λn. n + k) ` {i..j} = {i + k..j + k}"
  by (simp add: add.commute [of _ k])

lemma image_add_atLeastLessThan [simp]:
  "plus k ` {i..<j} = {i + k..<j + k}"
  by (simp add: image_set_diff atLeastLessThan_eq_atLeastAtMost_diff ac_simps)

lemma image_add_atLeastLessThan' [simp]:
  "(λn. n + k) ` {i..<j} = {i + k..<j + k}"
  by (simp add: add.commute [of _ k])

lemma image_add_greaterThanAtMost[simp]: "(+) c ` {a<..b} = {c + a<..c + b}"
  by (simp add: image_set_diff greaterThanAtMost_eq_atLeastAtMost_diff ac_simps)

end

context ordered_ab_group_add
begin

lemma
  fixes x :: 'a
  shows image_uminus_greaterThan[simp]: "uminus ` {x<..} = {..<-x}"
  and image_uminus_atLeast[simp]: "uminus ` {x..} = {..-x}"
proof safe
  fix y assume "y < -x"
  hence *:  "x < -y" using neg_less_iff_less[of "-y" x] by simp
  have "- (-y)  uminus ` {x<..}"
    by (rule imageI) (simp add: *)
  thus "y  uminus ` {x<..}" by simp
next
  fix y assume "y  -x"
  have "- (-y)  uminus ` {x..}"
    by (rule imageI) (use y  -x[THEN le_imp_neg_le] in simp)
  thus "y  uminus ` {x..}" by simp
qed simp_all

lemma
  fixes x :: 'a
  shows image_uminus_lessThan[simp]: "uminus ` {..<x} = {-x<..}"
  and image_uminus_atMost[simp]: "uminus ` {..x} = {-x..}"
proof -
  have "uminus ` {..<x} = uminus ` uminus ` {-x<..}"
    and "uminus ` {..x} = uminus ` uminus ` {-x..}" by simp_all
  thus "uminus ` {..<x} = {-x<..}" and "uminus ` {..x} = {-x..}"
    by (simp_all add: image_image
        del: image_uminus_greaterThan image_uminus_atLeast)
qed

lemma
  fixes x :: 'a
  shows image_uminus_atLeastAtMost[simp]: "uminus ` {x..y} = {-y..-x}"
  and image_uminus_greaterThanAtMost[simp]: "uminus ` {x<..y} = {-y..<-x}"
  and image_uminus_atLeastLessThan[simp]: "uminus ` {x..<y} = {-y<..-x}"
  and image_uminus_greaterThanLessThan[simp]: "uminus ` {x<..<y} = {-y<..<-x}"
  by (simp_all add: atLeastAtMost_def greaterThanAtMost_def atLeastLessThan_def
      greaterThanLessThan_def image_Int[OF inj_uminus] Int_commute)

lemma image_add_atMost[simp]: "(+) c ` {..a} = {..c + a}"
  by (auto intro!: image_eqI[where x="x - c" for x] simp: algebra_simps)

end

lemma image_Suc_atLeastAtMost [simp]:
  "Suc ` {i..j} = {Suc i..Suc j}"
  using image_add_atLeastAtMost [of 1 i j]
    by (simp only: plus_1_eq_Suc) simp

lemma image_Suc_atLeastLessThan [simp]:
  "Suc ` {i..<j} = {Suc i..<Suc j}"
  using image_add_atLeastLessThan [of 1 i j]
    by (simp only: plus_1_eq_Suc) simp

corollary image_Suc_atMost:
  "Suc ` {..n} = {1..Suc n}"
  by (simp add: atMost_atLeast0 atLeastLessThanSuc_atLeastAtMost)

corollary image_Suc_lessThan:
  "Suc ` {..<n} = {1..n}"
  by (simp add: lessThan_atLeast0 atLeastLessThanSuc_atLeastAtMost)

