src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
author hoelzl
Fri, 03 Dec 2010 00:36:01 +0100
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permissions -rw-r--r--
adapt proofs to changed set_plus_image (cf. ee8d0548c148);
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(*  Title:      HOL/Library/Convex_Euclidean_Space.thy
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    Author:     Robert Himmelmann, TU Muenchen
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    Author:     Bogdan Grechuk, University of Edinburgh
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*)
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header {* Convex sets, functions and related things. *}
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theory Convex_Euclidean_Space
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imports Topology_Euclidean_Space Convex Set_Algebras
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begin
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(* ------------------------------------------------------------------------- *)
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(* To be moved elsewhere                                                     *)
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(* ------------------------------------------------------------------------- *)
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lemma linear_scaleR: "linear (%(x :: 'n::euclidean_space). scaleR c x)"
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  by (metis linear_conv_bounded_linear scaleR.bounded_linear_right)
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lemma injective_scaleR: 
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assumes "(c :: real) ~= 0"
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shows "inj (%(x :: 'n::euclidean_space). scaleR c x)"
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  by (metis assms injI real_vector.scale_cancel_left)
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lemma linear_add_cmul:
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fixes f :: "('m::euclidean_space) => ('n::euclidean_space)" 
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assumes "linear f"
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shows "f(a *\<^sub>R x + b *\<^sub>R y) = a *\<^sub>R f x +  b *\<^sub>R f y"
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using linear_add[of f] linear_cmul[of f] assms by (simp) 
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lemma mem_convex_2:
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  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v=1"
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  shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
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  using assms convex_def[of S] by auto
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lemma mem_convex_alt:
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  assumes "convex S" "x : S" "y : S" "u>=0" "v>=0" "u+v>0"
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  shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) : S"
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apply (subst mem_convex_2) 
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using assms apply (auto simp add: algebra_simps zero_le_divide_iff)
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using add_divide_distrib[of u v "u+v"] by auto
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lemma card_ge1: assumes "d ~= {}" "finite d" shows "card d >= 1"
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by (metis Suc_eq_plus1 assms(1) assms(2) card_eq_0_iff fact_ge_one_nat fact_num_eq_if_nat one_le_mult_iff plus_nat.add_0)
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lemma inj_on_image_mem_iff: "inj_on f B ==> (A <= B) ==> (f a : f`A) ==> (a : B) ==> (a : A)" 
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by (blast dest: inj_onD)
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lemma independent_injective_on_span_image:
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  assumes iS: "independent (S::(_::euclidean_space) set)" 
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     and lf: "linear f" and fi: "inj_on f (span S)"
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  shows "independent (f ` S)"
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proof-
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  {fix a assume a: "a : S" "f a : span (f ` S - {f a})"
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    have eq: "f ` S - {f a} = f ` (S - {a})" using fi a span_inc
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      by (auto simp add: inj_on_def)
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    from a have "f a : f ` span (S -{a})"
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      unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
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    moreover have "span (S -{a}) <= span S" using span_mono[of "S-{a}" S] by auto
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    ultimately have "a : span (S -{a})" using fi a span_inc by (auto simp add: inj_on_def)
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    with a(1) iS  have False by (simp add: dependent_def) }
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  then show ?thesis unfolding dependent_def by blast
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qed
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lemma dim_image_eq:
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fixes f :: "'n::euclidean_space => 'm::euclidean_space"
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assumes lf: "linear f" and fi: "inj_on f (span S)" 
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shows "dim (f ` S) = dim (S:: ('n::euclidean_space) set)"
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proof-
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obtain B where B_def: "B<=S & independent B & S <= span B & card B = dim S" 
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  using basis_exists[of S] by auto
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hence "span S = span B" using span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
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hence "independent (f ` B)" using independent_injective_on_span_image[of B f] B_def assms by auto
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moreover have "card (f ` B) = card B" using assms card_image[of f B] subset_inj_on[of f "span S" B] 
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   B_def span_inc by auto
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moreover have "(f ` B) <= (f ` S)" using B_def by auto
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ultimately have "dim (f ` S) >= dim S" 
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  using independent_card_le_dim[of "f ` B" "f ` S"] B_def by auto
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from this show ?thesis using dim_image_le[of f S] assms by auto
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qed
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    81
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lemma linear_injective_on_subspace_0:
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fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
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assumes lf: "linear f" and "subspace S"
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  shows "inj_on f S <-> (!x : S. f x = 0 --> x = 0)"
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    86
proof-
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  have "inj_on f S <-> (!x : S. !y : S. f x = f y --> x = y)" by (simp add: inj_on_def)
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  also have "... <-> (!x : S. !y : S. f x - f y = 0 --> x - y = 0)" by simp
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  also have "... <-> (!x : S. !y : S. f (x - y) = 0 --> x - y = 0)"
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    by (simp add: linear_sub[OF lf])
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  also have "... <-> (! x : S. f x = 0 --> x = 0)" 
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    92
    using `subspace S` subspace_def[of S] subspace_sub[of S] by auto
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  finally show ?thesis .
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qed
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    95
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lemma subspace_Inter: "(!s : f. subspace s) ==> subspace (Inter f)"
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  unfolding subspace_def by auto 
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    98
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lemma span_eq[simp]: "(span s = s) <-> subspace s"
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proof-
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   101
  { fix f assume "f <= subspace"
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    hence "subspace (Inter f)" using subspace_Inter[of f] unfolding subset_eq mem_def by auto  }
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   103
  thus ?thesis using hull_eq[unfolded mem_def, of subspace s] span_def by auto
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   104
qed
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   105
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lemma basis_inj_on: "d \<subseteq> {..<DIM('n)} \<Longrightarrow> inj_on (basis :: nat => 'n::euclidean_space) d"
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  by(auto simp add: inj_on_def euclidean_eq[where 'a='n])
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   108
  
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lemma finite_substdbasis: "finite {basis i ::'n::euclidean_space |i. i : (d:: nat set)}" (is "finite ?S")
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proof-
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  have eq: "?S = basis ` d" by blast
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  show ?thesis unfolding eq apply(rule finite_subset[OF _ range_basis_finite]) by auto
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qed
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   114
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lemma card_substdbasis: assumes "d \<subseteq> {..<DIM('n::euclidean_space)}"
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  shows "card {basis i ::'n::euclidean_space | i. i : d} = card d" (is "card ?S = _")
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proof-
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  have eq: "?S = basis ` d" by blast
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  show ?thesis unfolding eq using card_image[OF basis_inj_on[of d]] assms by auto
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qed
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   121
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lemma substdbasis_expansion_unique: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
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   123
  shows "setsum (%i. f i *\<^sub>R basis i) d = (x::'a::euclidean_space)
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      <-> (!i<DIM('a). (i:d --> f i = x$$i) & (i ~: d --> x $$ i = 0))"
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   125
proof- have *:"\<And>x a b P. x * (if P then a else b) = (if P then x*a else x*b)" by auto
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   126
  have **:"finite d" apply(rule finite_subset[OF assms]) by fastsimp
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  have ***:"\<And>i. (setsum (%i. f i *\<^sub>R ((basis i)::'a)) d) $$ i = (\<Sum>x\<in>d. if x = i then f x else 0)"
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   128
    unfolding euclidean_component.setsum euclidean_scaleR basis_component *
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    apply(rule setsum_cong2) using assms by auto
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  show ?thesis unfolding euclidean_eq[where 'a='a] *** setsum_delta[OF **] using assms by auto
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   131
qed
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   132
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lemma independent_substdbasis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
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   134
  shows "independent {basis i ::'a::euclidean_space |i. i : (d :: nat set)}" (is "independent ?A")
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   135
proof -
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   136
  have *: "{basis i |i. i < DIM('a)} = basis ` {..<DIM('a)}" by auto
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   137
  show ?thesis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   138
    apply(intro independent_mono[of "{basis i ::'a |i. i : {..<DIM('a::euclidean_space)}}" "?A"] )
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   139
    using independent_basis[where 'a='a] assms by (auto simp: *)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   140
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   141
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   142
lemma dim_cball: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   143
assumes "0<e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   144
shows "dim (cball (0 :: 'n::euclidean_space) e) = DIM('n)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   145
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
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parents: 39302
diff changeset
   146
{ fix x :: "'n::euclidean_space" def y == "(e/norm x) *\<^sub>R x"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
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parents: 39302
diff changeset
   147
  hence "y : cball 0 e" using cball_def dist_norm[of 0 y] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   148
  moreover have "x = (norm x/e) *\<^sub>R y" using y_def assms by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   149
  moreover hence "x = (norm x/e) *\<^sub>R y"  by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   150
  ultimately have "x : span (cball 0 e)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   151
     using span_mul[of y "cball 0 e" "norm x/e"] span_inc[of "cball 0 e"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
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parents: 39302
diff changeset
   152
} hence "span (cball 0 e) = (UNIV :: ('n::euclidean_space) set)" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   153
from this show ?thesis using dim_span[of "cball (0 :: 'n::euclidean_space) e"] by (auto simp add: dim_UNIV)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   154
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   155
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   156
lemma indep_card_eq_dim_span:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   157
fixes B :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   158
assumes "independent B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   159
shows "finite B & card B = dim (span B)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   160
  using assms basis_card_eq_dim[of B "span B"] span_inc by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   161
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   162
lemma setsum_not_0: "setsum f A ~= 0 ==> EX a:A. f a ~= 0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   163
  apply(rule ccontr) by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   164
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   165
lemma translate_inj_on: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   166
fixes A :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   167
shows "inj_on (%x. a+x) A" unfolding inj_on_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   168
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   169
lemma translation_assoc:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   170
  fixes a b :: "'a::ab_group_add"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   171
  shows "(\<lambda>x. b+x) ` ((\<lambda>x. a+x) ` S) = (\<lambda>x. (a+b)+x) ` S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   172
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   173
lemma translation_invert:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   174
  fixes a :: "'a::ab_group_add"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   175
  assumes "(\<lambda>x. a+x) ` A = (\<lambda>x. a+x) ` B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   176
  shows "A=B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   177
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
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parents: 39302
diff changeset
   178
  have "(%x. -a+x) ` ((%x. a+x) ` A) = (%x. -a+x) ` ((%x. a+x) ` B)" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   179
  from this show ?thesis using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   180
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   181
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   182
lemma translation_galois:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   183
  fixes a :: "'a::ab_group_add"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   184
  shows "T=((\<lambda>x. a+x) ` S) <-> S=((\<lambda>x. (-a)+x) ` T)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   185
  using translation_assoc[of "-a" a S] apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   186
  using translation_assoc[of a "-a" T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   187
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   188
lemma translation_inverse_subset:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   189
  assumes "((%x. -a+x) ` V) <= (S :: 'n::ab_group_add set)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   190
  shows "V <= ((%x. a+x) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   191
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   192
{ fix x assume "x:V" hence "x-a : S" using assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   193
  hence "x : {a + v |v. v : S}" apply auto apply (rule exI[of _ "x-a"]) apply simp done 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   194
  hence "x : ((%x. a+x) ` S)" by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   195
  from this show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   196
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   197
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   198
lemma basis_0[simp]:"(basis i::'a::euclidean_space) = 0 \<longleftrightarrow> i\<ge>DIM('a)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   199
  using norm_basis[of i, where 'a='a] unfolding norm_eq_zero[where 'a='a,THEN sym] by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   200
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   201
lemma basis_to_basis_subspace_isomorphism:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   202
  assumes s: "subspace (S:: ('n::euclidean_space) set)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   203
  and t: "subspace (T :: ('m::euclidean_space) set)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
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parents: 39302
diff changeset
   204
  and d: "dim S = dim T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
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parents: 39302
diff changeset
   205
  and B: "B <= S" "independent B" "S <= span B" "card B = dim S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   206
  and C: "C <= T" "independent C" "T <= span C" "card C = dim T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   207
  shows "EX f. linear f & f ` B = C & f ` S = T & inj_on f S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   208
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   209
(* Proof is a modified copy of the proof of similar lemma subspace_isomorphism
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   210
*)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   211
  from B independent_bound have fB: "finite B" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   212
  from C independent_bound have fC: "finite C" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   213
  from B(4) C(4) card_le_inj[of B C] d obtain f where
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   214
    f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   215
  from linear_independent_extend[OF B(2)] obtain g where
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   216
    g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   217
  from inj_on_iff_eq_card[OF fB, of f] f(2)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   218
  have "card (f ` B) = card B" by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   219
  with B(4) C(4) have ceq: "card (f ` B) = card C" using d
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   220
    by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   221
  have "g ` B = f ` B" using g(2)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   222
    by (auto simp add: image_iff)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   223
  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   224
  finally have gBC: "g ` B = C" .
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   225
  have gi: "inj_on g B" using f(2) g(2)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   226
    by (auto simp add: inj_on_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   227
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   228
  {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   229
    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   230
    from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   231
    have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   232
    have "x=y" using g0[OF th1 th0] by simp }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   233
  then have giS: "inj_on g S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   234
    unfolding inj_on_def by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   235
  from span_subspace[OF B(1,3) s]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   236
  have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   237
  also have "\<dots> = span C" unfolding gBC ..
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   238
  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   239
  finally have gS: "g ` S = T" .
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   240
  from g(1) gS giS gBC show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   241
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   242
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   243
lemma closure_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   244
fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   245
assumes "linear f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   246
shows "f ` (closure S) <= closure (f ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   247
using image_closure_subset[of S f "closure (f ` S)"] assms linear_conv_bounded_linear[of f] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   248
linear_continuous_on[of f "closure S"] closed_closure[of "f ` S"] closure_subset[of "f ` S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   249
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   250
lemma closure_injective_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   251
fixes f :: "('n::euclidean_space) => ('n::euclidean_space)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   252
assumes "linear f" "inj f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   253
shows "f ` (closure S) = closure (f ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   254
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   255
obtain f' where f'_def: "linear f' & f o f' = id & f' o f = id" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   256
   using assms linear_injective_isomorphism[of f] isomorphism_expand by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   257
hence "f' ` closure (f ` S) <= closure (S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   258
   using closure_linear_image[of f' "f ` S"] image_compose[of f' f] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   259
hence "f ` f' ` closure (f ` S) <= f ` closure (S)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   260
hence "closure (f ` S) <= f ` closure (S)" using image_compose[of f f' "closure (f ` S)"] f'_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   261
from this show ?thesis using closure_linear_image[of f S] assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   262
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   263
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   264
lemma closure_direct_sum:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   265
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   266
fixes T :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   267
shows "closure (S <*> T) = closure S <*> closure T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   268
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   269
{ fix x assume "x : closure S <*> closure T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   270
  from this obtain xs xt where xst_def: "xs : closure S & xt : closure T & (xs,xt) = x" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   271
  { fix ee assume ee_def: "(ee :: real) > 0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   272
    def e == "ee/2" hence e_def: "(e :: real)>0 & 2*e=ee" using ee_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   273
    from this obtain e where e_def: "(e :: real)>0 & 2*e=ee" by auto
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    obtain ys where ys_def: "ys : S & (dist ys xs < e)"
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      using e_def xst_def closure_approachable[of xs S] by auto
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    obtain yt where yt_def: "yt : T & (dist yt xt < e)"
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      using e_def xst_def closure_approachable[of xt T] by auto
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   278
    from ys_def yt_def have "dist (ys,yt) (xs,xt) < sqrt (2*e^2)" 
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      unfolding dist_norm apply (auto simp add: norm_Pair) 
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      using mult_strict_mono'[of "norm (ys - xs)" e "norm (ys - xs)" e] e_def
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   281
      mult_strict_mono'[of "norm (yt - xt)" e "norm (yt - xt)" e] by (simp add: power2_eq_square)
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   282
    hence "((ys,yt) : S <*> T) & (dist (ys,yt) x < 2*e)"
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   283
      using e_def sqrt_add_le_add_sqrt[of "e^2" "e^2"] xst_def ys_def yt_def by auto
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   284
    hence "EX y: S <*> T. dist y x < ee" using e_def by auto
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   285
  } hence "x : closure (S <*> T)" using closure_approachable[of x "S <*> T"] by auto
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   286
}
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   287
hence "closure (S <*> T) >= closure S <*> closure T" by auto
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moreover have "closed (closure S <*> closure T)" using closed_Times by auto
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ultimately show ?thesis using closure_minimal[of "S <*> T" "closure S <*> closure T"]
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   290
  closure_subset[of S] closure_subset[of T] by auto
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   291
qed
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   292
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   293
lemma closure_scaleR: 
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   294
fixes S :: "('n::euclidean_space) set"
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   295
shows "(op *\<^sub>R c) ` (closure S) = closure ((op *\<^sub>R c) ` S)"
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   296
proof-
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   297
{ assume "c ~= 0" hence ?thesis using closure_injective_linear_image[of "(op *\<^sub>R c)" S]
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   298
      linear_scaleR injective_scaleR by auto 
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   299
}
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moreover
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   301
{ assume zero: "c=0 & S ~= {}"
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   302
  hence "closure S ~= {}" using closure_subset by auto
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   303
  hence "op *\<^sub>R c ` (closure S) = {0}" using zero by auto
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   304
  moreover have "op *\<^sub>R 0 ` S = {0}" using zero by auto
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hoelzl
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   305
  ultimately have ?thesis using zero by auto
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   306
}
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   307
moreover
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   308
{ assume "S={}" hence ?thesis by auto }
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   309
ultimately show ?thesis by blast
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qed
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diff changeset
   311
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   312
lemma fst_linear: "linear fst" unfolding linear_def by (simp add: algebra_simps)
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   313
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   314
lemma snd_linear: "linear snd" unfolding linear_def by (simp add: algebra_simps)
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diff changeset
   315
40897
1eb1b2f9d062 adapt proofs to changed set_plus_image (cf. ee8d0548c148);
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   316
lemma fst_snd_linear: "linear (%(x,y). x + y)" unfolding linear_def by (simp add: algebra_simps)
40377
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   317
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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lemma scaleR_2:
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   319
  fixes x :: "'a::real_vector"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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   320
  shows "scaleR 2 x = x + x"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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   321
unfolding one_add_one_is_two [symmetric] scaleR_left_distrib by simp
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
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diff changeset
   322
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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   323
declare euclidean_simps[simp]
33175
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37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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   325
lemma vector_choose_size: "0 <= c ==> \<exists>(x::'a::euclidean_space). norm x = c"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
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   326
  apply (rule exI[where x="c *\<^sub>R basis 0 ::'a"]) using DIM_positive[where 'a='a] by auto
33175
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himmelma
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diff changeset
   327
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himmelma
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   328
lemma setsum_delta_notmem: assumes "x\<notin>s"
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himmelma
parents:
diff changeset
   329
  shows "setsum (\<lambda>y. if (y = x) then P x else Q y) s = setsum Q s"
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himmelma
parents:
diff changeset
   330
        "setsum (\<lambda>y. if (x = y) then P x else Q y) s = setsum Q s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   331
        "setsum (\<lambda>y. if (y = x) then P y else Q y) s = setsum Q s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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   332
        "setsum (\<lambda>y. if (x = y) then P y else Q y) s = setsum Q s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
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   333
  apply(rule_tac [!] setsum_cong2) using assms by auto
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himmelma
parents:
diff changeset
   334
2083bde13ce1 distinguished session for multivariate analysis
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   335
lemma setsum_delta'':
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himmelma
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   336
  fixes s::"'a::real_vector set" assumes "finite s"
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himmelma
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diff changeset
   337
  shows "(\<Sum>x\<in>s. (if y = x then f x else 0) *\<^sub>R x) = (if y\<in>s then (f y) *\<^sub>R y else 0)"
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himmelma
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diff changeset
   338
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   339
  have *:"\<And>x y. (if y = x then f x else (0::real)) *\<^sub>R x = (if x=y then (f x) *\<^sub>R x else 0)" by auto
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himmelma
parents:
diff changeset
   340
  show ?thesis unfolding * using setsum_delta[OF assms, of y "\<lambda>x. f x *\<^sub>R x"] by auto
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himmelma
parents:
diff changeset
   341
qed
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himmelma
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diff changeset
   342
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himmelma
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   343
lemma if_smult:"(if P then x else (y::real)) *\<^sub>R v = (if P then x *\<^sub>R v else y *\<^sub>R v)" by auto
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himmelma
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diff changeset
   344
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
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   345
lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::'a::ordered_euclidean_space)) ` {a..b} =
33175
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   346
  (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})"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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diff changeset
   347
  using image_affinity_interval[of m 0 a b] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   348
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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   349
lemma dist_triangle_eq:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
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diff changeset
   350
  fixes x y z :: "'a::euclidean_space"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   351
  shows "dist x z = dist x y + dist y z \<longleftrightarrow> norm (x - y) *\<^sub>R (y - z) = norm (y - z) *\<^sub>R (x - y)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   352
proof- have *:"x - y + (y - z) = x - z" by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   353
  show ?thesis unfolding dist_norm norm_triangle_eq[of "x - y" "y - z", unfolded *]
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   354
    by(auto simp add:norm_minus_commute) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   355
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
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diff changeset
   356
lemma norm_minus_eqI:"x = - y \<Longrightarrow> norm x = norm y" by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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diff changeset
   357
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   358
lemma Min_grI: assumes "finite A" "A \<noteq> {}" "\<forall>a\<in>A. x < a" shows "x < Min A"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   359
  unfolding Min_gr_iff[OF assms(1,2)] using assms(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   360
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
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diff changeset
   361
lemma norm_lt: "norm x < norm y \<longleftrightarrow> inner x x < inner y y"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   362
  unfolding norm_eq_sqrt_inner by simp
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   363
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
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diff changeset
   364
lemma norm_le: "norm x \<le> norm y \<longleftrightarrow> inner x x \<le> inner y y"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   365
  unfolding norm_eq_sqrt_inner by simp
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   366
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   367
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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diff changeset
   368
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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   369
subsection {* Affine set and affine hull.*}
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   370
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   371
definition
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   372
  affine :: "'a::real_vector set \<Rightarrow> bool" where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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   373
  "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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diff changeset
   374
2083bde13ce1 distinguished session for multivariate analysis
himmelma
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   375
lemma affine_alt: "affine s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u::real. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
   376
unfolding affine_def by(metis eq_diff_eq')
33175
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himmelma
parents:
diff changeset
   377
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
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   378
lemma affine_empty[intro]: "affine {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   379
  unfolding affine_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   380
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   381
lemma affine_sing[intro]: "affine {x}"
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himmelma
parents:
diff changeset
   382
  unfolding affine_alt by (auto simp add: scaleR_left_distrib [symmetric])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   383
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   384
lemma affine_UNIV[intro]: "affine UNIV"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   385
  unfolding affine_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   386
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   387
lemma affine_Inter: "(\<forall>s\<in>f. affine s) \<Longrightarrow> affine (\<Inter> f)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   388
  unfolding affine_def by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   389
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   390
lemma affine_Int: "affine s \<Longrightarrow> affine t \<Longrightarrow> affine (s \<inter> t)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   391
  unfolding affine_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   392
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   393
lemma affine_affine_hull: "affine(affine hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   394
  unfolding hull_def using affine_Inter[of "{t \<in> affine. s \<subseteq> t}"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   395
  unfolding mem_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   396
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   397
lemma affine_hull_eq[simp]: "(affine hull s = s) \<longleftrightarrow> affine s"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
   398
by (metis affine_affine_hull hull_same mem_def)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   399
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   400
lemma setsum_restrict_set'': assumes "finite A"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   401
  shows "setsum f {x \<in> A. P x} = (\<Sum>x\<in>A. if P x  then f x else 0)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   402
  unfolding mem_def[of _ P, symmetric] unfolding setsum_restrict_set'[OF assms] ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   403
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   404
subsection {* Some explicit formulations (from Lars Schewe). *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   405
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   406
lemma affine: fixes V::"'a::real_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   407
  shows "affine V \<longleftrightarrow> (\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (setsum (\<lambda>x. (u x) *\<^sub>R x)) s \<in> V)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   408
unfolding affine_def apply rule apply(rule, rule, rule) apply(erule conjE)+ 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   409
defer apply(rule, rule, rule, rule, rule) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   410
  fix x y u v assume as:"x \<in> V" "y \<in> V" "u + v = (1::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   411
    "\<forall>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> V \<and> setsum u s = 1 \<longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   412
  thus "u *\<^sub>R x + v *\<^sub>R y \<in> V" apply(cases "x=y")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   413
    using as(4)[THEN spec[where x="{x,y}"], THEN spec[where x="\<lambda>w. if w = x then u else v"]] and as(1-3) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   414
    by(auto simp add: scaleR_left_distrib[THEN sym])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   415
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   416
  fix s u assume as:"\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   417
    "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = (1::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   418
  def n \<equiv> "card s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   419
  have "card s = 0 \<or> card s = 1 \<or> card s = 2 \<or> card s > 2" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   420
  thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" proof(auto simp only: disjE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   421
    assume "card s = 2" hence "card s = Suc (Suc 0)" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   422
    then obtain a b where "s = {a, b}" unfolding card_Suc_eq by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   423
    thus ?thesis using as(1)[THEN bspec[where x=a], THEN bspec[where x=b]] using as(4,5)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   424
      by(auto simp add: setsum_clauses(2))
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   425
  next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   426
      case (Suc n) fix s::"'a set" and u::"'a \<Rightarrow> real"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   427
      assume IA:"\<And>u s.  \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
34915
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34291
diff changeset
   428
               s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" and
7894c7dab132 Adapted to changes in induct method.
berghofe
parents: 34291
diff changeset
   429
        as:"Suc n = card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   430
           "finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   431
      have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   432
        assume " \<not> (\<exists>x\<in>s. u x \<noteq> 1)" hence "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   433
        thus False using as(7) and `card s > 2` by (metis Numeral1_eq1_nat less_0_number_of less_int_code(15)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   434
          less_nat_number_of not_less_iff_gr_or_eq of_nat_1 of_nat_eq_iff pos2 rel_simps(4)) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   435
      then obtain x where x:"x\<in>s" "u x \<noteq> 1" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   436
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   437
      have c:"card (s - {x}) = card s - 1" apply(rule card_Diff_singleton) using `x\<in>s` as(4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   438
      have *:"s = insert x (s - {x})" "finite (s - {x})" using `x\<in>s` and as(4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   439
      have **:"setsum u (s - {x}) = 1 - u x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   440
        using setsum_clauses(2)[OF *(2), of u x, unfolded *(1)[THEN sym] as(7)] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   441
      have ***:"inverse (1 - u x) * setsum u (s - {x}) = 1" unfolding ** using `u x \<noteq> 1` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   442
      have "(\<Sum>xa\<in>s - {x}. (inverse (1 - u x) * u xa) *\<^sub>R xa) \<in> V" proof(cases "card (s - {x}) > 2")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   443
        case True hence "s - {x} \<noteq> {}" "card (s - {x}) = n" unfolding c and as(1)[symmetric] proof(rule_tac ccontr) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   444
          assume "\<not> s - {x} \<noteq> {}" hence "card (s - {x}) = 0" unfolding card_0_eq[OF *(2)] by simp 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   445
          thus False using True by auto qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   446
        thus ?thesis apply(rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   447
        unfolding setsum_right_distrib[THEN sym] using as and *** and True by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   448
      next case False hence "card (s - {x}) = Suc (Suc 0)" using as(2) and c by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   449
        then obtain a b where "(s - {x}) = {a, b}" "a\<noteq>b" unfolding card_Suc_eq by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   450
        thus ?thesis using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   451
          using *** *(2) and `s \<subseteq> V` unfolding setsum_right_distrib by(auto simp add: setsum_clauses(2)) qed
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   452
      hence "u x + (1 - u x) = 1 \<Longrightarrow> u x *\<^sub>R x + (1 - u x) *\<^sub>R ((\<Sum>xa\<in>s - {x}. u xa *\<^sub>R xa) /\<^sub>R (1 - u x)) \<in> V"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   453
        apply-apply(rule as(3)[rule_format]) 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   454
        unfolding  RealVector.scaleR_right.setsum using x(1) as(6) by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   455
      thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   456
         apply(subst *) unfolding setsum_clauses(2)[OF *(2)]
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   457
         using `u x \<noteq> 1` by auto 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   458
    qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   459
  next assume "card s = 1" then obtain a where "s={a}" by(auto simp add: card_Suc_eq)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   460
    thus ?thesis using as(4,5) by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   461
  qed(insert `s\<noteq>{}` `finite s`, auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   462
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   463
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   464
lemma affine_hull_explicit:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   465
  "affine hull p = {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> setsum (\<lambda>v. (u v) *\<^sub>R v) s = y}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   466
  apply(rule hull_unique) apply(subst subset_eq) prefer 3 apply rule unfolding mem_Collect_eq and mem_def[of _ affine]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   467
  apply (erule exE)+ apply(erule conjE)+ prefer 2 apply rule proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   468
  fix x assume "x\<in>p" thus "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   469
    apply(rule_tac x="{x}" in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   470
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   471
  fix t x s u assume as:"p \<subseteq> t" "affine t" "finite s" "s \<noteq> {}" "s \<subseteq> p" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   472
  thus "x \<in> t" using as(2)[unfolded affine, THEN spec[where x=s], THEN spec[where x=u]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   473
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   474
  show "affine {y. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y}" unfolding affine_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   475
    apply(rule,rule,rule,rule,rule) unfolding mem_Collect_eq proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   476
    fix u v ::real assume uv:"u + v = 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   477
    fix x assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   478
    then obtain sx ux where x:"finite sx" "sx \<noteq> {}" "sx \<subseteq> p" "setsum ux sx = 1" "(\<Sum>v\<in>sx. ux v *\<^sub>R v) = x" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   479
    fix y assume "\<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   480
    then obtain sy uy where y:"finite sy" "sy \<noteq> {}" "sy \<subseteq> p" "setsum uy sy = 1" "(\<Sum>v\<in>sy. uy v *\<^sub>R v) = y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   481
    have xy:"finite (sx \<union> sy)" using x(1) y(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   482
    have **:"(sx \<union> sy) \<inter> sx = sx" "(sx \<union> sy) \<inter> sy = sy" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   483
    show "\<exists>s ua. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p \<and> setsum ua s = 1 \<and> (\<Sum>v\<in>s. ua v *\<^sub>R v) = u *\<^sub>R x + v *\<^sub>R y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   484
      apply(rule_tac x="sx \<union> sy" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   485
      apply(rule_tac x="\<lambda>a. (if a\<in>sx then u * ux a else 0) + (if a\<in>sy then v * uy a else 0)" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   486
      unfolding scaleR_left_distrib setsum_addf if_smult scaleR_zero_left  ** setsum_restrict_set[OF xy, THEN sym]
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   487
      unfolding scaleR_scaleR[THEN sym] RealVector.scaleR_right.setsum [symmetric] and setsum_right_distrib[THEN sym]
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   488
      unfolding x y using x(1-3) y(1-3) uv by simp qed qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   489
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   490
lemma affine_hull_finite:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   491
  assumes "finite s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   492
  shows "affine hull s = {y. \<exists>u. setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   493
  unfolding affine_hull_explicit and set_eq_iff and mem_Collect_eq apply (rule,rule)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   494
  apply(erule exE)+ apply(erule conjE)+ defer apply(erule exE) apply(erule conjE) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   495
  fix x u assume "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   496
  thus "\<exists>sa u. finite sa \<and> \<not> (\<forall>x. (x \<in> sa) = (x \<in> {})) \<and> sa \<subseteq> s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   497
    apply(rule_tac x=s in exI, rule_tac x=u in exI) using assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   498
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   499
  fix x t u assume "t \<subseteq> s" hence *:"s \<inter> t = t" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   500
  assume "finite t" "\<not> (\<forall>x. (x \<in> t) = (x \<in> {}))" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   501
  thus "\<exists>u. setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   502
    unfolding if_smult scaleR_zero_left and setsum_restrict_set[OF assms, THEN sym] and * by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   503
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   504
subsection {* Stepping theorems and hence small special cases. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   505
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   506
lemma affine_hull_empty[simp]: "affine hull {} = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   507
  apply(rule hull_unique) unfolding mem_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   508
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   509
lemma affine_hull_finite_step:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   510
  fixes y :: "'a::real_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   511
  shows "(\<exists>u. setsum u {} = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) {} = y) \<longleftrightarrow> w = 0 \<and> y = 0" (is ?th1)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   512
  "finite s \<Longrightarrow> (\<exists>u. setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y) \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   513
                (\<exists>v u. setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?as \<Longrightarrow> (?lhs = ?rhs)")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   514
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   515
  show ?th1 by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   516
  assume ?as 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   517
  { assume ?lhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   518
    then obtain u where u:"setsum u (insert a s) = w \<and> (\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   519
    have ?rhs proof(cases "a\<in>s")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   520
      case True hence *:"insert a s = s" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   521
      show ?thesis using u[unfolded *] apply(rule_tac x=0 in exI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   522
    next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   523
      case False thus ?thesis apply(rule_tac x="u a" in exI) using u and `?as` by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   524
    qed  } moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   525
  { assume ?rhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   526
    then obtain v u where vu:"setsum u s = w - v"  "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   527
    have *:"\<And>x M. (if x = a then v else M) *\<^sub>R x = (if x = a then v *\<^sub>R x else M *\<^sub>R x)" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   528
    have ?lhs proof(cases "a\<in>s")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   529
      case True thus ?thesis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   530
        apply(rule_tac x="\<lambda>x. (if x=a then v else 0) + u x" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   531
        unfolding setsum_clauses(2)[OF `?as`]  apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   532
        unfolding scaleR_left_distrib and setsum_addf 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   533
        unfolding vu and * and scaleR_zero_left
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   534
        by (auto simp add: setsum_delta[OF `?as`])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   535
    next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   536
      case False 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   537
      hence **:"\<And>x. x \<in> s \<Longrightarrow> u x = (if x = a then v else u x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   538
               "\<And>x. x \<in> s \<Longrightarrow> u x *\<^sub>R x = (if x = a then v *\<^sub>R x else u x *\<^sub>R x)" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   539
      from False show ?thesis
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   540
        apply(rule_tac x="\<lambda>x. if x=a then v else u x" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   541
        unfolding setsum_clauses(2)[OF `?as`] and * using vu
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   542
        using setsum_cong2[of s "\<lambda>x. u x *\<^sub>R x" "\<lambda>x. if x = a then v *\<^sub>R x else u x *\<^sub>R x", OF **(2)]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   543
        using setsum_cong2[of s u "\<lambda>x. if x = a then v else u x", OF **(1)] by auto  
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   544
    qed }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   545
  ultimately show "?lhs = ?rhs" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   546
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   547
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   548
lemma affine_hull_2:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   549
  fixes a b :: "'a::real_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   550
  shows "affine hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b| u v. (u + v = 1)}" (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   551
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   552
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   553
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   554
  have "?lhs = {y. \<exists>u. setsum u {a, b} = 1 \<and> (\<Sum>v\<in>{a, b}. u v *\<^sub>R v) = y}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   555
    using affine_hull_finite[of "{a,b}"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   556
  also have "\<dots> = {y. \<exists>v u. u b = 1 - v \<and> u b *\<^sub>R b = y - v *\<^sub>R a}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   557
    by(simp add: affine_hull_finite_step(2)[of "{b}" a]) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   558
  also have "\<dots> = ?rhs" unfolding * by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   559
  finally show ?thesis by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   560
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   561
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   562
lemma affine_hull_3:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   563
  fixes a b c :: "'a::real_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   564
  shows "affine hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c| u v w. u + v + w = 1}" (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   565
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   566
  have *:"\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::real)" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   567
         "\<And>x y z. z = x - y \<longleftrightarrow> y + z = (x::'a)" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   568
  show ?thesis apply(simp add: affine_hull_finite affine_hull_finite_step)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   569
    unfolding * apply auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   570
    apply(rule_tac x=v in exI) apply(rule_tac x=va in exI) apply auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   571
    apply(rule_tac x=u in exI) by(auto intro!: exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   572
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   573
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   574
lemma mem_affine:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   575
  assumes "affine S" "x : S" "y : S" "u+v=1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   576
  shows "(u *\<^sub>R x + v *\<^sub>R y) : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   577
  using assms affine_def[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   578
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   579
lemma mem_affine_3:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   580
  assumes "affine S" "x : S" "y : S" "z : S" "u+v+w=1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   581
  shows "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   582
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   583
have "(u *\<^sub>R x + v *\<^sub>R y + w *\<^sub>R z) : affine hull {x, y, z}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   584
  using affine_hull_3[of x y z] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   585
moreover have " affine hull {x, y, z} <= affine hull S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   586
  using hull_mono[of "{x, y, z}" "S"] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   587
moreover have "affine hull S = S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   588
  using assms affine_hull_eq[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   589
ultimately show ?thesis by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   590
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   591
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   592
lemma mem_affine_3_minus:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   593
  assumes "affine S" "x : S" "y : S" "z : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   594
  shows "x + v *\<^sub>R (y-z) : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   595
using mem_affine_3[of S x y z 1 v "-v"] assms by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   596
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   597
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   598
subsection {* Some relations between affine hull and subspaces. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   599
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   600
lemma affine_hull_insert_subset_span:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   601
  fixes a :: "'a::euclidean_space"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   602
  shows "affine hull (insert a s) \<subseteq> {a + v| v . v \<in> span {x - a | x . x \<in> s}}"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   603
  unfolding subset_eq Ball_def unfolding affine_hull_explicit span_explicit mem_Collect_eq
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   604
  apply(rule,rule) apply(erule exE)+ apply(erule conjE)+ proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   605
  fix x t u assume as:"finite t" "t \<noteq> {}" "t \<subseteq> insert a s" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   606
  have "(\<lambda>x. x - a) ` (t - {a}) \<subseteq> {x - a |x. x \<in> s}" using as(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   607
  thus "\<exists>v. x = a + v \<and> (\<exists>S u. finite S \<and> S \<subseteq> {x - a |x. x \<in> s} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = v)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   608
    apply(rule_tac x="x - a" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   609
    apply (rule conjI, simp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   610
    apply(rule_tac x="(\<lambda>x. x - a) ` (t - {a})" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   611
    apply(rule_tac x="\<lambda>x. u (x + a)" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   612
    apply (rule conjI) using as(1) apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   613
    apply (erule conjI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   614
    using as(1)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   615
    apply (simp add: setsum_reindex[unfolded inj_on_def] scaleR_right_diff_distrib setsum_subtractf scaleR_left.setsum[THEN sym] setsum_diff1 scaleR_left_diff_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   616
    unfolding as by simp qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   617
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   618
lemma affine_hull_insert_span:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   619
  fixes a :: "'a::euclidean_space"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   620
  assumes "a \<notin> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   621
  shows "affine hull (insert a s) =
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   622
            {a + v | v . v \<in> span {x - a | x.  x \<in> s}}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   623
  apply(rule, rule affine_hull_insert_subset_span) unfolding subset_eq Ball_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   624
  unfolding affine_hull_explicit and mem_Collect_eq proof(rule,rule,erule exE,erule conjE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   625
  fix y v assume "y = a + v" "v \<in> span {x - a |x. x \<in> s}"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   626
  then obtain t u where obt:"finite t" "t \<subseteq> {x - a |x. x \<in> s}" "a + (\<Sum>v\<in>t. u v *\<^sub>R v) = y" unfolding span_explicit by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   627
  def f \<equiv> "(\<lambda>x. x + a) ` t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   628
  have f:"finite f" "f \<subseteq> s" "(\<Sum>v\<in>f. u (v - a) *\<^sub>R (v - a)) = y - a" unfolding f_def using obt 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   629
    by(auto simp add: setsum_reindex[unfolded inj_on_def])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   630
  have *:"f \<inter> {a} = {}" "f \<inter> - {a} = f" using f(2) assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   631
  show "\<exists>sa u. finite sa \<and> sa \<noteq> {} \<and> sa \<subseteq> insert a s \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   632
    apply(rule_tac x="insert a f" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   633
    apply(rule_tac x="\<lambda>x. if x=a then 1 - setsum (\<lambda>x. u (x - a)) f else u (x - a)" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   634
    using assms and f unfolding setsum_clauses(2)[OF f(1)] and if_smult
35577
43b93e294522 Generalized setsum_cases
hoelzl
parents: 35542
diff changeset
   635
    unfolding setsum_cases[OF f(1), of "\<lambda>x. x = a"]
43b93e294522 Generalized setsum_cases
hoelzl
parents: 35542
diff changeset
   636
    by (auto simp add: setsum_subtractf scaleR_left.setsum algebra_simps *) qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   637
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   638
lemma affine_hull_span:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
   639
  fixes a :: "'a::euclidean_space"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   640
  assumes "a \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   641
  shows "affine hull s = {a + v | v. v \<in> span {x - a | x. x \<in> s - {a}}}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   642
  using affine_hull_insert_span[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   643
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   644
subsection{* Parallel Affine Sets *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   645
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   646
definition affine_parallel :: "'a::real_vector set => 'a::real_vector set => bool"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   647
where "affine_parallel S T = (? a. T = ((%x. a + x) ` S))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   648
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   649
lemma affine_parallel_expl_aux:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   650
   fixes S T :: "'a::real_vector set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   651
   assumes "!x. (x : S <-> (a+x) : T)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   652
   shows "T = ((%x. a + x) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   653
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   654
{ fix x assume "x : T" hence "(-a)+x : S" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   655
  hence " x : ((%x. a + x) ` S)" using imageI[of "-a+x" S "(%x. a+x)"] by auto}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   656
moreover have "T >= ((%x. a + x) ` S)" using assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   657
ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   658
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   659
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   660
lemma affine_parallel_expl: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   661
   "affine_parallel S T = (? a. !x. (x : S <-> (a+x) : T))" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   662
   unfolding affine_parallel_def using affine_parallel_expl_aux[of S _ T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   663
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   664
lemma affine_parallel_reflex: "affine_parallel S S" unfolding affine_parallel_def apply (rule exI[of _ "0"]) by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   665
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   666
lemma affine_parallel_commut:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   667
assumes "affine_parallel A B" shows "affine_parallel B A" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   668
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   669
from assms obtain a where "B=((%x. a + x) ` A)" unfolding affine_parallel_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   670
from this show ?thesis using translation_galois[of B a A] unfolding affine_parallel_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   671
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   672
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   673
lemma affine_parallel_assoc:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   674
assumes "affine_parallel A B" "affine_parallel B C"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   675
shows "affine_parallel A C" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   676
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   677
from assms obtain ab where "B=((%x. ab + x) ` A)" unfolding affine_parallel_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   678
moreover 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   679
from assms obtain bc where "C=((%x. bc + x) ` B)" unfolding affine_parallel_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   680
ultimately show ?thesis using translation_assoc[of bc ab A] unfolding affine_parallel_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   681
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   682
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   683
lemma affine_translation_aux:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   684
  fixes a :: "'a::real_vector"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   685
  assumes "affine ((%x. a + x) ` S)" shows "affine S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   686
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   687
{ fix x y u v
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   688
  assume xy: "x : S" "y : S" "(u :: real)+v=1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   689
  hence "(a+x):((%x. a + x) ` S)" "(a+y):((%x. a + x) ` S)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   690
  hence h1: "u *\<^sub>R  (a+x) + v *\<^sub>R (a+y) : ((%x. a + x) ` S)" using xy assms unfolding affine_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   691
  have "u *\<^sub>R (a+x) + v *\<^sub>R (a+y) = (u+v) *\<^sub>R a + (u *\<^sub>R x + v *\<^sub>R y)" by (simp add:algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   692
  also have "...= a + (u *\<^sub>R x + v *\<^sub>R y)" using `u+v=1` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   693
