src/HOL/Analysis/Ordered_Euclidean_Space.thy
 author nipkow Sat Dec 29 15:43:53 2018 +0100 (6 months ago) changeset 69529 4ab9657b3257 parent 69508 2a4c8a2a3f8e child 69631 6c3e6038e74c permissions -rw-r--r--
capitalize proper names in lemma names
```     1 theory Ordered_Euclidean_Space
```
```     2 imports
```
```     3   Cartesian_Euclidean_Space
```
```     4   "HOL-Library.Product_Order"
```
```     5 begin
```
```     6
```
```     7 subsection%important \<open>An ordering on euclidean spaces that will allow us to talk about intervals\<close>
```
```     8
```
```     9 class ordered_euclidean_space = ord + inf + sup + abs + Inf + Sup + euclidean_space +
```
```    10   assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i\<in>Basis. x \<bullet> i \<le> y \<bullet> i)"
```
```    11   assumes eucl_less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
```
```    12   assumes eucl_inf: "inf x y = (\<Sum>i\<in>Basis. inf (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)"
```
```    13   assumes eucl_sup: "sup x y = (\<Sum>i\<in>Basis. sup (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)"
```
```    14   assumes eucl_Inf: "Inf X = (\<Sum>i\<in>Basis. (INF x\<in>X. x \<bullet> i) *\<^sub>R i)"
```
```    15   assumes eucl_Sup: "Sup X = (\<Sum>i\<in>Basis. (SUP x\<in>X. x \<bullet> i) *\<^sub>R i)"
```
```    16   assumes eucl_abs: "\<bar>x\<bar> = (\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar> *\<^sub>R i)"
```
```    17 begin
```
```    18
```
```    19 subclass order
```
```    20   by standard
```
```    21     (auto simp: eucl_le eucl_less_le_not_le intro!: euclidean_eqI antisym intro: order.trans)
```
```    22
```
```    23 subclass ordered_ab_group_add_abs
```
```    24   by standard (auto simp: eucl_le inner_add_left eucl_abs abs_leI)
```
```    25
```
```    26 subclass ordered_real_vector
```
```    27   by standard (auto simp: eucl_le intro!: mult_left_mono mult_right_mono)
```
```    28
```
```    29 subclass lattice
```
```    30   by standard (auto simp: eucl_inf eucl_sup eucl_le)
```
```    31
```
```    32 subclass distrib_lattice
```
```    33   by standard (auto simp: eucl_inf eucl_sup sup_inf_distrib1 intro!: euclidean_eqI)
```
```    34
```
```    35 subclass conditionally_complete_lattice
```
```    36 proof%unimportant
```
```    37   fix z::'a and X::"'a set"
```
```    38   assume "X \<noteq> {}"
```
```    39   hence "\<And>i. (\<lambda>x. x \<bullet> i) ` X \<noteq> {}" by simp
```
```    40   thus "(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X" "(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
```
```    41     by (auto simp: eucl_Inf eucl_Sup eucl_le
```
```    42       intro!: cInf_greatest cSup_least)
```
```    43 qed (force intro!: cInf_lower cSup_upper
```
```    44       simp: bdd_below_def bdd_above_def preorder_class.bdd_below_def preorder_class.bdd_above_def
```
```    45         eucl_Inf eucl_Sup eucl_le)+
```
```    46
```
```    47 lemma%unimportant inner_Basis_inf_left: "i \<in> Basis \<Longrightarrow> inf x y \<bullet> i = inf (x \<bullet> i) (y \<bullet> i)"
```
```    48   and inner_Basis_sup_left: "i \<in> Basis \<Longrightarrow> sup x y \<bullet> i = sup (x \<bullet> i) (y \<bullet> i)"
```
```    49   by (simp_all add: eucl_inf eucl_sup inner_sum_left inner_Basis if_distrib comm_monoid_add_class.sum.delta
```
```    50       cong: if_cong)
```
```    51
```
```    52 lemma%unimportant inner_Basis_INF_left: "i \<in> Basis \<Longrightarrow> (INF x\<in>X. f x) \<bullet> i = (INF x\<in>X. f x \<bullet> i)"