lemma image_diff_atLeastAtMost [simp]:
  fixes d::"'a::linordered_idom" shows "((-) d ` {a..b}) = {d-b..d-a}"
proof
  show "{d - b..d - a}  (-) d ` {a..b}"
  proof
    fix x
    assume "x  {d - b..d - a}"
    then have "d - x  {a..b}" and "x = d - (d - x)"
      by auto
    then show "x  (-) d ` {a..b}"
      by (rule rev_image_eqI)
  qed
qed(auto)

lemma image_diff_atLeastLessThan [simp]:
  fixes a b c::"'a::linordered_idom"
  shows "(-) c ` {a..<b} = {c - b<..c - a}"
proof -
  have "(-) c ` {a..<b} = (+) c ` uminus ` {a ..<b}"
    unfolding image_image by simp
  also have " = {c - b<..c - a}" by simp
  finally show ?thesis by simp
qed

lemma image_minus_const_greaterThanAtMost[simp]:
  fixes a b c::"'a::linordered_idom"
  shows "(-) c ` {a<..b} = {c - b..<c - a}"
proof -
  have "(-) c ` {a<..b} = (+) c ` uminus ` {a<..b}"
    unfolding image_image by simp
  also have " = {c - b..<c - a}" by simp
  finally show ?thesis by simp
qed

lemma image_minus_const_atLeast[simp]:
  fixes a c::"'a::linordered_idom"
  shows "(-) c ` {a..} = {..c - a}"
proof -
  have "(-) c ` {a..} = (+) c ` uminus ` {a ..}"
    unfolding image_image by simp
  also have " = {..c - a}" by simp
  finally show ?thesis by simp
qed

lemma image_minus_const_AtMost[simp]:
  fixes b c::"'a::linordered_idom"
  shows "(-) c ` {..b} = {c - b..}"
proof -
  have "(-) c ` {..b} = (+) c ` uminus ` {..b}"
    unfolding image_image by simp
  also have " = {c - b..}" by simp
  finally show ?thesis by simp
qed

lemma image_minus_const_atLeastAtMost' [simp]:
  "(λt. t-d)`{a..b} = {a-d..b-d}" for d::"'a::linordered_idom"
  by (metis (no_types, lifting) diff_conv_add_uminus image_add_atLeastAtMost' image_cong)

context linordered_field
begin

lemma image_mult_atLeastAtMost [simp]:
  "((*) d ` {a..b}) = {d*a..d*b}" if "d>0"
  using that
  by (auto simp: field_simps mult_le_cancel_right intro: rev_image_eqI [where x="x/d" for x])

lemma image_divide_atLeastAtMost [simp]:
  "((λc. c / d) ` {a..b}) = {a/d..b/d}" if "d>0"
proof -
  from that have "inverse d > 0"
    by simp
  with image_mult_atLeastAtMost [of "inverse d" a b]
  have "(*) (inverse d) ` {a..b} = {inverse d * a..inverse d * b}"
    by blast
  moreover have "(*) (inverse d) = (λc. c / d)"
    by (simp add: fun_eq_iff field_simps)
  ultimately show ?thesis
    by simp
qed

lemma image_mult_atLeastAtMost_if:
  "(*) c ` {x .. y} =
    (if c > 0 then {c * x .. c * y} else if x  y then {c * y .. c * x} else {})"
proof (cases "c = 0  x > y")
  case True
  then show ?thesis
    by auto
next
  case False
  then have "x  y"
    by auto
  from False consider "c < 0"| "c > 0"
    by (auto simp add: neq_iff)
  then show ?thesis
  proof cases
    case 1
    have "(*) c ` {x..y} = {c * y..c * x}"
    proof (rule set_eqI)
      fix d
      from 1 have "inj (λz. z / c)"
        by (auto intro: injI)
      then have "d  (*) c ` {x..y}  d / c  (λz. z div c) ` (*) c ` {x..y}"
        by (subst inj_image_mem_iff) simp_all
      also have "  d / c  {x..y}"
        using 1 by (simp add: image_image)
      also have "  d  {c * y..c * x}"
        by (auto simp add: field_simps 1)
      finally show "d  (*) c ` {x..y}  d  {c * y..c * x}" .
    qed
    with x  y show ?thesis
      by auto
  qed (simp add: mult_left_mono_neg)
qed