  ultimately have "a + (u *\<^sub>R x + v *\<^sub>R y) : ((%x. a + x) ` S)" using h1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   694
  hence "u *\<^sub>R x + v *\<^sub>R y : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   695
} from this show ?thesis unfolding affine_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   696
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   697
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   698
lemma affine_translation:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   699
  fixes a :: "'a::real_vector"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   700
  shows "affine S <-> affine ((%x. a + x) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   701
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   702
have "affine S ==> affine ((%x. a + x) ` S)" using affine_translation_aux[of "-a" "((%x. a + x) ` S)"]  using translation_assoc[of "-a" a S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   703
from this show ?thesis using affine_translation_aux by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   704
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   705
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   706
lemma parallel_is_affine:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   707
fixes S T :: "'a::real_vector set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   708
assumes "affine S" "affine_parallel S T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   709
shows "affine T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   710
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   711
  from assms obtain a where "T=((%x. a + x) ` S)" unfolding affine_parallel_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   712
  from this show ?thesis using affine_translation assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   713
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   714
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   715
lemma subspace_imp_affine:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   716
  fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> affine s"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   717
  unfolding subspace_def affine_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   718
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   719
subsection{* Subspace Parallel to an Affine Set *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   720
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   721
lemma subspace_affine:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   722
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   723
  shows "subspace S <-> (affine S & 0 : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   724
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   725
have h0: "subspace S ==> (affine S & 0 : S)" using subspace_imp_affine[of S] subspace_0 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   726
{ assume assm: "affine S & 0 : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   727
  { fix c :: real 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   728
    fix x assume x_def: "x : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   729
    have "c *\<^sub>R x = (1-c) *\<^sub>R 0 + c *\<^sub>R x" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   730
    moreover have "(1-c) *\<^sub>R 0 + c *\<^sub>R x : S" using affine_alt[of S] assm x_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   731
    ultimately have "c *\<^sub>R x : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   732
  } hence h1: "!c. !x : S. c *\<^sub>R x : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   733
  { fix x y assume xy_def: "x : S" "y : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   734
    def u == "(1 :: real)/2"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   735
    have "(1/2) *\<^sub>R (x+y) = (1/2) *\<^sub>R (x+y)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   736
    moreover have "(1/2) *\<^sub>R (x+y)=(1/2) *\<^sub>R x + (1-(1/2)) *\<^sub>R y" by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   737
    moreover have "(1-u) *\<^sub>R x + u *\<^sub>R y : S" using affine_alt[of S] assm xy_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   738
    ultimately have "(1/2) *\<^sub>R (x+y) : S" using u_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   739
    moreover have "(x+y) = 2 *\<^sub>R ((1/2) *\<^sub>R (x+y))" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   740
    ultimately have "(x+y) : S" using h1[rule_format, of "(1/2) *\<^sub>R (x+y)" "2"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   741
  } hence "!x : S. !y : S. (x+y) : S" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   742
  hence "subspace S" using h1 assm unfolding subspace_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   743
} from this show ?thesis using h0 by metis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   744
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   745
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   746
lemma affine_diffs_subspace:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   747
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   748
  assumes "affine S" "a : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   749
  shows "subspace ((%x. (-a)+x) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   750
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   751
have "affine ((%x. (-a)+x) ` S)" using  affine_translation assms by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   752
moreover have "0 : ((%x. (-a)+x) ` S)" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   753
ultimately show ?thesis using subspace_affine by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   754
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   755
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   756
lemma parallel_subspace_explicit:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   757
fixes a :: "'n::euclidean_space"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   758
assumes "affine S" "a : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   759
assumes "L == {y. ? x : S. (-a)+x=y}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   760
shows "subspace L & affine_parallel S L" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   761
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   762
have par: "affine_parallel S L" unfolding affine_parallel_def using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   763
hence "affine L" using assms parallel_is_affine by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   764
moreover have "0 : L" using assms apply auto using exI[of "(%x. x:S & -a+x=0)" a] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   765
ultimately show ?thesis using subspace_affine par by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   766
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   767
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   768
lemma parallel_subspace_aux:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   769
fixes A B :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   770
assumes "subspace A" "subspace B" "affine_parallel A B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   771
shows "A>=B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   772
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   773
from assms obtain a where a_def: "!x. (x : A <-> (a+x) : B)" using affine_parallel_expl[of A B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   774
hence "-a : A" using assms subspace_0[of B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   775
hence "a : A" using assms subspace_neg[of A "-a"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   776
from this show ?thesis using assms a_def unfolding subspace_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   777
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   778
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   779
lemma parallel_subspace:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   780
fixes A B :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   781
assumes "subspace A" "subspace B" "affine_parallel A B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   782
shows "A=B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   783
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   784
have "A>=B" using assms parallel_subspace_aux by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   785
moreover have "A<=B" using assms parallel_subspace_aux[of B A] affine_parallel_commut by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   786
ultimately show ?thesis by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   787
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   788
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   789
lemma affine_parallel_subspace:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   790
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   791
assumes "affine S" "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   792
shows "?!L. subspace L & affine_parallel S L" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   793
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   794
have ex: "? L. subspace L & affine_parallel S L" using assms  parallel_subspace_explicit by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   795
{ fix L1 L2 assume ass: "subspace L1 & affine_parallel S L1" "subspace L2 & affine_parallel S L2"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   796
  hence "affine_parallel L1 L2" using affine_parallel_commut[of S L1] affine_parallel_assoc[of L1 S L2] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   797
  hence "L1=L2" using ass parallel_subspace by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   798
} from this show ?thesis using ex by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   799
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   800
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   801
subsection {* Cones. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   802
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   803
definition
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   804
  cone :: "'a::real_vector set \<Rightarrow> bool" where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   805
  "cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   806
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   807
lemma cone_empty[intro, simp]: "cone {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   808
  unfolding cone_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   809
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   810
lemma cone_univ[intro, simp]: "cone UNIV"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   811
  unfolding cone_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   812
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   813
lemma cone_Inter[intro]: "(\<forall>s\<in>f. cone s) \<Longrightarrow> cone(\<Inter> f)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   814
  unfolding cone_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   815
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   816
subsection {* Conic hull. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   817
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   818
lemma cone_cone_hull: "cone (cone hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   819
  unfolding hull_def using cone_Inter[of "{t \<in> conic. s \<subseteq> t}"] 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   820
  by (auto simp add: mem_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   821
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   822
lemma cone_hull_eq: "(cone hull s = s) \<longleftrightarrow> cone s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   823
  apply(rule hull_eq[unfolded mem_def])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   824
  using cone_Inter unfolding subset_eq by (auto simp add: mem_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   825
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   826
lemma mem_cone:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   827
  assumes "cone S" "x : S" "c>=0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   828
  shows "c *\<^sub>R x : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   829
  using assms cone_def[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   830
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   831
lemma cone_contains_0:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   832
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   833
assumes "cone S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   834
shows "(S ~= {}) <-> (0 : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   835
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   836
{ assume "S ~= {}" from this obtain a where "a:S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   837
  hence "0 : S" using assms mem_cone[of S a 0] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   838
} from this show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   839
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   840
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   841
lemma cone_0:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   842
shows "cone {(0 :: 'm::euclidean_space)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   843
unfolding cone_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   844
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   845
lemma cone_Union[intro]: "(!s:f. cone s) --> (cone (Union f))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   846
  unfolding cone_def by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   847
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   848
lemma cone_iff:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   849
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   850
assumes "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   851
shows "cone S <-> 0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   852
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   853
{ assume "cone S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   854
  { fix c assume "(c :: real)>0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   855
    { fix x assume "x : S" hence "x : (op *\<^sub>R c) ` S" unfolding image_def
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   856
        using `cone S` `c>0` mem_cone[of S x "1/c"]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   857
        exI[of "(%t. t:S & x = c *\<^sub>R t)" "(1 / c) *\<^sub>R x"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   858
    }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   859
    moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   860
    { fix x assume "x : (op *\<^sub>R c) ` S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   861
      (*from this obtain t where "t:S & x = c *\<^sub>R t" by auto*)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   862
      hence "x:S" using `cone S` `c>0` unfolding cone_def image_def `c>0` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   863
    }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   864
    ultimately have "((op *\<^sub>R c) ` S) = S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   865
  } hence "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)" using `cone S` cone_contains_0[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   866
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   867
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   868
{ assume a: "0:S & (!c. c>0 --> ((op *\<^sub>R c) ` S) = S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   869
  { fix x assume "x:S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   870
    fix c1 assume "(c1 :: real)>=0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   871
    hence "(c1=0) | (c1>0)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   872
    hence "c1 *\<^sub>R x : S" using a `x:S` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   873
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   874
 hence "cone S" unfolding cone_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   875
} ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   876
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   877
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   878
lemma cone_hull_empty:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   879
"cone hull {} = {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   880
by (metis cone_empty cone_hull_eq)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   881
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   882
lemma cone_hull_empty_iff:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   883
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   884
shows "(S = {}) <-> (cone hull S = {})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   885
by (metis bot_least cone_hull_empty hull_subset xtrans(5))
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   886
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   887
lemma cone_hull_contains_0: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   888
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   889
shows "(S ~= {}) <-> (0 : cone hull S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   890
using cone_cone_hull[of S] cone_contains_0[of "cone hull S"] cone_hull_empty_iff[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   891
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   892
lemma mem_cone_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   893
  assumes "x : S" "c>=0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   894
  shows "c *\<^sub>R x : cone hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   895
by (metis assms cone_cone_hull hull_inc mem_cone mem_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   896
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   897
lemma cone_hull_expl:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   898
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   899
shows "cone hull S = {c *\<^sub>R x | c x. c>=0 & x : S}" (is "?lhs = ?rhs")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   900
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   901
{ fix x assume "x : ?rhs"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   902
  from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   903
  fix c assume c_def: "(c :: real)>=0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   904
  hence "c *\<^sub>R x = (c*cx) *\<^sub>R xx" using x_def by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   905
  moreover have "(c*cx) >= 0" using c_def x_def using mult_nonneg_nonneg by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   906
  ultimately have "c *\<^sub>R x : ?rhs" using x_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   907
} hence "cone ?rhs" unfolding cone_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   908
  hence "?rhs : cone" unfolding mem_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   909
{ fix x assume "x : S" hence "1 *\<^sub>R x : ?rhs" apply auto apply(rule_tac x="1" in exI) by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   910
  hence "x : ?rhs" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   911
} hence "S <= ?rhs" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   912
hence "?lhs <= ?rhs" using `?rhs : cone` hull_minimal[of S "?rhs" "cone"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   913
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   914
{ fix x assume "x : ?rhs"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   915
  from this obtain cx xx where x_def: "x= cx *\<^sub>R xx & (cx :: real)>=0 & xx : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   916
  hence "xx : cone hull S" using hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   917
  hence "x : ?lhs" using x_def cone_cone_hull[of S] cone_def[of "cone hull S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   918
} ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   919
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   920
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   921
lemma cone_closure:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   922
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   923
assumes "cone S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   924
shows "cone (closure S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   925
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   926
{ assume "S = {}" hence ?thesis by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   927
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   928
{ assume "S ~= {}" hence "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   929
  hence "0:(closure S) & (!c. c>0 --> op *\<^sub>R c ` (closure S) = (closure S))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   930
     using closure_subset by (auto simp add: closure_scaleR)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   931
  hence ?thesis using cone_iff[of "closure S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   932
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   933
ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   934
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
   935
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   936
subsection {* Affine dependence and consequential theorems (from Lars Schewe). *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   937
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   938
definition
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   939
  affine_dependent :: "'a::real_vector set \<Rightarrow> bool" where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   940
  "affine_dependent s \<longleftrightarrow> (\<exists>x\<in>s. x \<in> (affine hull (s - {x})))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   941
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   942
lemma affine_dependent_explicit:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   943
  "affine_dependent p \<longleftrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   944
    (\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   945
    (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   946
  unfolding affine_dependent_def affine_hull_explicit mem_Collect_eq apply(rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   947
  apply(erule bexE,erule exE,erule exE) apply(erule conjE)+ defer apply(erule exE,erule exE) apply(erule conjE)+ apply(erule bexE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   948
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   949
  fix x s u assume as:"x \<in> p" "finite s" "s \<noteq> {}" "s \<subseteq> p - {x}" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   950
  have "x\<notin>s" using as(1,4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   951
  show "\<exists>s u. finite s \<and> s \<subseteq> p \<and> setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   952
    apply(rule_tac x="insert x s" in exI, rule_tac x="\<lambda>v. if v = x then - 1 else u v" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   953
    unfolding if_smult and setsum_clauses(2)[OF as(2)] and setsum_delta_notmem[OF `x\<notin>s`] and as using as by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   954
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   955
  fix s u v assume as:"finite s" "s \<subseteq> p" "setsum u s = 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" "v \<in> s" "u v \<noteq> 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   956
  have "s \<noteq> {v}" using as(3,6) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   957
  thus "\<exists>x\<in>p. \<exists>s u. finite s \<and> s \<noteq> {} \<and> s \<subseteq> p - {x} \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   958
    apply(rule_tac x=v in bexI, rule_tac x="s - {v}" in exI, rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   959
    unfolding scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] unfolding setsum_right_distrib[THEN sym] and setsum_diff1[OF as(1)] using as by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   960
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   961
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   962
lemma affine_dependent_explicit_finite:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   963
  fixes s :: "'a::real_vector set" assumes "finite s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   964
  shows "affine_dependent s \<longleftrightarrow> (\<exists>u. setsum u s = 0 \<and> (\<exists>v\<in>s. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = 0)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   965
  (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   966
proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   967
  have *:"\<And>vt u v. (if vt then u v else 0) *\<^sub>R v = (if vt then (u v) *\<^sub>R v else (0::'a))" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   968
  assume ?lhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   969
  then obtain t u v where "finite t" "t \<subseteq> s" "setsum u t = 0" "v\<in>t" "u v \<noteq> 0"  "(\<Sum>v\<in>t. u v *\<^sub>R v) = 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   970
    unfolding affine_dependent_explicit by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   971
  thus ?rhs apply(rule_tac x="\<lambda>x. if x\<in>t then u x else 0" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   972
    apply auto unfolding * and setsum_restrict_set[OF assms, THEN sym]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   973
    unfolding Int_absorb1[OF `t\<subseteq>s`] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   974
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   975
  assume ?rhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   976
  then obtain u v where "setsum u s = 0"  "v\<in>s" "u v \<noteq> 0" "(\<Sum>v\<in>s. u v *\<^sub>R v) = 0" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   977
  thus ?lhs unfolding affine_dependent_explicit using assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   978
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   979
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   980
subsection {* A general lemma. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   981
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   982
lemma convex_connected:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   983
  fixes s :: "'a::real_normed_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   984
  assumes "convex s" shows "connected s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   985
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   986
  { fix e1 e2 assume as:"open e1" "open e2" "e1 \<inter> e2 \<inter> s = {}" "s \<subseteq> e1 \<union> e2" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   987
    assume "e1 \<inter> s \<noteq> {}" "e2 \<inter> s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   988
    then obtain x1 x2 where x1:"x1\<in>e1" "x1\<in>s" and x2:"x2\<in>e2" "x2\<in>s" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   989
    hence n:"norm (x1 - x2) > 0" unfolding zero_less_norm_iff using as(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   990
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   991
    { fix x e::real assume as:"0 \<le> x" "x \<le> 1" "0 < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   992
      { fix y have *:"(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2) = (y - x) *\<^sub>R x1 - (y - x) *\<^sub>R x2"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   993
          by (simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   994
        assume "\<bar>y - x\<bar> < e / norm (x1 - x2)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   995
        hence "norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   996
          unfolding * and scaleR_right_diff_distrib[THEN sym]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   997
          unfolding less_divide_eq using n by auto  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   998
      hence "\<exists>d>0. \<forall>y. \<bar>y - x\<bar> < d \<longrightarrow> norm ((1 - x) *\<^sub>R x1 + x *\<^sub>R x2 - ((1 - y) *\<^sub>R x1 + y *\<^sub>R x2)) < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   999
        apply(rule_tac x="e / norm (x1 - x2)" in exI) using as
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1000
        apply auto unfolding zero_less_divide_iff using n by simp  }  note * = this
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1001
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1002
    have "\<exists>x\<ge>0. x \<le> 1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1 \<and> (1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1003
      apply(rule connected_real_lemma) apply (simp add: `x1\<in>e1` `x2\<in>e2` dist_commute)+
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1004
      using * apply(simp add: dist_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1005
      using as(1,2)[unfolded open_dist] apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1006
      using as(1,2)[unfolded open_dist] apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1007
      using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]] using x1 x2
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1008
      using as(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1009
    then obtain x where "x\<ge>0" "x\<le>1" "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e1"  "(1 - x) *\<^sub>R x1 + x *\<^sub>R x2 \<notin> e2" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1010
    hence False using as(4) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1011
      using assms[unfolded convex_alt, THEN bspec[where x=x1], THEN bspec[where x=x2]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1012
      using x1(2) x2(2) by auto  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1013
  thus ?thesis unfolding connected_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1014
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1015
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1016
subsection {* One rather trivial consequence. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1017
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  1018
lemma connected_UNIV[intro]: "connected (UNIV :: 'a::real_normed_vector set)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1019
  by(simp add: convex_connected convex_UNIV)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1020
36623
d26348b667f2 Moved Convex theory to library.
hoelzl
parents: 36590
diff changeset
  1021
subsection {* Balls, being convex, are connected. *}
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1022
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1023
lemma convex_box: fixes a::"'a::euclidean_space"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1024
  assumes "\<And>i. i<DIM('a) \<Longrightarrow> convex {x. P i x}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1025
  shows "convex {x. \<forall>i<DIM('a). P i (x$$i)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1026
  using assms unfolding convex_def by(auto simp add:euclidean_simps)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1027
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1028
lemma convex_positive_orthant: "convex {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 \<le> x$$i)}"
36623
d26348b667f2 Moved Convex theory to library.
hoelzl
parents: 36590
diff changeset
  1029
  by (rule convex_box) (simp add: atLeast_def[symmetric] convex_real_interval)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1030
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1031
lemma convex_local_global_minimum:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1032
  fixes s :: "'a::real_normed_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1033
  assumes "0<e" "convex_on s f" "ball x e \<subseteq> s" "\<forall>y\<in>ball x e. f x \<le> f y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1034
  shows "\<forall>y\<in>s. f x \<le> f y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1035
proof(rule ccontr)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1036
  have "x\<in>s" using assms(1,3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1037
  assume "\<not> (\<forall>y\<in>s. f x \<le> f y)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1038
  then obtain y where "y\<in>s" and y:"f x > f y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1039
  hence xy:"0 < dist x y" by (auto simp add: dist_nz[THEN sym])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1040
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1041
  then obtain u where "0 < u" "u \<le> 1" and u:"u < e / dist x y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1042
    using real_lbound_gt_zero[of 1 "e / dist x y"] using xy `e>0` and divide_pos_pos[of e "dist x y"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1043
  hence "f ((1-u) *\<^sub>R x + u *\<^sub>R y) \<le> (1-u) * f x + u * f y" using `x\<in>s` `y\<in>s`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1044
    using assms(2)[unfolded convex_on_def, THEN bspec[where x=x], THEN bspec[where x=y], THEN spec[where x="1-u"]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1045
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1046
  have *:"x - ((1 - u) *\<^sub>R x + u *\<^sub>R y) = u *\<^sub>R (x - y)" by (simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1047
  have "(1 - u) *\<^sub>R x + u *\<^sub>R y \<in> ball x e" unfolding mem_ball dist_norm unfolding * and norm_scaleR and abs_of_pos[OF `0<u`] unfolding dist_norm[THEN sym]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1048
    using u unfolding pos_less_divide_eq[OF xy] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1049
  hence "f x \<le> f ((1 - u) *\<^sub>R x + u *\<^sub>R y)" using assms(4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1050
  ultimately show False using mult_strict_left_mono[OF y `u>0`] unfolding left_diff_distrib by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1051
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1052
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1053
lemma convex_ball:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1054
  fixes x :: "'a::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1055
  shows "convex (ball x e)" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1056
proof(auto simp add: convex_def)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1057
  fix y z assume yz:"dist x y < e" "dist x z < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1058
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1059
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1060
    using convex_distance[of "ball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
36623
d26348b667f2 Moved Convex theory to library.
hoelzl
parents: 36590
diff changeset
  1061
  thus "dist x (u *\<^sub>R y + v *\<^sub>R z) < e" using convex_bound_lt[OF yz uv] by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1062
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1063
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1064
lemma convex_cball:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1065
  fixes x :: "'a::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1066
  shows "convex(cball x e)"
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  1067
proof(auto simp add: convex_def Ball_def)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1068
  fix y z assume yz:"dist x y \<le> e" "dist x z \<le> e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1069
  fix u v ::real assume uv:" 0 \<le> u" "0 \<le> v" "u + v = 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1070
  have "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> u * dist x y + v * dist x z" using uv yz
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1071
    using convex_distance[of "cball x e" x, unfolded convex_on_def, THEN bspec[where x=y], THEN bspec[where x=z]] by auto
36623
d26348b667f2 Moved Convex theory to library.
hoelzl
parents: 36590
diff changeset
  1072
  thus "dist x (u *\<^sub>R y + v *\<^sub>R z) \<le> e" using convex_bound_le[OF yz uv] by auto 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1073
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1074
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1075
lemma connected_ball:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1076
  fixes x :: "'a::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1077
  shows "connected (ball x e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1078
  using convex_connected convex_ball by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1079
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1080
lemma connected_cball:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1081
  fixes x :: "'a::real_normed_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1082
  shows "connected(cball x e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1083
  using convex_connected convex_cball by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1084
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1085
subsection {* Convex hull. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1086
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1087
lemma convex_convex_hull: "convex(convex hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1088
  unfolding hull_def using convex_Inter[of "{t\<in>convex. s\<subseteq>t}"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1089
  unfolding mem_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1090
34064
eee04bbbae7e avoid dependency on implicit dest rule predicate1D in proofs
haftmann
parents: 33758
diff changeset
  1091
lemma convex_hull_eq: "convex hull s = s \<longleftrightarrow> convex s"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1092
by (metis convex_convex_hull hull_same mem_def)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1093
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1094
lemma bounded_convex_hull:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1095
  fixes s :: "'a::real_normed_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1096
  assumes "bounded s" shows "bounded(convex hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1097
proof- from assms obtain B where B:"\<forall>x\<in>s. norm x \<le> B" unfolding bounded_iff by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1098
  show ?thesis apply(rule bounded_subset[OF bounded_cball, of _ 0 B])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1099
    unfolding subset_hull[unfolded mem_def, of convex, OF convex_cball]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1100
    unfolding subset_eq mem_cball dist_norm using B by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1101
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1102
lemma finite_imp_bounded_convex_hull:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1103
  fixes s :: "'a::real_normed_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1104
  shows "finite s \<Longrightarrow> bounded(convex hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1105
  using bounded_convex_hull finite_imp_bounded by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1106
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1107
subsection {* Convex hull is "preserved" by a linear function. *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1108
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1109
lemma convex_hull_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1110
  assumes "bounded_linear f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1111
  shows "f ` (convex hull s) = convex hull (f ` s)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1112
  apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1113
  apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1114
  apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1115
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1116
  interpret f: bounded_linear f by fact
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1117
  show "convex {x. f x \<in> convex hull f ` s}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1118
  unfolding convex_def by(auto simp add: f.scaleR f.add convex_convex_hull[unfolded convex_def, rule_format]) next
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1119
  interpret f: bounded_linear f by fact
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1120
  show "convex {x. x \<in> f ` (convex hull s)}" using  convex_convex_hull[unfolded convex_def, of s] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1121
    unfolding convex_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1122
qed auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1123
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1124
lemma in_convex_hull_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1125
  assumes "bounded_linear f" "x \<in> convex hull s"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1126
  shows "(f x) \<in> convex hull (f ` s)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1127
using convex_hull_linear_image[OF assms(1)] assms(2) by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1128
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1129
subsection {* Stepping theorems for convex hulls of finite sets. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1130
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1131
lemma convex_hull_empty[simp]: "convex hull {} = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1132
  apply(rule hull_unique) unfolding mem_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1133
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1134
lemma convex_hull_singleton[simp]: "convex hull {a} = {a}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1135
  apply(rule hull_unique) unfolding mem_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1136
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1137
lemma convex_hull_insert:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1138
  fixes s :: "'a::real_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1139
  assumes "s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1140
  shows "convex hull (insert a s) = {x. \<exists>u\<ge>0. \<exists>v\<ge>0. \<exists>b. (u + v = 1) \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1141
                                    b \<in> (convex hull s) \<and> (x = u *\<^sub>R a + v *\<^sub>R b)}" (is "?xyz = ?hull")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1142
 apply(rule,rule hull_minimal,rule) unfolding mem_def[of _ convex] and insert_iff prefer 3 apply rule proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1143
 fix x assume x:"x = a \<or> x \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1144
 thus "x\<in>?hull" apply rule unfolding mem_Collect_eq apply(rule_tac x=1 in exI) defer 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1145
   apply(rule_tac x=0 in exI) using assms hull_subset[of s convex] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1146
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1147
  fix x assume "x\<in>?hull"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1148
  then obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "x = u *\<^sub>R a + v *\<^sub>R b" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1149
  have "a\<in>convex hull insert a s" "b\<in>convex hull insert a s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1150
    using hull_mono[of s "insert a s" convex] hull_mono[of "{a}" "insert a s" convex] and obt(4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1151
  thus "x\<in> convex hull insert a s" unfolding obt(5) using convex_convex_hull[of "insert a s", unfolded convex_def]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1152
    apply(erule_tac x=a in ballE) apply(erule_tac x=b in ballE) apply(erule_tac x=u in allE) using obt by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1153
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1154
  show "convex ?hull" unfolding convex_def apply(rule,rule,rule,rule,rule,rule,rule) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1155
    fix x y u v assume as:"(0::real) \<le> u" "0 \<le> v" "u + v = 1" "x\<in>?hull" "y\<in>?hull"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1156
    from as(4) obtain u1 v1 b1 where obt1:"u1\<ge>0" "v1\<ge>0" "u1 + v1 = 1" "b1 \<in> convex hull s" "x = u1 *\<^sub>R a + v1 *\<^sub>R b1" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1157
    from as(5) obtain u2 v2 b2 where obt2:"u2\<ge>0" "v2\<ge>0" "u2 + v2 = 1" "b2 \<in> convex hull s" "y = u2 *\<^sub>R a + v2 *\<^sub>R b2" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1158
    have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1159
    have "\<exists>b \<in> convex hull s. u *\<^sub>R x + v *\<^sub>R y = (u * u1) *\<^sub>R a + (v * u2) *\<^sub>R a + (b - (u * u1) *\<^sub>R b - (v * u2) *\<^sub>R b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1160
    proof(cases "u * v1 + v * v2 = 0")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1161
      have *:"\<And>(x::'a) s1 s2. x - s1 *\<^sub>R x - s2 *\<^sub>R x = ((1::real) - (s1 + s2)) *\<^sub>R x" by (auto simp add: algebra_simps)
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1162
      case True hence **:"u * v1 = 0" "v * v2 = 0"
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1163
        using mult_nonneg_nonneg[OF `u\<ge>0` `v1\<ge>0`] mult_nonneg_nonneg[OF `v\<ge>0` `v2\<ge>0`] by arith+
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1164
      hence "u * u1 + v * u2 = 1" using as(3) obt1(3) obt2(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1165
      thus ?thesis unfolding obt1(5) obt2(5) * using assms hull_subset[of s convex] by(auto simp add: ** scaleR_right_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1166
    next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1167
      have "1 - (u * u1 + v * u2) = (u + v) - (u * u1 + v * u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1168
      also have "\<dots> = u * (v1 + u1 - u1) + v * (v2 + u2 - u2)" using as(3) obt1(3) obt2(3) by (auto simp add: field_simps) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1169
      also have "\<dots> = u * v1 + v * v2" by simp finally have **:"1 - (u * u1 + v * u2) = u * v1 + v * v2" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1170
      case False have "0 \<le> u * v1 + v * v2" "0 \<le> u * v1" "0 \<le> u * v1 + v * v2" "0 \<le> v * v2" apply -
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1171
        apply(rule add_nonneg_nonneg) prefer 4 apply(rule add_nonneg_nonneg) apply(rule_tac [!] mult_nonneg_nonneg)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1172
        using as(1,2) obt1(1,2) obt2(1,2) by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1173
      thus ?thesis unfolding obt1(5) obt2(5) unfolding * and ** using False
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1174
        apply(rule_tac x="((u * v1) / (u * v1 + v * v2)) *\<^sub>R b1 + ((v * v2) / (u * v1 + v * v2)) *\<^sub>R b2" in bexI) defer
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1175
        apply(rule convex_convex_hull[of s, unfolded convex_def, rule_format]) using obt1(4) obt2(4)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1176
        unfolding add_divide_distrib[THEN sym] and real_0_le_divide_iff
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1177
        by (auto simp add: scaleR_left_distrib scaleR_right_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1178
    qed note * = this
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1179
    have u1:"u1 \<le> 1" unfolding obt1(3)[THEN sym] and not_le using obt1(2) by auto
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1180
    have u2:"u2 \<le> 1" unfolding obt2(3)[THEN sym] and not_le using obt2(2) by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1181
    have "u1 * u + u2 * v \<le> (max u1 u2) * u + (max u1 u2) * v" apply(rule add_mono)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1182
      apply(rule_tac [!] mult_right_mono) using as(1,2) obt1(1,2) obt2(1,2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1183
    also have "\<dots> \<le> 1" unfolding mult.add_right[THEN sym] and as(3) using u1 u2 by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1184
    finally 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1185
    show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x="u * u1 + v * u2" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1186
      apply(rule conjI) defer apply(rule_tac x="1 - u * u1 - v * u2" in exI) unfolding Bex_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1187
      using as(1,2) obt1(1,2) obt2(1,2) * by(auto intro!: mult_nonneg_nonneg add_nonneg_nonneg simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1188
  qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1189
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1190
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1191
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1192
subsection {* Explicit expression for convex hull. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1193
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1194
lemma convex_hull_indexed:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1195
  fixes s :: "'a::real_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1196
  shows "convex hull s = {y. \<exists>k u x. (\<forall>i\<in>{1::nat .. k}. 0 \<le> u i \<and> x i \<in> s) \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1197
                            (setsum u {1..k} = 1) \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1198
                            (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} = y)}" (is "?xyz = ?hull")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1199
  apply(rule hull_unique) unfolding mem_def[of _ convex] apply(rule) defer
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1200
  apply(subst convex_def) apply(rule,rule,rule,rule,rule,rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1201
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1202
  fix x assume "x\<in>s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1203
  thus "x \<in> ?hull" unfolding mem_Collect_eq apply(rule_tac x=1 in exI, rule_tac x="\<lambda>x. 1" in exI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1204
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1205
  fix t assume as:"s \<subseteq> t" "convex t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1206
  show "?hull \<subseteq> t" apply(rule) unfolding mem_Collect_eq apply(erule exE | erule conjE)+ proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1207
    fix x k u y assume assm:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1208
    show "x\<in>t" unfolding assm(3)[THEN sym] apply(rule as(2)[unfolded convex, rule_format])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1209
      using assm(1,2) as(1) by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1210
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1211
  fix x y u v assume uv:"0\<le>u" "0\<le>v" "u+v=(1::real)" and xy:"x\<in>?hull" "y\<in>?hull"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1212
  from xy obtain k1 u1 x1 where x:"\<forall>i\<in>{1::nat..k1}. 0\<le>u1 i \<and> x1 i \<in> s" "setsum u1 {Suc 0..k1} = 1" "(\<Sum>i = Suc 0..k1. u1 i *\<^sub>R x1 i) = x" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1213
  from xy obtain k2 u2 x2 where y:"\<forall>i\<in>{1::nat..k2}. 0\<le>u2 i \<and> x2 i \<in> s" "setsum u2 {Suc 0..k2} = 1" "(\<Sum>i = Suc 0..k2. u2 i *\<^sub>R x2 i) = y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1214
  have *:"\<And>P (x1::'a) x2 s1 s2 i.(if P i then s1 else s2) *\<^sub>R (if P i then x1 else x2) = (if P i then s1 *\<^sub>R x1 else s2 *\<^sub>R x2)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1215
    "{1..k1 + k2} \<inter> {1..k1} = {1..k1}" "{1..k1 + k2} \<inter> - {1..k1} = (\<lambda>i. i + k1) ` {1..k2}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1216
    prefer 3 apply(rule,rule) unfolding image_iff apply(rule_tac x="x - k1" in bexI) by(auto simp add: not_le)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1217
  have inj:"inj_on (\<lambda>i. i + k1) {1..k2}" unfolding inj_on_def by auto  
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1218
  show "u *\<^sub>R x + v *\<^sub>R y \<in> ?hull" apply(rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1219
    apply(rule_tac x="k1 + k2" in exI, rule_tac x="\<lambda>i. if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1220
    apply(rule_tac x="\<lambda>i. if i \<in> {1..k1} then x1 i else x2 (i - k1)" in exI) apply(rule,rule) defer apply(rule)
35577
43b93e294522 Generalized setsum_cases
hoelzl
parents: 35542
diff changeset
  1221
    unfolding * and setsum_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and setsum_reindex[OF inj] and o_def Collect_mem_eq
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1222
    unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] setsum_right_distrib[THEN sym] proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1223
    fix i assume i:"i \<in> {1..k1+k2}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1224
    show "0 \<le> (if i \<in> {1..k1} then u * u1 i else v * u2 (i - k1)) \<and> (if i \<in> {1..k1} then x1 i else x2 (i - k1)) \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1225
    proof(cases "i\<in>{1..k1}")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1226
      case True thus ?thesis using mult_nonneg_nonneg[of u "u1 i"] and uv(1) x(1)[THEN bspec[where x=i]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1227
    next def j \<equiv> "i - k1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1228
      case False with i have "j \<in> {1..k2}" unfolding j_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1229
      thus ?thesis unfolding j_def[symmetric] using False
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1230
        using mult_nonneg_nonneg[of v "u2 j"] and uv(2) y(1)[THEN bspec[where x=j]] by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1231
  qed(auto simp add: not_le x(2,3) y(2,3) uv(3))
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1232
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1233
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1234
lemma convex_hull_finite:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1235
  fixes s :: "'a::real_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1236
  assumes "finite s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1237
  shows "convex hull s = {y. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1238
         setsum u s = 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y}" (is "?HULL = ?set")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1239
proof(rule hull_unique, auto simp add: mem_def[of _ convex] convex_def[of ?set])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1240
  fix x assume "x\<in>s" thus " \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>x\<in>s. u x *\<^sub>R x) = x" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1241
    apply(rule_tac x="\<lambda>y. if x=y then 1 else 0" in exI) apply auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1242
    unfolding setsum_delta'[OF assms] and setsum_delta''[OF assms] by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1243
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1244
  fix u v ::real assume uv:"0 \<le> u" "0 \<le> v" "u + v = 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1245
  fix ux assume ux:"\<forall>x\<in>s. 0 \<le> ux x" "setsum ux s = (1::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1246
  fix uy assume uy:"\<forall>x\<in>s. 0 \<le> uy x" "setsum uy s = (1::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1247
  { fix x assume "x\<in>s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1248
    hence "0 \<le> u * ux x + v * uy x" using ux(1)[THEN bspec[where x=x]] uy(1)[THEN bspec[where x=x]] and uv(1,2)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1249
      by (auto, metis add_nonneg_nonneg mult_nonneg_nonneg uv(1) uv(2))  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1250
  moreover have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1251
    unfolding setsum_addf and setsum_right_distrib[THEN sym] and ux(2) uy(2) using uv(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1252
  moreover have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1253
    unfolding scaleR_left_distrib and setsum_addf and scaleR_scaleR[THEN sym] and scaleR_right.setsum [symmetric] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1254
  ultimately show "\<exists>uc. (\<forall>x\<in>s. 0 \<le> uc x) \<and> setsum uc s = 1 \<and> (\<Sum>x\<in>s. uc x *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1255
    apply(rule_tac x="\<lambda>x. u * ux x + v * uy x" in exI) by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1256
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1257
  fix t assume t:"s \<subseteq> t" "convex t" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1258
  fix u assume u:"\<forall>x\<in>s. 0 \<le> u x" "setsum u s = (1::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1259
  thus "(\<Sum>x\<in>s. u x *\<^sub>R x) \<in> t" using t(2)[unfolded convex_explicit, THEN spec[where x=s], THEN spec[where x=u]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1260
    using assms and t(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1261
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1262
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1263
subsection {* Another formulation from Lars Schewe. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1264
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1265
lemma setsum_constant_scaleR:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1266
  fixes y :: "'a::real_vector"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1267
  shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1268
apply (cases "finite A")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1269
apply (induct set: finite)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1270
apply (simp_all add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1271
done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1272
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1273
lemma convex_hull_explicit:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1274
  fixes p :: "'a::real_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1275
  shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1276
             (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}" (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1277
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1278
  { fix x assume "x\<in>?lhs"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1279
    then obtain k u y where obt:"\<forall>i\<in>{1::nat..k}. 0 \<le> u i \<and> y i \<in> p" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R y i) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1280
      unfolding convex_hull_indexed by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1281
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1282
    have fin:"finite {1..k}" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1283
    have fin':"\<And>v. finite {i \<in> {1..k}. y i = v}" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1284
    { fix j assume "j\<in>{1..k}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1285
      hence "y j \<in> p" "0 \<le> setsum u {i. Suc 0 \<le> i \<and> i \<le> k \<and> y i = y j}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1286
        using obt(1)[THEN bspec[where x=j]] and obt(2) apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1287
        apply(rule setsum_nonneg) using obt(1) by auto } 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1288
    moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1289
    have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v}) = 1"  
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1290
      unfolding setsum_image_gen[OF fin, THEN sym] using obt(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1291
    moreover have "(\<Sum>v\<in>y ` {1..k}. setsum u {i \<in> {1..k}. y i = v} *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1292
      using setsum_image_gen[OF fin, of "\<lambda>i. u i *\<^sub>R y i" y, THEN sym]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1293
      unfolding scaleR_left.setsum using obt(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1294
    ultimately have "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1295
      apply(rule_tac x="y ` {1..k}" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1296
      apply(rule_tac x="\<lambda>v. setsum u {i\<in>{1..k}. y i = v}" in exI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1297
    hence "x\<in>?rhs" by auto  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1298
  moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1299
  { fix y assume "y\<in>?rhs"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1300
    then obtain s u where obt:"finite s" "s \<subseteq> p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1301
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1302
    obtain f where f:"inj_on f {1..card s}" "f ` {1..card s} = s" using ex_bij_betw_nat_finite_1[OF obt(1)] unfolding bij_betw_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1303
    
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1304
    { fix i::nat assume "i\<in>{1..card s}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1305
      hence "f i \<in> s"  apply(subst f(2)[THEN sym]) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1306
      hence "0 \<le> u (f i)" "f i \<in> p" using obt(2,3) by auto  }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1307
    moreover have *:"finite {1..card s}" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1308
    { fix y assume "y\<in>s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1309
      then obtain i where "i\<in>{1..card s}" "f i = y" using f using image_iff[of y f "{1..card s}"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1310
      hence "{x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = {i}" apply auto using f(1)[unfolded inj_on_def] apply(erule_tac x=x in ballE) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1311
      hence "card {x. Suc 0 \<le> x \<and> x \<le> card s \<and> f x = y} = 1" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1312
      hence "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x)) = u y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1313
            "(\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x) = u y *\<^sub>R y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1314
        by (auto simp add: setsum_constant_scaleR)   }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1315
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1316
    hence "(\<Sum>x = 1..card s. u (f x)) = 1" "(\<Sum>i = 1..card s. u (f i) *\<^sub>R f i) = y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1317
      unfolding setsum_image_gen[OF *(1), of "\<lambda>x. u (f x) *\<^sub>R f x" f] and setsum_image_gen[OF *(1), of "\<lambda>x. u (f x)" f] 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1318
      unfolding f using setsum_cong2[of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x) *\<^sub>R f x)" "\<lambda>v. u v *\<^sub>R v"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1319
      using setsum_cong2 [of s "\<lambda>y. (\<Sum>x\<in>{x \<in> {1..card s}. f x = y}. u (f x))" u] unfolding obt(4,5) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1320
    
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1321
    ultimately have "\<exists>k u x. (\<forall>i\<in>{1..k}. 0 \<le> u i \<and> x i \<in> p) \<and> setsum u {1..k} = 1 \<and> (\<Sum>i::nat = 1..k. u i *\<^sub>R x i) = y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1322
      apply(rule_tac x="card s" in exI) apply(rule_tac x="u \<circ> f" in exI) apply(rule_tac x=f in exI) by fastsimp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1323
    hence "y \<in> ?lhs" unfolding convex_hull_indexed by auto  }
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  1324
  ultimately show ?thesis unfolding set_eq_iff by blast
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1325
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1326
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1327
subsection {* A stepping theorem for that expansion. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1328
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1329
lemma convex_hull_finite_step:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1330
  fixes s :: "'a::real_vector set" assumes "finite s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1331
  shows "(\<exists>u. (\<forall>x\<in>insert a s. 0 \<le> u x) \<and> setsum u (insert a s) = w \<and> setsum (\<lambda>x. u x *\<^sub>R x) (insert a s) = y)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1332
     \<longleftrightarrow> (\<exists>v\<ge>0. \<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = w - v \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y - v *\<^sub>R a)" (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1333
proof(rule, case_tac[!] "a\<in>s")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1334
  assume "a\<in>s" hence *:"insert a s = s" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1335
  assume ?lhs thus ?rhs unfolding * apply(rule_tac x=0 in exI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1336
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1337
  assume ?lhs then obtain u where u:"\<forall>x\<in>insert a s. 0 \<le> u x" "setsum u (insert a s) = w" "(\<Sum>x\<in>insert a s. u x *\<^sub>R x) = y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1338
  assume "a\<notin>s" thus ?rhs apply(rule_tac x="u a" in exI) using u(1)[THEN bspec[where x=a]] apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1339
    apply(rule_tac x=u in exI) using u[unfolded setsum_clauses(2)[OF assms]] and `a\<notin>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1340
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1341
  assume "a\<in>s" hence *:"insert a s = s" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1342
  have fin:"finite (insert a s)" using assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1343
  assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1344
  show ?lhs apply(rule_tac x="\<lambda>x. (if a = x then v else 0) + u x" in exI) unfolding scaleR_left_distrib and setsum_addf and setsum_delta''[OF fin] and setsum_delta'[OF fin]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1345
    unfolding setsum_clauses(2)[OF assms] using uv and uv(2)[THEN bspec[where x=a]] and `a\<in>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1346
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1347
  assume ?rhs then obtain v u where uv:"v\<ge>0" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = w - v" "(\<Sum>x\<in>s. u x *\<^sub>R x) = y - v *\<^sub>R a" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1348
  moreover assume "a\<notin>s" moreover have "(\<Sum>x\<in>s. if a = x then v else u x) = setsum u s" "(\<Sum>x\<in>s. (if a = x then v else u x) *\<^sub>R x) = (\<Sum>x\<in>s. u x *\<^sub>R x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1349
    apply(rule_tac setsum_cong2) defer apply(rule_tac setsum_cong2) using `a\<notin>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1350
  ultimately show ?lhs apply(rule_tac x="\<lambda>x. if a = x then v else u x" in exI)  unfolding setsum_clauses(2)[OF assms] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1351
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1352
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1353
subsection {* Hence some special cases. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1354
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1355
lemma convex_hull_2:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1356
  "convex hull {a,b} = {u *\<^sub>R a + v *\<^sub>R b | u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v = 1}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1357
proof- have *:"\<And>u. (\<forall>x\<in>{a, b}. 0 \<le> u x) \<longleftrightarrow> 0 \<le> u a \<and> 0 \<le> u b" by auto have **:"finite {b}" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1358
show ?thesis apply(simp add: convex_hull_finite) unfolding convex_hull_finite_step[OF **, of a 1, unfolded * conj_assoc]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1359
  apply auto apply(rule_tac x=v in exI) apply(rule_tac x="1 - v" in exI) apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1360
  apply(rule_tac x=u in exI) apply simp apply(rule_tac x="\<lambda>x. v" in exI) by simp qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1361
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1362
lemma convex_hull_2_alt: "convex hull {a,b} = {a + u *\<^sub>R (b - a) | u.  0 \<le> u \<and> u \<le> 1}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1363
  unfolding convex_hull_2 unfolding Collect_def 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1364
proof(rule ext) have *:"\<And>x y ::real. x + y = 1 \<longleftrightarrow> x = 1 - y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1365
  fix x show "(\<exists>v u. x = v *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> v \<and> 0 \<le> u \<and> v + u = 1) = (\<exists>u. x = a + u *\<^sub>R (b - a) \<and> 0 \<le> u \<and> u \<le> 1)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1366
    unfolding * apply auto apply(rule_tac[!] x=u in exI) by (auto simp add: algebra_simps) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1367
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1368
lemma convex_hull_3:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1369
  "convex hull {a,b,c} = { u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c | u v w. 0 \<le> u \<and> 0 \<le> v \<and> 0 \<le> w \<and> u + v + w = 1}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1370
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1371
  have fin:"finite {a,b,c}" "finite {b,c}" "finite {c}" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1372
  have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1373
         "\<And>x y z ::_::euclidean_space. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by (auto simp add: field_simps)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1374
  show ?thesis unfolding convex_hull_finite[OF fin(1)] and Collect_def and convex_hull_finite_step[OF fin(2)] and *
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1375
    unfolding convex_hull_finite_step[OF fin(3)] apply(rule ext) apply simp apply auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1376
    apply(rule_tac x=va in exI) apply (rule_tac x="u c" in exI) apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1377
    apply(rule_tac x="1 - v - w" in exI) apply simp apply(rule_tac x=v in exI) apply simp apply(rule_tac x="\<lambda>x. w" in exI) by simp qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1378
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1379
lemma convex_hull_3_alt:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1380
  "convex hull {a,b,c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) | u v.  0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1381
proof- have *:"\<And>x y z ::real. x + y + z = 1 \<longleftrightarrow> x = 1 - y - z" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1382
  show ?thesis unfolding convex_hull_3 apply (auto simp add: *) apply(rule_tac x=v in exI) apply(rule_tac x=w in exI) apply (simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1383
    apply(rule_tac x=u in exI) apply(rule_tac x=v in exI) by (simp add: algebra_simps) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1384
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1385
subsection {* Relations among closure notions and corresponding hulls. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1386
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1387
text {* TODO: Generalize linear algebra concepts defined in @{text
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1388
Euclidean_Space.thy} so that we can generalize these lemmas. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1389
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1390
lemma affine_imp_convex: "affine s \<Longrightarrow> convex s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1391