```
```    53   and inner_Basis_SUP_left: "i \<in> Basis \<Longrightarrow> (SUP x\<in>X. f x) \<bullet> i = (SUP x\<in>X. f x \<bullet> i)"
```
```    54   using eucl_Sup [of "f ` X"] eucl_Inf [of "f ` X"] by (simp_all add: comp_def)
```
```    55
```
```    56 lemma%unimportant abs_inner: "i \<in> Basis \<Longrightarrow> \<bar>x\<bar> \<bullet> i = \<bar>x \<bullet> i\<bar>"
```
```    57   by (auto simp: eucl_abs)
```
```    58
```
```    59 lemma%unimportant
```
```    60   abs_scaleR: "\<bar>a *\<^sub>R b\<bar> = \<bar>a\<bar> *\<^sub>R \<bar>b\<bar>"
```
```    61   by (auto simp: eucl_abs abs_mult intro!: euclidean_eqI)
```
```    62
```
```    63 lemma%unimportant interval_inner_leI:
```
```    64   assumes "x \<in> {a .. b}" "0 \<le> i"
```
```    65   shows "a\<bullet>i \<le> x\<bullet>i" "x\<bullet>i \<le> b\<bullet>i"
```
```    66   using assms
```
```    67   unfolding euclidean_inner[of a i] euclidean_inner[of x i] euclidean_inner[of b i]
```
```    68   by (auto intro!: ordered_comm_monoid_add_class.sum_mono mult_right_mono simp: eucl_le)
```
```    69
```
```    70 lemma%unimportant inner_nonneg_nonneg:
```
```    71   shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a \<bullet> b"
```
```    72   using interval_inner_leI[of a 0 a b]
```
```    73   by auto
```
```    74
```
```    75 lemma%unimportant inner_Basis_mono:
```
```    76   shows "a \<le> b \<Longrightarrow> c \<in> Basis  \<Longrightarrow> a \<bullet> c \<le> b \<bullet> c"
```
```    77   by (simp add: eucl_le)
```
```    78
```
```    79 lemma%unimportant Basis_nonneg[intro, simp]: "i \<in> Basis \<Longrightarrow> 0 \<le> i"
```
```    80   by (auto simp: eucl_le inner_Basis)
```
```    81
```
```    82 lemma%unimportant Sup_eq_maximum_componentwise:
```
```    83   fixes s::"'a set"
```
```    84   assumes i: "\<And>b. b \<in> Basis \<Longrightarrow> X \<bullet> b = i b \<bullet> b"
```
```    85   assumes sup: "\<And>b x. b \<in> Basis \<Longrightarrow> x \<in> s \<Longrightarrow> x \<bullet> b \<le> X \<bullet> b"
```
```    86   assumes i_s: "\<And>b. b \<in> Basis \<Longrightarrow> (i b \<bullet> b) \<in> (\<lambda>x. x \<bullet> b) ` s"
```
```    87   shows "Sup s = X"
```
```    88   using assms
```
```    89   unfolding eucl_Sup euclidean_representation_sum
```
```    90   by (auto intro!: conditionally_complete_lattice_class.cSup_eq_maximum)
```
```    91
```
```    92 lemma%unimportant Inf_eq_minimum_componentwise:
```
```    93   assumes i: "\<And>b. b \<in> Basis \<Longrightarrow> X \<bullet> b = i b \<bullet> b"
```
```    94   assumes sup: "\<And>b x. b \<in> Basis \<Longrightarrow> x \<in> s \<Longrightarrow> X \<bullet> b \<le> x \<bullet> b"
```
```    95   assumes i_s: "\<And>b. b \<in> Basis \<Longrightarrow> (i b \<bullet> b) \<in> (\<lambda>x. x \<bullet> b) ` s"
```
```    96   shows "Inf s = X"
```
```    97   using assms
```
```    98   unfolding eucl_Inf euclidean_representation_sum
```
```    99   by (auto intro!: conditionally_complete_lattice_class.cInf_eq_minimum)
```
```   100
```
```   101 end
```
```   102
```
```   103 lemma%important
```
```   104   compact_attains_Inf_componentwise:
```
```   105   fixes b::"'a::ordered_euclidean_space"
```
```   106   assumes "b \<in> Basis" assumes "X \<noteq> {}" "compact X"
```
```   107   obtains x where "x \<in> X" "x \<bullet> b = Inf X \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> x \<bullet> b \<le> y \<bullet> b"
```
```   108 proof%unimportant atomize_elim