lemma image_mult_atLeastAtMost_if':
  "(λx. x * c) ` {x..y} =
    (if x  y then if c > 0 then {x * c .. y * c} else {y * c .. x * c} else {})"
  using image_mult_atLeastAtMost_if [of c x y] by (auto simp add: ac_simps)

lemma image_affinity_atLeastAtMost:
  "((λx. m * x + c) ` {a..b}) = (if {a..b} = {} then {}
            else if 0  m then {m * a + c .. m * b + c}
            else {m * b + c .. m * a + c})"
proof -
  have *: "(λx. m * x + c) = ((λx. x + c)  (*) m)"
    by (simp add: fun_eq_iff)
  show ?thesis by (simp only: * image_comp [symmetric] image_mult_atLeastAtMost_if)
    (auto simp add: mult_le_cancel_left)
qed

lemma image_affinity_atLeastAtMost_diff:
  "((λx. m*x - c) ` {a..b}) = (if {a..b}={} then {}
            else if 0  m then {m*a - c .. m*b - c}
            else {m*b - c .. m*a - c})"
  using image_affinity_atLeastAtMost [of m "-c" a b]
  by simp

lemma image_affinity_atLeastAtMost_div:
  "((λx. x/m + c) ` {a..b}) = (if {a..b}={} then {}
            else if 0  m then {a/m + c .. b/m + c}
            else {b/m + c .. a/m + c})"
  using image_affinity_atLeastAtMost [of "inverse m" c a b]
  by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)

lemma image_affinity_atLeastAtMost_div_diff:
  "((λx. x/m - c) ` {a..b}) = (if {a..b}={} then {}
            else if 0  m then {a/m - c .. b/m - c}
            else {b/m - c .. a/m - c})"
  using image_affinity_atLeastAtMost_diff [of "inverse m" c a b]
  by (simp add: field_class.field_divide_inverse algebra_simps inverse_eq_divide)

end

lemma atLeast1_lessThan_eq_remove0:
  "{Suc 0..<n} = {..<n} - {0}"
  by auto

lemma atLeast1_atMost_eq_remove0:
  "{Suc 0..n} = {..n} - {0}"
  by auto

lemma image_add_int_atLeastLessThan:
    "(λx. x + (l::int)) ` {0..<u-l} = {l..<u}"
  by safe auto

lemma image_minus_const_atLeastLessThan_nat:
  fixes c :: nat
  shows "(λi. i - c) ` {x ..< y} =
      (if c < y then {x - c ..< y - c} else if x < y then {0} else {})"
    (is "_ = ?right")
proof safe
  fix a assume a: "a  ?right"
  show "a  (λi. i - c) ` {x ..< y}"
  proof cases
    assume "c < y" with a show ?thesis
      by (auto intro!: image_eqI[of _ _ "a + c"])
  next
    assume "¬ c < y" with a show ?thesis
      by (auto intro!: image_eqI[of _ _ x] split: if_split_asm)
  qed
qed auto

lemma image_int_atLeastLessThan:
  "int ` {a..<b} = {int a..<int b}"
  by (auto intro!: image_eqI [where x = "nat x" for x])

lemma image_int_atLeastAtMost:
  "int ` {a..b} = {int a..int b}"
  by (auto intro!: image_eqI [where x = "nat x" for x])


subsubsection Finiteness

lemma finite_lessThan [iff]: fixes k :: nat shows "finite {..<k}"
  by (induct k) (simp_all add: lessThan_Suc)

lemma finite_atMost [iff]: fixes k :: nat shows "finite {..k}"
  by (induct k) (simp_all add: atMost_Suc)

lemma finite_greaterThanLessThan [iff]:
  fixes l :: nat shows "finite {l<..<u}"
  by (simp add: greaterThanLessThan_def)

lemma finite_atLeastLessThan [iff]:
  fixes l :: nat shows "finite {l..<u}"
  by (simp add: atLeastLessThan_def)

lemma finite_greaterThanAtMost [iff]:
  fixes l :: nat shows "finite {l<..u}"
  by (simp add: greaterThanAtMost_def)