  unfolding affine_def convex_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1392
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1393
lemma subspace_imp_convex:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1394
  fixes s :: "(_::euclidean_space) set" shows "subspace s \<Longrightarrow> convex s"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1395
  using subspace_imp_affine affine_imp_convex by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1396
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1397
lemma affine_hull_subset_span:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1398
  fixes s :: "(_::euclidean_space) set" shows "(affine hull s) \<subseteq> (span s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1399
by (metis hull_minimal mem_def span_inc subspace_imp_affine subspace_span)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1400
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1401
lemma convex_hull_subset_span:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1402
  fixes s :: "(_::euclidean_space) set" shows "(convex hull s) \<subseteq> (span s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1403
by (metis hull_minimal mem_def span_inc subspace_imp_convex subspace_span)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1404
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1405
lemma convex_hull_subset_affine_hull: "(convex hull s) \<subseteq> (affine hull s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1406
by (metis affine_affine_hull affine_imp_convex hull_minimal hull_subset mem_def)
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  1407
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1408
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1409
lemma affine_dependent_imp_dependent:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1410
  fixes s :: "(_::euclidean_space) set" shows "affine_dependent s \<Longrightarrow> dependent s"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1411
  unfolding affine_dependent_def dependent_def 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1412
  using affine_hull_subset_span by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1413
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1414
lemma dependent_imp_affine_dependent:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1415
  fixes s :: "(_::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1416
  assumes "dependent {x - a| x . x \<in> s}" "a \<notin> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1417
  shows "affine_dependent (insert a s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1418
proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1419
  from assms(1)[unfolded dependent_explicit] obtain S u v 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1420
    where obt:"finite S" "S \<subseteq> {x - a |x. x \<in> s}" "v\<in>S" "u v  \<noteq> 0" "(\<Sum>v\<in>S. u v *\<^sub>R v) = 0" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1421
  def t \<equiv> "(\<lambda>x. x + a) ` S"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1422
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1423
  have inj:"inj_on (\<lambda>x. x + a) S" unfolding inj_on_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1424
  have "0\<notin>S" using obt(2) assms(2) unfolding subset_eq by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1425
  have fin:"finite t" and  "t\<subseteq>s" unfolding t_def using obt(1,2) by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1426
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1427
  hence "finite (insert a t)" and "insert a t \<subseteq> insert a s" by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1428
  moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x)) = (\<Sum>x\<in>t. Q x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1429
    apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1430
  have "(\<Sum>x\<in>insert a t. if x = a then - (\<Sum>x\<in>t. u (x - a)) else u (x - a)) = 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1431
    unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` apply auto unfolding * by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1432
  moreover have "\<exists>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) \<noteq> 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1433
    apply(rule_tac x="v + a" in bexI) using obt(3,4) and `0\<notin>S` unfolding t_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1434
  moreover have *:"\<And>P Q. (\<Sum>x\<in>t. (if x = a then P x else Q x) *\<^sub>R x) = (\<Sum>x\<in>t. Q x *\<^sub>R x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1435
    apply(rule setsum_cong2) using `a\<notin>s` `t\<subseteq>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1436
  have "(\<Sum>x\<in>t. u (x - a)) *\<^sub>R a = (\<Sum>v\<in>t. u (v - a) *\<^sub>R v)" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1437
    unfolding scaleR_left.setsum unfolding t_def and setsum_reindex[OF inj] and o_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1438
    using obt(5) by (auto simp add: setsum_addf scaleR_right_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1439
  hence "(\<Sum>v\<in>insert a t. (if v = a then - (\<Sum>x\<in>t. u (x - a)) else u (v - a)) *\<^sub>R v) = 0"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1440
    unfolding setsum_clauses(2)[OF fin] using `a\<notin>s` `t\<subseteq>s` by (auto simp add: *)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1441
  ultimately show ?thesis unfolding affine_dependent_explicit
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1442
    apply(rule_tac x="insert a t" in exI) by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1443
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1444
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1445
lemma convex_cone:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1446
  "convex s \<and> cone s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. (x + y) \<in> s) \<and> (\<forall>x\<in>s. \<forall>c\<ge>0. (c *\<^sub>R x) \<in> s)" (is "?lhs = ?rhs")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1447
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1448
  { fix x y assume "x\<in>s" "y\<in>s" and ?lhs
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1449
    hence "2 *\<^sub>R x \<in>s" "2 *\<^sub>R y \<in> s" unfolding cone_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1450
    hence "x + y \<in> s" using `?lhs`[unfolded convex_def, THEN conjunct1]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1451
      apply(erule_tac x="2*\<^sub>R x" in ballE) apply(erule_tac x="2*\<^sub>R y" in ballE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1452
      apply(erule_tac x="1/2" in allE) apply simp apply(erule_tac x="1/2" in allE) by auto  }
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  1453
  thus ?thesis unfolding convex_def cone_def by blast
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1454
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1455
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1456
lemma affine_dependent_biggerset: fixes s::"('a::euclidean_space) set"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1457
  assumes "finite s" "card s \<ge> DIM('a) + 2"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1458
  shows "affine_dependent s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1459
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1460
  have "s\<noteq>{}" using assms by auto then obtain a where "a\<in>s" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1461
  have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1462
  have "card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1463
    apply(rule card_image) unfolding inj_on_def by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1464
  also have "\<dots> > DIM('a)" using assms(2)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1465
    unfolding card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1466
  finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1467
    apply(rule dependent_imp_affine_dependent)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1468
    apply(rule dependent_biggerset) by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1469
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1470
lemma affine_dependent_biggerset_general:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1471
  assumes "finite (s::('a::euclidean_space) set)" "card s \<ge> dim s + 2"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1472
  shows "affine_dependent s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1473
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1474
  from assms(2) have "s \<noteq> {}" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1475
  then obtain a where "a\<in>s" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1476
  have *:"{x - a |x. x \<in> s - {a}} = (\<lambda>x. x - a) ` (s - {a})" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1477
  have **:"card {x - a |x. x \<in> s - {a}} = card (s - {a})" unfolding * 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1478
    apply(rule card_image) unfolding inj_on_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1479
  have "dim {x - a |x. x \<in> s - {a}} \<le> dim s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1480
    apply(rule subset_le_dim) unfolding subset_eq
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1481
    using `a\<in>s` by (auto simp add:span_superset span_sub)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1482
  also have "\<dots> < dim s + 1" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1483
  also have "\<dots> \<le> card (s - {a})" using assms
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1484
    using card_Diff_singleton[OF assms(1) `a\<in>s`] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1485
  finally show ?thesis apply(subst insert_Diff[OF `a\<in>s`, THEN sym])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1486
    apply(rule dependent_imp_affine_dependent) apply(rule dependent_biggerset_general) unfolding ** by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1487
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1488
subsection {* Caratheodory's theorem. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1489
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1490
lemma convex_hull_caratheodory: fixes p::"('a::euclidean_space) set"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1491
  shows "convex hull p = {y. \<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and>
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1492
  (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> setsum (\<lambda>v. u v *\<^sub>R v) s = y}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  1493
  unfolding convex_hull_explicit set_eq_iff mem_Collect_eq
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1494
proof(rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1495
  fix y let ?P = "\<lambda>n. \<exists>s u. finite s \<and> card s = n \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1496
  assume "\<exists>s u. finite s \<and> s \<subseteq> p \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1497
  then obtain N where "?P N" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1498
  hence "\<exists>n\<le>N. (\<forall>k<n. \<not> ?P k) \<and> ?P n" apply(rule_tac ex_least_nat_le) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1499
  then obtain n where "?P n" and smallest:"\<forall>k<n. \<not> ?P k" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1500
  then obtain s u where obt:"finite s" "card s = n" "s\<subseteq>p" "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1"  "(\<Sum>v\<in>s. u v *\<^sub>R v) = y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1501
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1502
  have "card s \<le> DIM('a) + 1" proof(rule ccontr, simp only: not_le)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1503
    assume "DIM('a) + 1 < card s"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1504
    hence "affine_dependent s" using affine_dependent_biggerset[OF obt(1)] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1505
    then obtain w v where wv:"setsum w s = 0" "v\<in>s" "w v \<noteq> 0" "(\<Sum>v\<in>s. w v *\<^sub>R v) = 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1506
      using affine_dependent_explicit_finite[OF obt(1)] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1507
    def i \<equiv> "(\<lambda>v. (u v) / (- w v)) ` {v\<in>s. w v < 0}"  def t \<equiv> "Min i"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1508
    have "\<exists>x\<in>s. w x < 0" proof(rule ccontr, simp add: not_less)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1509
      assume as:"\<forall>x\<in>s. 0 \<le> w x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1510
      hence "setsum w (s - {v}) \<ge> 0" apply(rule_tac setsum_nonneg) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1511
      hence "setsum w s > 0" unfolding setsum_diff1'[OF obt(1) `v\<in>s`]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1512
        using as[THEN bspec[where x=v]] and `v\<in>s` using `w v \<noteq> 0` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1513
      thus False using wv(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1514
    qed hence "i\<noteq>{}" unfolding i_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1515
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1516
    hence "t \<ge> 0" using Min_ge_iff[of i 0 ] and obt(1) unfolding t_def i_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1517
      using obt(4)[unfolded le_less] apply auto unfolding divide_le_0_iff by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1518
    have t:"\<forall>v\<in>s. u v + t * w v \<ge> 0" proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1519
      fix v assume "v\<in>s" hence v:"0\<le>u v" using obt(4)[THEN bspec[where x=v]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1520
      show"0 \<le> u v + t * w v" proof(cases "w v < 0")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1521
        case False thus ?thesis apply(rule_tac add_nonneg_nonneg) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1522
          using v apply simp apply(rule mult_nonneg_nonneg) using `t\<ge>0` by auto next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1523
        case True hence "t \<le> u v / (- w v)" using `v\<in>s`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1524
          unfolding t_def i_def apply(rule_tac Min_le) using obt(1) by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1525
        thus ?thesis unfolding real_0_le_add_iff
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1526
          using pos_le_divide_eq[OF True[unfolded neg_0_less_iff_less[THEN sym]]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1527
      qed qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1528
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1529
    obtain a where "a\<in>s" and "t = (\<lambda>v. (u v) / (- w v)) a" and "w a < 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1530
      using Min_in[OF _ `i\<noteq>{}`] and obt(1) unfolding i_def t_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1531
    hence a:"a\<in>s" "u a + t * w a = 0" by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1532
    have *:"\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1533
      unfolding setsum_diff1'[OF obt(1) `a\<in>s`] by auto 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1534
    have "(\<Sum>v\<in>s. u v + t * w v) = 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1535
      unfolding setsum_addf wv(1) setsum_right_distrib[THEN sym] obt(5) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1536
    moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1537
      unfolding setsum_addf obt(6) scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] wv(4)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1538
      using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1539
    ultimately have "?P (n - 1)" apply(rule_tac x="(s - {a})" in exI)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1540
      apply(rule_tac x="\<lambda>v. u v + t * w v" in exI) using obt(1-3) and t and a
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1541
      by (auto simp add: * scaleR_left_distrib)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1542
    thus False using smallest[THEN spec[where x="n - 1"]] by auto qed
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1543
  thus "\<exists>s u. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1544
    \<and> (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s = 1 \<and> (\<Sum>v\<in>s. u v *\<^sub>R v) = y" using obt by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1545
qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1546
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1547
lemma caratheodory:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1548
 "convex hull p = {x::'a::euclidean_space. \<exists>s. finite s \<and> s \<subseteq> p \<and>
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1549
      card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  1550
  unfolding set_eq_iff apply(rule, rule) unfolding mem_Collect_eq proof-
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1551
  fix x assume "x \<in> convex hull p"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1552
  then obtain s u where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1553
     "\<forall>x\<in>s. 0 \<le> u x" "setsum u s = 1" "(\<Sum>v\<in>s. u v *\<^sub>R v) = x"unfolding convex_hull_caratheodory by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1554
  thus "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1555
    apply(rule_tac x=s in exI) using hull_subset[of s convex]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1556
  using convex_convex_hull[unfolded convex_explicit, of s, THEN spec[where x=s], THEN spec[where x=u]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1557
next
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1558
  fix x assume "\<exists>s. finite s \<and> s \<subseteq> p \<and> card s \<le> DIM('a) + 1 \<and> x \<in> convex hull s"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  1559
  then obtain s where "finite s" "s \<subseteq> p" "card s \<le> DIM('a) + 1" "x \<in> convex hull s" by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1560
  thus "x \<in> convex hull p" using hull_mono[OF `s\<subseteq>p`] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1561
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  1562
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1563
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1564
subsection {* Some Properties of Affine Dependent Sets *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1565
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1566
lemma affine_independent_empty: "~(affine_dependent {})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1567
  by (simp add: affine_dependent_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1568
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1569
lemma affine_independent_sing:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1570
fixes a :: "'n::euclidean_space" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1571
shows "~(affine_dependent {a})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1572
 by (simp add: affine_dependent_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1573
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1574
lemma affine_hull_translation:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1575
"affine hull ((%x. a + x) `  S) = (%x. a + x) ` (affine hull S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1576
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1577
have "affine ((%x. a + x) ` (affine hull S))" using affine_translation affine_affine_hull by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1578
moreover have "(%x. a + x) `  S <= (%x. a + x) ` (affine hull S)" using hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1579
ultimately have h1: "affine hull ((%x. a + x) `  S) <= (%x. a + x) ` (affine hull S)" by (metis hull_minimal mem_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1580
have "affine((%x. -a + x) ` (affine hull ((%x. a + x) `  S)))"  using affine_translation affine_affine_hull by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1581
moreover have "(%x. -a + x) ` (%x. a + x) `  S <= (%x. -a + x) ` (affine hull ((%x. a + x) `  S))" using hull_subset[of "(%x. a + x) `  S"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1582
moreover have "S=(%x. -a + x) ` (%x. a + x) `  S" using  translation_assoc[of "-a" a] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1583
ultimately have "(%x. -a + x) ` (affine hull ((%x. a + x) `  S)) >= (affine hull S)" by (metis hull_minimal mem_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1584
hence "affine hull ((%x. a + x) `  S) >= (%x. a + x) ` (affine hull S)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1585
from this show ?thesis using h1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1586
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1587
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1588
lemma affine_dependent_translation:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1589
  assumes "affine_dependent S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1590
  shows "affine_dependent ((%x. a + x) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1591
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1592
obtain x where x_def: "x : S & x : affine hull (S - {x})" using assms affine_dependent_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1593
have "op + a ` (S - {x}) = op + a ` S - {a + x}" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1594
hence "a+x : affine hull ((%x. a + x) ` S - {a+x})" using  affine_hull_translation[of a "S-{x}"] x_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1595
moreover have "a+x : (%x. a + x) ` S" using x_def by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1596
ultimately show ?thesis unfolding affine_dependent_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1597
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1598
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1599
lemma affine_dependent_translation_eq:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1600
  "affine_dependent S <-> affine_dependent ((%x. a + x) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1601
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1602
{ assume "affine_dependent ((%x. a + x) ` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1603
  hence "affine_dependent S" using affine_dependent_translation[of "((%x. a + x) ` S)" "-a"] translation_assoc[of "-a" a] by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1604
} from this show ?thesis using affine_dependent_translation by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1605
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1606
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1607
lemma affine_hull_0_dependent:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1608
  fixes S ::  "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1609
  assumes "0 : affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1610
  shows "dependent S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1611
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1612
obtain s u where s_u_def: "finite s & s ~= {} & s <= S & setsum u s = 1 & (SUM v:s. u v *\<^sub>R v) = 0" using assms affine_hull_explicit[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1613
hence "EX v:s. u v ~= 0" using setsum_not_0[of "u" "s"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1614
hence "finite s & s <= S & (EX v:s. u v ~= 0 & (SUM v:s. u v *\<^sub>R v) = 0)" using s_u_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1615
from this show ?thesis unfolding dependent_explicit[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1616
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1617
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1618
lemma affine_dependent_imp_dependent2:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1619
  fixes S :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1620
  assumes "affine_dependent (insert 0 S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1621
  shows "dependent S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1622
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1623
obtain x where x_def: "x:insert 0 S & x : affine hull (insert 0 S - {x})" using affine_dependent_def[of "(insert 0 S)"] assms by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1624
hence "x : span (insert 0 S - {x})" using affine_hull_subset_span by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1625
moreover have "span (insert 0 S - {x}) = span (S - {x})" using insert_Diff_if[of "0" S "{x}"] span_insert_0[of "S-{x}"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1626
ultimately have "x : span (S - {x})" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1627
hence "(x~=0) ==> dependent S" using x_def dependent_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1628
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1629
{ assume "x=0" hence "0 : affine hull S" using x_def hull_mono[of "S - {0}" S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1630
               hence "dependent S" using affine_hull_0_dependent by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1631
} ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1632
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1633
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1634
lemma affine_dependent_iff_dependent:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1635
  fixes S :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1636
  assumes "a ~: S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1637
  shows "affine_dependent (insert a S) <-> dependent ((%x. -a + x) ` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1638
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1639
have "(op + (- a) ` S)={x - a| x . x : S}" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1640
from this show ?thesis using affine_dependent_translation_eq[of "(insert a S)" "-a"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1641
      affine_dependent_imp_dependent2 assms 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1642
      dependent_imp_affine_dependent[of a S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1643
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1644
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1645
lemma affine_dependent_iff_dependent2:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1646
  fixes S :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1647
  assumes "a : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1648
  shows "affine_dependent S <-> dependent ((%x. -a + x) ` (S-{a}))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1649
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1650
have "insert a (S - {a})=S" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1651
from this show ?thesis using assms affine_dependent_iff_dependent[of a "S-{a}"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1652
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1653
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1654
lemma affine_hull_insert_span_gen:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1655
  fixes a :: "_::euclidean_space"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1656
  shows "affine hull (insert a s) = (%x. a+x) ` span ((%x. -a+x) ` s)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1657
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1658
have h1: "{x - a |x. x : s}=((%x. -a+x) ` s)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1659
{ assume "a ~: s" hence ?thesis using affine_hull_insert_span[of a s] h1 by auto}  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1660
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1661
{ assume a1: "a : s"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1662
  have "EX x. x:s & -a+x=0" apply (rule exI[of _ a]) using a1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1663
  hence "insert 0 ((%x. -a+x) ` (s - {a}))=(%x. -a+x) ` s" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1664
  hence "span ((%x. -a+x) ` (s - {a}))=span ((%x. -a+x) ` s)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1665
    using span_insert_0[of "op + (- a) ` (s - {a})"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1666
  moreover have "{x - a |x. x : (s - {a})}=((%x. -a+x) ` (s - {a}))" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1667
  moreover have "insert a (s - {a})=(insert a s)" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1668
  ultimately have ?thesis using assms affine_hull_insert_span[of "a" "s-{a}"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1669
} 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1670
ultimately show ?thesis by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1671
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1672
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1673
lemma affine_hull_span2:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1674
  fixes a :: "_::euclidean_space"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1675
  assumes "a : s"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1676
  shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` (s-{a}))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1677
  using affine_hull_insert_span_gen[of a "s - {a}", unfolded insert_Diff[OF assms]] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1678
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1679
lemma affine_hull_span_gen:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1680
  fixes a :: "_::euclidean_space"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1681
  assumes "a : affine hull s"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1682
  shows "affine hull s = (%x. a+x) ` span ((%x. -a+x) ` s)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1683
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1684
have "affine hull (insert a s) = affine hull s" using hull_redundant[of a affine s] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1685
from this show ?thesis using affine_hull_insert_span_gen[of a "s"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1686
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1687
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1688
lemma affine_hull_span_0:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1689
  assumes "(0 :: _::euclidean_space) : affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1690
  shows "affine hull S = span S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1691
using affine_hull_span_gen[of "0" S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1692
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1693
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1694
lemma extend_to_affine_basis:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1695
fixes S V :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1696
assumes "~(affine_dependent S)" "S <= V" "S~={}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1697
shows "? T. ~(affine_dependent T) & S<=T & T<=V & affine hull T = affine hull V"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1698
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1699
obtain a where a_def: "a : S" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1700
hence h0: "independent  ((%x. -a + x) ` (S-{a}))" using affine_dependent_iff_dependent2 assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1701
from this obtain B 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1702
   where B_def: "(%x. -a+x) ` (S - {a}) <= B & B <= (%x. -a+x) ` V & independent B & (%x. -a+x) ` V <= span B" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1703
   using maximal_independent_subset_extend[of "(%x. -a+x) ` (S-{a})" "(%x. -a + x) ` V"] assms by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1704
def T == "(%x. a+x) ` (insert 0 B)" hence "T=insert a ((%x. a+x) ` B)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1705
hence "affine hull T = (%x. a+x) ` span B" using affine_hull_insert_span_gen[of a "((%x. a+x) ` B)"] translation_assoc[of "-a" a B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1706
hence "V <= affine hull T" using B_def assms translation_inverse_subset[of a V "span B"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1707
moreover have "T<=V" using T_def B_def a_def assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1708
ultimately have "affine hull T = affine hull V" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1709
    by (metis Int_absorb1 Int_absorb2 Int_commute Int_lower2 assms hull_hull hull_mono) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1710
moreover have "S<=T" using T_def B_def translation_inverse_subset[of a "S-{a}" B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1711
moreover have "~(affine_dependent T)" using T_def affine_dependent_translation_eq[of "insert 0 B"] affine_dependent_imp_dependent2 B_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1712
ultimately show ?thesis using `T<=V` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1713
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1714
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1715
lemma affine_basis_exists: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1716
fixes V :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1717
shows "? B. B <= V & ~(affine_dependent B) & affine hull V = affine hull B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1718
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1719
{ assume empt: "V={}" have "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)" using empt affine_independent_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1720
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1721
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1722
{ assume nonempt: "V~={}" obtain x where "x:V" using nonempt by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1723
  hence "? B. B <= V & ~(affine_dependent B) & (affine hull V=affine hull B)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1724
  using affine_dependent_def[of "{x}"] extend_to_affine_basis[of "{x}:: ('n::euclidean_space) set" V] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1725
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1726
ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1727
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1728
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1729
subsection {* Affine Dimension of a Set *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1730
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1731
definition "aff_dim V = (SOME d :: int. ? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1732
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1733
lemma aff_dim_basis_exists:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1734
  fixes V :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1735
  shows "? B. (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1736
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1737
obtain B where B_def: "~(affine_dependent B) & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1738
from this show ?thesis unfolding aff_dim_def some_eq_ex[of "%d. ? (B :: ('n::euclidean_space) set). (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = d+1)"] apply auto apply (rule exI[of _ "int (card B)-(1 :: int)"]) apply (rule exI[of _ "B"]) by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1739
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1740
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1741
lemma affine_hull_nonempty: "(S ~= {}) <-> affine hull S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1742
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1743
fix S have "(S = {}) ==> affine hull S = {}"using affine_hull_empty by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1744
moreover have "affine hull S = {} ==> S = {}" unfolding hull_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1745
ultimately show "(S ~= {}) <-> affine hull S ~= {}" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1746
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1747
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1748
lemma aff_dim_parallel_subspace_aux:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1749
fixes B :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1750
assumes "~(affine_dependent B)" "a:B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1751
shows "finite B & ((card B) - 1 = dim (span ((%x. -a+x) ` (B-{a}))))" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1752
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1753
have "independent ((%x. -a + x) ` (B-{a}))" using affine_dependent_iff_dependent2 assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1754
hence fin: "dim (span ((%x. -a+x) ` (B-{a}))) = card ((%x. -a + x) ` (B-{a}))" "finite ((%x. -a + x) ` (B - {a}))"  using indep_card_eq_dim_span[of "(%x. -a+x) ` (B-{a})"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1755
{ assume emp: "(%x. -a + x) ` (B - {a}) = {}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1756
  have "B=insert a ((%x. a + x) ` (%x. -a + x) ` (B - {a}))" using translation_assoc[of "a" "-a" "(B - {a})"] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1757
  hence "B={a}" using emp by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1758
  hence ?thesis using assms fin by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1759
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1760
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1761
{ assume "(%x. -a + x) ` (B - {a}) ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1762
  hence "card ((%x. -a + x) ` (B - {a}))>0" using fin by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1763
  moreover have h1: "card ((%x. -a + x) ` (B-{a})) = card (B-{a})"  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1764
     apply (rule card_image) using translate_inj_on by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1765
  ultimately have "card (B-{a})>0" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1766
  hence "finite(B-{a})" using card_gt_0_iff[of "(B - {a})"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1767
  moreover hence "(card (B-{a})= (card B) - 1)" using card_Diff_singleton assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1768
  ultimately have ?thesis using fin h1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1769
} ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1770
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1771
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1772
lemma aff_dim_parallel_subspace:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1773
fixes V L :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1774
assumes "V ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1775
assumes "subspace L" "affine_parallel (affine hull V) L"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1776
shows "aff_dim V=int(dim L)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1777
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1778
obtain B where B_def: "affine hull B = affine hull V & ~ affine_dependent B & int (card B) = aff_dim V + 1" using aff_dim_basis_exists by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1779
hence "B~={}" using assms B_def  affine_hull_nonempty[of V] affine_hull_nonempty[of B] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1780
from this obtain a where a_def: "a : B" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1781
def Lb == "span ((%x. -a+x) ` (B-{a}))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1782
  moreover have "affine_parallel (affine hull B) Lb"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1783
     using Lb_def B_def assms affine_hull_span2[of a B] a_def  affine_parallel_commut[of "Lb" "(affine hull B)"] unfolding affine_parallel_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1784
  moreover have "subspace Lb" using Lb_def subspace_span by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1785
  moreover have "affine hull B ~= {}" using assms B_def affine_hull_nonempty[of V] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1786
  ultimately have "L=Lb" using assms affine_parallel_subspace[of "affine hull B"] affine_affine_hull[of B] B_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1787
  hence "dim L=dim Lb" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1788
  moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def B_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1789
(*  hence "card B=dim Lb+1" using `B~={}` card_gt_0_iff[of B] by auto *)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1790
  ultimately show ?thesis using B_def `B~={}` card_gt_0_iff[of B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1791
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1792
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1793
lemma aff_independent_finite:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1794
fixes B :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1795
assumes "~(affine_dependent B)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1796
shows "finite B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1797
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1798
{ assume "B~={}" from this obtain a where "a:B" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1799
  hence ?thesis using aff_dim_parallel_subspace_aux assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1800
} from this show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1801
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1802
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1803
lemma independent_finite:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1804
fixes B :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1805
assumes "independent B" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1806
shows "finite B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1807
using affine_dependent_imp_dependent[of B] aff_independent_finite[of B] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1808
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1809
lemma subspace_dim_equal:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1810
assumes "subspace (S :: ('n::euclidean_space) set)" "subspace T" "S <= T" "dim S >= dim T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1811
shows "S=T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1812
proof- 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1813
obtain B where B_def: "B <= S & independent B & S <= span B & (card B = dim S)" using basis_exists[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1814
hence "span B <= S" using span_mono[of B S] span_eq[of S] assms by metis 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1815
hence "span B = S" using B_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1816
have "dim S = dim T" using assms dim_subset[of S T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1817
hence "T <= span B" using card_eq_dim[of B T] B_def independent_finite assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1818
from this show ?thesis using assms `span B=S` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1819
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1820
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1821
lemma span_substd_basis:  assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1822
  shows "(span {basis i | i. i : d}) = {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1823
  (is "span ?A = ?B")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1824
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1825
have "?A <= ?B" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1826
moreover have s: "subspace ?B" using subspace_substandard[of "%i. i ~: d"] .
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1827
ultimately have "span ?A <= ?B" using span_mono[of "?A" "?B"] span_eq[of "?B"] by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1828
moreover have "card d <= dim (span ?A)" using independent_card_le_dim[of "?A" "span ?A"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1829
   independent_substdbasis[OF assms] card_substdbasis[OF assms] span_inc[of "?A"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1830
moreover hence "dim ?B <= dim (span ?A)" using dim_substandard[OF assms] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1831
ultimately show ?thesis using s subspace_dim_equal[of "span ?A" "?B"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1832
  subspace_span[of "?A"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1833
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1834
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1835
lemma basis_to_substdbasis_subspace_isomorphism:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1836
fixes B :: "('a::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1837
assumes "independent B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1838
shows "EX f d. card d = card B & linear f & f ` B = {basis i::'a |i. i : (d :: nat set)} & 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1839
       f ` span B = {x. ALL i<DIM('a). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B) \<and> d\<subseteq>{..<DIM('a)}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1840
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1841
  have B:"card B=dim B" using dim_unique[of B B "card B"] assms span_inc[of B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1842
  def d \<equiv> "{..<dim B}" have t:"card d = dim B" unfolding d_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1843
  have "dim B <= DIM('a)" using dim_subset_UNIV[of B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1844
  hence d:"d\<subseteq>{..<DIM('a)}" unfolding d_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1845
  let ?t = "{x::'a::euclidean_space. !i<DIM('a). i ~: d --> x$$i = 0}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1846
  have "EX f. linear f & f ` B = {basis i |i. i : d} &
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1847
    f ` span B = ?t & inj_on f (span B)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1848
    apply (rule basis_to_basis_subspace_isomorphism[of "span B" ?t B "{basis i |i. i : d}"])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1849
    apply(rule subspace_span) apply(rule subspace_substandard) defer
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1850
    apply(rule span_inc) apply(rule assms) defer unfolding dim_span[of B] apply(rule B)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1851
    unfolding span_substd_basis[OF d,THEN sym] card_substdbasis[OF d] apply(rule span_inc)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1852
    apply(rule independent_substdbasis[OF d]) apply(rule,assumption)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1853
    unfolding t[THEN sym] span_substd_basis[OF d] dim_substandard[OF d] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1854
  from this t `card B=dim B` show ?thesis using d by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1855
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1856
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1857
lemma aff_dim_empty:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1858
fixes S :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1859
shows "S = {} <-> aff_dim S = -1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1860
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1861
obtain B where "affine hull B = affine hull S & ~ affine_dependent B & int (card B) = aff_dim S + 1" using aff_dim_basis_exists by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1862
moreover hence "S={} <-> B={}" using affine_hull_nonempty[of B] affine_hull_nonempty[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1863
ultimately show ?thesis using aff_independent_finite[of B] card_gt_0_iff[of B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1864
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1865
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1866
lemma aff_dim_affine_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1867
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1868
shows "aff_dim (affine hull S)=aff_dim S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1869
unfolding aff_dim_def using hull_hull[of _ S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1870
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1871
lemma aff_dim_affine_hull2:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1872
fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1873
assumes "affine hull S=affine hull T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1874
shows "aff_dim S=aff_dim T" unfolding aff_dim_def using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1875
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1876
lemma aff_dim_unique: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1877
fixes B V :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1878
assumes "(affine hull B=affine hull V) & ~(affine_dependent B)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1879
shows "of_nat(card B) = aff_dim V+1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1880
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1881
{ assume "B={}" hence "V={}" using affine_hull_nonempty[of V] affine_hull_nonempty[of B] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1882
  hence "aff_dim V = (-1::int)"  using aff_dim_empty by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1883
  hence ?thesis using `B={}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1884
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1885
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1886
{ assume "B~={}" from this obtain a where a_def: "a:B" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1887
  def Lb == "span ((%x. -a+x) ` (B-{a}))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1888
  have "affine_parallel (affine hull B) Lb"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1889
     using Lb_def affine_hull_span2[of a B] a_def  affine_parallel_commut[of "Lb" "(affine hull B)"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1890
     unfolding affine_parallel_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1891
  moreover have "subspace Lb" using Lb_def subspace_span by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1892
  ultimately have "aff_dim B=int(dim Lb)" using aff_dim_parallel_subspace[of B Lb] `B~={}` by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1893
  moreover have "(card B) - 1 = dim Lb" "finite B" using Lb_def aff_dim_parallel_subspace_aux a_def assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1894
  ultimately have "(of_nat(card B) = aff_dim B+1)" using  `B~={}` card_gt_0_iff[of B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1895
  hence ?thesis using aff_dim_affine_hull2 assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1896
} ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1897
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1898
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1899
lemma aff_dim_affine_independent: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1900
fixes B :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1901
assumes "~(affine_dependent B)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1902
shows "of_nat(card B) = aff_dim B+1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1903
  using aff_dim_unique[of B B] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1904
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1905
lemma aff_dim_sing: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1906
fixes a :: "'n::euclidean_space" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1907
shows "aff_dim {a}=0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1908
  using aff_dim_affine_independent[of "{a}"] affine_independent_sing by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1909
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1910
lemma aff_dim_inner_basis_exists:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1911
  fixes V :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1912
  shows "? B. B<=V & (affine hull B=affine hull V) & ~(affine_dependent B) & (of_nat(card B) = aff_dim V+1)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1913
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1914
obtain B where B_def: "~(affine_dependent B) & B<=V & (affine hull B=affine hull V)" using affine_basis_exists[of V] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1915
moreover hence "of_nat(card B) = aff_dim V+1" using aff_dim_unique by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1916
ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1917
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1918
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1919
lemma aff_dim_le_card:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1920
fixes V :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1921
assumes "finite V"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1922
shows "aff_dim V <= of_nat(card V) - 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1923
 proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1924
 obtain B where B_def: "B<=V & (of_nat(card B) = aff_dim V+1)" using aff_dim_inner_basis_exists[of V] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1925
 moreover hence "card B <= card V" using assms card_mono by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1926
 ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1927
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1928
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1929
lemma aff_dim_parallel_eq:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1930
fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1931
assumes "affine_parallel (affine hull S) (affine hull T)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1932
shows "aff_dim S=aff_dim T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1933
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1934
{ assume "T~={}" "S~={}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1935
  from this obtain L where L_def: "subspace L & affine_parallel (affine hull T) L" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1936
       using affine_parallel_subspace[of "affine hull T"] affine_affine_hull[of T] affine_hull_nonempty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1937
  hence "aff_dim T = int(dim L)" using aff_dim_parallel_subspace `T~={}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1938
  moreover have "subspace L & affine_parallel (affine hull S) L" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1939
     using L_def affine_parallel_assoc[of "affine hull S" "affine hull T" L] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1940
  moreover hence "aff_dim S = int(dim L)" using aff_dim_parallel_subspace `S~={}` by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1941
  ultimately have ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1942
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1943
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1944
{ assume "S={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1945
  hence ?thesis using aff_dim_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1946
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1947
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1948
{ assume "T={}" hence "S={} & T={}" using assms affine_hull_nonempty unfolding affine_parallel_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1949
  hence ?thesis using aff_dim_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1950
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1951
ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1952
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1953
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1954
lemma aff_dim_translation_eq:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1955
fixes a :: "'n::euclidean_space"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1956
shows "aff_dim ((%x. a + x) ` S)=aff_dim S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1957
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1958
have "affine_parallel (affine hull S) (affine hull ((%x. a + x) ` S))" unfolding affine_parallel_def apply (rule exI[of _ "a"]) using affine_hull_translation[of a S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1959
from this show ?thesis using  aff_dim_parallel_eq[of S "(%x. a + x) ` S"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1960
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1961
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1962
lemma aff_dim_affine:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1963
fixes S L :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1964
assumes "S ~= {}" "affine S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1965
assumes "subspace L" "affine_parallel S L"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1966
shows "aff_dim S=int(dim L)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1967
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1968
have 1: "(affine hull S) = S" using assms affine_hull_eq[of S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1969
hence "affine_parallel (affine hull S) L" using assms by (simp add:1)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1970
from this show ?thesis using assms aff_dim_parallel_subspace[of S L] by blast 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1971
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1972
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1973
lemma dim_affine_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1974
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1975
shows "dim (affine hull S)=dim S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1976
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1977
have "dim (affine hull S)>=dim S" using dim_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1978
moreover have "dim(span S) >= dim (affine hull S)" using dim_subset affine_hull_subset_span by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1979
moreover have "dim(span S)=dim S" using dim_span by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1980
ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1981
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1982
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1983
lemma aff_dim_subspace:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1984
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1985
assumes "S ~= {}" "subspace S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1986
shows "aff_dim S=int(dim S)" using aff_dim_affine[of S S] assms subspace_imp_affine[of S] affine_parallel_reflex[of S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1987
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1988
lemma aff_dim_zero:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1989
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1990
assumes "0 : affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1991
shows "aff_dim S=int(dim S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1992
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1993
have "subspace(affine hull S)" using subspace_affine[of "affine hull S"] affine_affine_hull assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1994
hence "aff_dim (affine hull S) =int(dim (affine hull S))" using assms aff_dim_subspace[of "affine hull S"] by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1995
from this show ?thesis using aff_dim_affine_hull[of S] dim_affine_hull[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1996
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1997
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1998
lemma aff_dim_univ: "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  1999
  using aff_dim_subspace[of "(UNIV :: ('n::euclidean_space) set)"]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2000
    dim_UNIV[where 'a="'n::euclidean_space"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2001
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2002
lemma aff_dim_geq:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2003
  fixes V :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2004
  shows "aff_dim V >= -1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2005
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2006
obtain B where B_def: "affine hull B = affine hull V & ~ affine_dependent B & int (card B) = aff_dim V + 1" using aff_dim_basis_exists by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2007
from this show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2008
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2009
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2010
lemma independent_card_le_aff_dim: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2011
  assumes "(B::('n::euclidean_space) set) <= V"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2012
  assumes "~(affine_dependent B)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2013
  shows "int(card B) <= aff_dim V+1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2014
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2015
{ assume "B~={}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2016
  from this obtain T where T_def: "~(affine_dependent T) & B<=T & T<=V & affine hull T = affine hull V" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2017
  using assms extend_to_affine_basis[of B V] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2018
  hence "of_nat(card T) = aff_dim V+1" using aff_dim_unique by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2019
  hence ?thesis using T_def card_mono[of T B] aff_independent_finite[of T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2020
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2021
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2022
{ assume "B={}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2023
  moreover have "-1<= aff_dim V" using aff_dim_geq by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2024
  ultimately have ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2025
}  ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2026
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2027
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2028
lemma aff_dim_subset:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2029
  fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2030
  assumes "S <= T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2031
  shows "aff_dim S <= aff_dim T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2032
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2033
obtain B where B_def: "~(affine_dependent B) & B<=S & (affine hull B=affine hull S) & of_nat(card B) = aff_dim S+1" using aff_dim_inner_basis_exists[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2034
moreover hence "int (card B) <= aff_dim T + 1" using assms independent_card_le_aff_dim[of B T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2035
ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2036
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2037
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2038
lemma aff_dim_subset_univ:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2039
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2040
shows "aff_dim S <= int(DIM('n))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2041
proof - 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2042
  have "aff_dim (UNIV :: ('n::euclidean_space) set) = int(DIM('n))" using aff_dim_univ by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2043
  from this show "aff_dim (S:: ('n::euclidean_space) set) <= int(DIM('n))" using assms aff_dim_subset[of S "(UNIV :: ('n::euclidean_space) set)"] subset_UNIV by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2044
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2045
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2046
lemma affine_dim_equal:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2047
assumes "affine (S :: ('n::euclidean_space) set)" "affine T" "S ~= {}" "S <= T" "aff_dim S = aff_dim T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2048
shows "S=T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2049
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2050
obtain a where "a : S" using assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2051
hence "a : T" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2052
def LS == "{y. ? x : S. (-a)+x=y}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2053
hence ls: "subspace LS & affine_parallel S LS" using assms parallel_subspace_explicit[of S a LS] `a : S` by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2054
hence h1: "int(dim LS) = aff_dim S" using assms aff_dim_affine[of S LS] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2055
have "T ~= {}" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2056
def LT == "{y. ? x : T. (-a)+x=y}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2057
hence lt: "subspace LT & affine_parallel T LT" using assms parallel_subspace_explicit[of T a LT] `a : T` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2058
hence "int(dim LT) = aff_dim T" using assms aff_dim_affine[of T LT] `T ~= {}` by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2059
hence "dim LS = dim LT" using h1 assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2060
moreover have "LS <= LT" using LS_def LT_def assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2061
ultimately have "LS=LT" using subspace_dim_equal[of LS LT] ls lt by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2062
moreover have "S = {x. ? y : LS. a+y=x}" using LS_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2063
moreover have "T = {x. ? y : LT. a+y=x}" using LT_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2064
ultimately show ?thesis by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2065
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2066
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2067
lemma affine_hull_univ:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2068
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2069
assumes "aff_dim S = int(DIM('n))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2070
shows "affine hull S = (UNIV :: ('n::euclidean_space) set)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2071
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2072
have "S ~= {}" using assms aff_dim_empty[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2073
have h0: "S <= affine hull S" using hull_subset[of S _] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2074
have h1: "aff_dim (UNIV :: ('n::euclidean_space) set) = aff_dim S" using aff_dim_univ assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2075
hence h2: "aff_dim (affine hull S) <= aff_dim (UNIV :: ('n::euclidean_space) set)" using aff_dim_subset_univ[of "affine hull S"] assms h0 by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2076
have h3: "aff_dim S <= aff_dim (affine hull S)" using h0 aff_dim_subset[of S "affine hull S"] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2077
hence h4: "aff_dim (affine hull S) = aff_dim (UNIV :: ('n::euclidean_space) set)" using h0 h1 h2 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2078
from this show ?thesis using affine_dim_equal[of "affine hull S" "(UNIV :: ('n::euclidean_space) set)"] affine_affine_hull[of S] affine_UNIV assms h4 h0 `S ~= {}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2079
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2080
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2081
lemma aff_dim_convex_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2082
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2083
shows "aff_dim (convex hull S)=aff_dim S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2084
  using aff_dim_affine_hull[of S] convex_hull_subset_affine_hull[of S] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2085
  hull_subset[of S "convex"] aff_dim_subset[of S "convex hull S"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2086
  aff_dim_subset[of "convex hull S" "affine hull S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2087
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2088
lemma aff_dim_cball:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2089
fixes a :: "'n::euclidean_space" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2090
assumes "0<e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2091
shows "aff_dim (cball a e) = int (DIM('n))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2092
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2093
have "(%x. a + x) ` (cball 0 e)<=cball a e" unfolding cball_def dist_norm by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2094
hence "aff_dim (cball (0 :: 'n::euclidean_space) e) <= aff_dim (cball a e)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2095
  using aff_dim_translation_eq[of a "cball 0 e"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2096
        aff_dim_subset[of "op + a ` cball 0 e" "cball a e"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2097
moreover have "aff_dim (cball (0 :: 'n::euclidean_space) e) = int (DIM('n))" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2098
   using hull_inc[of "(0 :: 'n::euclidean_space)" "cball 0 e"] centre_in_cball[of "(0 :: 'n::euclidean_space)"] assms 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2099
   by (simp add: dim_cball[of e] aff_dim_zero[of "cball 0 e"])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2100
ultimately show ?thesis using aff_dim_subset_univ[of "cball a e"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2101
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2102
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2103
lemma aff_dim_open:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2104
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2105
assumes "open S" "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2106
shows "aff_dim S = int (DIM('n))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2107
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2108
obtain x where "x:S" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2109
from this obtain e where e_def: "e>0 & cball x e <= S" using open_contains_cball[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2110
from this have "aff_dim (cball x e) <= aff_dim S" using aff_dim_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2111
from this show ?thesis using aff_dim_cball[of e x] aff_dim_subset_univ[of S] e_def by auto     
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2112
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2113
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2114
lemma low_dim_interior:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2115
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2116
assumes "~(aff_dim S = int (DIM('n)))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2117
shows "interior S = {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2118
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2119
have "aff_dim(interior S) <= aff_dim S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2120
   using interior_subset aff_dim_subset[of "interior S" S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2121
from this show ?thesis using aff_dim_open[of "interior S"] aff_dim_subset_univ[of S] assms by auto   
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2122
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2123
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2124
subsection{* Relative Interior of a Set *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2125
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2126
definition "rel_interior S = {x. ? T. openin (subtopology euclidean (affine hull S)) T & x : T & T <= S}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2127
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2128
lemma rel_interior: "rel_interior S = {x : S. ? T. open T & x : T & (T Int (affine hull S)) <= S}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2129
  unfolding rel_interior_def[of S] openin_open[of "affine hull S"] apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2130
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2131
fix x T assume a: "x:S" "open T" "x : T" "(T Int (affine hull S)) <= S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2132