```
```   109   let ?proj = "(\<lambda>x. x \<bullet> b) ` X"
```
```   110   from assms have "compact ?proj" "?proj \<noteq> {}"
```
```   111     by (auto intro!: compact_continuous_image continuous_intros)
```
```   112   from compact_attains_inf[OF this]
```
```   113   obtain s x
```
```   114     where s: "s\<in>(\<lambda>x. x \<bullet> b) ` X" "\<And>t. t\<in>(\<lambda>x. x \<bullet> b) ` X \<Longrightarrow> s \<le> t"
```
```   115       and x: "x \<in> X" "s = x \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> x \<bullet> b \<le> y \<bullet> b"
```
```   116     by auto
```
```   117   hence "Inf ?proj = x \<bullet> b"
```
```   118     by (auto intro!: conditionally_complete_lattice_class.cInf_eq_minimum)
```
```   119   hence "x \<bullet> b = Inf X \<bullet> b"
```
```   120     by (auto simp: eucl_Inf inner_sum_left inner_Basis if_distrib \<open>b \<in> Basis\<close> sum.delta
```
```   121       cong: if_cong)
```
```   122   with x show "\<exists>x. x \<in> X \<and> x \<bullet> b = Inf X \<bullet> b \<and> (\<forall>y. y \<in> X \<longrightarrow> x \<bullet> b \<le> y \<bullet> b)" by blast
```
```   123 qed
```
```   124
```
```   125 lemma%important
```
```   126   compact_attains_Sup_componentwise:
```
```   127   fixes b::"'a::ordered_euclidean_space"
```
```   128   assumes "b \<in> Basis" assumes "X \<noteq> {}" "compact X"
```
```   129   obtains x where "x \<in> X" "x \<bullet> b = Sup X \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> y \<bullet> b \<le> x \<bullet> b"
```
```   130 proof%unimportant atomize_elim
```
```   131   let ?proj = "(\<lambda>x. x \<bullet> b) ` X"
```
```   132   from assms have "compact ?proj" "?proj \<noteq> {}"
```
```   133     by (auto intro!: compact_continuous_image continuous_intros)
```
```   134   from compact_attains_sup[OF this]
```
```   135   obtain s x
```
```   136     where s: "s\<in>(\<lambda>x. x \<bullet> b) ` X" "\<And>t. t\<in>(\<lambda>x. x \<bullet> b) ` X \<Longrightarrow> t \<le> s"
```
```   137       and x: "x \<in> X" "s = x \<bullet> b" "\<And>y. y \<in> X \<Longrightarrow> y \<bullet> b \<le> x \<bullet> b"
```
```   138     by auto
```
```   139   hence "Sup ?proj = x \<bullet> b"
```
```   140     by (auto intro!: cSup_eq_maximum)
```
```   141   hence "x \<bullet> b = Sup X \<bullet> b"
```
```   142     by (auto simp: eucl_Sup[where 'a='a] inner_sum_left inner_Basis if_distrib \<open>b \<in> Basis\<close> sum.delta
```
```   143       cong: if_cong)
```
```   144   with x show "\<exists>x. x \<in> X \<and> x \<bullet> b = Sup X \<bullet> b \<and> (\<forall>y. y \<in> X \<longrightarrow> y \<bullet> b \<le> x \<bullet> b)" by blast
```
```   145 qed
```
```   146
```
```   147 lemma%unimportant (in order) atLeastatMost_empty'[simp]:
```
```   148   "(\<not> a \<le> b) \<Longrightarrow> {a..b} = {}"
```
```   149   by (auto)
```
```   150
```
```   151 instance real :: ordered_euclidean_space
```
```   152   by standard auto
```
```   153
```
```   154 lemma%unimportant in_Basis_prod_iff:
```
```   155   fixes i::"'a::euclidean_space*'b::euclidean_space"
```
```   156   shows "i \<in> Basis \<longleftrightarrow> fst i = 0 \<and> snd i \<in> Basis \<or> snd i = 0 \<and> fst i \<in> Basis"
```
```   157   by (cases i) (auto simp: Basis_prod_def)
```
```   158
```
```   159 instantiation prod :: (abs, abs) abs
```
```   160 begin
```
```   161
```
```   162 definition "\<bar>x\<bar> = (\<bar>fst x\<bar>, \<bar>snd x\<bar>)"
```
```   163
```
```   164 instance ..