lemma finite_atLeastAtMost [iff]:
  fixes l :: nat shows "finite {l..u}"
  by (simp add: atLeastAtMost_def)

text A bounded set of natural numbers is finite.
lemma bounded_nat_set_is_finite: "(iN. i < (n::nat))  finite N"
  by (rule finite_subset [OF _ finite_lessThan]) auto

text A set of natural numbers is finite iff it is bounded.
lemma finite_nat_set_iff_bounded:
  "finite(N::nat set) = (m. nN. n<m)" (is "?F = ?B")
proof
  assume f:?F  show ?B
    using Max_ge[OF ?F, simplified less_Suc_eq_le[symmetric]] by blast
next
  assume ?B show ?F using ?B by(blast intro:bounded_nat_set_is_finite)
qed

lemma finite_nat_set_iff_bounded_le: "finite(N::nat set) = (m. nN. nm)"
  unfolding finite_nat_set_iff_bounded
  by (blast dest:less_imp_le_nat le_imp_less_Suc)

lemma finite_less_ub:
     "f::natnat. (!!n. n  f n)  finite {n. f n  u}"
  by (rule finite_subset[of _ "{..u}"])
    (auto intro: order_trans)

lemma bounded_Max_nat:
  fixes P :: "nat  bool"
  assumes x: "P x" and M: "x. P x  x  M"
  obtains m where "P m" "x. P x  x  m"
proof -
  have "finite {x. P x}"
    using M finite_nat_set_iff_bounded_le by auto
  then have "Max {x. P x}  {x. P x}"
    using Max_in x by auto
  then show ?thesis
    by (simp add: finite {x. P x} that)
qed


textAny subset of an interval of natural numbers the size of the
subset is exactly that interval.

lemma subset_card_intvl_is_intvl:
  assumes "A  {k..<k + card A}"
  shows "A = {k..<k + card A}"
proof (cases "finite A")
  case True
  from this and assms show ?thesis
  proof (induct A rule: finite_linorder_max_induct)
    case empty thus ?case by auto
  next
    case (insert b A)
    hence *: "b  A" by auto
    with insert have "A  {k..<k + card A}" and "b = k + card A"
      by fastforce+
    with insert * show ?case by auto
  qed
next
  case False
  with assms show ?thesis by simp
qed


subsubsection Proving Inclusions and Equalities between Unions

lemma UN_le_eq_Un0:
  "(in::nat. M i) = (i{1..n}. M i)  M 0" (is "?A = ?B")
proof
  show "?A  ?B"
  proof
    fix x assume "x  ?A"
    then obtain i where i: "in" "x  M i" by auto
    show "x  ?B"
    proof(cases i)
      case 0 with i show ?thesis by simp
    next
      case (Suc j) with i show ?thesis by auto
    qed
  qed
next
  show "?B  ?A" by fastforce
qed

lemma UN_le_add_shift:
  "(in::nat. M(i+k)) = (i{k..n+k}. M i)" (is "?A = ?B")
proof
  show "?A  ?B" by fastforce
next
  show "?B  ?A"
  proof
    fix x assume "x  ?B"
    then obtain i where i: "i  {k..n+k}" "x  M(i)" by auto
    hence "i-kn  x  M((i-k)+k)" by auto
    thus "x  ?A" by blast
  qed
qed

lemma UN_le_add_shift_strict:
  "(i<n::nat. M(i+k)) = (i{k..<n+k}. M i)" (is "?A = ?B")
proof
  show "?B  ?A"
  proof
    fix x assume "x  ?B"
    then obtain i where i: "i  {k..<n+k}" "x  M(i)" by auto
    then have "i - k < n  x  M((i-k) + k)" by auto
    then show "x  ?A" using UN_le_add_shift by blast
  qed
qed (fastforce)

lemma UN_UN_finite_eq: "(n::nat. i{0..<n}. A i) = (n. A n)"
  by (auto simp add: atLeast0LessThan)

lemma UN_finite_subset:
  "(n::nat. (i{0..<n}. A i)  C)  (n. A n)  C"
  by (subst UN_UN_finite_eq [symmetric]) blast