hence h1: "x : T Int affine hull S" using hull_inc by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2133
show "EX Tb. (EX Ta. open Ta & Tb = affine hull S Int Ta) & x : Tb & Tb <= S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2134
apply (rule_tac x="T Int (affine hull S)" in exI)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2135
using a h1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2136
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2137
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2138
lemma mem_rel_interior: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2139
     "x : rel_interior S <-> (? T. open T & x : (T Int S) & (T Int (affine hull S)) <= S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2140
     by (auto simp add: rel_interior)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2141
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2142
lemma mem_rel_interior_ball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((ball x e) Int (affine hull S)) <= S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2143
  apply (simp add: rel_interior, safe)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2144
  apply (force simp add: open_contains_ball)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2145
  apply (rule_tac x="ball x e" in exI)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2146
  apply (simp add: open_ball centre_in_ball)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2147
  done
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2148
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2149
lemma rel_interior_ball: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2150
      "rel_interior S = {x : S. ? e. e>0 & ((ball x e) Int (affine hull S)) <= S}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2151
      using mem_rel_interior_ball [of _ S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2152
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2153
lemma mem_rel_interior_cball: "x : rel_interior S <-> x : S & (? e. 0 < e & ((cball x e) Int (affine hull S)) <= S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2154
  apply (simp add: rel_interior, safe) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2155
  apply (force simp add: open_contains_cball)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2156
  apply (rule_tac x="ball x e" in exI)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2157
  apply (simp add: open_ball centre_in_ball subset_trans [OF ball_subset_cball])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2158
  apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2159
  done
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2160
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2161
lemma rel_interior_cball: "rel_interior S = {x : S. ? e. e>0 & ((cball x e) Int (affine hull S)) <= S}"       using mem_rel_interior_cball [of _ S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2162
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2163
lemma rel_interior_empty: "rel_interior {} = {}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2164
   by (auto simp add: rel_interior_def) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2165
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2166
lemma affine_hull_sing: "affine hull {a :: 'n::euclidean_space} = {a}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2167
by (metis affine_hull_eq affine_sing)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2168
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2169
lemma rel_interior_sing: "rel_interior {a :: 'n::euclidean_space} = {a}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2170
   unfolding rel_interior_ball affine_hull_sing apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2171
   apply(rule_tac x="1 :: real" in exI) apply simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2172
   done
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2173
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2174
lemma subset_rel_interior:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2175
fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2176
assumes "S<=T" "affine hull S=affine hull T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2177
shows "rel_interior S <= rel_interior T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2178
  using assms by (auto simp add: rel_interior_def)  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2179
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2180
lemma rel_interior_subset: "rel_interior S <= S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2181
   by (auto simp add: rel_interior_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2182
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2183
lemma rel_interior_subset_closure: "rel_interior S <= closure S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2184
   using rel_interior_subset by (auto simp add: closure_def) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2185
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2186
lemma interior_subset_rel_interior: "interior S <= rel_interior S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2187
   by (auto simp add: rel_interior interior_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2188
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2189
lemma interior_rel_interior:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2190
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2191
assumes "aff_dim S = int(DIM('n))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2192
shows "rel_interior S = interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2193
proof -
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2194
have "affine hull S = UNIV" using assms affine_hull_univ[of S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2195
from this show ?thesis unfolding rel_interior interior_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2196
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2197
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2198
lemma rel_interior_open:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2199
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2200
assumes "open S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2201
shows "rel_interior S = S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2202
by (metis assms interior_eq interior_subset_rel_interior rel_interior_subset set_eq_subset)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2203
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2204
lemma interior_rel_interior_gen:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2205
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2206
shows "interior S = (if aff_dim S = int(DIM('n)) then rel_interior S else {})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2207
by (metis interior_rel_interior low_dim_interior)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2208
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2209
lemma rel_interior_univ: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2210
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2211
shows "rel_interior (affine hull S) = affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2212
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2213
have h1: "rel_interior (affine hull S) <= affine hull S" using rel_interior_subset by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2214
{ fix x assume x_def: "x : affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2215
  obtain e :: real where "e=1" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2216
  hence "e>0 & ball x e Int affine hull (affine hull S) <= affine hull S" using hull_hull[of _ S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2217
  hence "x : rel_interior (affine hull S)" using x_def rel_interior_ball[of "affine hull S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2218
} from this show ?thesis using h1 by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2219
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2220
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2221
lemma rel_interior_univ2: "rel_interior (UNIV :: ('n::euclidean_space) set) = UNIV"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2222
by (metis open_UNIV rel_interior_open)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2223
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2224
lemma rel_interior_convex_shrink:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2225
  fixes S :: "('a::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2226
  assumes "convex S" "c : rel_interior S" "x : S" "0 < e" "e <= 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2227
  shows "x - e *\<^sub>R (x - c) : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2228
proof- 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2229
(* Proof is a modified copy of the proof of similar lemma mem_interior_convex_shrink 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2230
*)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2231
obtain d where "d>0" and d:"ball c d Int affine hull S <= S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2232
  using assms(2) unfolding  mem_rel_interior_ball by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2233
{   fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d & y : affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2234
    have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2235
    have "x : affine hull S" using assms hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2236
    moreover have "1 / e + - ((1 - e) / e) = 1" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2237
       using `e>0` mult_left.diff[of "1" "(1-e)" "1/e"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2238
    ultimately have **: "(1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x : affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2239
        using as affine_affine_hull[of S] mem_affine[of "affine hull S" y x "(1 / e)" "-((1 - e) / e)"] by (simp add: algebra_simps)     
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2240
    have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2241
      unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2242
      by(auto simp add:euclidean_eq[where 'a='a] field_simps) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2243
    also have "... = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2244
    also have "... < d" using as[unfolded dist_norm] and `e>0`
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2245
      by(auto simp add:pos_divide_less_eq[OF `e>0`] real_mult_commute)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2246
    finally have "y : S" apply(subst *) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2247
apply(rule assms(1)[unfolded convex_alt,rule_format])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2248
      apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) ** by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2249
} hence "ball (x - e *\<^sub>R (x - c)) (e*d) Int affine hull S <= S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2250
moreover have "0 < e*d" using `0<e` `0<d` using real_mult_order by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2251
moreover have "c : S" using assms rel_interior_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2252
moreover hence "x - e *\<^sub>R (x - c) : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2253
   using mem_convex[of S x c e] apply (simp add: algebra_simps) using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2254
ultimately show ?thesis 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2255
  using mem_rel_interior_ball[of "x - e *\<^sub>R (x - c)" S] `e>0` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2256
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2257
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2258
lemma interior_real_semiline:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2259
fixes a :: real
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2260
shows "interior {a..} = {a<..}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2261
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2262
{ fix y assume "a<y" hence "y : interior {a..}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2263
  apply (simp add: mem_interior) apply (rule_tac x="(y-a)" in exI) apply (auto simp add: dist_norm) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2264
  done }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2265
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2266
{ fix y assume "y : interior {a..}" (*hence "a<=y" using interior_subset by auto*)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2267
  from this obtain e where e_def: "e>0 & cball y e \<subseteq> {a..}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2268
     using mem_interior_cball[of y "{a..}"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2269
  moreover hence "y-e : cball y e" by (auto simp add: cball_def dist_norm) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2270
  ultimately have "a<=y-e" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2271
  hence "a<y" using e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2272
} ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2273
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2274
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2275
lemma rel_interior_real_interval:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2276
  fixes a b :: real assumes "a < b" shows "rel_interior {a..b} = {a<..<b}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2277
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2278
  have "{a<..<b} \<noteq> {}" using assms unfolding set_eq_iff by (auto intro!: exI[of _ "(a + b) / 2"])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2279
  then show ?thesis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2280
    using interior_rel_interior_gen[of "{a..b}", symmetric]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2281
    by (simp split: split_if_asm add: interior_closed_interval)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2282
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2283
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2284
lemma rel_interior_real_semiline:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2285
  fixes a :: real shows "rel_interior {a..} = {a<..}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2286
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2287
  have *: "{a<..} \<noteq> {}" unfolding set_eq_iff by (auto intro!: exI[of _ "a + 1"])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2288
  then show ?thesis using interior_real_semiline
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2289
     interior_rel_interior_gen[of "{a..}"]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2290
     by (auto split: split_if_asm)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2291
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2292
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2293
subsection "Relative open"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2294
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2295
definition "rel_open S <-> (rel_interior S) = S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2296
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2297
lemma rel_open: "rel_open S <-> openin (subtopology euclidean (affine hull S)) S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2298
 unfolding rel_open_def rel_interior_def apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2299
 using openin_subopen[of "subtopology euclidean (affine hull S)" S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2300
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2301
lemma opein_rel_interior: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2302
  "openin (subtopology euclidean (affine hull S)) (rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2303
  apply (simp add: rel_interior_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2304
  apply (subst openin_subopen) by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2305
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2306
lemma affine_rel_open: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2307
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2308
  assumes "affine S" shows "rel_open S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2309
  unfolding rel_open_def using assms rel_interior_univ[of S] affine_hull_eq[of S] by metis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2310
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2311
lemma affine_closed: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2312
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2313
  assumes "affine S" shows "closed S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2314
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2315
{ assume "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2316
  from this obtain L where L_def: "subspace L & affine_parallel S L"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2317
     using assms affine_parallel_subspace[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2318
  from this obtain "a" where a_def: "S=(op + a ` L)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2319
     using affine_parallel_def[of L S] affine_parallel_commut by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2320
  have "closed L" using L_def closed_subspace by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2321
  hence "closed S" using closed_translation a_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2322
} from this show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2323
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2324
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2325
lemma closure_affine_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2326
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2327
  shows "closure S <= affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2328
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2329
have "closure S <= closure (affine hull S)" using subset_closure by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2330
moreover have "closure (affine hull S) = affine hull S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2331
   using affine_affine_hull affine_closed[of "affine hull S"] closure_eq by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2332
ultimately show ?thesis by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2333
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2334
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2335
lemma closure_same_affine_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2336
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2337
  shows "affine hull (closure S) = affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2338
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2339
have "affine hull (closure S) <= affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2340
   using hull_mono[of "closure S" "affine hull S" "affine"] closure_affine_hull[of S] hull_hull[of "affine" S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2341
moreover have "affine hull (closure S) >= affine hull S"  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2342
   using hull_mono[of "S" "closure S" "affine"] closure_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2343
ultimately show ?thesis by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2344
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2345
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2346
lemma closure_aff_dim: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2347
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2348
  shows "aff_dim (closure S) = aff_dim S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2349
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2350
have "aff_dim S <= aff_dim (closure S)" using aff_dim_subset closure_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2351
moreover have "aff_dim (closure S) <= aff_dim (affine hull S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2352
  using aff_dim_subset closure_affine_hull by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2353
moreover have "aff_dim (affine hull S) = aff_dim S" using aff_dim_affine_hull by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2354
ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2355
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2356
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2357
lemma rel_interior_closure_convex_shrink:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2358
  fixes S :: "(_::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2359
  assumes "convex S" "c : rel_interior S" "x : closure S" "0 < e" "e <= 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2360
  shows "x - e *\<^sub>R (x - c) : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2361
proof- 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2362
(* Proof is a modified copy of the proof of similar lemma mem_interior_closure_convex_shrink
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2363
*)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2364
obtain d where "d>0" and d:"ball c d Int affine hull S <= S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2365
  using assms(2) unfolding mem_rel_interior_ball by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2366
have "EX y : S. norm (y - x) * (1 - e) < e * d" proof(cases "x : S")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2367
    case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2368
    case False hence x:"x islimpt S" using assms(3)[unfolded closure_def] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2369
    show ?thesis proof(cases "e=1")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2370
      case True obtain y where "y : S" "y ~= x" "dist y x < 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2371
        using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2372
      thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2373
      case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2374
        using `e<=1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2375
      then obtain y where "y : S" "y ~= x" "dist y x < e * d / (1 - e)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2376
        using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2377
      thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2378
  then obtain y where "y : S" and y:"norm (y - x) * (1 - e) < e * d" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2379
  def z == "c + ((1 - e) / e) *\<^sub>R (x - y)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2380
  have *:"x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)" unfolding z_def using `e>0` by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2381
  have zball: "z\<in>ball c d"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2382
    using mem_ball z_def dist_norm[of c] using y and assms(4,5) by (auto simp add:field_simps norm_minus_commute)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2383
  have "x : affine hull S" using closure_affine_hull assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2384
  moreover have "y : affine hull S" using `y : S` hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2385
  moreover have "c : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2386
  ultimately have "z : affine hull S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2387
    using z_def affine_affine_hull[of S] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2388
          mem_affine_3_minus [of "affine hull S" c x y "(1 - e) / e"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2389
          assms by (auto simp add: field_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2390
  hence "z : S" using d zball by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2391
  obtain d1 where "d1>0" and d1:"ball z d1 <= ball c d"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2392
    using zball open_ball[of c d] openE[of "ball c d" z] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2393
  hence "(ball z d1) Int (affine hull S) <= (ball c d) Int (affine hull S)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2394
  hence "(ball z d1) Int (affine hull S) <= S" using d by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2395
  hence "z : rel_interior S" using mem_rel_interior_ball using `d1>0` `z : S` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2396
  hence "y - e *\<^sub>R (y - z) : rel_interior S" using rel_interior_convex_shrink[of S z y e] assms`y : S` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2397
  thus ?thesis using * by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2398
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2399
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2400
subsection{* Relative interior preserves under linear transformations *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2401
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2402
lemma rel_interior_translation_aux:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2403
fixes a :: "'n::euclidean_space"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2404
shows "((%x. a + x) ` rel_interior S) <= rel_interior ((%x. a + x) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2405
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2406
{ fix x assume x_def: "x : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2407
  from this obtain T where T_def: "open T & x : (T Int S) & (T Int (affine hull S)) <= S" using mem_rel_interior[of x S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2408
  from this have "open ((%x. a + x) ` T)" and 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2409
    "(a + x) : (((%x. a + x) ` T) Int ((%x. a + x) ` S))" and 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2410
    "(((%x. a + x) ` T) Int (affine hull ((%x. a + x) ` S))) <= ((%x. a + x) ` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2411
    using affine_hull_translation[of a S] open_translation[of T a] x_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2412
  from this have "(a+x) : rel_interior ((%x. a + x) ` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2413
    using mem_rel_interior[of "a+x" "((%x. a + x) ` S)"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2414
} from this show ?thesis by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2415
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2416
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2417
lemma rel_interior_translation:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2418
fixes a :: "'n::euclidean_space"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2419
shows "rel_interior ((%x. a + x) ` S) = ((%x. a + x) ` rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2420
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2421
have "(%x. (-a) + x) ` rel_interior ((%x. a + x) ` S) <= rel_interior S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2422
   using rel_interior_translation_aux[of "-a" "(%x. a + x) ` S"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2423
         translation_assoc[of "-a" "a"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2424
hence "((%x. a + x) ` rel_interior S) >= rel_interior ((%x. a + x) ` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2425
   using translation_inverse_subset[of a "rel_interior (op + a ` S)" "rel_interior S"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2426
   by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2427
from this show ?thesis using  rel_interior_translation_aux[of a S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2428
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2429
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2430
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2431
lemma affine_hull_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2432
assumes "bounded_linear f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2433
shows "f ` (affine hull s) = affine hull f ` s"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2434
(* Proof is a modified copy of the proof of similar lemma convex_hull_linear_image
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2435
*)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2436
  apply rule unfolding subset_eq ball_simps apply(rule_tac[!] hull_induct, rule hull_inc) prefer 3  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2437
  apply(erule imageE)apply(rule_tac x=xa in image_eqI) apply assumption
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2438
  apply(rule hull_subset[unfolded subset_eq, rule_format]) apply assumption
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2439
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2440
  interpret f: bounded_linear f by fact
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2441
  show "affine {x. f x : affine hull f ` s}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2442
  unfolding affine_def by(auto simp add: f.scaleR f.add affine_affine_hull[unfolded affine_def, rule_format]) next
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2443
  interpret f: bounded_linear f by fact
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2444
  show "affine {x. x : f ` (affine hull s)}" using affine_affine_hull[unfolded affine_def, of s] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2445
    unfolding affine_def by (auto simp add: f.scaleR [symmetric] f.add [symmetric])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2446
qed auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2447
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2448
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2449
lemma rel_interior_injective_on_span_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2450
fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2451
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2452
assumes "bounded_linear f" and "inj_on f (span S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2453
shows "rel_interior (f ` S) = f ` (rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2454
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2455
{ fix z assume z_def: "z : rel_interior (f ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2456
  have "z : f ` S" using z_def rel_interior_subset[of "f ` S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2457
  from this obtain x where x_def: "x : S & (f x = z)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2458
  obtain e2 where e2_def: "e2>0 & cball z e2 Int affine hull (f ` S) <= (f ` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2459
    using z_def rel_interior_cball[of "f ` S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2460
  obtain K where K_def: "K>0 & (! x. norm (f x) <= norm x * K)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2461
   using assms RealVector.bounded_linear.pos_bounded[of f] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2462
  def e1 == "1/K" hence e1_def: "e1>0 & (! x. e1 * norm (f x) <= norm x)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2463
   using K_def pos_le_divide_eq[of e1] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2464
  def e == "e1 * e2" hence "e>0" using e1_def e2_def real_mult_order by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2465
  { fix y assume y_def: "y : cball x e Int affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2466
    from this have h1: "f y : affine hull (f ` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2467
      using affine_hull_linear_image[of f S] assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2468
    from y_def have "norm (x-y)<=e1 * e2" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2469
      using cball_def[of x e] dist_norm[of x y] e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2470
    moreover have "(f x)-(f y)=f (x-y)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2471
       using assms linear_sub[of f x y] linear_conv_bounded_linear[of f] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2472
    moreover have "e1 * norm (f (x-y)) <= norm (x-y)" using e1_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2473
    ultimately have "e1 * norm ((f x)-(f y)) <= e1 * e2" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2474
    hence "(f y) : (cball z e2)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2475
      using cball_def[of "f x" e2] dist_norm[of "f x" "f y"] e1_def x_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2476
    hence "f y : (f ` S)" using y_def e2_def h1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2477
    hence "y : S" using assms y_def hull_subset[of S] affine_hull_subset_span 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2478
         inj_on_image_mem_iff[of f "span S" S y] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2479
  } 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2480
  hence "z : f ` (rel_interior S)" using mem_rel_interior_cball[of x S] `e>0` x_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2481
} 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2482
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2483
{ fix x assume x_def: "x : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2484
  from this obtain e2 where e2_def: "e2>0 & cball x e2 Int affine hull S <= S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2485
    using rel_interior_cball[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2486
  have "x : S" using x_def rel_interior_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2487
  hence *: "f x : f ` S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2488
  have "! x:span S. f x = 0 --> x = 0" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2489
    using assms subspace_span linear_conv_bounded_linear[of f] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2490
          linear_injective_on_subspace_0[of f "span S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2491
  from this obtain e1 where e1_def: "e1>0 & (! x : span S. e1 * norm x <= norm (f x))" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2492
   using assms injective_imp_isometric[of "span S" f] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2493
         subspace_span[of S] closed_subspace[of "span S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2494
  def e == "e1 * e2" hence "e>0" using e1_def e2_def real_mult_order by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2495
  { fix y assume y_def: "y : cball (f x) e Int affine hull (f ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2496
    from this have "y : f ` (affine hull S)" using affine_hull_linear_image[of f S] assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2497
    from this obtain xy where xy_def: "xy : affine hull S & (f xy = y)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2498
    from this y_def have "norm ((f x)-(f xy))<=e1 * e2" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2499
      using cball_def[of "f x" e] dist_norm[of "f x" y] e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2500
    moreover have "(f x)-(f xy)=f (x-xy)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2501
       using assms linear_sub[of f x xy] linear_conv_bounded_linear[of f] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2502
    moreover have "x-xy : span S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2503
       using subspace_sub[of "span S" x xy] subspace_span `x : S` xy_def 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2504
             affine_hull_subset_span[of S] span_inc by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2505
    moreover hence "e1 * norm (x-xy) <= norm (f (x-xy))" using e1_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2506
    ultimately have "e1 * norm (x-xy) <= e1 * e2" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2507
    hence "xy : (cball x e2)"  using cball_def[of x e2] dist_norm[of x xy] e1_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2508
    hence "y : (f ` S)" using xy_def e2_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2509
  } 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2510
  hence "(f x) : rel_interior (f ` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2511
     using mem_rel_interior_cball[of "(f x)" "(f ` S)"] * `e>0` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2512
} 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2513
ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2514
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2515
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2516
lemma rel_interior_injective_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2517
fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2518
assumes "bounded_linear f" and "inj f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2519
shows "rel_interior (f ` S) = f ` (rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2520
using assms rel_interior_injective_on_span_linear_image[of f S] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2521
      subset_inj_on[of f "UNIV" "span S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2522
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2523
subsection{* Some Properties of subset of standard basis *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2524
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2525
lemma affine_hull_substd_basis: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2526
  shows "affine hull (insert 0 {basis i | i. i : d}) =
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2527
  {x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2528
 (is "affine hull (insert 0 ?A) = ?B")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2529
proof- have *:"\<And>A. op + (0\<Colon>'a) ` A = A" "\<And>A. op + (- (0\<Colon>'a)) ` A = A" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2530
  show ?thesis unfolding affine_hull_insert_span_gen span_substd_basis[OF assms,THEN sym] * ..
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2531
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2532
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2533
lemma affine_hull_convex_hull: "affine hull (convex hull S) = affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2534
by (metis Int_absorb1 Int_absorb2 convex_hull_subset_affine_hull hull_hull hull_mono hull_subset)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  2535
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2536
subsection {* Openness and compactness are preserved by convex hull operation. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2537
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  2538
lemma open_convex_hull[intro]:
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2539
  fixes s :: "'a::real_normed_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2540
  assumes "open s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2541
  shows "open(convex hull s)"
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  2542
  unfolding open_contains_cball convex_hull_explicit unfolding mem_Collect_eq ball_simps(10)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2543
proof(rule, rule) fix a
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2544
  assume "\<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = a"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2545
  then obtain t u where obt:"finite t" "t\<subseteq>s" "\<forall>x\<in>t. 0 \<le> u x" "setsum u t = 1" "(\<Sum>v\<in>t. u v *\<^sub>R v) = a" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2546
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2547
  from assms[unfolded open_contains_cball] obtain b where b:"\<forall>x\<in>s. 0 < b x \<and> cball x (b x) \<subseteq> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2548
    using bchoice[of s "\<lambda>x e. e>0 \<and> cball x e \<subseteq> s"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2549
  have "b ` t\<noteq>{}" unfolding i_def using obt by auto  def i \<equiv> "b ` t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2550
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2551
  show "\<exists>e>0. cball a e \<subseteq> {y. \<exists>sa u. finite sa \<and> sa \<subseteq> s \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2552
    apply(rule_tac x="Min i" in exI) unfolding subset_eq apply rule defer apply rule unfolding mem_Collect_eq
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2553
  proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2554
    show "0 < Min i" unfolding i_def and Min_gr_iff[OF finite_imageI[OF obt(1)] `b \` t\<noteq>{}`]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2555
      using b apply simp apply rule apply(erule_tac x=x in ballE) using `t\<subseteq>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2556
  next  fix y assume "y \<in> cball a (Min i)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2557
    hence y:"norm (a - y) \<le> Min i" unfolding dist_norm[THEN sym] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2558
    { fix x assume "x\<in>t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2559
      hence "Min i \<le> b x" unfolding i_def apply(rule_tac Min_le) using obt(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2560
      hence "x + (y - a) \<in> cball x (b x)" using y unfolding mem_cball dist_norm by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2561
      moreover from `x\<in>t` have "x\<in>s" using obt(2) by auto
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  2562
      ultimately have "x + (y - a) \<in> s" using y and b[THEN bspec[where x=x]] unfolding subset_eq by fast }
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2563
    moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2564
    have *:"inj_on (\<lambda>v. v + (y - a)) t" unfolding inj_on_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2565
    have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a))) = 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2566
      unfolding setsum_reindex[OF *] o_def using obt(4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2567
    moreover have "(\<Sum>v\<in>(\<lambda>v. v + (y - a)) ` t. u (v - (y - a)) *\<^sub>R v) = y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2568
      unfolding setsum_reindex[OF *] o_def using obt(4,5)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2569
      by (simp add: setsum_addf setsum_subtractf scaleR_left.setsum[THEN sym] scaleR_right_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2570
    ultimately show "\<exists>sa u. finite sa \<and> (\<forall>x\<in>sa. x \<in> s) \<and> (\<forall>x\<in>sa. 0 \<le> u x) \<and> setsum u sa = 1 \<and> (\<Sum>v\<in>sa. u v *\<^sub>R v) = y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2571
      apply(rule_tac x="(\<lambda>v. v + (y - a)) ` t" in exI) apply(rule_tac x="\<lambda>v. u (v - (y - a))" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2572
      using obt(1, 3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2573
  qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2574
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2575
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2576
lemma compact_convex_combinations:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2577
  fixes s t :: "'a::real_normed_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2578
  assumes "compact s" "compact t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2579
  shows "compact { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2580
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2581
  let ?X = "{0..1} \<times> s \<times> t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2582
  let ?h = "(\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2583
  have *:"{ (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> t} = ?h ` ?X"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  2584
    apply(rule set_eqI) unfolding image_iff mem_Collect_eq
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2585
    apply rule apply auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2586
    apply (rule_tac x=u in rev_bexI, simp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2587
    apply (erule rev_bexI, erule rev_bexI, simp)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2588
    by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2589
  have "continuous_on ({0..1} \<times> s \<times> t)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2590
     (\<lambda>z. (1 - fst z) *\<^sub>R fst (snd z) + fst z *\<^sub>R snd (snd z))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2591
    unfolding continuous_on by (rule ballI) (intro tendsto_intros)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2592
  thus ?thesis unfolding *
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2593
    apply (rule compact_continuous_image)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2594
    apply (intro compact_Times compact_interval assms)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2595
    done
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2596
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2597
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2598
lemma compact_convex_hull: fixes s::"('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2599
  assumes "compact s"  shows "compact(convex hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2600
proof(cases "s={}")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2601
  case True thus ?thesis using compact_empty by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2602
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2603
  case False then obtain w where "w\<in>s" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2604
  show ?thesis unfolding caratheodory[of s]
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2605
  proof(induct ("DIM('a) + 1"))
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2606
    have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}" 
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  2607
      using compact_empty by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2608
    case 0 thus ?case unfolding * by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2609
  next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2610
    case (Suc n)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2611
    show ?case proof(cases "n=0")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2612
      case True have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} = s"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  2613
        unfolding set_eq_iff and mem_Collect_eq proof(rule, rule)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2614
        fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2615
        then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2616
        show "x\<in>s" proof(cases "card t = 0")
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  2617
          case True thus ?thesis using t(4) unfolding card_0_eq[OF t(1)] by simp
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2618
        next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2619
          case False hence "card t = Suc 0" using t(3) `n=0` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2620
          then obtain a where "t = {a}" unfolding card_Suc_eq by auto
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  2621
          thus ?thesis using t(2,4) by simp
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2622
        qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2623
      next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2624
        fix x assume "x\<in>s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2625
        thus "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2626
          apply(rule_tac x="{x}" in exI) unfolding convex_hull_singleton by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2627
      qed thus ?thesis using assms by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2628
    next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2629
      case False have "{x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t} =
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2630
        { (1 - u) *\<^sub>R x + u *\<^sub>R y | x y u. 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2631
        0 \<le> u \<and> u \<le> 1 \<and> x \<in> s \<and> y \<in> {x. \<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> x \<in> convex hull t}}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  2632
        unfolding set_eq_iff and mem_Collect_eq proof(rule,rule)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2633
        fix x assume "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2634
          0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2635
        then obtain u v c t where obt:"x = (1 - c) *\<^sub>R u + c *\<^sub>R v"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2636
          "0 \<le> c \<and> c \<le> 1" "u \<in> s" "finite t" "t \<subseteq> s" "card t \<le> n"  "v \<in> convex hull t" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2637
        moreover have "(1 - c) *\<^sub>R u + c *\<^sub>R v \<in> convex hull insert u t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2638
          apply(rule mem_convex) using obt(2) and convex_convex_hull and hull_subset[of "insert u t" convex]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2639
          using obt(7) and hull_mono[of t "insert u t"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2640
        ultimately show "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2641
          apply(rule_tac x="insert u t" in exI) by (auto simp add: card_insert_if)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2642
      next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2643
        fix x assume "\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> Suc n \<and> x \<in> convex hull t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2644
        then obtain t where t:"finite t" "t \<subseteq> s" "card t \<le> Suc n" "x \<in> convex hull t" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2645
        let ?P = "\<exists>u v c. x = (1 - c) *\<^sub>R u + c *\<^sub>R v \<and>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2646
          0 \<le> c \<and> c \<le> 1 \<and> u \<in> s \<and> (\<exists>t. finite t \<and> t \<subseteq> s \<and> card t \<le> n \<and> v \<in> convex hull t)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2647
        show ?P proof(cases "card t = Suc n")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2648
          case False hence "card t \<le> n" using t(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2649
          thus ?P apply(rule_tac x=w in exI, rule_tac x=x in exI, rule_tac x=1 in exI) using `w\<in>s` and t
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2650
            by(auto intro!: exI[where x=t])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2651
        next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2652
          case True then obtain a u where au:"t = insert a u" "a\<notin>u" apply(drule_tac card_eq_SucD) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2653
          show ?P proof(cases "u={}")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2654
            case True hence "x=a" using t(4)[unfolded au] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2655
            show ?P unfolding `x=a` apply(rule_tac x=a in exI, rule_tac x=a in exI, rule_tac x=1 in exI)
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  2656
              using t and `n\<noteq>0` unfolding au by(auto intro!: exI[where x="{a}"])
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2657
          next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2658
            case False obtain ux vx b where obt:"ux\<ge>0" "vx\<ge>0" "ux + vx = 1" "b \<in> convex hull u" "x = ux *\<^sub>R a + vx *\<^sub>R b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2659
              using t(4)[unfolded au convex_hull_insert[OF False]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2660
            have *:"1 - vx = ux" using obt(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2661
            show ?P apply(rule_tac x=a in exI, rule_tac x=b in exI, rule_tac x=vx in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2662
              using obt and t(1-3) unfolding au and * using card_insert_disjoint[OF _ au(2)]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2663
              by(auto intro!: exI[where x=u])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2664
          qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2665
        qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2666
      qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2667
      thus ?thesis using compact_convex_combinations[OF assms Suc] by simp 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2668
    qed
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  2669
  qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2670
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2671
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2672
lemma finite_imp_compact_convex_hull:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2673
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2674
  shows "finite s \<Longrightarrow> compact(convex hull s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  2675
by (metis compact_convex_hull finite_imp_compact)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2676
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2677
subsection {* Extremal points of a simplex are some vertices. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2678
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2679
lemma dist_increases_online:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2680
  fixes a b d :: "'a::real_inner"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2681
  assumes "d \<noteq> 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2682
  shows "dist a (b + d) > dist a b \<or> dist a (b - d) > dist a b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2683
proof(cases "inner a d - inner b d > 0")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2684
  case True hence "0 < inner d d + (inner a d * 2 - inner b d * 2)" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2685
    apply(rule_tac add_pos_pos) using assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2686
  thus ?thesis apply(rule_tac disjI2) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2687
    by (simp add: algebra_simps inner_commute)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2688
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2689
  case False hence "0 < inner d d + (inner b d * 2 - inner a d * 2)" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2690
    apply(rule_tac add_pos_nonneg) using assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2691
  thus ?thesis apply(rule_tac disjI1) unfolding dist_norm and norm_eq_sqrt_inner and real_sqrt_less_iff
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2692
    by (simp add: algebra_simps inner_commute)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2693
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2694
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2695
lemma norm_increases_online:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2696
  fixes d :: "'a::real_inner"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2697
  shows "d \<noteq> 0 \<Longrightarrow> norm(a + d) > norm a \<or> norm(a - d) > norm a"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2698
  using dist_increases_online[of d a 0] unfolding dist_norm by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2699
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2700
lemma simplex_furthest_lt:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2701
  fixes s::"'a::real_inner set" assumes "finite s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2702
  shows "\<forall>x \<in> (convex hull s).  x \<notin> s \<longrightarrow> (\<exists>y\<in>(convex hull s). norm(x - a) < norm(y - a))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2703
proof(induct_tac rule: finite_induct[of s])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2704
  fix x s assume as:"finite s" "x\<notin>s" "\<forall>x\<in>convex hull s. x \<notin> s \<longrightarrow> (\<exists>y\<in>convex hull s. norm (x - a) < norm (y - a))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2705
  show "\<forall>xa\<in>convex hull insert x s. xa \<notin> insert x s \<longrightarrow> (\<exists>y\<in>convex hull insert x s. norm (xa - a) < norm (y - a))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2706
  proof(rule,rule,cases "s = {}")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2707
    case False fix y assume y:"y \<in> convex hull insert x s" "y \<notin> insert x s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2708
    obtain u v b where obt:"u\<ge>0" "v\<ge>0" "u + v = 1" "b \<in> convex hull s" "y = u *\<^sub>R x + v *\<^sub>R b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2709
      using y(1)[unfolded convex_hull_insert[OF False]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2710
    show "\<exists>z\<in>convex hull insert x s. norm (y - a) < norm (z - a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2711
    proof(cases "y\<in>convex hull s")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2712
      case True then obtain z where "z\<in>convex hull s" "norm (y - a) < norm (z - a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2713
        using as(3)[THEN bspec[where x=y]] and y(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2714
      thus ?thesis apply(rule_tac x=z in bexI) unfolding convex_hull_insert[OF False] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2715
    next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2716
      case False show ?thesis  using obt(3) proof(cases "u=0", case_tac[!] "v=0")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2717
        assume "u=0" "v\<noteq>0" hence "y = b" using obt by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2718
        thus ?thesis using False and obt(4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2719
      next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2720
        assume "u\<noteq>0" "v=0" hence "y = x" using obt by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2721
        thus ?thesis using y(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2722
      next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2723
        assume "u\<noteq>0" "v\<noteq>0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2724
        then obtain w where w:"w>0" "w<u" "w<v" using real_lbound_gt_zero[of u v] and obt(1,2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2725
        have "x\<noteq>b" proof(rule ccontr) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2726
          assume "\<not> x\<noteq>b" hence "y=b" unfolding obt(5)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2727
            using obt(3) by(auto simp add: scaleR_left_distrib[THEN sym])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2728
          thus False using obt(4) and False by simp qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2729
        hence *:"w *\<^sub>R (x - b) \<noteq> 0" using w(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2730
        show ?thesis using dist_increases_online[OF *, of a y]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2731
        proof(erule_tac disjE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2732
          assume "dist a y < dist a (y + w *\<^sub>R (x - b))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2733
          hence "norm (y - a) < norm ((u + w) *\<^sub>R x + (v - w) *\<^sub>R b - a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2734
            unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2735
          moreover have "(u + w) *\<^sub>R x + (v - w) *\<^sub>R b \<in> convex hull insert x s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2736
            unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2737
            apply(rule_tac x="u + w" in exI) apply rule defer 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2738
            apply(rule_tac x="v - w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2739
          ultimately show ?thesis by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2740
        next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2741
          assume "dist a y < dist a (y - w *\<^sub>R (x - b))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2742
          hence "norm (y - a) < norm ((u - w) *\<^sub>R x + (v + w) *\<^sub>R b - a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2743
            unfolding dist_commute[of a] unfolding dist_norm obt(5) by (simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2744
          moreover have "(u - w) *\<^sub>R x + (v + w) *\<^sub>R b \<in> convex hull insert x s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2745
            unfolding convex_hull_insert[OF `s\<noteq>{}`] and mem_Collect_eq
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2746
            apply(rule_tac x="u - w" in exI) apply rule defer 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2747
            apply(rule_tac x="v + w" in exI) using `u\<ge>0` and w and obt(3,4) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2748
          ultimately show ?thesis by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2749
        qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2750
      qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2751
    qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2752
  qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2753
qed (auto simp add: assms)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2754
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2755
lemma simplex_furthest_le:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2756
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2757
  assumes "finite s" "s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2758
  shows "\<exists>y\<in>s. \<forall>x\<in>(convex hull s). norm(x - a) \<le> norm(y - a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2759
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2760
  have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2761
  then obtain x where x:"x\<in>convex hull s" "\<forall>y\<in>convex hull s. norm (y - a) \<le> norm (x - a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2762
    using distance_attains_sup[OF finite_imp_compact_convex_hull[OF assms(1)], of a]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2763
    unfolding dist_commute[of a] unfolding dist_norm by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2764
  thus ?thesis proof(cases "x\<in>s")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2765
    case False then obtain y where "y\<in>convex hull s" "norm (x - a) < norm (y - a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2766
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=x]] and x(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2767
    thus ?thesis using x(2)[THEN bspec[where x=y]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2768
  qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2769
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2770
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2771
lemma simplex_furthest_le_exists:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2772
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2773
  shows "finite s \<Longrightarrow> (\<forall>x\<in>(convex hull s). \<exists>y\<in>s. norm(x - a) \<le> norm(y - a))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2774
  using simplex_furthest_le[of s] by (cases "s={}")auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2775
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2776
lemma simplex_extremal_le:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2777
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2778
  assumes "finite s" "s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2779
  shows "\<exists>u\<in>s. \<exists>v\<in>s. \<forall>x\<in>convex hull s. \<forall>y \<in> convex hull s. norm(x - y) \<le> norm(u - v)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2780
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2781
  have "convex hull s \<noteq> {}" using hull_subset[of s convex] and assms(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2782
  then obtain u v where obt:"u\<in>convex hull s" "v\<in>convex hull s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2783
    "\<forall>x\<in>convex hull s. \<forall>y\<in>convex hull s. norm (x - y) \<le> norm (u - v)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2784
    using compact_sup_maxdistance[OF finite_imp_compact_convex_hull[OF assms(1)]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2785
  thus ?thesis proof(cases "u\<notin>s \<or> v\<notin>s", erule_tac disjE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2786
    assume "u\<notin>s" then obtain y where "y\<in>convex hull s" "norm (u - v) < norm (y - v)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2787
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=u]] and obt(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2788
    thus ?thesis using obt(3)[THEN bspec[where x=y], THEN bspec[where x=v]] and obt(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2789
  next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2790
    assume "v\<notin>s" then obtain y where "y\<in>convex hull s" "norm (v - u) < norm (y - u)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2791
      using simplex_furthest_lt[OF assms(1), THEN bspec[where x=v]] and obt(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2792
    thus ?thesis using obt(3)[THEN bspec[where x=u], THEN bspec[where x=y]] and obt(1)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2793
      by (auto simp add: norm_minus_commute)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2794
  qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2795
qed 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2796
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2797
lemma simplex_extremal_le_exists:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2798
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2799
  shows "finite s \<Longrightarrow> x \<in> convex hull s \<Longrightarrow> y \<in> convex hull s
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2800
  \<Longrightarrow> (\<exists>u\<in>s. \<exists>v\<in>s. norm(x - y) \<le> norm(u - v))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2801
  using convex_hull_empty simplex_extremal_le[of s] by(cases "s={}")auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2802
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2803
subsection {* Closest point of a convex set is unique, with a continuous projection. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2804
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2805
definition
36337
87b6c83d7ed7 generalize constant closest_point
huffman
parents: 36071
diff changeset
  2806
  closest_point :: "'a::{real_inner,heine_borel} set \<Rightarrow> 'a \<Rightarrow> 'a" where
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2807
 "closest_point s a = (SOME x. x \<in> s \<and> (\<forall>y\<in>s. dist a x \<le> dist a y))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2808
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2809
lemma closest_point_exists:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2810
  assumes "closed s" "s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2811
  shows  "closest_point s a \<in> s" "\<forall>y\<in>s. dist a (closest_point s a) \<le> dist a y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2812
  unfolding closest_point_def apply(rule_tac[!] someI2_ex) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2813
  using distance_attains_inf[OF assms(1,2), of a] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2814
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2815
lemma closest_point_in_set:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2816
  "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s a) \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2817
  by(meson closest_point_exists)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2818
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2819
lemma closest_point_le:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2820
  "closed s \<Longrightarrow> x \<in> s \<Longrightarrow> dist a (closest_point s a) \<le> dist a x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2821
  using closest_point_exists[of s] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2822
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2823
lemma closest_point_self:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2824
  assumes "x \<in> s"  shows "closest_point s x = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2825
  unfolding closest_point_def apply(rule some1_equality, rule ex1I[of _ x]) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2826
  using assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2827
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2828
lemma closest_point_refl:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2829
 "closed s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> (closest_point s x = x \<longleftrightarrow> x \<in> s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2830
  using closest_point_in_set[of s x] closest_point_self[of x s] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2831
36337
87b6c83d7ed7 generalize constant closest_point
huffman
parents: 36071
diff changeset
  2832
lemma closer_points_lemma:
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2833
  assumes "inner y z > 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2834
  shows "\<exists>u>0. \<forall>v>0. v \<le> u \<longrightarrow> norm(v *\<^sub>R z - y) < norm y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2835
proof- have z:"inner z z > 0" unfolding inner_gt_zero_iff using assms by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2836
  thus ?thesis using assms apply(rule_tac x="inner y z / inner z z" in exI) apply(rule) defer proof(rule+)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2837
    fix v assume "0<v" "v \<le> inner y z / inner z z"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2838
    thus "norm (v *\<^sub>R z - y) < norm y" unfolding norm_lt using z and assms
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2839
      by (simp add: field_simps inner_diff inner_commute mult_strict_left_mono[OF _ `0<v`])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2840
  qed(rule divide_pos_pos, auto) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2841
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2842
lemma closer_point_lemma:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2843
  assumes "inner (y - x) (z - x) > 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2844
  shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2845
proof- obtain u where "u>0" and u:"\<forall>v>0. v \<le> u \<longrightarrow> norm (v *\<^sub>R (z - x) - (y - x)) < norm (y - x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2846
    using closer_points_lemma[OF assms] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2847
  show ?thesis apply(rule_tac x="min u 1" in exI) using u[THEN spec[where x="min u 1"]] and `u>0`
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2848
    unfolding dist_norm by(auto simp add: norm_minus_commute field_simps) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2849
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2850
lemma any_closest_point_dot:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2851
  assumes "convex s" "closed s" "x \<in> s" "y \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2852
  shows "inner (a - x) (y - x) \<le> 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2853
proof(rule ccontr) assume "\<not> inner (a - x) (y - x) \<le> 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2854
  then obtain u where u:"u>0" "u\<le>1" "dist (x + u *\<^sub>R (y - x)) a < dist x a" using closer_point_lemma[of a x y] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2855
  let ?z = "(1 - u) *\<^sub>R x + u *\<^sub>R y" have "?z \<in> s" using mem_convex[OF assms(1,3,4), of u] using u by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2856
  thus False using assms(5)[THEN bspec[where x="?z"]] and u(3) by (auto simp add: dist_commute algebra_simps) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2857