```
```   165
```
```   166 end
```
```   167
```
```   168 instance prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
```
```   169   by standard
```
```   170     (auto intro!: add_mono simp add: euclidean_representation_sum'  Ball_def inner_prod_def
```
```   171       in_Basis_prod_iff inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left Inf_prod_def
```
```   172       inner_Basis_SUP_left Sup_prod_def less_prod_def less_eq_prod_def eucl_le[where 'a='a]
```
```   173       eucl_le[where 'a='b] abs_prod_def abs_inner)
```
```   174
```
```   175 text\<open>Instantiation for intervals on \<open>ordered_euclidean_space\<close>\<close>
```
```   176
```
```   177 lemma%important
```
```   178   fixes a :: "'a::ordered_euclidean_space"
```
```   179   shows cbox_interval: "cbox a b = {a..b}"
```
```   180     and interval_cbox: "{a..b} = cbox a b"
```
```   181     and eucl_le_atMost: "{x. \<forall>i\<in>Basis. x \<bullet> i <= a \<bullet> i} = {..a}"
```
```   182     and eucl_le_atLeast: "{x. \<forall>i\<in>Basis. a \<bullet> i <= x \<bullet> i} = {a..}"
```
```   183     by%unimportant (auto simp: eucl_le[where 'a='a] eucl_less_def box_def cbox_def)
```
```   184
```
```   185 lemma%unimportant vec_nth_real_1_iff_cbox [simp]:
```
```   186   fixes a b :: real
```
```   187   shows "(\<lambda>x::real^1. x \$ 1) ` S = {a..b} \<longleftrightarrow> S = cbox (vec a) (vec b)"
```
```   188   by (metis interval_cbox vec_nth_1_iff_cbox)
```
```   189
```
```   190 lemma%unimportant closed_eucl_atLeastAtMost[simp, intro]:
```
```   191   fixes a :: "'a::ordered_euclidean_space"
```
```   192   shows "closed {a..b}"
```
```   193   by (simp add: cbox_interval[symmetric] closed_cbox)
```
```   194
```
```   195 lemma%unimportant closed_eucl_atMost[simp, intro]:
```
```   196   fixes a :: "'a::ordered_euclidean_space"
```
```   197   shows "closed {..a}"
```
```   198   by (simp add: closed_interval_left eucl_le_atMost[symmetric])
```
```   199
```
```   200 lemma%unimportant closed_eucl_atLeast[simp, intro]:
```
```   201   fixes a :: "'a::ordered_euclidean_space"
```
```   202   shows "closed {a..}"
```
```   203   by (simp add: closed_interval_right eucl_le_atLeast[symmetric])
```
```   204
```
```   205 lemma%unimportant bounded_closed_interval [simp]:
```
```   206   fixes a :: "'a::ordered_euclidean_space"
```
```   207   shows "bounded {a .. b}"
```
```   208   using bounded_cbox[of a b]
```
```   209   by (metis interval_cbox)
```
```   210
```
```   211 lemma%unimportant convex_closed_interval [simp]:
```
```   212   fixes a :: "'a::ordered_euclidean_space"
```
```   213   shows "convex {a .. b}"
```
```   214   using convex_box[of a b]
```
```   215   by (metis interval_cbox)
```
```   216
```
```   217 lemma%unimportant image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a .. b} =
```
```   218   (if {a .. b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a .. m *\<^sub>R b} else {m *\<^sub>R b .. m *\<^sub>R a})"
```
```   219   using image_smult_cbox[of m a b]
```
```   220   by (simp add: cbox_interval)
```
```   221
```
```   222 lemma%unimportant [simp]:
```
```   223   fixes a b::"'a::ordered_euclidean_space" and r s::real
```
```   224   shows is_interval_io: "is_interval {..<r}"
```
```   225     and is_interval_ic: "is_interval {..a}"
```
```   226     and is_interval_oi: "is_interval {r<..}"
```
```   227     and is_interval_ci: "is_interval {a..}"
```
```   228     and is_interval_oo: "is_interval {r<..<s}"
```
```   229     and is_interval_oc: "is_interval {r<..s}"
```
```   230     and is_interval_co: "is_interval {r..<s}"
```
```   231     and is_interval_cc: "is_interval {b..a}"
```
```   232   by (force simp: is_interval_def eucl_le[where 'a='a])+
```
```   233
```
```   234 lemma connected_interval [simp]:
```