lemma UN_finite2_subset:
  assumes "n::nat. (i{0..<n}. A i)  (i{0..<n + k}. B i)"
  shows "(n. A n)  (n. B n)"
proof (rule UN_finite_subset, rule subsetI)
  fix n and a
  from assms have "(i{0..<n}. A i)  (i{0..<n + k}. B i)" .
  moreover assume "a  (i{0..<n}. A i)"
  ultimately have "a  (i{0..<n + k}. B i)" by blast
  then show "a  (i. B i)" by (auto simp add: UN_UN_finite_eq)
qed

lemma UN_finite2_eq:
  assumes "(n::nat. (i{0..<n}. A i) = (i{0..<n + k}. B i))"
  shows "(n. A n) = (n. B n)"
proof (rule subset_antisym [OF UN_finite_subset UN_finite2_subset])
  fix n
  show " (A ` {0..<n})  (n. B n)"
    using assms by auto
next
  fix n
  show " (B ` {0..<n})   (A ` {0..<n + k})"
    using assms by (force simp add: atLeastLessThan_add_Un [of 0])+
qed


subsubsection Cardinality

lemma card_lessThan [simp]: "card {..<u} = u"
  by (induct u, simp_all add: lessThan_Suc)

lemma card_atMost [simp]: "card {..u} = Suc u"
  by (simp add: lessThan_Suc_atMost [THEN sym])

lemma card_atLeastLessThan [simp]: "card {l..<u} = u - l"
proof -
  have "(λx. x + l) ` {..<u - l}  {l..<u}"
    by auto
  moreover have "{l..<u}  (λx. x + l) ` {..<u-l}"
  proof
    fix x
    assume *: "x  {l..<u}"
    then have "x - l  {..< u -l}"
      by auto
    then have "(x - l) + l  (λx. x + l) ` {..< u -l}"
      by auto
    then show "x  (λx. x + l) ` {..<u - l}"
      using * by auto
  qed
  ultimately have "{l..<u} = (λx. x + l) ` {..<u-l}"
    by auto
  then have "card {l..<u} = card {..<u-l}"
    by (simp add: card_image inj_on_def)
  then show ?thesis
    by simp
qed

lemma card_atLeastAtMost [simp]: "card {l..u} = Suc u - l"
  by (subst atLeastLessThanSuc_atLeastAtMost [THEN sym], simp)

lemma card_greaterThanAtMost [simp]: "card {l<..u} = u - l"
  by (subst atLeastSucAtMost_greaterThanAtMost [THEN sym], simp)

lemma card_greaterThanLessThan [simp]: "card {l<..<u} = u - Suc l"
  by (subst atLeastSucLessThan_greaterThanLessThan [THEN sym], simp)

lemma subset_eq_atLeast0_lessThan_finite:
  fixes n :: nat
  assumes "N  {0..<n}"
  shows "finite N"
  using assms finite_atLeastLessThan by (rule finite_subset)

lemma subset_eq_atLeast0_atMost_finite:
  fixes n :: nat
  assumes "N  {0..n}"
  shows "finite N"
  using assms finite_atLeastAtMost by (rule finite_subset)

lemma ex_bij_betw_nat_finite:
  "finite M  h. bij_betw h {0..<card M} M"
  apply(drule finite_imp_nat_seg_image_inj_on)
  apply(auto simp:atLeast0LessThan[symmetric] lessThan_def[symmetric] card_image bij_betw_def)
  done

lemma ex_bij_betw_finite_nat:
  "finite M  h. bij_betw h M {0..<card M}"
  by (blast dest: ex_bij_betw_nat_finite bij_betw_inv)

lemma finite_same_card_bij:
  "finite A  finite B  card A = card B  h. bij_betw h A B"
  apply(drule ex_bij_betw_finite_nat)
  apply(drule ex_bij_betw_nat_finite)
  apply(auto intro!:bij_betw_trans)
  done

lemma ex_bij_betw_nat_finite_1:
  "finite M  h. bij_betw h {1 .. card M} M"
  by (rule finite_same_card_bij) auto