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2858
lemma any_closest_point_unique:
36337
87b6c83d7ed7 generalize constant closest_point
huffman
parents: 36071
diff changeset
  2859
  fixes x :: "'a::real_inner"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2860
  assumes "convex s" "closed s" "x \<in> s" "y \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2861
  "\<forall>z\<in>s. dist a x \<le> dist a z" "\<forall>z\<in>s. dist a y \<le> dist a z"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2862
  shows "x = y" using any_closest_point_dot[OF assms(1-4,5)] and any_closest_point_dot[OF assms(1-2,4,3,6)]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2863
  unfolding norm_pths(1) and norm_le_square
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2864
  by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2865
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2866
lemma closest_point_unique:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2867
  assumes "convex s" "closed s" "x \<in> s" "\<forall>z\<in>s. dist a x \<le> dist a z"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2868
  shows "x = closest_point s a"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2869
  using any_closest_point_unique[OF assms(1-3) _ assms(4), of "closest_point s a"] 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2870
  using closest_point_exists[OF assms(2)] and assms(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2871
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2872
lemma closest_point_dot:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2873
  assumes "convex s" "closed s" "x \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2874
  shows "inner (a - closest_point s a) (x - closest_point s a) \<le> 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2875
  apply(rule any_closest_point_dot[OF assms(1,2) _ assms(3)])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2876
  using closest_point_exists[OF assms(2)] and assms(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2877
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2878
lemma closest_point_lt:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2879
  assumes "convex s" "closed s" "x \<in> s" "x \<noteq> closest_point s a"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2880
  shows "dist a (closest_point s a) < dist a x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2881
  apply(rule ccontr) apply(rule_tac notE[OF assms(4)])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2882
  apply(rule closest_point_unique[OF assms(1-3), of a])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2883
  using closest_point_le[OF assms(2), of _ a] by fastsimp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2884
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2885
lemma closest_point_lipschitz:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2886
  assumes "convex s" "closed s" "s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2887
  shows "dist (closest_point s x) (closest_point s y) \<le> dist x y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2888
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2889
  have "inner (x - closest_point s x) (closest_point s y - closest_point s x) \<le> 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2890
       "inner (y - closest_point s y) (closest_point s x - closest_point s y) \<le> 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2891
    apply(rule_tac[!] any_closest_point_dot[OF assms(1-2)])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2892
    using closest_point_exists[OF assms(2-3)] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2893
  thus ?thesis unfolding dist_norm and norm_le
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2894
    using inner_ge_zero[of "(x - closest_point s x) - (y - closest_point s y)"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2895
    by (simp add: inner_add inner_diff inner_commute) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2896
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2897
lemma continuous_at_closest_point:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2898
  assumes "convex s" "closed s" "s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2899
  shows "continuous (at x) (closest_point s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2900
  unfolding continuous_at_eps_delta 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2901
  using le_less_trans[OF closest_point_lipschitz[OF assms]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2902
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2903
lemma continuous_on_closest_point:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2904
  assumes "convex s" "closed s" "s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2905
  shows "continuous_on t (closest_point s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  2906
by(metis continuous_at_imp_continuous_on continuous_at_closest_point[OF assms])
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2907
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2908
subsection {* Various point-to-set separating/supporting hyperplane theorems. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2909
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2910
lemma supporting_hyperplane_closed_point:
36337
87b6c83d7ed7 generalize constant closest_point
huffman
parents: 36071
diff changeset
  2911
  fixes z :: "'a::{real_inner,heine_borel}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2912
  assumes "convex s" "closed s" "s \<noteq> {}" "z \<notin> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2913
  shows "\<exists>a b. \<exists>y\<in>s. inner a z < b \<and> (inner a y = b) \<and> (\<forall>x\<in>s. inner a x \<ge> b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2914
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2915
  from distance_attains_inf[OF assms(2-3)] obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2916
  show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) y" in exI, rule_tac x=y in bexI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2917
    apply rule defer apply rule defer apply(rule, rule ccontr) using `y\<in>s` proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2918
    show "inner (y - z) z < inner (y - z) y" apply(subst diff_less_iff(1)[THEN sym])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2919
      unfolding inner_diff_right[THEN sym] and inner_gt_zero_iff using `y\<in>s` `z\<notin>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2920
  next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2921
    fix x assume "x\<in>s" have *:"\<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> dist z y \<le> dist z ((1 - u) *\<^sub>R y + u *\<^sub>R x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2922
      using assms(1)[unfolded convex_alt] and y and `x\<in>s` and `y\<in>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2923
    assume "\<not> inner (y - z) y \<le> inner (y - z) x" then obtain v where
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2924
      "v>0" "v\<le>1" "dist (y + v *\<^sub>R (x - y)) z < dist y z" using closer_point_lemma[of z y x] apply - by (auto simp add: inner_diff)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2925
    thus False using *[THEN spec[where x=v]] by(auto simp add: dist_commute algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2926
  qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2927
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2928
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2929
lemma separating_hyperplane_closed_point:
36337
87b6c83d7ed7 generalize constant closest_point
huffman
parents: 36071
diff changeset
  2930
  fixes z :: "'a::{real_inner,heine_borel}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2931
  assumes "convex s" "closed s" "z \<notin> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2932
  shows "\<exists>a b. inner a z < b \<and> (\<forall>x\<in>s. inner a x > b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2933
proof(cases "s={}")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2934
  case True thus ?thesis apply(rule_tac x="-z" in exI, rule_tac x=1 in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2935
    using less_le_trans[OF _ inner_ge_zero[of z]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2936
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2937
  case False obtain y where "y\<in>s" and y:"\<forall>x\<in>s. dist z y \<le> dist z x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2938
    using distance_attains_inf[OF assms(2) False] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2939
  show ?thesis apply(rule_tac x="y - z" in exI, rule_tac x="inner (y - z) z + (norm(y - z))\<twosuperior> / 2" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2940
    apply rule defer apply rule proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2941
    fix x assume "x\<in>s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2942
    have "\<not> 0 < inner (z - y) (x - y)" apply(rule_tac notI) proof(drule closer_point_lemma)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2943
      assume "\<exists>u>0. u \<le> 1 \<and> dist (y + u *\<^sub>R (x - y)) z < dist y z"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2944
      then obtain u where "u>0" "u\<le>1" "dist (y + u *\<^sub>R (x - y)) z < dist y z" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2945
      thus False using y[THEN bspec[where x="y + u *\<^sub>R (x - y)"]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2946
        using assms(1)[unfolded convex_alt, THEN bspec[where x=y]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2947
        using `x\<in>s` `y\<in>s` by (auto simp add: dist_commute algebra_simps) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2948
    moreover have "0 < norm (y - z) ^ 2" using `y\<in>s` `z\<notin>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2949
    hence "0 < inner (y - z) (y - z)" unfolding power2_norm_eq_inner by simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2950
    ultimately show "inner (y - z) z + (norm (y - z))\<twosuperior> / 2 < inner (y - z) x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2951
      unfolding power2_norm_eq_inner and not_less by (auto simp add: field_simps inner_commute inner_diff)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2952
  qed(insert `y\<in>s` `z\<notin>s`, auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2953
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2954
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2955
lemma separating_hyperplane_closed_0:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2956
  assumes "convex (s::('a::euclidean_space) set)" "closed s" "0 \<notin> s"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2957
  shows "\<exists>a b. a \<noteq> 0 \<and> 0 < b \<and> (\<forall>x\<in>s. inner a x > b)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2958
  proof(cases "s={}")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2959
  case True have "norm ((basis 0)::'a) = 1" by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2960
  hence "norm ((basis 0)::'a) = 1" "basis 0 \<noteq> (0::'a)" defer
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2961
    apply(subst norm_le_zero_iff[THEN sym]) by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2962
  thus ?thesis apply(rule_tac x="basis 0" in exI, rule_tac x=1 in exI)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2963
    using True using DIM_positive[where 'a='a] by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2964
next case False thus ?thesis using False using separating_hyperplane_closed_point[OF assms]
35542
8f97d8caabfd replaced \<bullet> with inner
himmelma
parents: 34964
diff changeset
  2965
    apply - apply(erule exE)+ unfolding inner.zero_right apply(rule_tac x=a in exI, rule_tac x=b in exI) by auto qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2966
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2967
subsection {* Now set-to-set for closed/compact sets. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2968
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2969
lemma separating_hyperplane_closed_compact:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2970
  assumes "convex (s::('a::euclidean_space) set)" "closed s" "convex t" "compact t" "t \<noteq> {}" "s \<inter> t = {}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2971
  shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2972
proof(cases "s={}")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2973
  case True
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2974
  obtain b where b:"b>0" "\<forall>x\<in>t. norm x \<le> b" using compact_imp_bounded[OF assms(4)] unfolding bounded_pos by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  2975
  obtain z::"'a" where z:"norm z = b + 1" using vector_choose_size[of "b + 1"] and b(1) by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2976
  hence "z\<notin>t" using b(2)[THEN bspec[where x=z]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2977
  then obtain a b where ab:"inner a z < b" "\<forall>x\<in>t. b < inner a x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2978
    using separating_hyperplane_closed_point[OF assms(3) compact_imp_closed[OF assms(4)], of z] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2979
  thus ?thesis using True by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2980
next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2981
  case False then obtain y where "y\<in>s" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2982
  obtain a b where "0 < b" "\<forall>x\<in>{x - y |x y. x \<in> s \<and> y \<in> t}. b < inner a x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2983
    using separating_hyperplane_closed_point[OF convex_differences[OF assms(1,3)], of 0]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2984
    using closed_compact_differences[OF assms(2,4)] using assms(6) by(auto, blast)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2985
  hence ab:"\<forall>x\<in>s. \<forall>y\<in>t. b + inner a y < inner a x" apply- apply(rule,rule) apply(erule_tac x="x - y" in ballE) by (auto simp add: inner_diff)
33270
paulson
parents: 33175
diff changeset
  2986
  def k \<equiv> "Sup ((\<lambda>x. inner a x) ` t)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2987
  show ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-(k + b / 2)" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2988
    apply(rule,rule) defer apply(rule) unfolding inner_minus_left and neg_less_iff_less proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2989
    from ab have "((\<lambda>x. inner a x) ` t) *<= (inner a y - b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2990
      apply(erule_tac x=y in ballE) apply(rule setleI) using `y\<in>s` by auto
33270
paulson
parents: 33175
diff changeset
  2991
    hence k:"isLub UNIV ((\<lambda>x. inner a x) ` t) k" unfolding k_def apply(rule_tac Sup) using assms(5) by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2992
    fix x assume "x\<in>t" thus "inner a x < (k + b / 2)" using `0<b` and isLubD2[OF k, of "inner a x"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2993
  next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2994
    fix x assume "x\<in>s" 
33270
paulson
parents: 33175
diff changeset
  2995
    hence "k \<le> inner a x - b" unfolding k_def apply(rule_tac Sup_least) using assms(5)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2996
      using ab[THEN bspec[where x=x]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2997
    thus "k + b / 2 < inner a x" using `0 < b` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2998
  qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  2999
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3000
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3001
lemma separating_hyperplane_compact_closed:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3002
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3003
  assumes "convex s" "compact s" "s \<noteq> {}" "convex t" "closed t" "s \<inter> t = {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3004
  shows "\<exists>a b. (\<forall>x\<in>s. inner a x < b) \<and> (\<forall>x\<in>t. inner a x > b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3005
proof- obtain a b where "(\<forall>x\<in>t. inner a x < b) \<and> (\<forall>x\<in>s. b < inner a x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3006
    using separating_hyperplane_closed_compact[OF assms(4-5,1-2,3)] and assms(6) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3007
  thus ?thesis apply(rule_tac x="-a" in exI, rule_tac x="-b" in exI) by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3008
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3009
subsection {* General case without assuming closure and getting non-strict separation. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3010
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3011
lemma separating_hyperplane_set_0:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3012
  assumes "convex s" "(0::'a::euclidean_space) \<notin> s"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3013
  shows "\<exists>a. a \<noteq> 0 \<and> (\<forall>x\<in>s. 0 \<le> inner a x)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3014
proof- let ?k = "\<lambda>c. {x::'a. 0 \<le> inner c x}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3015
  have "frontier (cball 0 1) \<inter> (\<Inter> (?k ` s)) \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3016
    apply(rule compact_imp_fip) apply(rule compact_frontier[OF compact_cball])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3017
    defer apply(rule,rule,erule conjE) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3018
    fix f assume as:"f \<subseteq> ?k ` s" "finite f"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3019
    obtain c where c:"f = ?k ` c" "c\<subseteq>s" "finite c" using finite_subset_image[OF as(2,1)] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3020
    then obtain a b where ab:"a \<noteq> 0" "0 < b"  "\<forall>x\<in>convex hull c. b < inner a x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3021
      using separating_hyperplane_closed_0[OF convex_convex_hull, of c]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3022
      using finite_imp_compact_convex_hull[OF c(3), THEN compact_imp_closed] and assms(2)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3023
      using subset_hull[unfolded mem_def, of convex, OF assms(1), THEN sym, of c] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3024
    hence "\<exists>x. norm x = 1 \<and> (\<forall>y\<in>c. 0 \<le> inner y x)" apply(rule_tac x="inverse(norm a) *\<^sub>R a" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3025
       using hull_subset[of c convex] unfolding subset_eq and inner_scaleR
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3026
       apply- apply rule defer apply rule apply(rule mult_nonneg_nonneg)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3027
       by(auto simp add: inner_commute elim!: ballE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3028
    thus "frontier (cball 0 1) \<inter> \<Inter>f \<noteq> {}" unfolding c(1) frontier_cball dist_norm by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3029
  qed(insert closed_halfspace_ge, auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3030
  then obtain x where "norm x = 1" "\<forall>y\<in>s. x\<in>?k y" unfolding frontier_cball dist_norm by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3031
  thus ?thesis apply(rule_tac x=x in exI) by(auto simp add: inner_commute) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3032
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3033
lemma separating_hyperplane_sets:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3034
  assumes "convex s" "convex (t::('a::euclidean_space) set)" "s \<noteq> {}" "t \<noteq> {}" "s \<inter> t = {}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3035
  shows "\<exists>a b. a \<noteq> 0 \<and> (\<forall>x\<in>s. inner a x \<le> b) \<and> (\<forall>x\<in>t. inner a x \<ge> b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3036
proof- from separating_hyperplane_set_0[OF convex_differences[OF assms(2,1)]]
33270
paulson
parents: 33175
diff changeset
  3037
  obtain a where "a\<noteq>0" "\<forall>x\<in>{x - y |x y. x \<in> t \<and> y \<in> s}. 0 \<le> inner a x" 
paulson
parents: 33175
diff changeset
  3038
    using assms(3-5) by auto 
paulson
parents: 33175
diff changeset
  3039
  hence "\<forall>x\<in>t. \<forall>y\<in>s. inner a y \<le> inner a x"
paulson
parents: 33175
diff changeset
  3040
    by (force simp add: inner_diff)
paulson
parents: 33175
diff changeset
  3041
  thus ?thesis
paulson
parents: 33175
diff changeset
  3042
    apply(rule_tac x=a in exI, rule_tac x="Sup ((\<lambda>x. inner a x) ` s)" in exI) using `a\<noteq>0`
paulson
parents: 33175
diff changeset
  3043
    apply auto
paulson
parents: 33175
diff changeset
  3044
    apply (rule Sup[THEN isLubD2]) 
paulson
parents: 33175
diff changeset
  3045
    prefer 4
paulson
parents: 33175
diff changeset
  3046
    apply (rule Sup_least) 
paulson
parents: 33175
diff changeset
  3047
     using assms(3-5) apply (auto simp add: setle_def)
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3048
    apply metis
33270
paulson
parents: 33175
diff changeset
  3049
    done
paulson
parents: 33175
diff changeset
  3050
qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3051
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3052
subsection {* More convexity generalities. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3053
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3054
lemma convex_closure:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3055
  fixes s :: "'a::real_normed_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3056
  assumes "convex s" shows "convex(closure s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3057
  unfolding convex_def Ball_def closure_sequential
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3058
  apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3059
  apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3060
  apply(rule assms[unfolded convex_def, rule_format]) prefer 6
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3061
  apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3062
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3063
lemma convex_interior:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3064
  fixes s :: "'a::real_normed_vector set"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3065
  assumes "convex s" shows "convex(interior s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3066
  unfolding convex_alt Ball_def mem_interior apply(rule,rule,rule,rule,rule,rule) apply(erule exE | erule conjE)+ proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3067
  fix x y u assume u:"0 \<le> u" "u \<le> (1::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3068
  fix e d assume ed:"ball x e \<subseteq> s" "ball y d \<subseteq> s" "0<d" "0<e" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3069
  show "\<exists>e>0. ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) e \<subseteq> s" apply(rule_tac x="min d e" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3070
    apply rule unfolding subset_eq defer apply rule proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3071
    fix z assume "z \<in> ball ((1 - u) *\<^sub>R x + u *\<^sub>R y) (min d e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3072
    hence "(1- u) *\<^sub>R (z - u *\<^sub>R (y - x)) + u *\<^sub>R (z + (1 - u) *\<^sub>R (y - x)) \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3073
      apply(rule_tac assms[unfolded convex_alt, rule_format])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3074
      using ed(1,2) and u unfolding subset_eq mem_ball Ball_def dist_norm by(auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3075
    thus "z \<in> s" using u by (auto simp add: algebra_simps) qed(insert u ed(3-4), auto) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3076
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  3077
lemma convex_hull_eq_empty[simp]: "convex hull s = {} \<longleftrightarrow> s = {}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3078
  using hull_subset[of s convex] convex_hull_empty by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3079
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3080
subsection {* Moving and scaling convex hulls. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3081
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3082
lemma convex_hull_translation_lemma:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3083
  "convex hull ((\<lambda>x. a + x) ` s) \<subseteq> (\<lambda>x. a + x) ` (convex hull s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3084
by (metis convex_convex_hull convex_translation hull_minimal hull_subset image_mono mem_def)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3085
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3086
lemma convex_hull_bilemma: fixes neg
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3087
  assumes "(\<forall>s a. (convex hull (up a s)) \<subseteq> up a (convex hull s))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3088
  shows "(\<forall>s. up a (up (neg a) s) = s) \<and> (\<forall>s. up (neg a) (up a s) = s) \<and> (\<forall>s t a. s \<subseteq> t \<longrightarrow> up a s \<subseteq> up a t)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3089
  \<Longrightarrow> \<forall>s. (convex hull (up a s)) = up a (convex hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3090
  using assms by(metis subset_antisym) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3091
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3092
lemma convex_hull_translation:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3093
  "convex hull ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (convex hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3094
  apply(rule convex_hull_bilemma[rule_format, of _ _ "\<lambda>a. -a"], rule convex_hull_translation_lemma) unfolding image_image by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3095
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3096
lemma convex_hull_scaling_lemma:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3097
 "(convex hull ((\<lambda>x. c *\<^sub>R x) ` s)) \<subseteq> (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3098
by (metis convex_convex_hull convex_scaling hull_subset mem_def subset_hull subset_image_iff)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3099
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3100
lemma convex_hull_scaling:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3101
  "convex hull ((\<lambda>x. c *\<^sub>R x) ` s) = (\<lambda>x. c *\<^sub>R x) ` (convex hull s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3102
  apply(cases "c=0") defer apply(rule convex_hull_bilemma[rule_format, of _ _ inverse]) apply(rule convex_hull_scaling_lemma)
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  3103
  unfolding image_image scaleR_scaleR by(auto simp add:image_constant_conv)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3104
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3105
lemma convex_hull_affinity:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3106
  "convex hull ((\<lambda>x. a + c *\<^sub>R x) ` s) = (\<lambda>x. a + c *\<^sub>R x) ` (convex hull s)"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3107
by(simp only: image_image[THEN sym] convex_hull_scaling convex_hull_translation)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3108
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3109
subsection {* Convexity of cone hulls *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3110
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3111
lemma convex_cone_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3112
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3113
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3114
shows "convex (cone hull S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3115
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3116
{ fix x y assume xy_def: "x : cone hull S & y : cone hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3117
  hence "S ~= {}" using cone_hull_empty_iff[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3118
  fix u v assume uv_def: "u>=0 & v>=0 & (u :: real)+v=1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3119
  hence *: "u *\<^sub>R x : cone hull S & v *\<^sub>R y : cone hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3120
     using cone_cone_hull[of S] xy_def cone_def[of "cone hull S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3121
  from * obtain cx xx where x_def: "u *\<^sub>R x = cx *\<^sub>R xx & (cx :: real)>=0 & xx : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3122
     using cone_hull_expl[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3123
  from * obtain cy yy where y_def: "v *\<^sub>R y = cy *\<^sub>R yy & (cy :: real)>=0 & yy : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3124
     using cone_hull_expl[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3125
  { assume "cx+cy<=0" hence "u *\<^sub>R x=0 & v *\<^sub>R y=0" using x_def y_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3126
    hence "u *\<^sub>R x+ v *\<^sub>R y = 0" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3127
    hence "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" using cone_hull_contains_0[of S] `S ~= {}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3128
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3129
  moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3130
  { assume "cx+cy>0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3131
    hence "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3132
      using assms mem_convex_alt[of S xx yy cx cy] x_def y_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3133
    hence "cx *\<^sub>R xx + cy *\<^sub>R yy : cone hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3134
      using mem_cone_hull[of "(cx/(cx+cy)) *\<^sub>R xx + (cy/(cx+cy)) *\<^sub>R yy" S "cx+cy"]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3135
      `cx+cy>0` by (auto simp add: scaleR_right_distrib)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3136
    hence "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" using x_def y_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3137
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3138
  moreover have "(cx+cy<=0) | (cx+cy>0)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3139
  ultimately have "u *\<^sub>R x+ v *\<^sub>R y : cone hull S" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3140
} from this show ?thesis unfolding convex_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3141
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3142
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3143
lemma cone_convex_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3144
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3145
assumes "cone S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3146
shows "cone (convex hull S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3147
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3148
{ assume "S = {}" hence ?thesis by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3149
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3150
{ assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3151
  { fix c assume "(c :: real)>0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3152
    hence "op *\<^sub>R c ` (convex hull S) = convex hull (op *\<^sub>R c ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3153
       using convex_hull_scaling[of _ S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3154
    also have "...=convex hull S" using * `c>0` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3155
    finally have "op *\<^sub>R c ` (convex hull S) = convex hull S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3156
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3157
  hence "0 : convex hull S & (!c. c>0 --> (op *\<^sub>R c ` (convex hull S)) = (convex hull S))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3158
     using * hull_subset[of S convex] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3159
  hence ?thesis using `S ~= {}` cone_iff[of "convex hull S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3160
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3161
ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3162
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  3163
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3164
subsection {* Convex set as intersection of halfspaces. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3165
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3166
lemma convex_halfspace_intersection:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3167
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3168
  assumes "closed s" "convex s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3169
  shows "s = \<Inter> {h. s \<subseteq> h \<and> (\<exists>a b. h = {x. inner a x \<le> b})}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  3170
  apply(rule set_eqI, rule) unfolding Inter_iff Ball_def mem_Collect_eq apply(rule,rule,erule conjE) proof- 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3171
  fix x  assume "\<forall>xa. s \<subseteq> xa \<and> (\<exists>a b. xa = {x. inner a x \<le> b}) \<longrightarrow> x \<in> xa"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3172
  hence "\<forall>a b. s \<subseteq> {x. inner a x \<le> b} \<longrightarrow> x \<in> {x. inner a x \<le> b}" by blast
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3173
  thus "x\<in>s" apply(rule_tac ccontr) apply(drule separating_hyperplane_closed_point[OF assms(2,1)])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3174
    apply(erule exE)+ apply(erule_tac x="-a" in allE, erule_tac x="-b" in allE) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3175
qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3176
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3177
subsection {* Radon's theorem (from Lars Schewe). *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3178
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3179
lemma radon_ex_lemma:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3180
  assumes "finite c" "affine_dependent c"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3181
  shows "\<exists>u. setsum u c = 0 \<and> (\<exists>v\<in>c. u v \<noteq> 0) \<and> setsum (\<lambda>v. u v *\<^sub>R v) c = 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3182
proof- from assms(2)[unfolded affine_dependent_explicit] guess s .. then guess u ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3183
  thus ?thesis apply(rule_tac x="\<lambda>v. if v\<in>s then u v else 0" in exI) unfolding if_smult scaleR_zero_left
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3184
    and setsum_restrict_set[OF assms(1), THEN sym] by(auto simp add: Int_absorb1) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3185
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3186
lemma radon_s_lemma:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3187
  assumes "finite s" "setsum f s = (0::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3188
  shows "setsum f {x\<in>s. 0 < f x} = - setsum f {x\<in>s. f x < 0}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3189
proof- have *:"\<And>x. (if f x < 0 then f x else 0) + (if 0 < f x then f x else 0) = f x" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3190
  show ?thesis unfolding real_add_eq_0_iff[THEN sym] and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3191
    using assms(2) by assumption qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3192
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3193
lemma radon_v_lemma:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3194
  assumes "finite s" "setsum f s = 0" "\<forall>x. g x = (0::real) \<longrightarrow> f x = (0::'a::euclidean_space)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3195
  shows "(setsum f {x\<in>s. 0 < g x}) = - setsum f {x\<in>s. g x < 0}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3196
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3197
  have *:"\<And>x. (if 0 < g x then f x else 0) + (if g x < 0 then f x else 0) = f x" using assms(3) by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3198
  show ?thesis unfolding eq_neg_iff_add_eq_0 and setsum_restrict_set''[OF assms(1)] and setsum_addf[THEN sym] and *
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3199
    using assms(2) by assumption qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3200
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3201
lemma radon_partition:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3202
  assumes "finite c" "affine_dependent c"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3203
  shows "\<exists>m p. m \<inter> p = {} \<and> m \<union> p = c \<and> (convex hull m) \<inter> (convex hull p) \<noteq> {}" proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3204
  obtain u v where uv:"setsum u c = 0" "v\<in>c" "u v \<noteq> 0"  "(\<Sum>v\<in>c. u v *\<^sub>R v) = 0" using radon_ex_lemma[OF assms] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3205
  have fin:"finite {x \<in> c. 0 < u x}" "finite {x \<in> c. 0 > u x}" using assms(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3206
  def z \<equiv> "(inverse (setsum u {x\<in>c. u x > 0})) *\<^sub>R setsum (\<lambda>x. u x *\<^sub>R x) {x\<in>c. u x > 0}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3207
  have "setsum u {x \<in> c. 0 < u x} \<noteq> 0" proof(cases "u v \<ge> 0")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3208
    case False hence "u v < 0" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3209
    thus ?thesis proof(cases "\<exists>w\<in>{x \<in> c. 0 < u x}. u w > 0") 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3210
      case True thus ?thesis using setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3211
    next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3212
      case False hence "setsum u c \<le> setsum (\<lambda>x. if x=v then u v else 0) c" apply(rule_tac setsum_mono) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3213
      thus ?thesis unfolding setsum_delta[OF assms(1)] using uv(2) and `u v < 0` and uv(1) by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3214
  qed (insert setsum_nonneg_eq_0_iff[of _ u, OF fin(1)] uv(2-3), auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3215
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 36725
diff changeset
  3216
  hence *:"setsum u {x\<in>c. u x > 0} > 0" unfolding less_le apply(rule_tac conjI, rule_tac setsum_nonneg) by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3217
  moreover have "setsum u ({x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}) = setsum u c"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3218
    "(\<Sum>x\<in>{x \<in> c. 0 < u x} \<union> {x \<in> c. u x < 0}. u x *\<^sub>R x) = (\<Sum>x\<in>c. u x *\<^sub>R x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3219
    using assms(1) apply(rule_tac[!] setsum_mono_zero_left) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3220
  hence "setsum u {x \<in> c. 0 < u x} = - setsum u {x \<in> c. 0 > u x}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3221
   "(\<Sum>x\<in>{x \<in> c. 0 < u x}. u x *\<^sub>R x) = - (\<Sum>x\<in>{x \<in> c. 0 > u x}. u x *\<^sub>R x)" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3222
    unfolding eq_neg_iff_add_eq_0 using uv(1,4) by (auto simp add:  setsum_Un_zero[OF fin, THEN sym]) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3223
  moreover have "\<forall>x\<in>{v \<in> c. u v < 0}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * - u x" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3224
    apply (rule) apply (rule mult_nonneg_nonneg) using * by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3225
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3226
  ultimately have "z \<in> convex hull {v \<in> c. u v \<le> 0}" unfolding convex_hull_explicit mem_Collect_eq
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3227
    apply(rule_tac x="{v \<in> c. u v < 0}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3228
    using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3229
    by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3230
  moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3231
    apply (rule) apply (rule mult_nonneg_nonneg) using * by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3232
  hence "z \<in> convex hull {v \<in> c. u v > 0}" unfolding convex_hull_explicit mem_Collect_eq
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3233
    apply(rule_tac x="{v \<in> c. 0 < u v}" in exI, rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * u y" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3234
    using assms(1) unfolding scaleR_scaleR[THEN sym] scaleR_right.setsum [symmetric] and z_def using *
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3235
    by(auto simp add: setsum_negf mult_right.setsum[THEN sym])
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3236
  ultimately show ?thesis apply(rule_tac x="{v\<in>c. u v \<le> 0}" in exI, rule_tac x="{v\<in>c. u v > 0}" in exI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3237
qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3238
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3239
lemma radon: assumes "affine_dependent c"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3240
  obtains m p where "m\<subseteq>c" "p\<subseteq>c" "m \<inter> p = {}" "(convex hull m) \<inter> (convex hull p) \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3241
proof- from assms[unfolded affine_dependent_explicit] guess s .. then guess u ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3242
  hence *:"finite s" "affine_dependent s" and s:"s \<subseteq> c" unfolding affine_dependent_explicit by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3243
  from radon_partition[OF *] guess m .. then guess p ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3244
  thus ?thesis apply(rule_tac that[of p m]) using s by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3245
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3246
subsection {* Helly's theorem. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3247
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3248
lemma helly_induct: fixes f::"('a::euclidean_space) set set"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3249
  assumes "card f = n" "n \<ge> DIM('a) + 1"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3250
  "\<forall>s\<in>f. convex s" "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3251
  shows "\<Inter> f \<noteq> {}"
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33714
diff changeset
  3252
using assms proof(induct n arbitrary: f)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3253
case (Suc n)
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33714
diff changeset
  3254
have "finite f" using `card f = Suc n` by (auto intro: card_ge_0_finite)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3255
show "\<Inter> f \<noteq> {}" apply(cases "n = DIM('a)") apply(rule Suc(5)[rule_format])
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33714
diff changeset
  3256
  unfolding `card f = Suc n` proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3257
  assume ng:"n \<noteq> DIM('a)" hence "\<exists>X. \<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" apply(rule_tac bchoice) unfolding ex_in_conv
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33714
diff changeset
  3258
    apply(rule, rule Suc(1)[rule_format]) unfolding card_Diff_singleton_if[OF `finite f`] `card f = Suc n`
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33714
diff changeset
  3259
    defer defer apply(rule Suc(4)[rule_format]) defer apply(rule Suc(5)[rule_format]) using Suc(3) `finite f` by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3260
  then obtain X where X:"\<forall>s\<in>f. X s \<in> \<Inter>(f - {s})" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3261
  show ?thesis proof(cases "inj_on X f")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3262
    case False then obtain s t where st:"s\<noteq>t" "s\<in>f" "t\<in>f" "X s = X t" unfolding inj_on_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3263
    hence *:"\<Inter> f = \<Inter> (f - {s}) \<inter> \<Inter> (f - {t})" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3264
    show ?thesis unfolding * unfolding ex_in_conv[THEN sym] apply(rule_tac x="X s" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3265
      apply(rule, rule X[rule_format]) using X st by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3266
  next case True then obtain m p where mp:"m \<inter> p = {}" "m \<union> p = X ` f" "convex hull m \<inter> convex hull p \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3267
      using radon_partition[of "X ` f"] and affine_dependent_biggerset[of "X ` f"]
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33714
diff changeset
  3268
      unfolding card_image[OF True] and `card f = Suc n` using Suc(3) `finite f` and ng by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3269
    have "m \<subseteq> X ` f" "p \<subseteq> X ` f" using mp(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3270
    then obtain g h where gh:"m = X ` g" "p = X ` h" "g \<subseteq> f" "h \<subseteq> f" unfolding subset_image_iff by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3271
    hence "f \<union> (g \<union> h) = f" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3272
    hence f:"f = g \<union> h" using inj_on_Un_image_eq_iff[of X f "g \<union> h"] and True
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3273
      unfolding mp(2)[unfolded image_Un[THEN sym] gh] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3274
    have *:"g \<inter> h = {}" using mp(1) unfolding gh using inj_on_image_Int[OF True gh(3,4)] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3275
    have "convex hull (X ` h) \<subseteq> \<Inter> g" "convex hull (X ` g) \<subseteq> \<Inter> h"
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33714
diff changeset
  3276
      apply(rule_tac [!] hull_minimal) using Suc gh(3-4)  unfolding mem_def unfolding subset_eq
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3277
      apply(rule_tac [2] convex_Inter, rule_tac [4] convex_Inter) apply rule prefer 3 apply rule proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3278
      fix x assume "x\<in>X ` g" then guess y unfolding image_iff ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3279
      thus "x\<in>\<Inter>h" using X[THEN bspec[where x=y]] using * f by auto next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3280
      fix x assume "x\<in>X ` h" then guess y unfolding image_iff ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3281
      thus "x\<in>\<Inter>g" using X[THEN bspec[where x=y]] using * f by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3282
    qed(auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3283
    thus ?thesis unfolding f using mp(3)[unfolded gh] by blast qed
37647
a5400b94d2dd minimize dependencies on Numeral_Type
huffman
parents: 37489
diff changeset
  3284
qed(auto) qed(auto)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3285
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3286
lemma helly: fixes f::"('a::euclidean_space) set set"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3287
  assumes "card f \<ge> DIM('a) + 1" "\<forall>s\<in>f. convex s"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3288
          "\<forall>t\<subseteq>f. card t = DIM('a) + 1 \<longrightarrow> \<Inter> t \<noteq> {}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3289
  shows "\<Inter> f \<noteq>{}"
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33714
diff changeset
  3290
  apply(rule helly_induct) using assms by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3291
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3292
subsection {* Homeomorphism of all convex compact sets with nonempty interior. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3293
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3294
lemma compact_frontier_line_lemma:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3295
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3296
  assumes "compact s" "0 \<in> s" "x \<noteq> 0" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3297
  obtains u where "0 \<le> u" "(u *\<^sub>R x) \<in> frontier s" "\<forall>v>u. (v *\<^sub>R x) \<notin> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3298
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3299
  obtain b where b:"b>0" "\<forall>x\<in>s. norm x \<le> b" using compact_imp_bounded[OF assms(1), unfolded bounded_pos] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3300
  let ?A = "{y. \<exists>u. 0 \<le> u \<and> u \<le> b / norm(x) \<and> (y = u *\<^sub>R x)}"
36431
340755027840 move definitions and theorems for type real^1 to separate theory file
huffman
parents: 36365
diff changeset
  3301
  have A:"?A = (\<lambda>u. u *\<^sub>R x) ` {0 .. b / norm x}"
340755027840 move definitions and theorems for type real^1 to separate theory file
huffman
parents: 36365
diff changeset
  3302
    by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3303
  have *:"\<And>x A B. x\<in>A \<Longrightarrow> x\<in>B \<Longrightarrow> A\<inter>B \<noteq> {}" by blast
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3304
  have "compact ?A" unfolding A apply(rule compact_continuous_image, rule continuous_at_imp_continuous_on)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3305
    apply(rule, rule continuous_vmul)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3306
    apply(rule continuous_at_id) by(rule compact_interval)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3307
  moreover have "{y. \<exists>u\<ge>0. u \<le> b / norm x \<and> y = u *\<^sub>R x} \<inter> s \<noteq> {}" apply(rule *[OF _ assms(2)])
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3308
    unfolding mem_Collect_eq using `b>0` assms(3) by(auto intro!: divide_nonneg_pos)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3309
  ultimately obtain u y where obt: "u\<ge>0" "u \<le> b / norm x" "y = u *\<^sub>R x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3310
    "y\<in>?A" "y\<in>s" "\<forall>z\<in>?A \<inter> s. dist 0 z \<le> dist 0 y" using distance_attains_sup[OF compact_inter[OF _ assms(1), of ?A], of 0] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3311
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3312
  have "norm x > 0" using assms(3)[unfolded zero_less_norm_iff[THEN sym]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3313
  { fix v assume as:"v > u" "v *\<^sub>R x \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3314
    hence "v \<le> b / norm x" using b(2)[rule_format, OF as(2)] 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3315
      using `u\<ge>0` unfolding pos_le_divide_eq[OF `norm x > 0`] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3316
    hence "norm (v *\<^sub>R x) \<le> norm y" apply(rule_tac obt(6)[rule_format, unfolded dist_0_norm]) apply(rule IntI) defer 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3317
      apply(rule as(2)) unfolding mem_Collect_eq apply(rule_tac x=v in exI) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3318
      using as(1) `u\<ge>0` by(auto simp add:field_simps) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3319
    hence False unfolding obt(3) using `u\<ge>0` `norm x > 0` `v>u` by(auto simp add:field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3320
  } note u_max = this
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3321
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3322
  have "u *\<^sub>R x \<in> frontier s" unfolding frontier_straddle apply(rule,rule,rule) apply(rule_tac x="u *\<^sub>R x" in bexI) unfolding obt(3)[THEN sym]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3323
    prefer 3 apply(rule_tac x="(u + (e / 2) / norm x) *\<^sub>R x" in exI) apply(rule, rule) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3324
    fix e  assume "0 < e" and as:"(u + e / 2 / norm x) *\<^sub>R x \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3325
    hence "u + e / 2 / norm x > u" using`norm x > 0` by(auto simp del:zero_less_norm_iff intro!: divide_pos_pos)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3326
    thus False using u_max[OF _ as] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3327
  qed(insert `y\<in>s`, auto simp add: dist_norm scaleR_left_distrib obt(3))
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3328
  thus ?thesis by(metis that[of u] u_max obt(1))
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3329
qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3330
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3331
lemma starlike_compact_projective:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3332
  assumes "compact s" "cball (0::'a::euclidean_space) 1 \<subseteq> s "
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3333
  "\<forall>x\<in>s. \<forall>u. 0 \<le> u \<and> u < 1 \<longrightarrow> (u *\<^sub>R x) \<in> (s - frontier s )"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3334
  shows "s homeomorphic (cball (0::'a::euclidean_space) 1)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3335
proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3336
  have fs:"frontier s \<subseteq> s" apply(rule frontier_subset_closed) using compact_imp_closed[OF assms(1)] by simp
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3337
  def pi \<equiv> "\<lambda>x::'a. inverse (norm x) *\<^sub>R x"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3338
  have "0 \<notin> frontier s" unfolding frontier_straddle apply(rule ccontr) unfolding not_not apply(erule_tac x=1 in allE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3339
    using assms(2)[unfolded subset_eq Ball_def mem_cball] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3340
  have injpi:"\<And>x y. pi x = pi y \<and> norm x = norm y \<longleftrightarrow> x = y" unfolding pi_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3341
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3342
  have contpi:"continuous_on (UNIV - {0}) pi" apply(rule continuous_at_imp_continuous_on)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3343
    apply rule unfolding pi_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3344
    apply (rule continuous_mul)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3345
    apply (rule continuous_at_inv[unfolded o_def])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3346
    apply (rule continuous_at_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3347
    apply simp
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3348
    apply (rule continuous_at_id)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3349
    done
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3350
  def sphere \<equiv> "{x::'a. norm x = 1}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3351
  have pi:"\<And>x. x \<noteq> 0 \<Longrightarrow> pi x \<in> sphere" "\<And>x u. u>0 \<Longrightarrow> pi (u *\<^sub>R x) = pi x" unfolding pi_def sphere_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3352
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3353
  have "0\<in>s" using assms(2) and centre_in_cball[of 0 1] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3354
  have front_smul:"\<forall>x\<in>frontier s. \<forall>u\<ge>0. u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" proof(rule,rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3355
    fix x u assume x:"x\<in>frontier s" and "(0::real)\<le>u"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3356
    hence "x\<noteq>0" using `0\<notin>frontier s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3357
    obtain v where v:"0 \<le> v" "v *\<^sub>R x \<in> frontier s" "\<forall>w>v. w *\<^sub>R x \<notin> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3358
      using compact_frontier_line_lemma[OF assms(1) `0\<in>s` `x\<noteq>0`] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3359
    have "v=1" apply(rule ccontr) unfolding neq_iff apply(erule disjE) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3360
      assume "v<1" thus False using v(3)[THEN spec[where x=1]] using x and fs by auto next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3361
      assume "v>1" thus False using assms(3)[THEN bspec[where x="v *\<^sub>R x"], THEN spec[where x="inverse v"]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3362
        using v and x and fs unfolding inverse_less_1_iff by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3363
    show "u *\<^sub>R x \<in> s \<longleftrightarrow> u \<le> 1" apply rule  using v(3)[unfolded `v=1`, THEN spec[where x=u]] proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3364
      assume "u\<le>1" thus "u *\<^sub>R x \<in> s" apply(cases "u=1")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3365
        using assms(3)[THEN bspec[where x=x], THEN spec[where x=u]] using `0\<le>u` and x and fs by auto qed auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3366
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3367
  have "\<exists>surf. homeomorphism (frontier s) sphere pi surf"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3368
    apply(rule homeomorphism_compact) apply(rule compact_frontier[OF assms(1)])
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  3369
    apply(rule continuous_on_subset[OF contpi]) defer apply(rule set_eqI,rule) 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3370
    unfolding inj_on_def prefer 3 apply(rule,rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3371
  proof- fix x assume "x\<in>pi ` frontier s" then obtain y where "y\<in>frontier s" "x = pi y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3372
    thus "x \<in> sphere" using pi(1)[of y] and `0 \<notin> frontier s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3373
  next fix x assume "x\<in>sphere" hence "norm x = 1" "x\<noteq>0" unfolding sphere_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3374
    then obtain u where "0 \<le> u" "u *\<^sub>R x \<in> frontier s" "\<forall>v>u. v *\<^sub>R x \<notin> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3375
      using compact_frontier_line_lemma[OF assms(1) `0\<in>s`, of x] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3376
    thus "x \<in> pi ` frontier s" unfolding image_iff le_less pi_def apply(rule_tac x="u *\<^sub>R x" in bexI) using `norm x = 1` `0\<notin>frontier s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3377
  next fix x y assume as:"x \<in> frontier s" "y \<in> frontier s" "pi x = pi y"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3378
    hence xys:"x\<in>s" "y\<in>s" using fs by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3379
    from as(1,2) have nor:"norm x \<noteq> 0" "norm y \<noteq> 0" using `0\<notin>frontier s` by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3380
    from nor have x:"x = norm x *\<^sub>R ((inverse (norm y)) *\<^sub>R y)" unfolding as(3)[unfolded pi_def, THEN sym] by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3381
    from nor have y:"y = norm y *\<^sub>R ((inverse (norm x)) *\<^sub>R x)" unfolding as(3)[unfolded pi_def] by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3382
    have "0 \<le> norm y * inverse (norm x)" "0 \<le> norm x * inverse (norm y)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3383
      unfolding divide_inverse[THEN sym] apply(rule_tac[!] divide_nonneg_pos) using nor by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3384
    hence "norm x = norm y" apply(rule_tac ccontr) unfolding neq_iff
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3385
      using x y and front_smul[THEN bspec, OF as(1), THEN spec[where x="norm y * (inverse (norm x))"]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3386
      using front_smul[THEN bspec, OF as(2), THEN spec[where x="norm x * (inverse (norm y))"]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3387
      using xys nor by(auto simp add:field_simps divide_le_eq_1 divide_inverse[THEN sym])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3388
    thus "x = y" apply(subst injpi[THEN sym]) using as(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3389
  qed(insert `0 \<notin> frontier s`, auto)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3390
  then obtain surf where surf:"\<forall>x\<in>frontier s. surf (pi x) = x"  "pi ` frontier s = sphere" "continuous_on (frontier s) pi"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3391
    "\<forall>y\<in>sphere. pi (surf y) = y" "surf ` sphere = frontier s" "continuous_on sphere surf" unfolding homeomorphism_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3392
  
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3393
  have cont_surfpi:"continuous_on (UNIV -  {0}) (surf \<circ> pi)" apply(rule continuous_on_compose, rule contpi)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3394
    apply(rule continuous_on_subset[of sphere], rule surf(6)) using pi(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3395
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3396
  { fix x assume as:"x \<in> cball (0::'a) 1"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3397
    have "norm x *\<^sub>R surf (pi x) \<in> s" proof(cases "x=0 \<or> norm x = 1") 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3398
      case False hence "pi x \<in> sphere" "norm x < 1" using pi(1)[of x] as by(auto simp add: dist_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3399
      thus ?thesis apply(rule_tac assms(3)[rule_format, THEN DiffD1])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3400
        apply(rule_tac fs[unfolded subset_eq, rule_format])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3401
        unfolding surf(5)[THEN sym] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3402
    next case True thus ?thesis apply rule defer unfolding pi_def apply(rule fs[unfolded subset_eq, rule_format])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3403
        unfolding  surf(5)[unfolded sphere_def, THEN sym] using `0\<in>s` by auto qed } note hom = this
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3404
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3405
  { fix x assume "x\<in>s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3406
    hence "x \<in> (\<lambda>x. norm x *\<^sub>R surf (pi x)) ` cball 0 1" proof(cases "x=0")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3407
      case True show ?thesis unfolding image_iff True apply(rule_tac x=0 in bexI) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3408
    next let ?a = "inverse (norm (surf (pi x)))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3409
      case False hence invn:"inverse (norm x) \<noteq> 0" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3410
      from False have pix:"pi x\<in>sphere" using pi(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3411
      hence "pi (surf (pi x)) = pi x" apply(rule_tac surf(4)[rule_format]) by assumption
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3412
      hence **:"norm x *\<^sub>R (?a *\<^sub>R surf (pi x)) = x" apply(rule_tac scaleR_left_imp_eq[OF invn]) unfolding pi_def using invn by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3413
      hence *:"?a * norm x > 0" and"?a > 0" "?a \<noteq> 0" using surf(5) `0\<notin>frontier s` apply -
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3414
        apply(rule_tac mult_pos_pos) using False[unfolded zero_less_norm_iff[THEN sym]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3415
      have "norm (surf (pi x)) \<noteq> 0" using ** False by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3416
      hence "norm x = norm ((?a * norm x) *\<^sub>R surf (pi x))"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3417
        unfolding norm_scaleR abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3418
      moreover have "pi x = pi ((inverse (norm (surf (pi x))) * norm x) *\<^sub>R surf (pi x))" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3419
        unfolding pi(2)[OF *] surf(4)[rule_format, OF pix] ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3420
      moreover have "surf (pi x) \<in> frontier s" using surf(5) pix by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3421
      hence "dist 0 (inverse (norm (surf (pi x))) *\<^sub>R x) \<le> 1" unfolding dist_norm
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3422
        using ** and * using front_smul[THEN bspec[where x="surf (pi x)"], THEN spec[where x="norm x * ?a"]]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3423
        using False `x\<in>s` by(auto simp add:field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3424
      ultimately show ?thesis unfolding image_iff apply(rule_tac x="inverse (norm (surf(pi x))) *\<^sub>R x" in bexI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3425
        apply(subst injpi[THEN sym]) unfolding abs_mult abs_norm_cancel abs_of_pos[OF `?a > 0`]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3426
        unfolding pi(2)[OF `?a > 0`] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3427
    qed } note hom2 = this
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3428
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3429
  show ?thesis apply(subst homeomorphic_sym) apply(rule homeomorphic_compact[where f="\<lambda>x. norm x *\<^sub>R surf (pi x)"])
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  3430
    apply(rule compact_cball) defer apply(rule set_eqI, rule, erule imageE, drule hom)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3431
    prefer 4 apply(rule continuous_at_imp_continuous_on, rule) apply(rule_tac [3] hom2) proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3432
    fix x::"'a" assume as:"x \<in> cball 0 1"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3433
    thus "continuous (at x) (\<lambda>x. norm x *\<^sub>R surf (pi x))" proof(cases "x=0")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3434
      case False thus ?thesis apply(rule_tac continuous_mul, rule_tac continuous_at_norm)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3435
        using cont_surfpi unfolding continuous_on_eq_continuous_at[OF open_delete[OF open_UNIV]] o_def by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3436
    next obtain B where B:"\<forall>x\<in>s. norm x \<le> B" using compact_imp_bounded[OF assms(1)] unfolding bounded_iff by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3437
      hence "B > 0" using assms(2) unfolding subset_eq apply(erule_tac x="basis 0" in ballE) defer 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3438
        apply(erule_tac x="basis 0" in ballE)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3439
        unfolding Ball_def mem_cball dist_norm using DIM_positive[where 'a='a]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3440
        by(auto simp add:norm_basis[unfolded One_nat_def])
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3441
      case True show ?thesis unfolding True continuous_at Lim_at apply(rule,rule) apply(rule_tac x="e / B" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3442
        apply(rule) apply(rule divide_pos_pos) prefer 3 apply(rule,rule,erule conjE)
36586
4fa71a69d5b2 remove redundant lemma norm_0
huffman
parents: 36583
diff changeset
  3443
        unfolding norm_zero scaleR_zero_left dist_norm diff_0_right norm_scaleR abs_norm_cancel proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3444
        fix e and x::"'a" assume as:"norm x < e / B" "0 < norm x" "0<e"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3445
        hence "surf (pi x) \<in> frontier s" using pi(1)[of x] unfolding surf(5)[THEN sym] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3446
        hence "norm (surf (pi x)) \<le> B" using B fs by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3447
        hence "norm x * norm (surf (pi x)) \<le> norm x * B" using as(2) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3448
        also have "\<dots> < e / B * B" apply(rule mult_strict_right_mono) using as(1) `B>0` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3449
        also have "\<dots> = e" using `B>0` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3450
        finally show "norm x * norm (surf (pi x)) < e" by assumption
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3451
      qed(insert `B>0`, auto) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3452
  next { fix x assume as:"surf (pi x) = 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3453
      have "x = 0" proof(rule ccontr)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3454
        assume "x\<noteq>0" hence "pi x \<in> sphere" using pi(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3455
        hence "surf (pi x) \<in> frontier s" using surf(5) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3456
        thus False using `0\<notin>frontier s` unfolding as by simp qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3457
    } note surf_0 = this
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3458
    show "inj_on (\<lambda>x. norm x *\<^sub>R surf (pi x)) (cball 0 1)" unfolding inj_on_def proof(rule,rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3459
      fix x y assume as:"x \<in> cball 0 1" "y \<in> cball 0 1" "norm x *\<^sub>R surf (pi x) = norm y *\<^sub>R surf (pi y)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3460
      thus "x=y" proof(cases "x=0 \<or> y=0") 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3461
        case True thus ?thesis using as by(auto elim: surf_0) next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3462
        case False
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3463
        hence "pi (surf (pi x)) = pi (surf (pi y))" using as(3)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3464
          using pi(2)[of "norm x" "surf (pi x)"] pi(2)[of "norm y" "surf (pi y)"] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3465
        moreover have "pi x \<in> sphere" "pi y \<in> sphere" using pi(1) False by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3466
        ultimately have *:"pi x = pi y" using surf(4)[THEN bspec[where x="pi x"]] surf(4)[THEN bspec[where x="pi y"]] by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3467
        moreover have "norm x = norm y" using as(3)[unfolded *] using False by(auto dest:surf_0)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3468
        ultimately show ?thesis using injpi by auto qed qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3469
  qed auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3470
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3471
lemma homeomorphic_convex_compact_lemma: fixes s::"('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3472
  assumes "convex s" "compact s" "cball 0 1 \<subseteq> s"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3473
  shows "s homeomorphic (cball (0::'a) 1)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3474
  apply(rule starlike_compact_projective[OF assms(2-3)]) proof(rule,rule,rule,erule conjE)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3475
  fix x u assume as:"x \<in> s" "0 \<le> u" "u < (1::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3476
  hence "u *\<^sub>R x \<in> interior s" unfolding interior_def mem_Collect_eq
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3477
    apply(rule_tac x="ball (u *\<^sub>R x) (1 - u)" in exI) apply(rule, rule open_ball)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3478
    unfolding centre_in_ball apply rule defer apply(rule) unfolding mem_ball proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3479
    fix y assume "dist (u *\<^sub>R x) y < 1 - u"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3480
    hence "inverse (1 - u) *\<^sub>R (y - u *\<^sub>R x) \<in> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3481
      using assms(3) apply(erule_tac subsetD) unfolding mem_cball dist_commute dist_norm
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3482
      unfolding group_add_class.diff_0 group_add_class.diff_0_right norm_minus_cancel norm_scaleR
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3483
      apply (rule mult_left_le_imp_le[of "1 - u"])
36844
5f9385ecc1a7 Removed usage of normalizating locales.