```   235   fixes a b::"'a::ordered_euclidean_space"
```
```   236   shows "connected {a..b}"
```
```   237   using is_interval_cc is_interval_connected by blast
```
```   238
```
```   239 lemma path_connected_interval [simp]:
```
```   240   fixes a b::"'a::ordered_euclidean_space"
```
```   241   shows "path_connected {a..b}"
```
```   242   using is_interval_cc is_interval_path_connected by blast
```
```   243
```
```   244 lemma%unimportant is_interval_real_ereal_oo: "is_interval (real_of_ereal ` {N<..<M::ereal})"
```
```   245   by (auto simp: real_atLeastGreaterThan_eq)
```
```   246
```
```   247 lemma%unimportant compact_interval [simp]:
```
```   248   fixes a b::"'a::ordered_euclidean_space"
```
```   249   shows "compact {a .. b}"
```
```   250   by (metis compact_cbox interval_cbox)
```
```   251
```
```   252 lemma%unimportant homeomorphic_closed_intervals:
```
```   253   fixes a :: "'a::euclidean_space" and b and c :: "'a::euclidean_space" and d
```
```   254   assumes "box a b \<noteq> {}" and "box c d \<noteq> {}"
```
```   255     shows "(cbox a b) homeomorphic (cbox c d)"
```
```   256 apply (rule homeomorphic_convex_compact)
```
```   257 using assms apply auto
```
```   258 done
```
```   259
```
```   260 lemma%unimportant homeomorphic_closed_intervals_real:
```
```   261   fixes a::real and b and c::real and d
```
```   262   assumes "a<b" and "c<d"
```
```   263   shows "{a..b} homeomorphic {c..d}"
```
```   264   using assms by (auto intro: homeomorphic_convex_compact)
```
```   265
```
```   266 no_notation
```
```   267   eucl_less (infix "<e" 50)
```
```   268
```
```   269 lemma%unimportant One_nonneg: "0 \<le> (\<Sum>Basis::'a::ordered_euclidean_space)"
```
```   270   by (auto intro: sum_nonneg)
```
```   271
```
```   272 lemma%unimportant
```
```   273   fixes a b::"'a::ordered_euclidean_space"
```
```   274   shows bdd_above_cbox[intro, simp]: "bdd_above (cbox a b)"
```
```   275     and bdd_below_cbox[intro, simp]: "bdd_below (cbox a b)"
```
```   276     and bdd_above_box[intro, simp]: "bdd_above (box a b)"
```
```   277     and bdd_below_box[intro, simp]: "bdd_below (box a b)"
```
```   278   unfolding atomize_conj
```
```   279   by (metis bdd_above_Icc bdd_above_mono bdd_below_Icc bdd_below_mono bounded_box
```
```   280             bounded_subset_cbox_symmetric interval_cbox)
```
```   281
```
```   282 instantiation vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
```
```   283 begin
```
```   284
```
```   285 definition%important "inf x y = (\<chi> i. inf (x \$ i) (y \$ i))"
```
```   286 definition%important "sup x y = (\<chi> i. sup (x \$ i) (y \$ i))"
```
```   287 definition%important "Inf X = (\<chi> i. (INF x\<in>X. x \$ i))"
```
```   288 definition%important "Sup X = (\<chi> i. (SUP x\<in>X. x \$ i))"
```
```   289 definition%important "\<bar>x\<bar> = (\<chi> i. \<bar>x \$ i\<bar>)"
```
```   290
```
```   291 instance
```
```   292   apply standard
```
```   293   unfolding euclidean_representation_sum'
```
```   294   apply (auto simp: less_eq_vec_def inf_vec_def sup_vec_def Inf_vec_def Sup_vec_def inner_axis
```
```   295     Basis_vec_def inner_Basis_inf_left inner_Basis_sup_left inner_Basis_INF_left
```
```   296     inner_Basis_SUP_left eucl_le[where 'a='a] less_le_not_le abs_vec_def abs_inner)
```
```   297   done
```
```   298
```
```   299 end
```
```   300
```
```   301 lemma%unimportant ANR_interval [iff]:
```
```   302     fixes a :: "'a::ordered_euclidean_space"
```
```   303     shows "ANR{a..b}"
```
```   304 by (simp add: interval_cbox)
```
```   305
```
```   306 lemma%unimportant ENR_interval [iff]:
```
```   307     fixes a :: "'a::ordered_euclidean_space"
```
```   308     shows "ENR{a..b}"
```
```   309   by (auto simp: interval_cbox)
```
```   310
```
```   311 end
```
```   312
```