lemma bij_betw_iff_card:
  assumes "finite A" "finite B"
  shows "(f. bij_betw f A B)  (card A = card B)"
proof
  assume "card A = card B"
  moreover obtain f where "bij_betw f A {0 ..< card A}"
    using assms ex_bij_betw_finite_nat by blast
  moreover obtain g where "bij_betw g {0 ..< card B} B"
    using assms ex_bij_betw_nat_finite by blast
  ultimately have "bij_betw (g  f) A B"
    by (auto simp: bij_betw_trans)
  thus "(f. bij_betw f A B)" by blast
qed (auto simp: bij_betw_same_card)

lemma subset_eq_atLeast0_lessThan_card:
  fixes n :: nat
  assumes "N  {0..<n}"
  shows "card N  n"
proof -
  from assms finite_lessThan have "card N  card {0..<n}"
    using card_mono by blast
  then show ?thesis by simp
qed

text Relational version of @{thm [source] card_inj_on_le}:
lemma card_le_if_inj_on_rel:
assumes "finite B"
  "a. a  A  b. bB  r a b"
  "a1 a2 b.  a1  A;  a2  A;  b  B;  r a1 b;  r a2 b   a1 = a2"
shows "card A  card B"
proof -
  let ?P = "λa b. b  B  r a b"
  let ?f = "λa. SOME b. ?P a b"
  have 1: "?f ` A  B"  by (auto intro: someI2_ex[OF assms(2)])
  have "inj_on ?f A"
    unfolding inj_on_def
  proof safe
    fix a1 a2 assume asms: "a1  A" "a2  A" "?f a1 = ?f a2"
    have 0: "?f a1  B" using "1" a1  A by blast
    have 1: "r a1 (?f a1)" using someI_ex[OF assms(2)[OF a1  A]] by blast
    have 2: "r a2 (?f a1)" using someI_ex[OF assms(2)[OF a2  A]] asms(3) by auto
    show "a1 = a2" using assms(3)[OF asms(1,2) 0 1 2] .
  qed
  with 1 show ?thesis using card_inj_on_le[of ?f A B] assms(1) by simp
qed

lemma inj_on_funpow_least: contributor Lars Noschinski
  inj_on (λk. (f ^^ k) s) {0..<n}
  if (f ^^ n) s = s m. 0 < m  m < n  (f ^^ m) s  s
proof -
  { fix k l assume A: "k < n" "l < n" "k  l" "(f ^^ k) s = (f ^^ l) s"
    define k' l' where "k' = min k l" and "l' = max k l"
    with A have A': "k' < l'" "(f ^^ k') s = (f ^^ l') s" "l' < n"
      by (auto simp: min_def max_def)

    have "s = (f ^^ ((n - l') + l')) s" using that l' < n by simp
    also have " = (f ^^ (n - l')) ((f ^^ l') s)" by (simp add: funpow_add)
    also have "(f ^^ l') s = (f ^^ k') s" by (simp add: A')
    also have "(f ^^ (n - l'))  = (f ^^ (n - l' + k')) s" by (simp add: funpow_add)
    finally have "(f ^^ (n - l' + k')) s = s" by simp
    moreover have "n - l' + k' < n" "0 < n - l' + k'"using A' by linarith+
    ultimately have False using that(2) by auto
  }
  then show ?thesis by (intro inj_onI) auto
qed


subsection Intervals of integers

lemma atLeastLessThanPlusOne_atLeastAtMost_int: "{l..<u+1} = {l..(u::int)}"
  by (auto simp add: atLeastAtMost_def atLeastLessThan_def)

lemma atLeastPlusOneAtMost_greaterThanAtMost_int: "{l+1..u} = {l<..(u::int)}"
  by (auto simp add: atLeastAtMost_def greaterThanAtMost_def)

lemma atLeastPlusOneLessThan_greaterThanLessThan_int:
    "{l+1..<u} = {l<..<u::int}"
  by (auto simp add: atLeastLessThan_def greaterThanLessThan_def)

subsubsection Finiteness

lemma image_atLeastZeroLessThan_int:
  assumes "0  u"
  shows "{(0::int)..<u}