hoelzl
parents: 36778
diff changeset
  3484
      unfolding mult_assoc[symmetric] using `u<1` by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3485
    thus "y \<in> s" using assms(1)[unfolded convex_def, rule_format, of "inverse(1 - u) *\<^sub>R (y - u *\<^sub>R x)" x "1 - u" u]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3486
      using as unfolding scaleR_scaleR by auto qed auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3487
  thus "u *\<^sub>R x \<in> s - frontier s" using frontier_def and interior_subset by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3488
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3489
lemma homeomorphic_convex_compact_cball: fixes e::real and s::"('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3490
  assumes "convex s" "compact s" "interior s \<noteq> {}" "0 < e"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3491
  shows "s homeomorphic (cball (b::'a) e)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3492
proof- obtain a where "a\<in>interior s" using assms(3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3493
  then obtain d where "d>0" and d:"cball a d \<subseteq> s" unfolding mem_interior_cball by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3494
  let ?d = "inverse d" and ?n = "0::'a"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3495
  have "cball ?n 1 \<subseteq> (\<lambda>x. inverse d *\<^sub>R (x - a)) ` s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3496
    apply(rule, rule_tac x="d *\<^sub>R x + a" in image_eqI) defer
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3497
    apply(rule d[unfolded subset_eq, rule_format]) using `d>0` unfolding mem_cball dist_norm
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3498
    by(auto simp add: mult_right_le_one_le)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3499
  hence "(\<lambda>x. inverse d *\<^sub>R (x - a)) ` s homeomorphic cball ?n 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3500
    using homeomorphic_convex_compact_lemma[of "(\<lambda>x. ?d *\<^sub>R -a + ?d *\<^sub>R x) ` s", OF convex_affinity compact_affinity]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3501
    using assms(1,2) by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3502
  thus ?thesis apply(rule_tac homeomorphic_trans[OF _ homeomorphic_balls(2)[of 1 _ ?n]])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3503
    apply(rule homeomorphic_trans[OF homeomorphic_affinity[of "?d" s "?d *\<^sub>R -a"]])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3504
    using `d>0` `e>0` by(auto simp add: uminus_add_conv_diff scaleR_right_diff_distrib) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3505
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3506
lemma homeomorphic_convex_compact: fixes s::"('a::euclidean_space) set" and t::"('a) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3507
  assumes "convex s" "compact s" "interior s \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3508
          "convex t" "compact t" "interior t \<noteq> {}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3509
  shows "s homeomorphic t"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3510
  using assms by(meson zero_less_one homeomorphic_trans homeomorphic_convex_compact_cball homeomorphic_sym)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3511
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3512
subsection {* Epigraphs of convex functions. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3513
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3514
definition "epigraph s (f::_ \<Rightarrow> real) = {xy. fst xy \<in> s \<and> f (fst xy) \<le> snd xy}"
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3515
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3516
lemma mem_epigraph: "(x, y) \<in> epigraph s f \<longleftrightarrow> x \<in> s \<and> f x \<le> y" unfolding epigraph_def by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3517
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  3518
(** This might break sooner or later. In fact it did already once. **)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3519
lemma convex_epigraph: 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3520
  "convex(epigraph s f) \<longleftrightarrow> convex_on s f \<and> convex s"
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3521
  unfolding convex_def convex_on_def
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3522
  unfolding Ball_def split_paired_All epigraph_def
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3523
  unfolding mem_Collect_eq fst_conv snd_conv fst_add snd_add fst_scaleR snd_scaleR Ball_def[symmetric]
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  3524
  apply safe defer apply(erule_tac x=x in allE,erule_tac x="f x" in allE) apply safe
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  3525
  apply(erule_tac x=xa in allE,erule_tac x="f xa" in allE) prefer 3
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3526
  apply(rule_tac y="u * f a + v * f aa" in order_trans) defer by(auto intro!:mult_left_mono add_mono)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3527
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3528
lemma convex_epigraphI:
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3529
  "convex_on s f \<Longrightarrow> convex s \<Longrightarrow> convex(epigraph s f)"
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3530
unfolding convex_epigraph by auto
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3531
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3532
lemma convex_epigraph_convex:
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3533
  "convex s \<Longrightarrow> convex_on s f \<longleftrightarrow> convex(epigraph s f)"
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3534
by(simp add: convex_epigraph)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3535
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3536
subsection {* Use this to derive general bound property of convex function. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3537
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3538
lemma convex_on:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3539
  assumes "convex s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3540
  shows "convex_on s f \<longleftrightarrow> (\<forall>k u x. (\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> x i \<in> s) \<and> setsum u {1..k} = 1 \<longrightarrow>
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3541
   f (setsum (\<lambda>i. u i *\<^sub>R x i) {1..k} ) \<le> setsum (\<lambda>i. u i * f(x i)) {1..k} ) "
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3542
  unfolding convex_epigraph_convex[OF assms] convex epigraph_def Ball_def mem_Collect_eq
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3543
  unfolding fst_setsum snd_setsum fst_scaleR snd_scaleR
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3544
  apply safe
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3545
  apply (drule_tac x=k in spec)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3546
  apply (drule_tac x=u in spec)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3547
  apply (drule_tac x="\<lambda>i. (x i, f (x i))" in spec)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3548
  apply simp
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3549
  using assms[unfolded convex] apply simp
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 36725
diff changeset
  3550
  apply(rule_tac y="\<Sum>i = 1..k. u i * f (fst (x i))" in order_trans)
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3551
  defer apply(rule setsum_mono) apply(erule_tac x=i in allE) unfolding real_scaleR_def
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  3552
  apply(rule mult_left_mono)using assms[unfolded convex] by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3553
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3554
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3555
subsection {* Convexity of general and special intervals. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3556
37732
6432bf0d7191 generalize type of is_interval to class euclidean_space
huffman
parents: 37673
diff changeset
  3557
lemma convexI: (* TODO: move to Library/Convex.thy *)
6432bf0d7191 generalize type of is_interval to class euclidean_space
huffman
parents: 37673
diff changeset
  3558
  assumes "\<And>x y u v. \<lbrakk>x \<in> s; y \<in> s; 0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s"
6432bf0d7191 generalize type of is_interval to class euclidean_space
huffman
parents: 37673
diff changeset
  3559
  shows "convex s"
6432bf0d7191 generalize type of is_interval to class euclidean_space
huffman
parents: 37673
diff changeset
  3560
using assms unfolding convex_def by fast
6432bf0d7191 generalize type of is_interval to class euclidean_space
huffman
parents: 37673
diff changeset
  3561
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3562
lemma is_interval_convex:
37732
6432bf0d7191 generalize type of is_interval to class euclidean_space
huffman
parents: 37673
diff changeset
  3563
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3564
  assumes "is_interval s" shows "convex s"
37732
6432bf0d7191 generalize type of is_interval to class euclidean_space
huffman
parents: 37673
diff changeset
  3565
proof (rule convexI)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3566
  fix x y u v assume as:"x \<in> s" "y \<in> s" "0 \<le> u" "0 \<le> v" "u + v = (1::real)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3567
  hence *:"u = 1 - v" "1 - v \<ge> 0" and **:"v = 1 - u" "1 - u \<ge> 0" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3568
  { fix a b assume "\<not> b \<le> u * a + v * b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3569
    hence "u * a < (1 - v) * b" unfolding not_le using as(4) by(auto simp add: field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3570
    hence "a < b" unfolding * using as(4) *(2) apply(rule_tac mult_left_less_imp_less[of "1 - v"]) by(auto simp add: field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3571
    hence "a \<le> u * a + v * b" unfolding * using as(4) by (auto simp add: field_simps intro!:mult_right_mono)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3572
  } moreover
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3573
  { fix a b assume "\<not> u * a + v * b \<le> a"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3574
    hence "v * b > (1 - u) * a" unfolding not_le using as(4) by(auto simp add: field_simps)
36350
bc7982c54e37 dropped group_simps, ring_simps, field_eq_simps
haftmann
parents: 36341
diff changeset
  3575
    hence "a < b" unfolding * using as(4) apply(rule_tac mult_left_less_imp_less) by(auto simp add: field_simps)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3576
    hence "u * a + v * b \<le> b" unfolding ** using **(2) as(3) by(auto simp add: field_simps intro!:mult_right_mono) }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3577
  ultimately show "u *\<^sub>R x + v *\<^sub>R y \<in> s" apply- apply(rule assms[unfolded is_interval_def, rule_format, OF as(1,2)])
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3578
    using as(3-) DIM_positive[where 'a='a] by(auto simp add:euclidean_simps) qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3579
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3580
lemma is_interval_connected:
37732
6432bf0d7191 generalize type of is_interval to class euclidean_space
huffman
parents: 37673
diff changeset
  3581
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3582
  shows "is_interval s \<Longrightarrow> connected s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3583
  using is_interval_convex convex_connected by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3584
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3585
lemma convex_interval: "convex {a .. b}" "convex {a<..<b::'a::ordered_euclidean_space}"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3586
  apply(rule_tac[!] is_interval_convex) using is_interval_interval by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3587
36431
340755027840 move definitions and theorems for type real^1 to separate theory file
huffman
parents: 36365
diff changeset
  3588
(* FIXME: rewrite these lemmas without using vec1
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3589
subsection {* On @{text "real^1"}, @{text "is_interval"}, @{text "convex"} and @{text "connected"} are all equivalent. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3590
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3591
lemma is_interval_1:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3592
  "is_interval s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall> x. dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b \<longrightarrow> x \<in> s)"
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  3593
  unfolding is_interval_def forall_1 by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3594
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3595
lemma is_interval_connected_1: "is_interval s \<longleftrightarrow> connected (s::(real^1) set)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3596
  apply(rule, rule is_interval_connected, assumption) unfolding is_interval_1
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3597
  apply(rule,rule,rule,rule,erule conjE,rule ccontr) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3598
  fix a b x assume as:"connected s" "a \<in> s" "b \<in> s" "dest_vec1 a \<le> dest_vec1 x" "dest_vec1 x \<le> dest_vec1 b" "x\<notin>s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3599
  hence *:"dest_vec1 a < dest_vec1 x" "dest_vec1 x < dest_vec1 b" apply(rule_tac [!] ccontr) unfolding not_less by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3600
  let ?halfl = "{z. inner (basis 1) z < dest_vec1 x} " and ?halfr = "{z. inner (basis 1) z > dest_vec1 x} "
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3601
  { fix y assume "y \<in> s" have "y \<in> ?halfr \<union> ?halfl" apply(rule ccontr)
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  3602
    using as(6) `y\<in>s` by (auto simp add: inner_vector_def) }
34964
4e8be3c04d37 Replaced vec1 and dest_vec1 by abbreviation.
hoelzl
parents: 34915
diff changeset
  3603
  moreover have "a\<in>?halfl" "b\<in>?halfr" using * by (auto simp add: inner_vector_def)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3604
  hence "?halfl \<inter> s \<noteq> {}" "?halfr \<inter> s \<noteq> {}"  using as(2-3) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3605
  ultimately show False apply(rule_tac notE[OF as(1)[unfolded connected_def]])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3606
    apply(rule_tac x="?halfl" in exI, rule_tac x="?halfr" in exI) 
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3607
    apply(rule, rule open_halfspace_lt, rule, rule open_halfspace_gt)
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3608
    by(auto simp add: field_simps) qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3609
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3610
lemma is_interval_convex_1:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3611
  "is_interval s \<longleftrightarrow> convex (s::(real^1) set)" 
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3612
by(metis is_interval_convex convex_connected is_interval_connected_1)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3613
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3614
lemma convex_connected_1:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3615
  "connected s \<longleftrightarrow> convex (s::(real^1) set)" 
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3616
by(metis is_interval_convex convex_connected is_interval_connected_1)
36431
340755027840 move definitions and theorems for type real^1 to separate theory file
huffman
parents: 36365
diff changeset
  3617
*)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3618
subsection {* Another intermediate value theorem formulation. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3619
37673
f69f4b079275 generalize more lemmas from ordered_euclidean_space to euclidean_space
huffman
parents: 37647
diff changeset
  3620
lemma ivt_increasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3621
  assumes "a \<le> b" "continuous_on {a .. b} f" "(f a)$$k \<le> y" "y \<le> (f b)$$k"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3622
  shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3623
proof- have "f a \<in> f ` {a..b}" "f b \<in> f ` {a..b}" apply(rule_tac[!] imageI) 
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3624
    using assms(1) by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3625
  thus ?thesis using connected_ivt_component[of "f ` {a..b}" "f a" "f b" k y]
36431
340755027840 move definitions and theorems for type real^1 to separate theory file
huffman
parents: 36365
diff changeset
  3626
    using connected_continuous_image[OF assms(2) convex_connected[OF convex_real_interval(5)]]
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3627
    using assms by(auto intro!: imageI) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3628
37673
f69f4b079275 generalize more lemmas from ordered_euclidean_space to euclidean_space
huffman
parents: 37647
diff changeset
  3629
lemma ivt_increasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
36431
340755027840 move definitions and theorems for type real^1 to separate theory file
huffman
parents: 36365
diff changeset
  3630
  shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3631
   \<Longrightarrow> f a$$k \<le> y \<Longrightarrow> y \<le> f b$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3632
by(rule ivt_increasing_component_on_1)
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3633
  (auto simp add: continuous_at_imp_continuous_on)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3634
37673
f69f4b079275 generalize more lemmas from ordered_euclidean_space to euclidean_space
huffman
parents: 37647
diff changeset
  3635
lemma ivt_decreasing_component_on_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3636
  assumes "a \<le> b" "continuous_on {a .. b} f" "(f b)$$k \<le> y" "y \<le> (f a)$$k"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3637
  shows "\<exists>x\<in>{a..b}. (f x)$$k = y"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3638
  apply(subst neg_equal_iff_equal[THEN sym])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3639
  using ivt_increasing_component_on_1[of a b "\<lambda>x. - f x" k "- y"] using assms using continuous_on_neg
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3640
  by (auto simp add:euclidean_simps)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3641
37673
f69f4b079275 generalize more lemmas from ordered_euclidean_space to euclidean_space
huffman
parents: 37647
diff changeset
  3642
lemma ivt_decreasing_component_1: fixes f::"real \<Rightarrow> 'a::euclidean_space"
36431
340755027840 move definitions and theorems for type real^1 to separate theory file
huffman
parents: 36365
diff changeset
  3643
  shows "a \<le> b \<Longrightarrow> \<forall>x\<in>{a .. b}. continuous (at x) f
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3644
    \<Longrightarrow> f b$$k \<le> y \<Longrightarrow> y \<le> f a$$k \<Longrightarrow> \<exists>x\<in>{a..b}. (f x)$$k = y"
36071
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3645
by(rule ivt_decreasing_component_on_1)
c8ae8e56d42e tuned many proofs a bit
nipkow
parents: 35577
diff changeset
  3646
  (auto simp: continuous_at_imp_continuous_on)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3647
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3648
subsection {* A bound within a convex hull, and so an interval. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3649
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3650
lemma convex_on_convex_hull_bound:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3651
  assumes "convex_on (convex hull s) f" "\<forall>x\<in>s. f x \<le> b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3652
  shows "\<forall>x\<in> convex hull s. f x \<le> b" proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3653
  fix x assume "x\<in>convex hull s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3654
  then obtain k u v where obt:"\<forall>i\<in>{1..k::nat}. 0 \<le> u i \<and> v i \<in> s" "setsum u {1..k} = 1" "(\<Sum>i = 1..k. u i *\<^sub>R v i) = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3655
    unfolding convex_hull_indexed mem_Collect_eq by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3656
  have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b" using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3657
    unfolding setsum_left_distrib[THEN sym] obt(2) mult_1 apply(drule_tac meta_mp) apply(rule mult_left_mono)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3658
    using assms(2) obt(1) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3659
  thus "f x \<le> b" using assms(1)[unfolded convex_on[OF convex_convex_hull], rule_format, of k u v]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3660
    unfolding obt(2-3) using obt(1) and hull_subset[unfolded subset_eq, rule_format, of _ s] by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3661
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3662
lemma unit_interval_convex_hull:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3663
  "{0::'a::ordered_euclidean_space .. (\<chi>\<chi> i. 1)} = convex hull {x. \<forall>i<DIM('a). (x$$i = 0) \<or> (x$$i = 1)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3664
  (is "?int = convex hull ?points")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3665
proof- have 01:"{0,(\<chi>\<chi> i. 1)} \<subseteq> convex hull ?points" apply rule apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3666
  { fix n x assume "x\<in>{0::'a::ordered_euclidean_space .. \<chi>\<chi> i. 1}" "n \<le> DIM('a)" "card {i. i<DIM('a) \<and> x$$i \<noteq> 0} \<le> n" 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3667
  hence "x\<in>convex hull ?points" proof(induct n arbitrary: x)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3668
    case 0 hence "x = 0" apply(subst euclidean_eq) apply rule by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3669
    thus "x\<in>convex hull ?points" using 01 by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3670
  next
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3671
    case (Suc n) show "x\<in>convex hull ?points" proof(cases "{i. i<DIM('a) \<and> x$$i \<noteq> 0} = {}")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3672
      case True hence "x = 0" apply(subst euclidean_eq) by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3673
      thus "x\<in>convex hull ?points" using 01 by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3674
    next
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3675
      case False def xi \<equiv> "Min ((\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0})"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3676
      have "xi \<in> (\<lambda>i. x$$i) ` {i. i<DIM('a) \<and> x$$i \<noteq> 0}" unfolding xi_def apply(rule Min_in) using False by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3677
      then obtain i where i':"x$$i = xi" "x$$i \<noteq> 0" "i<DIM('a)" by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3678
      have i:"\<And>j. j<DIM('a) \<Longrightarrow> x$$j > 0 \<Longrightarrow> x$$i \<le> x$$j"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3679
        unfolding i'(1) xi_def apply(rule_tac Min_le) unfolding image_iff
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3680
        defer apply(rule_tac x=j in bexI) using i' by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3681
      have i01:"x$$i \<le> 1" "x$$i > 0" using Suc(2)[unfolded mem_interval,rule_format,of i]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3682
        using i'(2-) `x$$i \<noteq> 0` by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3683
      show ?thesis proof(cases "x$$i=1")
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3684
        case True have "\<forall>j\<in>{i. i<DIM('a) \<and> x$$i \<noteq> 0}. x$$j = 1" apply(rule, rule ccontr) unfolding mem_Collect_eq
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3685
        proof(erule conjE) fix j assume as:"x $$ j \<noteq> 0" "x $$ j \<noteq> 1" "j<DIM('a)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3686
          hence j:"x$$j \<in> {0<..<1}" using Suc(2) by(auto simp add: eucl_le[where 'a='a] elim!:allE[where x=j])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3687
          hence "x$$j \<in> op $$ x ` {i. i<DIM('a) \<and> x $$ i \<noteq> 0}" using as(3) by auto 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3688
          hence "x$$j \<ge> x$$i" unfolding i'(1) xi_def apply(rule_tac Min_le) by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3689
          thus False using True Suc(2) j by(auto simp add: elim!:ballE[where x=j]) qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3690
        thus "x\<in>convex hull ?points" apply(rule_tac hull_subset[unfolded subset_eq, rule_format])
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  3691
          by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3692
      next let ?y = "\<lambda>j. if x$$j = 0 then 0 else (x$$j - x$$i) / (1 - x$$i)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3693
        case False hence *:"x = x$$i *\<^sub>R (\<chi>\<chi> j. if x$$j = 0 then 0 else 1) + (1 - x$$i) *\<^sub>R (\<chi>\<chi> j. ?y j)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3694
          apply(subst euclidean_eq) by(auto simp add: field_simps euclidean_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3695
        { fix j assume j:"j<DIM('a)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3696
          have "x$$j \<noteq> 0 \<Longrightarrow> 0 \<le> (x $$ j - x $$ i) / (1 - x $$ i)" "(x $$ j - x $$ i) / (1 - x $$ i) \<le> 1"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3697
            apply(rule_tac divide_nonneg_pos) using i(1)[of j] using False i01
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3698
            using Suc(2)[unfolded mem_interval, rule_format, of j] using j
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3699
            by(auto simp add:field_simps euclidean_simps)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3700
          hence "0 \<le> ?y j \<and> ?y j \<le> 1" by auto }
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3701
        moreover have "i\<in>{j. j<DIM('a) \<and> x$$j \<noteq> 0} - {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3702
          using i01 using i'(3) by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3703
        hence "{j. j<DIM('a) \<and> x$$j \<noteq> 0} \<noteq> {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0}" using i'(3) by blast
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3704
        hence **:"{j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<subset> {j. j<DIM('a) \<and> x$$j \<noteq> 0}" apply - apply rule 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3705
          by( auto simp add:euclidean_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3706
        have "card {j. j<DIM('a) \<and> ((\<chi>\<chi> j. ?y j)::'a) $$ j \<noteq> 0} \<le> n"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3707
          using less_le_trans[OF psubset_card_mono[OF _ **] Suc(4)] by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3708
        ultimately show ?thesis apply(subst *) apply(rule convex_convex_hull[unfolded convex_def, rule_format])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3709
          apply(rule_tac hull_subset[unfolded subset_eq, rule_format]) defer apply(rule Suc(1))
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  3710
          unfolding mem_interval using i01 Suc(3) by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3711
      qed qed qed } note * = this
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3712
  have **:"DIM('a) = card {..<DIM('a)}" by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3713
  show ?thesis apply rule defer apply(rule hull_minimal) unfolding subset_eq prefer 3 apply rule 
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3714
    apply(rule_tac n2="DIM('a)" in *) prefer 3 apply(subst(2) **) 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3715
    apply(rule card_mono) using 01 and convex_interval(1) prefer 5 apply - apply rule
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3716
    unfolding mem_interval apply rule unfolding mem_Collect_eq apply(erule_tac x=i in allE)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3717
    by(auto simp add: mem_def[of _ convex]) qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3718
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3719
subsection {* And this is a finite set of vertices. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3720
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3721
lemma unit_cube_convex_hull: obtains s where "finite s" "{0 .. (\<chi>\<chi> i. 1)::'a::ordered_euclidean_space} = convex hull s"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3722
  apply(rule that[of "{x::'a. \<forall>i<DIM('a). x$$i=0 \<or> x$$i=1}"])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3723
  apply(rule finite_subset[of _ "(\<lambda>s. (\<chi>\<chi> i. if i\<in>s then 1::real else 0)::'a) ` Pow {..<DIM('a)}"])
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3724
  prefer 3 apply(rule unit_interval_convex_hull) apply rule unfolding mem_Collect_eq proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3725
  fix x::"'a" assume as:"\<forall>i<DIM('a). x $$ i = 0 \<or> x $$ i = 1"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3726
  show "x \<in> (\<lambda>s. \<chi>\<chi> i. if i \<in> s then 1 else 0) ` Pow {..<DIM('a)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3727
    apply(rule image_eqI[where x="{i. i<DIM('a) \<and> x$$i = 1}"])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3728
    using as apply(subst euclidean_eq) by auto qed auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3729
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3730
subsection {* Hence any cube (could do any nonempty interval). *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3731
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3732
lemma cube_convex_hull:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3733
  assumes "0 < d" obtains s::"('a::ordered_euclidean_space) set" where
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3734
  "finite s" "{x - (\<chi>\<chi> i. d) .. x + (\<chi>\<chi> i. d)} = convex hull s" proof-
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3735
  let ?d = "(\<chi>\<chi> i. d)::'a"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  3736
  have *:"{x - ?d .. x + ?d} = (\<lambda>y. x - ?d + (2 * d) *\<^sub>R y) ` {0 .. \<chi>\<chi> i. 1}" apply(rule set_eqI, rule)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3737
    unfolding image_iff defer apply(erule bexE) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3738
    fix y assume as:"y\<in>{x - ?d .. x + ?d}"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3739
    { fix i assume i:"i<DIM('a)" have "x $$ i \<le> d + y $$ i" "y $$ i \<le> d + x $$ i"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3740
        using as[unfolded mem_interval, THEN spec[where x=i]] i
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3741
        by(auto simp add:euclidean_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3742
      hence "1 \<ge> inverse d * (x $$ i - y $$ i)" "1 \<ge> inverse d * (y $$ i - x $$ i)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3743
        apply(rule_tac[!] mult_left_le_imp_le[OF _ assms]) unfolding mult_assoc[THEN sym]
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  3744
        using assms by(auto simp add: field_simps)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3745
      hence "inverse d * (x $$ i * 2) \<le> 2 + inverse d * (y $$ i * 2)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3746
            "inverse d * (y $$ i * 2) \<le> 2 + inverse d * (x $$ i * 2)" by(auto simp add:field_simps) }
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3747
    hence "inverse (2 * d) *\<^sub>R (y - (x - ?d)) \<in> {0..\<chi>\<chi> i.1}" unfolding mem_interval using assms
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3748
      by(auto simp add: euclidean_simps field_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3749
    thus "\<exists>z\<in>{0..\<chi>\<chi> i.1}. y = x - ?d + (2 * d) *\<^sub>R z" apply- apply(rule_tac x="inverse (2 * d) *\<^sub>R (y - (x - ?d))" in bexI) 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3750
      using assms by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3751
  next
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3752
    fix y z assume as:"z\<in>{0..\<chi>\<chi> i.1}" "y = x - ?d + (2*d) *\<^sub>R z" 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3753
    have "\<And>i. i<DIM('a) \<Longrightarrow> 0 \<le> d * z $$ i \<and> d * z $$ i \<le> d"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3754
      using assms as(1)[unfolded mem_interval] apply(erule_tac x=i in allE)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3755
      apply rule apply(rule mult_nonneg_nonneg) prefer 3 apply(rule mult_right_le_one_le)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3756
      using assms by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3757
    thus "y \<in> {x - ?d..x + ?d}" unfolding as(2) mem_interval apply- apply rule using as(1)[unfolded mem_interval]
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3758
      apply(erule_tac x=i in allE) using assms by(auto simp add: euclidean_simps) qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3759
  obtain s where "finite s" "{0::'a..\<chi>\<chi> i.1} = convex hull s" using unit_cube_convex_hull by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3760
  thus ?thesis apply(rule_tac that[of "(\<lambda>y. x - ?d + (2 * d) *\<^sub>R y)` s"]) unfolding * and convex_hull_affinity by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3761
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3762
subsection {* Bounded convex function on open set is continuous. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3763
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3764
lemma convex_on_bounded_continuous:
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3765
  fixes s :: "('a::real_normed_vector) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3766
  assumes "open s" "convex_on s f" "\<forall>x\<in>s. abs(f x) \<le> b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3767
  shows "continuous_on s f"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3768
  apply(rule continuous_at_imp_continuous_on) unfolding continuous_at_real_range proof(rule,rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3769
  fix x e assume "x\<in>s" "(0::real) < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3770
  def B \<equiv> "abs b + 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3771
  have B:"0 < B" "\<And>x. x\<in>s \<Longrightarrow> abs (f x) \<le> B"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3772
    unfolding B_def defer apply(drule assms(3)[rule_format]) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3773
  obtain k where "k>0"and k:"cball x k \<subseteq> s" using assms(1)[unfolded open_contains_cball, THEN bspec[where x=x]] using `x\<in>s` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3774
  show "\<exists>d>0. \<forall>x'. norm (x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3775
    apply(rule_tac x="min (k / 2) (e / (2 * B) * k)" in exI) apply rule defer proof(rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3776
    fix y assume as:"norm (y - x) < min (k / 2) (e / (2 * B) * k)" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3777
    show "\<bar>f y - f x\<bar> < e" proof(cases "y=x")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3778
      case False def t \<equiv> "k / norm (y - x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3779
      have "2 < t" "0<t" unfolding t_def using as False and `k>0` by(auto simp add:field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3780
      have "y\<in>s" apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3781
        apply(rule order_trans[of _ "2 * norm (x - y)"]) using as by(auto simp add: field_simps norm_minus_commute) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3782
      { def w \<equiv> "x + t *\<^sub>R (y - x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3783
        have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3784
          unfolding t_def using `k>0` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3785
        have "(1 / t) *\<^sub>R x + - x + ((t - 1) / t) *\<^sub>R x = (1 / t - 1 + (t - 1) / t) *\<^sub>R x" by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3786
        also have "\<dots> = 0"  using `t>0` by(auto simp add:field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3787
        finally have w:"(1 / t) *\<^sub>R w + ((t - 1) / t) *\<^sub>R x = y" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3788
        have  "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3789
        hence "(f w - f x) / t < e"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3790
          using B(2)[OF `w\<in>s`] and B(2)[OF `x\<in>s`] using `t>0` by(auto simp add:field_simps) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3791
        hence th1:"f y - f x < e" apply- apply(rule le_less_trans) defer apply assumption
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3792
          using assms(2)[unfolded convex_on_def,rule_format,of w x "1/t" "(t - 1)/t", unfolded w]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3793
          using `0<t` `2<t` and `x\<in>s` `w\<in>s` by(auto simp add:field_simps) }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3794
      moreover 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3795
      { def w \<equiv> "x - t *\<^sub>R (y - x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3796
        have "w\<in>s" unfolding w_def apply(rule k[unfolded subset_eq,rule_format]) unfolding mem_cball dist_norm 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3797
          unfolding t_def using `k>0` by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3798
        have "(1 / (1 + t)) *\<^sub>R x + (t / (1 + t)) *\<^sub>R x = (1 / (1 + t) + t / (1 + t)) *\<^sub>R x" by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3799
        also have "\<dots>=x" using `t>0` by (auto simp add:field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3800
        finally have w:"(1 / (1+t)) *\<^sub>R w + (t / (1 + t)) *\<^sub>R y = x" unfolding w_def using False and `t>0` by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3801
        have  "2 * B < e * t" unfolding t_def using `0<e` `0<k` `B>0` and as and False by (auto simp add:field_simps) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3802
        hence *:"(f w - f y) / t < e" using B(2)[OF `w\<in>s`] and B(2)[OF `y\<in>s`] using `t>0` by(auto simp add:field_simps) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3803
        have "f x \<le> 1 / (1 + t) * f w + (t / (1 + t)) * f y" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3804
          using assms(2)[unfolded convex_on_def,rule_format,of w y "1/(1+t)" "t / (1+t)",unfolded w]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3805
          using `0<t` `2<t` and `y\<in>s` `w\<in>s` by (auto simp add:field_simps)
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 36725
diff changeset
  3806
        also have "\<dots> = (f w + t * f y) / (1 + t)" using `t>0` unfolding divide_inverse by (auto simp add:field_simps)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3807
        also have "\<dots> < e + f y" using `t>0` * `e>0` by(auto simp add:field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3808
        finally have "f x - f y < e" by auto }
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3809
      ultimately show ?thesis by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3810
    qed(insert `0<e`, auto) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3811
  qed(insert `0<e` `0<k` `0<B`, auto simp add:field_simps intro!:mult_pos_pos) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3812
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3813
subsection {* Upper bound on a ball implies upper and lower bounds. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3814
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3815
lemma convex_bounds_lemma:
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3816
  fixes x :: "'a::real_normed_vector"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3817
  assumes "convex_on (cball x e) f"  "\<forall>y \<in> cball x e. f y \<le> b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3818
  shows "\<forall>y \<in> cball x e. abs(f y) \<le> b + 2 * abs(f x)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3819
  apply(rule) proof(cases "0 \<le> e") case True
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3820
  fix y assume y:"y\<in>cball x e" def z \<equiv> "2 *\<^sub>R x - y"
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3821
  have *:"x - (2 *\<^sub>R x - y) = y - x" by (simp add: scaleR_2)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3822
  have z:"z\<in>cball x e" using y unfolding z_def mem_cball dist_norm * by(auto simp add: norm_minus_commute)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3823
  have "(1 / 2) *\<^sub>R y + (1 / 2) *\<^sub>R z = x" unfolding z_def by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3824
  thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using assms(1)[unfolded convex_on_def,rule_format, OF y z, of "1/2" "1/2"]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3825
    using assms(2)[rule_format,OF y] assms(2)[rule_format,OF z] by(auto simp add:field_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3826
next case False fix y assume "y\<in>cball x e" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3827
  hence "dist x y < 0" using False unfolding mem_cball not_le by (auto simp del: dist_not_less_zero)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3828
  thus "\<bar>f y\<bar> \<le> b + 2 * \<bar>f x\<bar>" using zero_le_dist[of x y] by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3829
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3830
subsection {* Hence a convex function on an open set is continuous. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3831
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3832
lemma convex_on_continuous:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3833
  assumes "open (s::('a::ordered_euclidean_space) set)" "convex_on s f" 
37673
f69f4b079275 generalize more lemmas from ordered_euclidean_space to euclidean_space
huffman
parents: 37647
diff changeset
  3834
  (* FIXME: generalize to euclidean_space *)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3835
  shows "continuous_on s f"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3836
  unfolding continuous_on_eq_continuous_at[OF assms(1)] proof
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3837
  note dimge1 = DIM_positive[where 'a='a]
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3838
  fix x assume "x\<in>s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3839
  then obtain e where e:"cball x e \<subseteq> s" "e>0" using assms(1) unfolding open_contains_cball by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3840
  def d \<equiv> "e / real DIM('a)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3841
  have "0 < d" unfolding d_def using `e>0` dimge1 by(rule_tac divide_pos_pos, auto) 
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3842
  let ?d = "(\<chi>\<chi> i. d)::'a"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3843
  obtain c where c:"finite c" "{x - ?d..x + ?d} = convex hull c" using cube_convex_hull[OF `d>0`, of x] by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3844
  have "x\<in>{x - ?d..x + ?d}" using `d>0` unfolding mem_interval by(auto simp add:euclidean_simps)
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  3845
  hence "c\<noteq>{}" using c by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3846
  def k \<equiv> "Max (f ` c)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3847
  have "convex_on {x - ?d..x + ?d} f" apply(rule convex_on_subset[OF assms(2)])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3848
    apply(rule subset_trans[OF _ e(1)]) unfolding subset_eq mem_cball proof 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3849
    fix z assume z:"z\<in>{x - ?d..x + ?d}"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3850
    have e:"e = setsum (\<lambda>i. d) {..<DIM('a)}" unfolding setsum_constant d_def using dimge1
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3851
      unfolding real_eq_of_nat by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3852
    show "dist x z \<le> e" unfolding dist_norm e apply(rule_tac order_trans[OF norm_le_l1], rule setsum_mono)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3853
      using z[unfolded mem_interval] apply(erule_tac x=i in allE) by(auto simp add:euclidean_simps) qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3854
  hence k:"\<forall>y\<in>{x - ?d..x + ?d}. f y \<le> k" unfolding c(2) apply(rule_tac convex_on_convex_hull_bound) apply assumption
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3855
    unfolding k_def apply(rule, rule Max_ge) using c(1) by auto
37647
a5400b94d2dd minimize dependencies on Numeral_Type
huffman
parents: 37489
diff changeset
  3856
  have "d \<le> e" unfolding d_def apply(rule mult_imp_div_pos_le) using `e>0` dimge1 unfolding mult_le_cancel_left1 by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3857
  hence dsube:"cball x d \<subseteq> cball x e" unfolding subset_eq Ball_def mem_cball by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3858
  have conv:"convex_on (cball x d) f" apply(rule convex_on_subset, rule convex_on_subset[OF assms(2)]) apply(rule e(1)) using dsube by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3859
  hence "\<forall>y\<in>cball x d. abs (f y) \<le> k + 2 * abs (f x)" apply(rule_tac convex_bounds_lemma) apply assumption proof
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3860
    fix y assume y:"y\<in>cball x d"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3861
    { fix i assume "i<DIM('a)" hence "x $$ i - d \<le> y $$ i"  "y $$ i \<le> x $$ i + d" 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3862
        using order_trans[OF component_le_norm y[unfolded mem_cball dist_norm], of i] by(auto simp add:euclidean_simps)  }
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3863
    thus "f y \<le> k" apply(rule_tac k[rule_format]) unfolding mem_cball mem_interval dist_norm 
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3864
      by(auto simp add:euclidean_simps) qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3865
  hence "continuous_on (ball x d) f" apply(rule_tac convex_on_bounded_continuous)
33270
paulson
parents: 33175
diff changeset
  3866
    apply(rule open_ball, rule convex_on_subset[OF conv], rule ball_subset_cball)
paulson
parents: 33175
diff changeset
  3867
    apply force
paulson
parents: 33175
diff changeset
  3868
    done
paulson
parents: 33175
diff changeset
  3869
  thus "continuous (at x) f" unfolding continuous_on_eq_continuous_at[OF open_ball]
paulson
parents: 33175
diff changeset
  3870
    using `d>0` by auto 
paulson
parents: 33175
diff changeset
  3871
qed
paulson
parents: 33175
diff changeset
  3872
paulson
parents: 33175
diff changeset
  3873
subsection {* Line segments, Starlike Sets, etc.*}
paulson
parents: 33175
diff changeset
  3874
paulson
parents: 33175
diff changeset
  3875
(* Use the same overloading tricks as for intervals, so that 
paulson
parents: 33175
diff changeset
  3876
   segment[a,b] is closed and segment(a,b) is open relative to affine hull. *)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3877
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3878
definition
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3879
  midpoint :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a" where
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3880
  "midpoint a b = (inverse (2::real)) *\<^sub>R (a + b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3881
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3882
definition
36341
2623a1987e1d generalize more constants and lemmas
huffman
parents: 36340
diff changeset
  3883
  open_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3884
  "open_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real.  0 < u \<and> u < 1}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3885
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3886
definition
36341
2623a1987e1d generalize more constants and lemmas
huffman
parents: 36340
diff changeset
  3887
  closed_segment :: "'a::real_vector \<Rightarrow> 'a \<Rightarrow> 'a set" where
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3888
  "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b | u::real. 0 \<le> u \<and> u \<le> 1}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3889
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3890
definition "between = (\<lambda> (a,b). closed_segment a b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3891
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3892
lemmas segment = open_segment_def closed_segment_def
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3893
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3894
definition "starlike s \<longleftrightarrow> (\<exists>a\<in>s. \<forall>x\<in>s. closed_segment a x \<subseteq> s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3895
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3896
lemma midpoint_refl: "midpoint x x = x"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3897
  unfolding midpoint_def unfolding scaleR_right_distrib unfolding scaleR_left_distrib[THEN sym] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3898
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3899
lemma midpoint_sym: "midpoint a b = midpoint b a" unfolding midpoint_def by (auto simp add: scaleR_right_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3900
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3901
lemma midpoint_eq_iff: "midpoint a b = c \<longleftrightarrow> a + b = c + c"
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3902
proof -
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3903
  have "midpoint a b = c \<longleftrightarrow> scaleR 2 (midpoint a b) = scaleR 2 c"
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3904
    by simp
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3905
  thus ?thesis
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3906
    unfolding midpoint_def scaleR_2 [symmetric] by simp
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3907
qed
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3908
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3909
lemma dist_midpoint:
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3910
  fixes a b :: "'a::real_normed_vector" shows
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3911
  "dist a (midpoint a b) = (dist a b) / 2" (is ?t1)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3912
  "dist b (midpoint a b) = (dist a b) / 2" (is ?t2)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3913
  "dist (midpoint a b) a = (dist a b) / 2" (is ?t3)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3914
  "dist (midpoint a b) b = (dist a b) / 2" (is ?t4)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3915
proof-
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3916
  have *: "\<And>x y::'a. 2 *\<^sub>R x = - y \<Longrightarrow> norm x = (norm y) / 2" unfolding equation_minus_iff by auto
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3917
  have **:"\<And>x y::'a. 2 *\<^sub>R x =   y \<Longrightarrow> norm x = (norm y) / 2" by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3918
  note scaleR_right_distrib [simp]
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3919
  show ?t1 unfolding midpoint_def dist_norm apply (rule **)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3920
    by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3921
  show ?t2 unfolding midpoint_def dist_norm apply (rule *)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3922
    by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3923
  show ?t3 unfolding midpoint_def dist_norm apply (rule *)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3924
    by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3925
  show ?t4 unfolding midpoint_def dist_norm apply (rule **)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3926
    by (simp add: scaleR_right_diff_distrib, simp add: scaleR_2)
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3927
qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3928
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3929
lemma midpoint_eq_endpoint:
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3930
  "midpoint a b = a \<longleftrightarrow> a = b"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3931
  "midpoint a b = b \<longleftrightarrow> a = b"
36338
7808fbc9c3b4 generalize more constants and lemmas
huffman
parents: 36337
diff changeset
  3932
  unfolding midpoint_eq_iff by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3933
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3934
lemma convex_contains_segment:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3935
  "convex s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. closed_segment a b \<subseteq> s)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3936
  unfolding convex_alt closed_segment_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3937
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3938
lemma convex_imp_starlike:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3939
  "convex s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> starlike s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3940
  unfolding convex_contains_segment starlike_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3941
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3942
lemma segment_convex_hull:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3943
 "closed_segment a b = convex hull {a,b}" proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3944
  have *:"\<And>x. {x} \<noteq> {}" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3945
  have **:"\<And>u v. u + v = 1 \<longleftrightarrow> u = 1 - (v::real)" by auto
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  3946
  show ?thesis unfolding segment convex_hull_insert[OF *] convex_hull_singleton apply(rule set_eqI)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3947
    unfolding mem_Collect_eq apply(rule,erule exE) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3948
    apply(rule_tac x="1 - u" in exI) apply rule defer apply(rule_tac x=u in exI) defer
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3949
    apply(erule exE, (erule conjE)?)+ apply(rule_tac x="1 - u" in exI) unfolding ** by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3950
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3951
lemma convex_segment: "convex (closed_segment a b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3952
  unfolding segment_convex_hull by(rule convex_convex_hull)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3953
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3954
lemma ends_in_segment: "a \<in> closed_segment a b" "b \<in> closed_segment a b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3955
  unfolding segment_convex_hull apply(rule_tac[!] hull_subset[unfolded subset_eq, rule_format]) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3956
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3957
lemma segment_furthest_le:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3958
  fixes a b x y :: "'a::euclidean_space"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3959
  assumes "x \<in> closed_segment a b" shows "norm(y - x) \<le> norm(y - a) \<or>  norm(y - x) \<le> norm(y - b)" proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3960
  obtain z where "z\<in>{a, b}" "norm (x - y) \<le> norm (z - y)" using simplex_furthest_le[of "{a, b}" y]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3961
    using assms[unfolded segment_convex_hull] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3962
  thus ?thesis by(auto simp add:norm_minus_commute) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3963
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3964
lemma segment_bound:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3965
  fixes x a b :: "'a::euclidean_space"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3966
  assumes "x \<in> closed_segment a b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3967
  shows "norm(x - a) \<le> norm(b - a)" "norm(x - b) \<le> norm(b - a)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3968
  using segment_furthest_le[OF assms, of a]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3969
  using segment_furthest_le[OF assms, of b]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3970
  by (auto simp add:norm_minus_commute) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3971
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3972
lemma segment_refl:"closed_segment a a = {a}" unfolding segment by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3973
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3974
lemma between_mem_segment: "between (a,b) x \<longleftrightarrow> x \<in> closed_segment a b"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3975
  unfolding between_def mem_def by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3976
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3977
lemma between:"between (a,b) (x::'a::euclidean_space) \<longleftrightarrow> dist a b = (dist a x) + (dist x b)"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3978
proof(cases "a = b")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3979
  case True thus ?thesis unfolding between_def split_conv mem_def[of x, symmetric]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3980
    by(auto simp add:segment_refl dist_commute) next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3981
  case False hence Fal:"norm (a - b) \<noteq> 0" and Fal2: "norm (a - b) > 0" by auto 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3982
  have *:"\<And>u. a - ((1 - u) *\<^sub>R a + u *\<^sub>R b) = u *\<^sub>R (a - b)" by (auto simp add: algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3983
  show ?thesis unfolding between_def split_conv mem_def[of x, symmetric] closed_segment_def mem_Collect_eq
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3984
    apply rule apply(erule exE, (erule conjE)+) apply(subst dist_triangle_eq) proof-
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3985
      fix u assume as:"x = (1 - u) *\<^sub>R a + u *\<^sub>R b" "0 \<le> u" "u \<le> 1" 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3986
      hence *:"a - x = u *\<^sub>R (a - b)" "x - b = (1 - u) *\<^sub>R (a - b)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3987
        unfolding as(1) by(auto simp add:algebra_simps)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3988
      show "norm (a - x) *\<^sub>R (x - b) = norm (x - b) *\<^sub>R (a - x)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3989
        unfolding norm_minus_commute[of x a] * using as(2,3)
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  3990
        by(auto simp add: field_simps)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3991
    next assume as:"dist a b = dist a x + dist x b"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3992
      have "norm (a - x) / norm (a - b) \<le> 1" unfolding divide_le_eq_1_pos[OF Fal2]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3993
        unfolding as[unfolded dist_norm] norm_ge_zero by auto 
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  3994
      thus "\<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b \<and> 0 \<le> u \<and> u \<le> 1" apply(rule_tac x="dist a x / dist a b" in exI)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3995
        unfolding dist_norm apply(subst euclidean_eq) apply rule defer apply(rule, rule divide_nonneg_pos) prefer 4
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3996
      proof(rule,rule) fix i assume i:"i<DIM('a)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3997
          have "((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i =
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3998
            ((norm (a - b) - norm (a - x)) * (a $$ i) + norm (a - x) * (b $$ i)) / norm (a - b)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  3999
            using Fal by(auto simp add: field_simps euclidean_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4000
          also have "\<dots> = x$$i" apply(rule divide_eq_imp[OF Fal])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4001
            unfolding as[unfolded dist_norm] using as[unfolded dist_triangle_eq] apply-
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4002
            apply(subst (asm) euclidean_eq) using i apply(erule_tac x=i in allE) by(auto simp add:field_simps euclidean_simps)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4003
          finally show "x $$ i = ((1 - norm (a - x) / norm (a - b)) *\<^sub>R a + (norm (a - x) / norm (a - b)) *\<^sub>R b) $$ i" 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4004
            by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4005
        qed(insert Fal2, auto) qed qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4006
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4007
lemma between_midpoint: fixes a::"'a::euclidean_space" shows
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4008
  "between (a,b) (midpoint a b)" (is ?t1) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4009
  "between (b,a) (midpoint a b)" (is ?t2)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4010
proof- have *:"\<And>x y z. x = (1/2::real) *\<^sub>R z \<Longrightarrow> y = (1/2) *\<^sub>R z \<Longrightarrow> norm z = norm x + norm y" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4011
  show ?t1 ?t2 unfolding between midpoint_def dist_norm apply(rule_tac[!] *)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4012
    unfolding euclidean_eq[where 'a='a]
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4013
    by(auto simp add:field_simps euclidean_simps) qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4014
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4015
lemma between_mem_convex_hull:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4016
  "between (a,b) x \<longleftrightarrow> x \<in> convex hull {a,b}"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4017
  unfolding between_mem_segment segment_convex_hull ..
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4018
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4019
subsection {* Shrinking towards the interior of a convex set. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4020
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4021
lemma mem_interior_convex_shrink:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4022
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4023
  assumes "convex s" "c \<in> interior s" "x \<in> s" "0 < e" "e \<le> 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4024
  shows "x - e *\<^sub>R (x - c) \<in> interior s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4025
proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4026
  show ?thesis unfolding mem_interior apply(rule_tac x="e*d" in exI)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4027
    apply(rule) defer unfolding subset_eq Ball_def mem_ball proof(rule,rule)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4028
    fix y assume as:"dist (x - e *\<^sub>R (x - c)) y < e * d"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4029
    have *:"y = (1 - (1 - e)) *\<^sub>R ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) + (1 - e) *\<^sub>R x" using `e>0` by (auto simp add: scaleR_left_diff_distrib scaleR_right_diff_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4030
    have "dist c ((1 / e) *\<^sub>R y - ((1 - e) / e) *\<^sub>R x) = abs(1/e) * norm (e *\<^sub>R c - y + (1 - e) *\<^sub>R x)"
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4031
      unfolding dist_norm unfolding norm_scaleR[THEN sym] apply(rule arg_cong[where f=norm]) using `e>0`
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4032
      by(auto simp add: euclidean_simps euclidean_eq[where 'a='a] field_simps) 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4033
    also have "\<dots> = abs(1/e) * norm (x - e *\<^sub>R (x - c) - y)" by(auto intro!:arg_cong[where f=norm] simp add: algebra_simps)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4034
    also have "\<dots> < d" using as[unfolded dist_norm] and `e>0`
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 36725
diff changeset
  4035
      by(auto simp add:pos_divide_less_eq[OF `e>0`] mult_commute)
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4036
    finally show "y \<in> s" apply(subst *) apply(rule assms(1)[unfolded convex_alt,rule_format])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4037
      apply(rule d[unfolded subset_eq,rule_format]) unfolding mem_ball using assms(3-5) by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4038
  qed(rule mult_pos_pos, insert `e>0` `d>0`, auto) qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4039
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4040
lemma mem_interior_closure_convex_shrink:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4041
  fixes s :: "('a::euclidean_space) set"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4042
  assumes "convex s" "c \<in> interior s" "x \<in> closure s" "0 < e" "e \<le> 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4043
  shows "x - e *\<^sub>R (x - c) \<in> interior s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4044
proof- obtain d where "d>0" and d:"ball c d \<subseteq> s" using assms(2) unfolding mem_interior by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4045
  have "\<exists>y\<in>s. norm (y - x) * (1 - e) < e * d" proof(cases "x\<in>s")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4046
    case True thus ?thesis using `e>0` `d>0` by(rule_tac bexI[where x=x], auto intro!: mult_pos_pos) next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4047
    case False hence x:"x islimpt s" using assms(3)[unfolded closure_def] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4048
    show ?thesis proof(cases "e=1")
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4049
      case True obtain y where "y\<in>s" "y \<noteq> x" "dist y x < 1"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4050
        using x[unfolded islimpt_approachable,THEN spec[where x=1]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4051
      thus ?thesis apply(rule_tac x=y in bexI) unfolding True using `d>0` by auto next
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4052
      case False hence "0 < e * d / (1 - e)" and *:"1 - e > 0"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4053
        using `e\<le>1` `e>0` `d>0` by(auto intro!:mult_pos_pos divide_pos_pos)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4054
      then obtain y where "y\<in>s" "y \<noteq> x" "dist y x < e * d / (1 - e)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4055
        using x[unfolded islimpt_approachable,THEN spec[where x="e*d / (1 - e)"]] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4056
      thus ?thesis apply(rule_tac x=y in bexI) unfolding dist_norm using pos_less_divide_eq[OF *] by auto qed qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4057
  then obtain y where "y\<in>s" and y:"norm (y - x) * (1 - e) < e * d" by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4058
  def z \<equiv> "c + ((1 - e) / e) *\<^sub>R (x - y)"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4059
  have *:"x - e *\<^sub>R (x - c) = y - e *\<^sub>R (y - z)" unfolding z_def using `e>0` by (auto simp add: scaleR_right_diff_distrib scaleR_right_distrib scaleR_left_diff_distrib)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4060
  have "z\<in>interior s" apply(rule subset_interior[OF d,unfolded subset_eq,rule_format])
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4061
    unfolding interior_open[OF open_ball] mem_ball z_def dist_norm using y and assms(4,5)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4062
    by(auto simp add:field_simps norm_minus_commute)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4063
  thus ?thesis unfolding * apply - apply(rule mem_interior_convex_shrink) 
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4064
    using assms(1,4-5) `y\<in>s` by auto qed
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4065
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4066
subsection {* Some obvious but surprisingly hard simplex lemmas. *}
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4067
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4068
lemma simplex:
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4069
  assumes "finite s" "0 \<notin> s"
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4070
  shows "convex hull (insert 0 s) =  { y. (\<exists>u. (\<forall>x\<in>s. 0 \<le> u x) \<and> setsum u s \<le> 1 \<and> setsum (\<lambda>x. u x *\<^sub>R x) s = y)}"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  4071
  unfolding convex_hull_finite[OF finite.insertI[OF assms(1)]] apply(rule set_eqI, rule) unfolding mem_Collect_eq
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4072
  apply(erule_tac[!] exE) apply(erule_tac[!] conjE)+ unfolding setsum_clauses(2)[OF assms(1)]
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4073
  apply(rule_tac x=u in exI) defer apply(rule_tac x="\<lambda>x. if x = 0 then 1 - setsum u s else u x" in exI) using assms(2)
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4074
  unfolding if_smult and setsum_delta_notmem[OF assms(2)] by auto
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4075
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4076
lemma substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4077
  shows "convex hull (insert 0 { basis i | i. i : d}) =
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4078
        {x::'a::euclidean_space . (!i<DIM('a). 0 <= x$$i) & setsum (%i. x$$i) d <= 1 &
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4079
  (!i<DIM('a). i ~: d --> x$$i = 0)}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4080
  (is "convex hull (insert 0 ?p) = ?s")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4081
(* Proof is a modified copy of the proof of similar lemma std_simplex in Convex_Euclidean_Space.thy *)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4082
proof- let ?D = d (*"{..<DIM('a::euclidean_space)}"*)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4083
  have "0 ~: ?p" using assms by (auto simp: image_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4084
  have "{(basis i)::'n::euclidean_space |i. i \<in> ?D} = basis ` ?D" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4085
  note sumbas = this setsum_reindex[OF basis_inj_on[of d], unfolded o_def, OF assms]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4086
  show ?thesis unfolding simplex[OF finite_substdbasis `0 ~: ?p`] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4087
    apply(rule set_eqI) unfolding mem_Collect_eq apply rule
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4088
    apply(erule exE, (erule conjE)+) apply(erule_tac[2] conjE)+ proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4089
    fix x::"'a::euclidean_space" and u assume as: "\<forall>x\<in>{basis i |i. i \<in>?D}. 0 \<le> u x"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4090
      "setsum u {basis i |i. i \<in> ?D} \<le> 1" "(\<Sum>x\<in>{basis i |i. i \<in>?D}. u x *\<^sub>R x) = x"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4091
    have *:"\<forall>i<DIM('a). i:d --> u (basis i) = x$$i" and "(!i<DIM('a). i ~: d --> x $$ i = 0)" using as(3) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4092
      unfolding sumbas unfolding substdbasis_expansion_unique[OF assms] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4093
    hence **:"setsum u {basis i |i. i \<in> ?D} = setsum (op $$ x) ?D" unfolding sumbas 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4094
      apply-apply(rule setsum_cong2) using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4095
    have " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4096
      apply - proof(rule,rule,rule)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4097
      fix i assume i:"i<DIM('a)" have "i : d ==> 0 \<le> x$$i" unfolding *[rule_format,OF i,THEN sym] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4098
         apply(rule_tac as(1)[rule_format]) by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4099
      moreover have "i ~: d ==> 0 \<le> x$$i" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4100
        using `(!i<DIM('a). i ~: d --> x $$ i = 0)`[rule_format, OF i] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4101
      ultimately show "0 \<le> x$$i" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4102
    qed(insert as(2)[unfolded **], auto)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4103
    from this show " (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (op $$ x) ?D \<le> 1 & (!i<DIM('a). i ~: d --> x $$ i = 0)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4104
      using `(!i<DIM('a). i ~: d --> x $$ i = 0)` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4105
  next fix x::"'a::euclidean_space" assume as:"\<forall>i<DIM('a). 0 \<le> x $$ i" "setsum (op $$ x) ?D \<le> 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4106
      "(!i<DIM('a). i ~: d --> x $$ i = 0)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4107
    show "\<exists>u. (\<forall>x\<in>{basis i |i. i \<in> ?D}. 0 \<le> u x) \<and>
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4108
      setsum u {basis i |i. i \<in> ?D} \<le> 1 \<and> (\<Sum>x\<in>{basis i |i. i \<in> ?D}. u x *\<^sub>R x) = x"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4109
      apply(rule_tac x="\<lambda>y. inner y x" in exI) apply(rule,rule) unfolding mem_Collect_eq apply(erule exE)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4110
      using as(1) apply(erule_tac x=i in allE) unfolding sumbas apply safe unfolding not_less basis_zero
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4111
      unfolding substdbasis_expansion_unique[OF assms] euclidean_component_def[THEN sym]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4112
      using as(2,3) by(auto simp add:dot_basis not_less  basis_zero) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4113
  qed qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4114
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4115
lemma std_simplex:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4116
  "convex hull (insert 0 { basis i | i. i<DIM('a)}) =
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4117
        {x::'a::euclidean_space . (\<forall>i<DIM('a). 0 \<le> x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} \<le> 1 }"
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4118
  using substd_simplex[of "{..<DIM('a)}"] by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4119
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4120
lemma interior_std_simplex:
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4121
  "interior (convex hull (insert 0 { basis i| i. i<DIM('a)})) =
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4122
  {x::'a::euclidean_space. (\<forall>i<DIM('a). 0 < x$$i) \<and> setsum (\<lambda>i. x$$i) {..<DIM('a)} < 1 }"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
  4123
  apply(rule set_eqI) unfolding mem_interior std_simplex unfolding subset_eq mem_Collect_eq Ball_def mem_ball
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4124
  unfolding Ball_def[symmetric] apply rule apply(erule exE, (erule conjE)+) defer apply(erule conjE) proof-
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4125
  fix x::"'a" and e assume "0<e" and as:"\<forall>xa. dist x xa < e \<longrightarrow> (\<forall>x<DIM('a). 0 \<le> xa $$ x) \<and> setsum (op $$ xa) {..<DIM('a)} \<le> 1"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4126
  show "(\<forall>xa<DIM('a). 0 < x $$ xa) \<and> setsum (op $$ x) {..<DIM('a)} < 1" apply(safe) proof-
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4127
    fix i assume i:"i<DIM('a)" thus "0 < x $$ i" using as[THEN spec[where x="x - (e / 2) *\<^sub>R basis i"]] and `e>0`
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4128
      unfolding dist_norm  by(auto simp add: inner_simps euclidean_component_def dot_basis elim!:allE[where x=i])
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4129
  next have **:"dist x (x + (e / 2) *\<^sub>R basis 0) < e" using  `e>0`
38642
8fa437809c67 dropped type classes mult_mono and mult_mono1; tuned names of technical rule duplicates
haftmann
parents: 37732
diff changeset
  4130
      unfolding dist_norm by(auto intro!: mult_strict_left_mono)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4131
    have "\<And>i. i<DIM('a) \<Longrightarrow> (x + (e / 2) *\<^sub>R basis 0) $$ i = x$$i + (if i = 0 then e/2 else 0)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4132
      unfolding euclidean_component_def by(auto simp add:inner_simps dot_basis)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4133
    hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)} = setsum (\<lambda>i. x$$i + (if 0 = i then e/2 else 0)) {..<DIM('a)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4134
      apply(rule_tac setsum_cong) by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4135
    have "setsum (op $$ x) {..<DIM('a)} < setsum (op $$ (x + (e / 2) *\<^sub>R basis 0)) {..<DIM('a)}" unfolding * setsum_addf
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4136
      using `0<e` DIM_positive[where 'a='a] apply(subst setsum_delta') by auto
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4137
    also have "\<dots> \<le> 1" using ** apply(drule_tac as[rule_format]) by auto
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4138
    finally show "setsum (op $$ x) {..<DIM('a)} < 1" by auto qed
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4139
next fix x::"'a" assume as:"\<forall>i<DIM('a). 0 < x $$ i" "setsum (op $$ x) {..<DIM('a)} < 1"
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4140
  guess a using UNIV_witness[where 'a='b] ..
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4141
  let ?d = "(1 - setsum (op $$ x) {..<DIM('a)}) / real (DIM('a))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4142
  have "Min ((op $$ x) ` {..<DIM('a)}) > 0" apply(rule Min_grI) using as(1) by auto
37647
a5400b94d2dd minimize dependencies on Numeral_Type
huffman
parents: 37489
diff changeset
  4143
  moreover have"?d > 0" apply(rule divide_pos_pos) using as(2) by(auto simp add: Suc_le_eq)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4144
  ultimately show "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> (\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4145
    apply(rule_tac x="min (Min ((op $$ x) ` {..<DIM('a)})) ?D" in exI) apply rule defer apply(rule,rule) proof-
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4146
    fix y assume y:"dist x y < min (Min (op $$ x ` {..<DIM('a)})) ?d"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4147
    have "setsum (op $$ y) {..<DIM('a)} \<le> setsum (\<lambda>i. x$$i + ?d) {..<DIM('a)}" proof(rule setsum_mono)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4148
      fix i assume "i\<in>{..<DIM('a)}" hence "abs (y$$i - x$$i) < ?d" apply-apply(rule le_less_trans)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4149
        using component_le_norm[of "y - x" i]
36362
06475a1547cb fix lots of looping simp calls and other warnings
huffman
parents: 36341
diff changeset
  4150
        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2] by(auto simp add: norm_minus_commute)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4151
      thus "y $$ i \<le> x $$ i + ?d" by auto qed
37647
a5400b94d2dd minimize dependencies on Numeral_Type
huffman
parents: 37489
diff changeset
  4152
    also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat by(auto simp add: Suc_le_eq)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4153
    finally show "(\<forall>i<DIM('a). 0 \<le> y $$ i) \<and> setsum (op $$ y) {..<DIM('a)} \<le> 1" 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4154
    proof safe fix i assume i:"i<DIM('a)"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4155
      have "norm (x - y) < x$$i" apply(rule less_le_trans) 
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4156
        apply(rule y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]) using i by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4157
      thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format, of i] by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4158
    qed qed auto qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4159
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4160
lemma interior_std_simplex_nonempty: obtains a::"'a::euclidean_space" where
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4161
  "a \<in> interior(convex hull (insert 0 {basis i | i . i<DIM('a)}))" proof-
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4162
  let ?D = "{..<DIM('a)}" let ?a = "setsum (\<lambda>b::'a. inverse (2 * real DIM('a)) *\<^sub>R b) {(basis i) | i. i<DIM('a)}"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4163
  have *:"{basis i :: 'a | i. i<DIM('a)} = basis ` ?D" by auto
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4164
  { fix i assume i:"i<DIM('a)" have "?a $$ i = inverse (2 * real DIM('a))"
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4165
      unfolding euclidean_component.setsum * and setsum_reindex[OF basis_inj] and o_def
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4166
      apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real DIM('a)) else 0) ?D"]) apply(rule setsum_cong2)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4167
      defer apply(subst setsum_delta') unfolding euclidean_component_def using i by(auto simp add:dot_basis) }
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4168
  note ** = this
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4169
  show ?thesis apply(rule that[of ?a]) unfolding interior_std_simplex mem_Collect_eq proof safe
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4170
    fix i assume i:"i<DIM('a)" show "0 < ?a $$ i" unfolding **[OF i] by(auto simp add: Suc_le_eq)
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4171
  next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real DIM('a))) ?D" apply(rule setsum_cong2, rule **) by auto
36778
739a9379e29b avoid using real-specific versions of generic lemmas
huffman
parents: 36725
diff changeset
  4172
    also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat divide_inverse[THEN sym] by (auto simp add:field_simps)
37489
44e42d392c6e Introduce a type class for euclidean spaces, port most lemmas from real^'n to this type class.
hoelzl
parents: 36844
diff changeset
  4173
    finally show "setsum (op $$ ?a) ?D < 1" by auto qed qed
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  4174
40377
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4175
lemma rel_interior_substd_simplex: assumes "d\<subseteq>{..<DIM('a::euclidean_space)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4176
  shows "rel_interior (convex hull (insert 0 { basis i| i. i : d})) =
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4177
  {x::'a::euclidean_space. (\<forall>i\<in>d. 0 < x$$i) & setsum (%i. x$$i) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4178
  (is "rel_interior (convex hull (insert 0 ?p)) = ?s")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4179
(* Proof is a modified copy of the proof of similar lemma interior_std_simplex in Convex_Euclidean_Space.thy *)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4180
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4181
have "finite d" apply(rule finite_subset) using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4182
{ assume "d={}" hence ?thesis using rel_interior_sing using euclidean_eq[of _ 0] by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4183
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4184
{ assume "d~={}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4185
have h0: "affine hull (convex hull (insert 0 ?p))={x::'a::euclidean_space. (!i<DIM('a). i ~: d --> x$$i = 0)}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4186
   using affine_hull_convex_hull affine_hull_substd_basis assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4187
have aux: "!x::'n::euclidean_space. !i. ((! i:d. 0 <= x$$i) & (!i. i ~: d --> x$$i = 0))--> 0 <= x$$i" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4188
{ fix x::"'a::euclidean_space" assume x_def: "x : rel_interior (convex hull (insert 0 ?p))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4189
  from this obtain e where e0: "e>0" and 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4190
       "ball x e Int {xa. (!i<DIM('a). i ~: d --> xa$$i = 0)} <= convex hull (insert 0 ?p)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4191
       using mem_rel_interior_ball[of x "convex hull (insert 0 ?p)"] h0 by auto   
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4192
  hence as: "ALL xa. (dist x xa < e & (!i<DIM('a). i ~: d --> xa$$i = 0)) -->
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4193
    (!i : d. 0 <= xa $$ i) & setsum (op $$ xa) d <= 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4194
    unfolding ball_def unfolding substd_simplex[OF assms] using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4195
  have x0: "(!i<DIM('a). i ~: d --> x$$i = 0)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4196
    using x_def rel_interior_subset  substd_simplex[OF assms] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4197
  have "(!i : d. 0 < x $$ i) & setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" apply(rule,rule) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4198
  proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4199
    fix i::nat assume "i:d" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4200
    hence "\<forall>ia\<in>d. 0 \<le> (x - (e / 2) *\<^sub>R basis i) $$ ia" apply-apply(rule as[rule_format,THEN conjunct1])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4201
      unfolding dist_norm using assms `e>0` x0 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4202
    thus "0 < x $$ i" apply(erule_tac x=i in ballE) using `e>0` `i\<in>d` assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4203
  next obtain a where a:"a:d" using `d ~= {}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4204
    have **:"dist x (x + (e / 2) *\<^sub>R basis a) < e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4205
      using  `e>0` and Euclidean_Space.norm_basis[of a]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4206
      unfolding dist_norm by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4207
    have "\<And>i. (x + (e / 2) *\<^sub>R basis a) $$ i = x$$i + (if i = a then e/2 else 0)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4208
      unfolding euclidean_simps using a assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4209
    hence *:"setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d =
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4210
      setsum (\<lambda>i. x$$i + (if a = i then e/2 else 0)) d" by(rule_tac setsum_cong, auto)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4211
    have h1: "(ALL i<DIM('a). i ~: d --> (x + (e / 2) *\<^sub>R basis a) $$ i = 0)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4212
      using as[THEN spec[where x="x + (e / 2) *\<^sub>R basis a"]] `a:d` using x0
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4213
      by(auto simp add: norm_basis elim:allE[where x=a]) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4214
    have "setsum (op $$ x) d < setsum (op $$ (x + (e / 2) *\<^sub>R basis a)) d" unfolding * setsum_addf
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4215
      using `0<e` `a:d` using `finite d` by(auto simp add: setsum_delta')
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4216
    also have "\<dots> \<le> 1" using ** h1 as[rule_format, of "x + (e / 2) *\<^sub>R basis a"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4217
    finally show "setsum (op $$ x) d < 1 & (!i<DIM('a). i ~: d --> x$$i = 0)" using x0 by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4218
  qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4219
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4220
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4221
{
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4222
  fix x::"'a::euclidean_space" assume as: "x : ?s"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4223
  have "!i. ((0<x$$i) | (0=x$$i) --> 0<=x$$i)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4224
  moreover have "!i. (i:d) | (i ~: d)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4225
  ultimately 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4226
  have "!i. ( (ALL i:d. 0 < x$$i) & (ALL i. i ~: d --> x$$i = 0) ) --> 0 <= x$$i" by metis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4227
  hence h2: "x : convex hull (insert 0 ?p)" using as assms 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4228
    unfolding substd_simplex[OF assms] by fastsimp 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4229
  obtain a where a:"a:d" using `d ~= {}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4230
  let ?d = "(1 - setsum (op $$ x) d) / real (card d)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4231
  have "card d >= 1" using `d ~={}` card_ge1[of d] using `finite d` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4232
  have "Min ((op $$ x) ` d) > 0" apply(rule Min_grI) using as `card d >= 1` `finite d` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4233
  moreover have "?d > 0" apply(rule divide_pos_pos) using as using `card d >= 1` by(auto simp add: Suc_le_eq)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4234
  ultimately have h3: "min (Min ((op $$ x) ` d)) ?d > 0" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4235
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4236
  have "x : rel_interior (convex hull (insert 0 ?p))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4237
    unfolding rel_interior_ball mem_Collect_eq h0 apply(rule,rule h2)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4238
    unfolding substd_simplex[OF assms]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4239
    apply(rule_tac x="min (Min ((op $$ x) ` d)) ?d" in exI) apply(rule,rule h3) apply safe unfolding mem_ball
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4240
  proof- fix y::'a assume y:"dist x y < min (Min (op $$ x ` d)) ?d" and y2:"(!i<DIM('a). i ~: d --> y$$i = 0)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4241
    have "setsum (op $$ y) d \<le> setsum (\<lambda>i. x$$i + ?d) d" proof(rule setsum_mono)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4242
      fix i assume i:"i\<in>d"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4243
      have "abs (y$$i - x$$i) < ?d" apply(rule le_less_trans) using component_le_norm[of "y - x" i]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4244
        using y[unfolded min_less_iff_conj dist_norm, THEN conjunct2]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4245
        by(auto simp add: norm_minus_commute)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4246
      thus "y $$ i \<le> x $$ i + ?d" by auto qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4247
    also have "\<dots> \<le> 1" unfolding setsum_addf setsum_constant card_enum real_eq_of_nat
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4248
      using `card d >= 1` by(auto simp add: Suc_le_eq)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4249
    finally show "setsum (op $$ y) d \<le> 1" .
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4250
     
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4251
    fix i assume "i<DIM('a)" thus "0 \<le> y$$i" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4252
    proof(cases "i\<in>d") case True
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4253
      have "norm (x - y) < x$$i" using y[unfolded min_less_iff_conj dist_norm, THEN conjunct1]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4254
        using Min_gr_iff[of "op $$ x ` d" "norm (x - y)"] `card d >= 1` `i:d`
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4255
        apply auto by (metis Suc_n_not_le_n True card_eq_0_iff finite_imageI)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4256
      thus "0 \<le> y$$i" using component_le_norm[of "x - y" i] and as(1)[rule_format] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4257
    qed(insert y2, auto)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4258
  qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4259
} ultimately have
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4260
    "!!x :: 'a::euclidean_space. (x : rel_interior (convex hull insert 0 {basis i |i. i : d})) =
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4261
    (x : {x. (ALL i:d. 0 < x $$ i) &
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4262
    setsum (op $$ x) d < 1 & (ALL i<DIM('a). i ~: d --> x $$ i = 0)})" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4263
from this have ?thesis by (rule set_eqI)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4264
} ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4265
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4266
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4267
lemma rel_interior_substd_simplex_nonempty: assumes "d ~={}" "d\<subseteq>{..<DIM('a::euclidean_space)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4268
  obtains a::"'a::euclidean_space" where
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4269
  "a : rel_interior(convex hull (insert 0 {basis i | i . i : d}))" proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4270
(* Proof is a modified copy of the proof of similar lemma interior_std_simplex_nonempty in Convex_Euclidean_Space.thy *)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4271
  let ?D = d let ?a = "setsum (\<lambda>b::'a::euclidean_space. inverse (2 * real (card d)) *\<^sub>R b) {(basis i) | i. i \<in> ?D}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4272
  have *:"{basis i :: 'a | i. i \<in> ?D} = basis ` ?D" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4273
  have "finite d" apply(rule finite_subset) using assms(2) by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4274
  hence d1: "real(card d) >= 1" using `d ~={}` card_ge1[of d] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4275
  { fix i assume "i:d" have "?a $$ i = inverse (2 * real (card d))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4276
      unfolding * setsum_reindex[OF basis_inj_on, OF assms(2)] o_def
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4277
      apply(rule trans[of _ "setsum (\<lambda>j. if i = j then inverse (2 * real (card d)) else 0) ?D"]) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4278
      unfolding euclidean_component.setsum
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4279
      apply(rule setsum_cong2)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4280
      using `i:d` `finite d` setsum_delta'[of d i "(%k. inverse (2 * real (card d)))"] d1 assms(2)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4281
      by (auto simp add: Euclidean_Space.basis_component[of i])}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4282
  note ** = this
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4283
  show ?thesis apply(rule that[of ?a]) unfolding rel_interior_substd_simplex[OF assms(2)] mem_Collect_eq
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4284
  proof safe fix i assume "i:d" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4285
    have "0 < inverse (2 * real (card d))" using d1 by(auto simp add: Suc_le_eq)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4286
    also have "...=?a $$ i" using **[of i] `i:d` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4287
    finally show "0 < ?a $$ i" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4288
  next have "setsum (op $$ ?a) ?D = setsum (\<lambda>i. inverse (2 * real (card d))) ?D" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4289
      by(rule setsum_cong2, rule **) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4290
    also have "\<dots> < 1" unfolding setsum_constant card_enum real_eq_of_nat real_divide_def[THEN sym]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4291
      by (auto simp add:field_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4292
    finally show "setsum (op $$ ?a) ?D < 1" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4293
  next fix i assume "i<DIM('a)" and "i~:d"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4294
    have "?a : (span {basis i | i. i : d})" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4295
      apply (rule span_setsum[of "{basis i |i. i : d}" "(%b. b /\<^sub>R (2 * real (card d)))" "{basis i |i. i : d}"]) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4296
      using finite_substdbasis[of d] apply blast 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4297
    proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4298
      { fix x assume "(x :: 'a::euclidean_space): {basis i |i. i : d}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4299
        hence "x : span {basis i |i. i : d}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4300
          using span_superset[of _ "{basis i |i. i : d}"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4301
        hence "(x /\<^sub>R (2 * real (card d))) : (span {basis i |i. i : d})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4302
          using span_mul[of x "{basis i |i. i : d}" "(inverse (real (card d)) / 2)"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4303
      } thus "\<forall>x\<in>{basis i |i. i \<in> d}. x /\<^sub>R (2 * real (card d)) \<in> span {basis i ::'a |i. i \<in> d}" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4304
    qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4305
    thus "?a $$ i = 0 " using `i~:d` unfolding span_substd_basis[OF assms(2)] using `i<DIM('a)` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4306
  qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4307
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4308
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4309
subsection{* Relative Interior of Convex Set *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4310
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4311
lemma rel_interior_convex_nonempty_aux: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4312
fixes S :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4313
assumes "convex S" and "0 : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4314
shows "rel_interior S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4315
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4316
{ assume "S = {0}" hence ?thesis using rel_interior_sing by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4317
moreover { 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4318
assume "S ~= {0}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4319
obtain B where B_def: "independent B & B<=S & (S <= span B) & card B = dim S" using basis_exists[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4320
hence "B~={}" using B_def assms `S ~= {0}` span_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4321
have "insert 0 B <= span B" using subspace_span[of B] subspace_0[of "span B"] span_inc by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4322
hence "span (insert 0 B) <= span B" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4323
    using span_span[of B] span_mono[of "insert 0 B" "span B"] by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4324
hence "convex hull insert 0 B <= span B" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4325
    using convex_hull_subset_span[of "insert 0 B"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4326
hence "span (convex hull insert 0 B) <= span B"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4327
    using span_span[of B] span_mono[of "convex hull insert 0 B" "span B"] by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4328
hence *: "span (convex hull insert 0 B) = span B" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4329
    using span_mono[of B "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4330
hence "span (convex hull insert 0 B) = span S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4331
    using B_def span_mono[of B S] span_mono[of S "span B"] span_span[of B] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4332
moreover have "0 : affine hull (convex hull insert 0 B)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4333
    using hull_subset[of "convex hull insert 0 B"] hull_subset[of "insert 0 B"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4334
ultimately have **: "affine hull (convex hull insert 0 B) = affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4335
    using affine_hull_span_0[of "convex hull insert 0 B"] affine_hull_span_0[of "S"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4336
    assms  hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4337
obtain d and f::"'n=>'n" where fd: "card d = card B & linear f & f ` B = {basis i |i. i : (d :: nat set)} & 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4338
       f ` span B = {x. ALL i<DIM('n). i ~: d --> x $$ i = (0::real)} &  inj_on f (span B)" and d:"d\<subseteq>{..<DIM('n)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4339
    using basis_to_substdbasis_subspace_isomorphism[of B,OF _ ] B_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4340
hence "bounded_linear f" using linear_conv_bounded_linear by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4341
have "d ~={}" using fd B_def `B ~={}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4342
have "(insert 0 {basis i |i. i : d}) = f ` (insert 0 B)" using fd linear_0 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4343
hence "(convex hull (insert 0 {basis i |i. i : d})) = f ` (convex hull (insert 0 B))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4344
   using convex_hull_linear_image[of f "(insert 0 {basis i |i. i : d})"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4345
   convex_hull_linear_image[of f "(insert 0 B)"] `bounded_linear f` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4346
moreover have "rel_interior (f ` (convex hull insert 0 B)) = 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4347
   f ` rel_interior (convex hull insert 0 B)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4348
   apply (rule  rel_interior_injective_on_span_linear_image[of f "(convex hull insert 0 B)"])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4349
   using `bounded_linear f` fd * by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4350
ultimately have "rel_interior (convex hull insert 0 B) ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4351
   using rel_interior_substd_simplex_nonempty[OF `d~={}` d] apply auto by blast 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4352
moreover have "convex hull (insert 0 B) <= S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4353
   using B_def assms hull_mono[of "insert 0 B" "S" "convex"] convex_hull_eq by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4354
ultimately have ?thesis using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4355
} ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4356
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4357
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4358
lemma rel_interior_convex_nonempty:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4359
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4360
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4361
shows "rel_interior S = {} <-> S = {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4362
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4363
{ assume "S ~= {}" from this obtain a where "a : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4364
  hence "0 : op + (-a) ` S" using assms exI[of "(%x. x:S & -a+x=0)" a] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4365
  hence "rel_interior (op + (-a) ` S) ~= {}"  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4366
    using rel_interior_convex_nonempty_aux[of "op + (-a) ` S"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4367
          convex_translation[of S "-a"] assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4368
  hence "rel_interior S ~= {}" using rel_interior_translation by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4369
} from this show ?thesis using rel_interior_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4370
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4371
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4372
lemma convex_rel_interior:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4373
fixes S :: "(_::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4374
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4375
shows "convex (rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4376
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4377
{ fix "x" "y" "u"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4378
  assume assm: "x:rel_interior S" "y:rel_interior S" "0<=u" "(u :: real) <= 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4379
  hence "x:S" using rel_interior_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4380
  have "x - u *\<^sub>R (x-y) : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4381
  proof(cases "0=u")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4382
     case False hence "0<u" using assm by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4383
        thus ?thesis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4384
        using assm rel_interior_convex_shrink[of S y x u] assms `x:S` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4385
     next
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4386
     case True thus ?thesis using assm by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4387
  qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4388
  hence "(1-u) *\<^sub>R x + u *\<^sub>R y : rel_interior S" by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4389
} from this show ?thesis unfolding convex_alt by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4390
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4391
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4392
lemma convex_closure_rel_interior: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4393
fixes S :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4394
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4395
shows "closure(rel_interior S) = closure S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4396
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4397
have h1: "closure(rel_interior S) <= closure S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4398
   using subset_closure[of "rel_interior S" S] rel_interior_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4399
{ assume "S ~= {}" from this obtain a where a_def: "a : rel_interior S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4400
    using rel_interior_convex_nonempty assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4401
  { fix x assume x_def: "x : closure S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4402
    { assume "x=a" hence "x : closure(rel_interior S)" using a_def unfolding closure_def by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4403
    moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4404
    { assume "x ~= a"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4405
       { fix e :: real assume e_def: "e>0" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4406
         def e1 == "min 1 (e/norm (x - a))" hence e1_def: "e1>0 & e1<=1 & e1*norm(x-a)<=e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4407
            using `x ~= a` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(x-a)"] by simp 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4408
         hence *: "x - e1 *\<^sub>R (x - a) : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4409
            using rel_interior_closure_convex_shrink[of S a x e1] assms x_def a_def e1_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4410
         have "EX y. y:rel_interior S & y ~= x & (dist y x) <= e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4411
            apply (rule_tac x="x - e1 *\<^sub>R (x - a)" in exI)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4412
            using * e1_def dist_norm[of "x - e1 *\<^sub>R (x - a)" x] `x ~= a` by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4413
      } hence "x islimpt rel_interior S" unfolding islimpt_approachable_le by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4414
      hence "x : closure(rel_interior S)" unfolding closure_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4415
    } ultimately have "x : closure(rel_interior S)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4416
  } hence ?thesis using h1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4417
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4418
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4419
{ assume "S = {}" hence "rel_interior S = {}" using rel_interior_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4420
  hence "closure(rel_interior S) = {}" using closure_empty by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4421
  hence ?thesis using `S={}` by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4422
} ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4423
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4424
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4425
lemma rel_interior_same_affine_hull:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4426
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4427
  assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4428
  shows "affine hull (rel_interior S) = affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4429
by (metis assms closure_same_affine_hull convex_closure_rel_interior)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4430
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4431
lemma rel_interior_aff_dim: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4432
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4433
  assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4434
  shows "aff_dim (rel_interior S) = aff_dim S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4435
by (metis aff_dim_affine_hull2 assms rel_interior_same_affine_hull)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4436
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4437
lemma rel_interior_rel_interior:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4438
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4439
  assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4440
  shows "rel_interior (rel_interior S) = rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4441
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4442
have "openin (subtopology euclidean (affine hull (rel_interior S))) (rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4443
  using opein_rel_interior[of S] rel_interior_same_affine_hull[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4444
from this show ?thesis using rel_interior_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4445
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4446
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4447
lemma rel_interior_rel_open:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4448
  fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4449
  assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4450
  shows "rel_open (rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4451
unfolding rel_open_def using rel_interior_rel_interior assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4452
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4453
lemma convex_rel_interior_closure_aux:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4454
  fixes x y z :: "_::euclidean_space"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4455
  assumes "0 < a" "0 < b" "(a+b) *\<^sub>R z = a *\<^sub>R x + b *\<^sub>R y"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4456
  obtains e where "0 < e" "e <= 1" "z = y - e *\<^sub>R (y-x)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4457
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4458
def e == "a/(a+b)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4459
have "z = (1 / (a + b)) *\<^sub>R ((a + b) *\<^sub>R z)" apply auto using assms by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4460
also have "... = (1 / (a + b)) *\<^sub>R (a *\<^sub>R x + b *\<^sub>R y)" using assms
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4461
   scaleR_cancel_left[of "1/(a+b)" "(a + b) *\<^sub>R z" "a *\<^sub>R x + b *\<^sub>R y"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4462
also have "... = y - e *\<^sub>R (y-x)" using e_def apply (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4463
   using scaleR_left_distrib[of "a/(a+b)" "b/(a+b)" y] assms add_divide_distrib[of a b "a+b"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4464
finally have "z = y - e *\<^sub>R (y-x)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4465
moreover have "0<e" using e_def assms divide_pos_pos[of a "a+b"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4466
moreover have "e<=1" using e_def assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4467
ultimately show ?thesis using that[of e] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4468
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4469
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4470
lemma convex_rel_interior_closure: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4471
  fixes S :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4472
  assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4473
  shows "rel_interior (closure S) = rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4474
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4475
{ assume "S={}" hence ?thesis using assms rel_interior_convex_nonempty by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4476
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4477
{ assume "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4478
  have "rel_interior (closure S) >= rel_interior S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4479
    using subset_rel_interior[of S "closure S"] closure_same_affine_hull closure_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4480
  moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4481
  { fix z assume z_def: "z : rel_interior (closure S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4482
    obtain x where x_def: "x : rel_interior S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4483
      using `S ~= {}` assms rel_interior_convex_nonempty by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4484
    { assume "x=z" hence "z : rel_interior S" using x_def by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4485
    moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4486
    { assume "x ~= z"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4487
      obtain e where e_def: "e > 0 & cball z e Int affine hull closure S <= closure S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4488
        using z_def rel_interior_cball[of "closure S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4489
      hence *: "0 < e/norm(z-x)" using e_def `x ~= z` divide_pos_pos[of e "norm(z-x)"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4490
      def y == "z + (e/norm(z-x)) *\<^sub>R (z-x)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4491
      have yball: "y : cball z e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4492
        using mem_cball y_def dist_norm[of z y] e_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4493
      have "x : affine hull closure S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4494
        using x_def rel_interior_subset_closure hull_inc[of x "closure S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4495
      moreover have "z : affine hull closure S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4496
        using z_def rel_interior_subset hull_subset[of "closure S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4497
      ultimately have "y : affine hull closure S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4498
        using y_def affine_affine_hull[of "closure S"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4499
          mem_affine_3_minus [of "affine hull closure S" z z x "e/norm(z-x)"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4500
      hence "y : closure S" using e_def yball by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4501
      have "(1+(e/norm(z-x))) *\<^sub>R z = (e/norm(z-x)) *\<^sub>R x + y"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4502
        using y_def by (simp add: algebra_simps) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4503
      from this obtain e1 where "0 < e1 & e1 <= 1 & z = y - e1 *\<^sub>R (y - x)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4504
        using * convex_rel_interior_closure_aux[of "e / norm (z - x)" 1 z x y] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4505
          by (auto simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4506
      hence "z : rel_interior S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4507
        using rel_interior_closure_convex_shrink assms x_def `y : closure S` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4508
    } ultimately have "z : rel_interior S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4509
  } ultimately have ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4510
} ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4511
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4512
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4513
lemma convex_interior_closure: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4514
fixes S :: "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4515
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4516
shows "interior (closure S) = interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4517
using closure_aff_dim[of S] interior_rel_interior_gen[of S] interior_rel_interior_gen[of "closure S"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4518
      convex_rel_interior_closure[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4519
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4520
lemma closure_eq_rel_interior_eq:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4521
fixes S1 S2 ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4522
assumes "convex S1" "convex S2"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4523
shows "(closure S1 = closure S2) <-> (rel_interior S1 = rel_interior S2)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4524
 by (metis convex_rel_interior_closure convex_closure_rel_interior assms)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4525
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4526
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4527
lemma closure_eq_between:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4528
fixes S1 S2 ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4529
assumes "convex S1" "convex S2"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4530
shows "(closure S1 = closure S2) <-> 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4531
      ((rel_interior S1 <= S2) & (S2 <= closure S1))" (is "?A <-> ?B")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4532
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4533
have "?A --> ?B" by (metis assms closure_subset convex_rel_interior_closure rel_interior_subset)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4534
moreover have "?B --> (closure S1 <= closure S2)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4535
     by (metis assms(1) convex_closure_rel_interior subset_closure)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4536
moreover have "?B --> (closure S1 >= closure S2)" by (metis closed_closure closure_minimal)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4537
ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4538
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4539
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4540
lemma open_inter_closure_rel_interior:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4541
fixes S A ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4542
assumes "convex S" "open A"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4543
shows "((A Int closure S) = {}) <-> ((A Int rel_interior S) = {})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4544
by (metis assms convex_closure_rel_interior open_inter_closure_eq_empty) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4545
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4546
definition "rel_frontier S = closure S - rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4547
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4548
lemma closed_affine_hull: "closed (affine hull ((S :: ('n::euclidean_space) set)))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4549
by (metis affine_affine_hull affine_closed)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4550
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4551
lemma closed_rel_frontier: "closed(rel_frontier (S :: ('n::euclidean_space) set))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4552
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4553
have *: "closedin (subtopology euclidean (affine hull S)) (closure S - rel_interior S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4554
apply (rule closedin_diff[of "subtopology euclidean (affine hull S)""closure S" "rel_interior S"])  using closed_closedin_trans[of "affine hull S" "closure S"] closed_affine_hull[of S] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4555
  closure_affine_hull[of S] opein_rel_interior[of S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4556
show ?thesis apply (rule closedin_closed_trans[of "affine hull S" "rel_frontier S"]) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4557
  unfolding rel_frontier_def using * closed_affine_hull by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4558
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4559
 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4560
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4561
lemma convex_rel_frontier_aff_dim:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4562
fixes S1 S2 ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4563
assumes "convex S1" "convex S2" "S2 ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4564
assumes "S1 <= rel_frontier S2"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4565
shows "aff_dim S1 < aff_dim S2" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4566
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4567
have "S1 <= closure S2" using assms unfolding rel_frontier_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4568
hence *: "affine hull S1 <= affine hull S2" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4569
   using hull_mono[of "S1" "closure S2"] closure_same_affine_hull[of S2] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4570
hence "aff_dim S1 <= aff_dim S2" using * aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4571
    aff_dim_subset[of "affine hull S1" "affine hull S2"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4572
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4573
{ assume eq: "aff_dim S1 = aff_dim S2"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4574
  hence "S1 ~= {}" using aff_dim_empty[of S1] aff_dim_empty[of S2] `S2 ~= {}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4575
  have **: "affine hull S1 = affine hull S2" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4576
     apply (rule affine_dim_equal) using * affine_affine_hull apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4577
     using `S1 ~= {}` hull_subset[of S1] apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4578
     using eq aff_dim_affine_hull[of S1] aff_dim_affine_hull[of S2] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4579
  obtain a where a_def: "a : rel_interior S1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4580
     using  `S1 ~= {}` rel_interior_convex_nonempty assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4581
  obtain T where T_def: "open T & a : T Int S1 & T Int affine hull S1 <= S1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4582
     using mem_rel_interior[of a S1] a_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4583
  hence "a : T Int closure S2" using a_def assms unfolding rel_frontier_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4584
  from this obtain b where b_def: "b : T Int rel_interior S2" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4585
     using open_inter_closure_rel_interior[of S2 T] assms T_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4586
  hence "b : affine hull S1" using rel_interior_subset hull_subset[of S2] ** by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4587
  hence "b : S1" using T_def b_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4588
  hence False using b_def assms unfolding rel_frontier_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4589
} ultimately show ?thesis using zless_le by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4590
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4591
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4592
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4593
lemma convex_rel_interior_if:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4594
fixes S ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4595
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4596
assumes "z : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4597
shows "(!x:affine hull S. EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S ))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4598
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4599
obtain e1 where e1_def: "e1>0 & cball z e1 Int affine hull S <= S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4600
    using mem_rel_interior_cball[of z S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4601
{ fix x assume x_def: "x:affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4602
  { assume "x ~= z"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4603
    def m == "1+e1/norm(x-z)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4604
    hence "m>1" using e1_def `x ~= z` divide_pos_pos[of e1 "norm (x - z)"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4605
    { fix e assume e_def: "e>1 & e<=m"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4606
      have "z : affine hull S" using assms rel_interior_subset hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4607
      hence *: "(1-e)*\<^sub>R x+ e *\<^sub>R z : affine hull S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4608
         using mem_affine[of "affine hull S" x z "(1-e)" e] affine_affine_hull[of S] x_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4609
      have "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) = norm ((e - 1) *\<^sub>R (x-z))" by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4610
      also have "...= (e - 1) * norm(x-z)" using norm_scaleR e_def by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4611
      also have "...<=(m - 1) * norm(x-z)" using e_def mult_right_mono[of _ _ "norm(x-z)"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4612
      also have "...= (e1 / norm (x - z)) * norm (x - z)" using m_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4613
      also have "...=e1" using `x ~= z` e1_def by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4614
      finally have **: "norm (z + e *\<^sub>R x - (x + e *\<^sub>R z)) <= e1" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4615
      have "(1-e)*\<^sub>R x+ e *\<^sub>R z : cball z e1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4616
         using m_def ** unfolding cball_def dist_norm by (auto simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4617
      hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def * e1_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4618
    } hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" using `m>1` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4619
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4620
  moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4621
  { assume "x=z" def m == "1+e1" hence "m>1" using e1_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4622
    { fix e assume e_def: "e>1 & e<=m"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4623
      hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4624
        using e1_def x_def `x=z` by (auto simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4625
      hence "(1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4626
    } hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" using `m>1` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4627
  } ultimately have "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4628
} from this show ?thesis by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4629
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4630
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4631
lemma convex_rel_interior_if2:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4632
fixes S ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4633
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4634
assumes "z : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4635
shows "(!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4636
using convex_rel_interior_if[of S z] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4637
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4638
lemma convex_rel_interior_only_if:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4639
fixes S ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4640
assumes "convex S" "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4641
assumes "(!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4642
shows "z : rel_interior S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4643
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4644
obtain x where x_def: "x : rel_interior S" using rel_interior_convex_nonempty assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4645
hence "x:S" using rel_interior_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4646
from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4647
def y == "(1 - e) *\<^sub>R x + e *\<^sub>R z" hence "y:S" using e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4648
def e1 == "1/e" hence "0<e1 & e1<1" using e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4649
hence "z=y-(1-e1)*\<^sub>R (y-x)" using e1_def y_def by (auto simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4650
from this show ?thesis 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4651
    using rel_interior_convex_shrink[of S x y "1-e1"] `0<e1 & e1<1` `y:S` x_def assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4652
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4653
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4654
lemma convex_rel_interior_iff:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4655
fixes S ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4656
assumes "convex S" "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4657
shows "z : rel_interior S <-> (!x:S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4658
using assms hull_subset[of S "affine"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4659
      convex_rel_interior_if[of S z] convex_rel_interior_only_if[of S z] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4660
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4661
lemma convex_rel_interior_iff2:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4662
fixes S ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4663
assumes "convex S" "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4664
shows "z : rel_interior S <-> (!x:affine hull S. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4665
using assms hull_subset[of S] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4666
      convex_rel_interior_if2[of S z] convex_rel_interior_only_if[of S z] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4667
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4668
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4669
lemma convex_interior_iff:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4670
fixes S ::  "('n::euclidean_space) set" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4671
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4672
shows "z : interior S <-> (!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4673
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4674
{ assume a: "~(aff_dim S = int DIM('n))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4675
  { assume "z : interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4676
    hence False using a interior_rel_interior_gen[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4677
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4678
  moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4679
  { assume r: "!x. EX e. e>0 & z+ e *\<^sub>R x : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4680
    { fix x obtain e1 where e1_def: "e1>0 & z+ e1 *\<^sub>R (x-z) : S" using r by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4681
      obtain e2 where e2_def: "e2>0 & z+ e2 *\<^sub>R (z-x) : S" using r by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4682
      def x1 == "z+ e1 *\<^sub>R (x-z)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4683
         hence x1: "x1 : affine hull S" using e1_def hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4684
      def x2 == "z+ e2 *\<^sub>R (z-x)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4685
         hence x2: "x2 : affine hull S" using e2_def hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4686
      have *: "e1/(e1+e2) + e2/(e1+e2) = 1" using divide.add[of e1 e2 "e1+e2"] e1_def e2_def by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4687
      hence "z = (e2/(e1+e2)) *\<^sub>R x1 + (e1/(e1+e2)) *\<^sub>R x2"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4688
         using x1_def x2_def apply (auto simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4689
         using scaleR_left_distrib[of "e1/(e1+e2)" "e2/(e1+e2)" z] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4690
      hence z: "z : affine hull S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4691
         using mem_affine[of "affine hull S" x1 x2 "e2/(e1+e2)" "e1/(e1+e2)"]  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4692
         x1 x2 affine_affine_hull[of S] * by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4693
      have "x1-x2 = (e1+e2) *\<^sub>R (x-z)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4694
         using x1_def x2_def by (auto simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4695
      hence "x=z+(1/(e1+e2)) *\<^sub>R (x1-x2)" using e1_def e2_def by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4696
      hence "x : affine hull S" using mem_affine_3_minus[of "affine hull S" z x1 x2 "1/(e1+e2)"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4697
          x1 x2 z affine_affine_hull[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4698
    } hence "affine hull S = UNIV" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4699
    hence "aff_dim S = int DIM('n)" using aff_dim_affine_hull[of S] by (simp add: aff_dim_univ)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4700
    hence False using a by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4701
  } ultimately have ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4702
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4703
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4704
{ assume a: "aff_dim S = int DIM('n)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4705
  hence "S ~= {}" using aff_dim_empty[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4706
  have *: "affine hull S=UNIV" using a affine_hull_univ by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4707
  { assume "z : interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4708
    hence "z : rel_interior S" using a interior_rel_interior_gen[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4709
    hence **: "(!x. EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4710
      using convex_rel_interior_iff2[of S z] assms `S~={}` * by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4711
    fix x obtain e1 where e1_def: "e1>1 & (1-e1)*\<^sub>R (z-x)+ e1 *\<^sub>R z : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4712
      using **[rule_format, of "z-x"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4713
    def e == "e1 - 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4714
    hence "(1-e1)*\<^sub>R (z-x)+ e1 *\<^sub>R z = z+ e *\<^sub>R x" by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4715
    hence "e>0 & z+ e *\<^sub>R x : S" using e1_def e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4716
    hence "EX e. e>0 & z+ e *\<^sub>R x : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4717
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4718
  moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4719
  { assume r: "(!x. EX e. e>0 & z+ e *\<^sub>R x : S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4720
    { fix x obtain e1 where e1_def: "e1>0 & z + e1*\<^sub>R (z-x) : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4721
         using r[rule_format, of "z-x"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4722
      def e == "e1 + 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4723
      hence "z + e1*\<^sub>R (z-x) = (1-e)*\<^sub>R x+ e *\<^sub>R z" by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4724
      hence "e > 1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S" using e1_def e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4725
      hence "EX e. e>1 & (1-e)*\<^sub>R x+ e *\<^sub>R z : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4726
    }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4727
    hence "z : rel_interior S" using convex_rel_interior_iff2[of S z] assms `S~={}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4728
    hence "z : interior S" using a interior_rel_interior_gen[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4729
  } ultimately have ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4730
} ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4731
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4732
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4733
subsection{* Relative interior and closure under commom operations *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4734
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4735
lemma rel_interior_inter_aux: "Inter {rel_interior S |S. S : I} <= Inter I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4736
proof- 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4737
{ fix y assume "y : Inter {rel_interior S |S. S : I}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4738
  hence y_def: "!S : I. y : rel_interior S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4739
  { fix S assume "S : I" hence "y : S" using rel_interior_subset y_def by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4740
  hence "y : Inter I" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4741
} thus ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4742
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4743
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4744
lemma closure_inter: "closure (Inter I) <= Inter {closure S |S. S : I}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4745
proof- 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4746
{ fix y assume "y : Inter I" hence y_def: "!S : I. y : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4747
  { fix S assume "S : I" hence "y : closure S" using closure_subset y_def by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4748
  hence "y : Inter {closure S |S. S : I}" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4749
} hence "Inter I <= Inter {closure S |S. S : I}" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4750
moreover have "Inter {closure S |S. S : I} : closed" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4751
  unfolding mem_def closed_Inter closed_closure by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4752
ultimately show ?thesis using closure_hull[of "Inter I"]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4753
  hull_minimal[of "Inter I" "Inter {closure S |S. S : I}" "closed"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4754
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4755
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4756
lemma convex_closure_rel_interior_inter: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4757
assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4758
assumes "Inter {rel_interior S |S. S : I} ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4759
shows "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4760
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4761
obtain x where x_def: "!S : I. x : rel_interior S" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4762
{ fix y assume "y : Inter {closure S |S. S : I}" hence y_def: "!S : I. y : closure S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4763
  { assume "y = x" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4764
    hence "y : closure (Inter {rel_interior S |S. S : I})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4765
       using x_def closure_subset[of "Inter {rel_interior S |S. S : I}"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4766
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4767
  moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4768
  { assume "y ~= x"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4769
    { fix e :: real assume e_def: "0 < e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4770
      def e1 == "min 1 (e/norm (y - x))" hence e1_def: "e1>0 & e1<=1 & e1*norm(y-x)<=e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4771
        using `y ~= x` `e>0` divide_pos_pos[of e] le_divide_eq[of e1 e "norm(y-x)"] by simp 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4772
      def z == "y - e1 *\<^sub>R (y - x)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4773
      { fix S assume "S : I" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4774
        hence "z : rel_interior S" using rel_interior_closure_convex_shrink[of S x y e1] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4775
           assms x_def y_def e1_def z_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4776
      } hence *: "z : Inter {rel_interior S |S. S : I}" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4777
      have "EX z. z:Inter {rel_interior S |S. S : I} & z ~= y & (dist z y) <= e"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4778
           apply (rule_tac x="z" in exI) using `y ~= x` z_def * e1_def e_def dist_norm[of z y] by simp
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4779
    } hence "y islimpt Inter {rel_interior S |S. S : I}" unfolding islimpt_approachable_le by blast 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4780
    hence "y : closure (Inter {rel_interior S |S. S : I})" unfolding closure_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4781
  } ultimately have "y : closure (Inter {rel_interior S |S. S : I})" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4782
} from this show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4783
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4784
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4785
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4786
lemma convex_closure_inter: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4787
assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4788
assumes "Inter {rel_interior S |S. S : I} ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4789
shows "closure (Inter I) = Inter {closure S |S. S : I}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4790
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4791
have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4792
  using convex_closure_rel_interior_inter assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4793
moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4794
    using rel_interior_inter_aux 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4795
          subset_closure[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4796
ultimately show ?thesis using closure_inter[of I] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4797
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4798
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4799
lemma convex_inter_rel_interior_same_closure: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4800
assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4801
assumes "Inter {rel_interior S |S. S : I} ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4802
shows "closure (Inter {rel_interior S |S. S : I}) = closure (Inter I)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4803
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4804
have "Inter {closure S |S. S : I} <= closure (Inter {rel_interior S |S. S : I})" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4805
  using convex_closure_rel_interior_inter assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4806
moreover have "closure (Inter {rel_interior S |S. S : I}) <= closure (Inter I)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4807
    using rel_interior_inter_aux 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4808
          subset_closure[of "Inter {rel_interior S |S. S : I}" "Inter I"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4809
ultimately show ?thesis using closure_inter[of I] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4810
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4811
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4812
lemma convex_rel_interior_inter: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4813
assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4814
assumes "Inter {rel_interior S |S. S : I} ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4815
shows "rel_interior (Inter I) <= Inter {rel_interior S |S. S : I}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4816
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4817
have "convex(Inter I)" using assms convex_Inter by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4818
moreover have "convex(Inter {rel_interior S |S. S : I})" apply (rule convex_Inter)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4819
   using assms convex_rel_interior by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4820
ultimately have "rel_interior (Inter {rel_interior S |S. S : I}) = rel_interior (Inter I)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4821
   using convex_inter_rel_interior_same_closure assms 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4822
   closure_eq_rel_interior_eq[of "Inter {rel_interior S |S. S : I}" "Inter I"] by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4823
from this show ?thesis using rel_interior_subset[of "Inter {rel_interior S |S. S : I}"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4824
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4825
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4826
lemma convex_rel_interior_finite_inter: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4827
assumes "!S : I. convex (S :: ('n::euclidean_space) set)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4828
assumes "Inter {rel_interior S |S. S : I} ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4829
assumes "finite I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4830
shows "rel_interior (Inter I) = Inter {rel_interior S |S. S : I}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4831
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4832
have "Inter I ~= {}" using assms rel_interior_inter_aux[of I] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4833
have "convex (Inter I)" using convex_Inter assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4834
{ assume "I={}" hence ?thesis using Inter_empty rel_interior_univ2 by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4835
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4836
{ assume "I ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4837
{ fix z assume z_def: "z : Inter {rel_interior S |S. S : I}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4838
  { fix x assume x_def: "x : Inter I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4839
    { fix S assume S_def: "S : I" hence "z : rel_interior S" "x : S" using z_def x_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4840
      (*from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : S"*)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4841
      hence "EX m. m>1 & (!e. (e>1 & e<=m) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S )"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4842
         using convex_rel_interior_if[of S z] S_def assms hull_subset[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4843
    } from this obtain mS where mS_def: "!S : I. (mS(S) > (1 :: real) & 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4844
         (!e. (e>1 & e<=mS(S)) --> (1-e)*\<^sub>R x+ e *\<^sub>R z : S))" by metis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4845
    obtain e where e_def: "e=Min (mS ` I)" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4846
    have "e : (mS ` I)" using e_def assms `I ~= {}` by (simp add: Min_in) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4847
    hence "e>(1 :: real)" using mS_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4848
    moreover have "!S : I. e<=mS(S)" using e_def assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4849
    ultimately have "EX e>1. (1 - e) *\<^sub>R x + e *\<^sub>R z : Inter I" using mS_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4850
  } hence "z : rel_interior (Inter I)" using convex_rel_interior_iff[of "Inter I" z]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4851
       `Inter I ~= {}` `convex (Inter I)` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4852
} from this have ?thesis using convex_rel_interior_inter[of I] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4853
} ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4854
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4855
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4856
lemma convex_closure_inter_two: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4857
fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4858
assumes "convex S" "convex T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4859
assumes "(rel_interior S) Int (rel_interior T) ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4860
shows "closure (S Int T) = (closure S) Int (closure T)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4861
using convex_closure_inter[of "{S,T}"] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4862
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4863
lemma convex_rel_interior_inter_two: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4864
fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4865
assumes "convex S" "convex T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4866
assumes "(rel_interior S) Int (rel_interior T) ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4867
shows "rel_interior (S Int T) = (rel_interior S) Int (rel_interior T)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4868
using convex_rel_interior_finite_inter[of "{S,T}"] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4869
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4870
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4871
lemma convex_affine_closure_inter: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4872
fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4873
assumes "convex S" "affine T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4874
assumes "(rel_interior S) Int T ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4875
shows "closure (S Int T) = (closure S) Int T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4876
proof- 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4877
have "affine hull T = T" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4878
hence "rel_interior T = T" using rel_interior_univ[of T] by metis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4879
moreover have "closure T = T" using assms affine_closed[of T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4880
ultimately show ?thesis using convex_closure_inter_two[of S T] assms affine_imp_convex by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4881
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4882
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4883
lemma convex_affine_rel_interior_inter: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4884
fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4885
assumes "convex S" "affine T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4886
assumes "(rel_interior S) Int T ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4887
shows "rel_interior (S Int T) = (rel_interior S) Int T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4888
proof- 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4889
have "affine hull T = T" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4890
hence "rel_interior T = T" using rel_interior_univ[of T] by metis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4891
moreover have "closure T = T" using assms affine_closed[of T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4892
ultimately show ?thesis using convex_rel_interior_inter_two[of S T] assms affine_imp_convex by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4893
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4894
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4895
lemma subset_rel_interior_convex:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4896
fixes S T :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4897
assumes "convex S" "convex T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4898
assumes "S <= closure T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4899
assumes "~(S <= rel_frontier T)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4900
shows "rel_interior S <= rel_interior T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4901
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4902
have *: "S Int closure T = S" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4903
have "~(rel_interior S <= rel_frontier T)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4904
     using subset_closure[of "rel_interior S" "rel_frontier T"] closed_rel_frontier[of T] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4905
     closure_closed convex_closure_rel_interior[of S] closure_subset[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4906
hence "(rel_interior S) Int (rel_interior (closure T)) ~= {}" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4907
     using assms rel_frontier_def[of T] rel_interior_subset convex_rel_interior_closure[of T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4908
hence "rel_interior S Int rel_interior T = rel_interior (S Int closure T)" using assms convex_closure  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4909
     convex_rel_interior_inter_two[of S "closure T"] convex_rel_interior_closure[of T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4910
also have "...=rel_interior (S)" using * by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4911
finally show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4912
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4913
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4914
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4915
lemma rel_interior_convex_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4916
fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4917
assumes "linear f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4918
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4919
shows "f ` (rel_interior S) = rel_interior (f ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4920
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4921
{ assume "S = {}" hence ?thesis using assms rel_interior_empty rel_interior_convex_nonempty by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4922
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4923
{ assume "S ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4924
have *: "f ` (rel_interior S) <= f ` S" unfolding image_mono using rel_interior_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4925
have "f ` S <= f ` (closure S)" unfolding image_mono using closure_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4926
also have "... = f ` (closure (rel_interior S))" using convex_closure_rel_interior assms by auto  
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4927
also have "... <= closure (f ` (rel_interior S))" using closure_linear_image assms by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4928
finally have "closure (f ` S) = closure (f ` rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4929
   using subset_closure[of "f ` S" "closure (f ` rel_interior S)"] closure_closure 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4930
         subset_closure[of "f ` rel_interior S" "f ` S"] * by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4931
hence "rel_interior (f ` S) = rel_interior (f ` rel_interior S)" using assms convex_rel_interior
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4932
   linear_conv_bounded_linear[of f] convex_linear_image[of S] convex_linear_image[of "rel_interior S"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4933
   closure_eq_rel_interior_eq[of "f ` S" "f ` rel_interior S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4934
hence "rel_interior (f ` S) <= f ` rel_interior S" using rel_interior_subset by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4935
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4936
{ fix z assume z_def: "z : f ` rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4937
  from this obtain z1 where z1_def: "z1 : rel_interior S & (f z1 = z)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4938
  { fix x assume "x : f ` S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4939
    from this obtain x1 where x1_def: "x1 : S & (f x1 = x)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4940
    from this obtain e where e_def: "e>1 & (1 - e) *\<^sub>R x1 + e *\<^sub>R z1 : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4941
       using convex_rel_interior_iff[of S z1] `convex S` x1_def z1_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4942
    moreover have "f ((1 - e) *\<^sub>R x1 + e *\<^sub>R z1) = (1 - e) *\<^sub>R x + e *\<^sub>R z"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4943
        using x1_def z1_def `linear f` by (simp add: linear_add_cmul)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4944
    ultimately have "(1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4945
        using imageI[of "(1 - e) *\<^sub>R x1 + e *\<^sub>R z1" S f] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4946
    hence "EX e. (e>1 & (1 - e) *\<^sub>R x + e *\<^sub>R z : f ` S)" using e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4947
  } from this have "z : rel_interior (f ` S)" using convex_rel_interior_iff[of "f ` S" z] `convex S` 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4948
       `linear f` `S ~= {}` convex_linear_image[of S f]  linear_conv_bounded_linear[of f] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4949
} ultimately have ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4950
} ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4951
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4952
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4953
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4954
lemma convex_linear_preimage:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4955
  assumes c:"convex S" and l:"bounded_linear f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4956
  shows "convex(f -` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4957
proof(auto simp add: convex_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4958
  interpret f: bounded_linear f by fact
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4959
  fix x y assume xy:"f x : S" "f y : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4960
  fix u v ::real assume uv:"0 <= u" "0 <= v" "u + v = 1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4961
  show "f (u *\<^sub>R x + v *\<^sub>R y) : S" unfolding image_iff
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4962
    using bexI[of _ "u *\<^sub>R x + v *\<^sub>R y"] f.add f.scaleR
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4963
      c[unfolded convex_def] xy uv by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4964
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4965
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4966
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4967
lemma rel_interior_convex_linear_preimage:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4968
fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4969
assumes "linear f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4970
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4971
assumes "f -` (rel_interior S) ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4972
shows "rel_interior (f -` S) = f -` (rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4973
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4974
have "S ~= {}" using assms rel_interior_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4975
have nonemp: "f -` S ~= {}" by (metis assms(3) rel_interior_subset subset_empty vimage_mono) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4976
hence "S Int (range f) ~= {}" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4977
have conv: "convex (f -` S)" using convex_linear_preimage assms linear_conv_bounded_linear by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4978
hence "convex (S Int (range f))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4979
  by (metis assms(1) assms(2) convex_Int subspace_UNIV subspace_imp_convex subspace_linear_image)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4980
{ fix z assume "z : f -` (rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4981
  hence z_def: "f z : rel_interior S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4982
  { fix x assume "x : f -` S" from this have x_def: "f x : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4983
    from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4984
      using convex_rel_interior_iff[of S "f z"] z_def assms `S ~= {}` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4985
    moreover have "(1-e)*\<^sub>R (f x)+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R x + e *\<^sub>R z)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4986
      using `linear f` by (simp add: linear_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4987
    ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R z : f -` S" using e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4988
  } hence "z : rel_interior (f -` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4989
       using convex_rel_interior_iff[of "f -` S" z] conv nonemp by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4990
} 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4991
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4992
{ fix z assume z_def: "z : rel_interior (f -` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4993
  { fix x assume x_def: "x: S Int (range f)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4994
    from this obtain y where y_def: "(f y = x) & (y : f -` S)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4995
    from this obtain e where e_def: "e>1 & (1-e)*\<^sub>R y+ e *\<^sub>R z : f -` S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4996
      using convex_rel_interior_iff[of "f -` S" z] z_def conv by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4997
    moreover have "(1-e)*\<^sub>R x+ e *\<^sub>R (f z) = f ((1-e)*\<^sub>R y + e *\<^sub>R z)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4998
      using `linear f` y_def by (simp add: linear_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  4999
    ultimately have "EX e. e>1 & (1-e)*\<^sub>R x + e *\<^sub>R (f z) : S Int (range f)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5000
      using e_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5001
  } hence "f z : rel_interior (S Int (range f))" using `convex (S Int (range f))`
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5002
    `S Int (range f) ~= {}` convex_rel_interior_iff[of "S Int (range f)" "f z"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5003
  moreover have "affine (range f)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5004
    by (metis assms(1) subspace_UNIV subspace_imp_affine subspace_linear_image)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5005
  ultimately have "f z : rel_interior S" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5006
    using convex_affine_rel_interior_inter[of S "range f"] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5007
  hence "z : f -` (rel_interior S)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5008
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5009
ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5010
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5011
    
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5012
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5013
lemma convex_direct_sum:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5014
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5015
fixes T :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5016
assumes "convex S" "convex T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5017
shows "convex (S <*> T)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5018
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5019
{
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5020
fix x assume "x : S <*> T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5021
from this obtain xs xt where xst_def: "xs : S & xt : T & (xs,xt) = x" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5022
fix y assume "y : S <*> T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5023
from this obtain ys yt where yst_def: "ys : S & yt : T & (ys,yt) = y" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5024
fix u v assume uv_def: "(u :: real)>=0 & (v :: real)>=0 & u+v=1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5025
have "u *\<^sub>R x + v *\<^sub>R y = (u *\<^sub>R xs + v *\<^sub>R ys, u *\<^sub>R xt + v *\<^sub>R yt)" using xst_def yst_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5026
moreover have "u *\<^sub>R xs + v *\<^sub>R ys : S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5027
   using uv_def xst_def yst_def convex_def[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5028
moreover have "u *\<^sub>R xt + v *\<^sub>R yt : T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5029
   using uv_def xst_def yst_def convex_def[of T] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5030
ultimately have "u *\<^sub>R x + v *\<^sub>R y : S <*> T" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5031
} from this show ?thesis unfolding convex_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5032
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5033
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5034
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5035
lemma convex_hull_direct_sum:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5036
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5037
fixes T :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5038
shows "convex hull (S <*> T) = (convex hull S) <*> (convex hull T)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5039
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5040
{ fix x assume "x : (convex hull S) <*> (convex hull T)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5041
  from this obtain xs xt where xst_def: "xs : convex hull S & xt : convex hull T & (xs,xt) = x" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5042
  from xst_def obtain sI su where s: "finite sI & sI <= S & (ALL x:sI. 0 <= su x) & setsum su sI = 1 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5043
     & (SUM v:sI. su v *\<^sub>R v) = xs" using convex_hull_explicit[of S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5044
  from xst_def obtain tI tu where t: "finite tI & tI <= T & (ALL x:tI. 0 <= tu x) & setsum tu tI = 1 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5045
     & (SUM v:tI. tu v *\<^sub>R v) = xt" using convex_hull_explicit[of T] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5046
  def I == "(sI <*> tI)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5047
  def u == "(%i. (su (fst i))*(tu(snd i)))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5048
  have "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5049
     (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vs)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5050
     using fst_setsum[of "(%v. (su (fst v) * tu (snd v)) *\<^sub>R v)" "sI <*> tI"]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5051
     by (simp add: split_def scaleR_prod_def setsum_cartesian_product)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5052
  also have "...=(SUM vt:tI. tu vt *\<^sub>R (SUM vs:sI. su vs *\<^sub>R vs))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5053
     using setsum_commute[of "(%vt vs. (su vs * tu vt) *\<^sub>R vs)" sI tI]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5054
     by (simp add: mult_commute scaleR_right.setsum)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5055
  also have "...=(SUM vt:tI. tu vt *\<^sub>R xs)" using s by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5056
  also have "...=(SUM vt:tI. tu vt) *\<^sub>R xs" by (simp add: scaleR_left.setsum)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5057
  also have "...=xs" using t by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5058
  finally have h1: "fst (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xs" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5059
  have "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5060
     (SUM vs:sI. SUM vt:tI. (su vs * tu vt) *\<^sub>R vt)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5061
     using snd_setsum[of "(%v. (su (fst v) * tu (snd v)) *\<^sub>R v)" "sI <*> tI"]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5062
     by (simp add: split_def scaleR_prod_def setsum_cartesian_product)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5063
  also have "...=(SUM vs:sI. su vs *\<^sub>R (SUM vt:tI. tu vt *\<^sub>R vt))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5064
     by (simp add: mult_commute scaleR_right.setsum)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5065
  also have "...=(SUM vs:sI. su vs *\<^sub>R xt)" using t by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5066
  also have "...=(SUM vs:sI. su vs) *\<^sub>R xt" by (simp add: scaleR_left.setsum)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5067
  also have "...=xt" using s by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5068
  finally have h2: "snd (SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v)=xt" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5069
  from h1 h2 have "(SUM v:sI <*> tI. (su (fst v) * tu (snd v)) *\<^sub>R v) = x" using xst_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5070
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5071
  moreover have "finite I & (I <= S <*> T)" using s t I_def by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5072
  moreover have "!i:I. 0 <= u i" using s t I_def u_def by (simp add: mult_nonneg_nonneg)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5073
  moreover have "setsum u I = 1" using u_def I_def setsum_cartesian_product[of "(% x y. (su x)*(tu y))"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5074
     s t setsum_product[of su sI tu tI] by (auto simp add: split_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5075
  ultimately have "x : convex hull (S <*> T)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5076
     apply (subst convex_hull_explicit[of "S <*> T"]) apply rule
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5077
     apply (rule_tac x="I" in exI) apply (rule_tac x="u" in exI)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5078
     using I_def u_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5079
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5080
hence "convex hull (S <*> T) >= (convex hull S) <*> (convex hull T)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5081
moreover have "(convex hull S) <*> (convex hull T) : convex" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5082
   unfolding mem_def by (simp add: convex_direct_sum convex_convex_hull)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5083
ultimately show ?thesis 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5084
   using hull_minimal[of "S <*> T" "(convex hull S) <*> (convex hull T)" "convex"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5085
         hull_subset[of S convex] hull_subset[of T convex] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5086
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5087
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5088
lemma rel_interior_direct_sum:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5089
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5090
fixes T :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5091
assumes "convex S" "convex T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5092
shows "rel_interior (S <*> T) = rel_interior S <*> rel_interior T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5093
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5094
{ assume "S={}" hence ?thesis apply auto using rel_interior_empty by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5095
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5096
{ assume "T={}" hence ?thesis apply auto using rel_interior_empty by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5097
moreover {
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5098
assume "S ~={}" "T ~={}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5099
hence ri: "rel_interior S ~= {}" "rel_interior T ~= {}" using rel_interior_convex_nonempty assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5100
hence "fst -` rel_interior S ~= {}" using fst_vimage_eq_Times[of "rel_interior S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5101
hence "rel_interior ((fst :: 'n * 'm => 'n) -` S) = fst -` rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5102
  using fst_linear `convex S` rel_interior_convex_linear_preimage[of fst S] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5103
hence s: "rel_interior (S <*> (UNIV :: 'm set)) = rel_interior S <*> UNIV" by (simp add: fst_vimage_eq_Times)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5104
from ri have "snd -` rel_interior T ~= {}" using snd_vimage_eq_Times[of "rel_interior T"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5105
hence "rel_interior ((snd :: 'n * 'm => 'm) -` T) = snd -` rel_interior T"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5106
  using snd_linear `convex T` rel_interior_convex_linear_preimage[of snd T] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5107
hence t: "rel_interior ((UNIV :: 'n set) <*> T) = UNIV <*> rel_interior T" by (simp add: snd_vimage_eq_Times)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5108
from s t have *: "rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5109
  = rel_interior S <*> rel_interior T" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5110
have "(S <*> T) = (S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5111
hence "rel_interior (S <*> T) = rel_interior ((S <*> (UNIV :: 'm set)) Int ((UNIV :: 'n set) <*> T))" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5112
also have "...=rel_interior (S <*> (UNIV :: 'm set)) Int rel_interior ((UNIV :: 'n set) <*> T)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5113
   apply (subst convex_rel_interior_inter_two[of "S <*> (UNIV :: 'm set)" "(UNIV :: 'n set) <*> T"]) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5114
   using * ri assms convex_direct_sum by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5115
finally have ?thesis using * by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5116
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5117
ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5118
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5119
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5120
lemma rel_interior_scaleR: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5121
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5122
assumes "c ~= 0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5123
shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5124
using rel_interior_injective_linear_image[of "(op *\<^sub>R c)" S]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5125
      linear_conv_bounded_linear[of "op *\<^sub>R c"] linear_scaleR injective_scaleR[of c] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5126
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5127
lemma rel_interior_convex_scaleR: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5128
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5129
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5130
shows "(op *\<^sub>R c) ` (rel_interior S) = rel_interior ((op *\<^sub>R c) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5131
by (metis assms linear_scaleR rel_interior_convex_linear_image)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5132
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5133
lemma convex_rel_open_scaleR: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5134
fixes S :: "('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5135
assumes "convex S" "rel_open S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5136
shows "convex ((op *\<^sub>R c) ` S) & rel_open ((op *\<^sub>R c) ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5137
by (metis assms convex_scaling rel_interior_convex_scaleR rel_open_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5138
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5139
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5140
lemma convex_rel_open_finite_inter: 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5141
assumes "!S : I. (convex (S :: ('n::euclidean_space) set) & rel_open S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5142
assumes "finite I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5143
shows "convex (Inter I) & rel_open (Inter I)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5144
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5145
{ assume "Inter {rel_interior S |S. S : I} = {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5146
  hence "Inter I = {}" using assms unfolding rel_open_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5147
  hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5148
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5149
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5150
{ assume "Inter {rel_interior S |S. S : I} ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5151
  hence "rel_open (Inter I)" using assms unfolding rel_open_def
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5152
    using convex_rel_interior_finite_inter[of I] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5153
  hence ?thesis using convex_Inter assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5154
} ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5155
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5156
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5157
lemma convex_rel_open_linear_image:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5158
fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5159
assumes "linear f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5160
assumes "convex S" "rel_open S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5161
shows "convex (f ` S) & rel_open (f ` S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5162
by (metis assms convex_linear_image rel_interior_convex_linear_image 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5163
   linear_conv_bounded_linear rel_open_def)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5164
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5165
lemma convex_rel_open_linear_preimage:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5166
fixes f :: "('m::euclidean_space) => ('n::euclidean_space)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5167
assumes "linear f"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5168
assumes "convex S" "rel_open S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5169
shows "convex (f -` S) & rel_open (f -` S)" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5170
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5171
{ assume "f -` (rel_interior S) = {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5172
  hence "f -` S = {}" using assms unfolding rel_open_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5173
  hence ?thesis unfolding rel_open_def using rel_interior_empty by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5174
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5175
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5176
{ assume "f -` (rel_interior S) ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5177
  hence "rel_open (f -` S)" using assms unfolding rel_open_def
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5178
    using rel_interior_convex_linear_preimage[of f S] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5179
  hence ?thesis using convex_linear_preimage assms linear_conv_bounded_linear by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5180
} ultimately show ?thesis by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5181
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5182
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5183
lemma rel_interior_projection:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5184
fixes S :: "('m::euclidean_space*'n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5185
fixes f :: "'m::euclidean_space => ('n::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5186
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5187
assumes "f = (%y. {z. (y,z) : S})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5188
shows "(y,z) : rel_interior S <-> (y : rel_interior {y. (f y ~= {})} & z : rel_interior (f y))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5189
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5190
{ fix y assume "y : {y. (f y ~= {})}" from this obtain z where "(y,z) : S" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5191
  hence "EX x. x : S & y = fst x" apply (rule_tac x="(y,z)" in exI) by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5192
  from this obtain x where "x : S & y = fst x" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5193
  hence "y : fst ` S" unfolding image_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5194
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5195
hence "fst ` S = {y. (f y ~= {})}" unfolding fst_def using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5196
hence h1: "fst ` rel_interior S = rel_interior {y. (f y ~= {})}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5197
   using rel_interior_convex_linear_image[of fst S] assms fst_linear by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5198
{ fix y assume "y : rel_interior {y. (f y ~= {})}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5199
  hence "y : fst ` rel_interior S" using h1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5200
  hence *: "rel_interior S Int fst -` {y} ~= {}" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5201
  moreover have aff: "affine (fst -` {y})" unfolding affine_alt by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5202
  ultimately have **: "rel_interior (S Int fst -` {y}) = rel_interior S Int fst -` {y}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5203
    using convex_affine_rel_interior_inter[of S "fst -` {y}"] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5204
  have conv: "convex (S Int fst -` {y})" using convex_Int assms aff affine_imp_convex by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5205
  { fix x assume "x : f y"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5206
    hence "(y,x) : S Int (fst -` {y})" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5207
    moreover have "x = snd (y,x)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5208
    ultimately have "x : snd ` (S Int fst -` {y})" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5209
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5210
  hence "snd ` (S Int fst -` {y}) = f y" using assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5211
  hence ***: "rel_interior (f y) = snd ` rel_interior (S Int fst -` {y})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5212
    using rel_interior_convex_linear_image[of snd "S Int fst -` {y}"] snd_linear conv by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5213
  { fix z assume "z : rel_interior (f y)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5214
    hence "z : snd ` rel_interior (S Int fst -` {y})" using *** by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5215
    moreover have "{y} = fst ` rel_interior (S Int fst -` {y})" using * ** rel_interior_subset by auto   
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5216
    ultimately have "(y,z) : rel_interior (S Int fst -` {y})" by force
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5217
    hence "(y,z) : rel_interior S" using ** by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5218
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5219
  moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5220
  { fix z assume "(y,z) : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5221
    hence "(y,z) : rel_interior (S Int fst -` {y})" using ** by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5222
    hence "z : snd ` rel_interior (S Int fst -` {y})" by (metis Range_iff snd_eq_Range) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5223
    hence "z : rel_interior (f y)" using *** by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5224
  }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5225
  ultimately have "!!z. (y,z) : rel_interior S <-> z : rel_interior (f y)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5226
} 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5227
hence h2: "!!y z. y : rel_interior {t. f t ~= {}} ==> ((y, z) : rel_interior S) = (z : rel_interior (f y))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5228
  by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5229
{ fix y z assume asm: "(y, z) : rel_interior S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5230
  hence "y : fst ` rel_interior S" by (metis Domain_iff fst_eq_Domain)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5231
  hence "y : rel_interior {t. f t ~= {}}" using h1 by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5232
  hence "y : rel_interior {t. f t ~= {}} & (z : rel_interior (f y))" using h2 asm by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5233
} from this show ?thesis using h2 by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5234
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5235
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5236
subsection{* Relative interior of convex cone *}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5237
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5238
lemma cone_rel_interior:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5239
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5240
assumes "cone S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5241
shows "cone ({0} Un (rel_interior S))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5242
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5243
{ assume "S = {}" hence ?thesis by (simp add: rel_interior_empty cone_0) }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5244
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5245
{ assume "S ~= {}" hence *: "0:S & (!c. c>0 --> op *\<^sub>R c ` S = S)" using cone_iff[of S] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5246
  hence *: "0:({0} Un (rel_interior S)) &
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5247
           (!c. c>0 --> op *\<^sub>R c ` ({0} Un rel_interior S) = ({0} Un rel_interior S))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5248
           by (auto simp add: rel_interior_scaleR)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5249
  hence ?thesis using cone_iff[of "{0} Un rel_interior S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5250
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5251
ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5252
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5253
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5254
lemma rel_interior_convex_cone_aux:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5255
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5256
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5257
shows "(c,x) : rel_interior (cone hull ({(1 :: real)} <*> S)) <-> 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5258
       c>0 & x : ((op *\<^sub>R c) ` (rel_interior S))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5259
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5260
{ assume "S={}" hence ?thesis by (simp add: rel_interior_empty cone_hull_empty) } 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5261
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5262
{ assume "S ~= {}" from this obtain s where "s : S" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5263
have conv: "convex ({(1 :: real)} <*> S)" using convex_direct_sum[of "{(1 :: real)}" S] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5264
   assms convex_singleton[of "1 :: real"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5265
def f == "(%y. {z. (y,z) : cone hull ({(1 :: real)} <*> S)})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5266
hence *: "(c, x) : rel_interior (cone hull ({(1 :: real)} <*> S)) =
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5267
      (c : rel_interior {y. f y ~= {}} & x : rel_interior (f c))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5268
  apply (subst rel_interior_projection[of "cone hull ({(1 :: real)} <*> S)" f c x])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5269
  using convex_cone_hull[of "{(1 :: real)} <*> S"] conv by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5270
{ fix y assume "(y :: real)>=0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5271
  hence "y *\<^sub>R (1,s) : cone hull ({(1 :: real)} <*> S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5272
     using cone_hull_expl[of "{(1 :: real)} <*> S"] `s:S` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5273
  hence "f y ~= {}" using f_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5274
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5275
hence "{y. f y ~= {}} = {0..}" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5276
hence **: "rel_interior {y. f y ~= {}} = {0<..}" using rel_interior_real_semiline by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5277
{ fix c assume "c>(0 :: real)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5278
  hence "f c = (op *\<^sub>R c ` S)" using f_def cone_hull_expl[of "{(1 :: real)} <*> S"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5279
  hence "rel_interior (f c)= (op *\<^sub>R c ` rel_interior S)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5280
     using rel_interior_convex_scaleR[of S c] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5281
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5282
hence ?thesis using * ** by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5283
} ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5284
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5285
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5286
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5287
lemma rel_interior_convex_cone:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5288
fixes S :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5289
assumes "convex S"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5290
shows "rel_interior (cone hull ({(1 :: real)} <*> S)) = 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5291
       {(c,c *\<^sub>R x) |c x. c>0 & x : (rel_interior S)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5292
(is "?lhs=?rhs")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5293
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5294
{ fix z assume "z:?lhs" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5295
  have *: "z=(fst z,snd z)" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5296
  have "z:?rhs" using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms `z:?lhs` apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5297
     apply (rule_tac x="fst z" in exI) apply (rule_tac x="x" in exI) using * by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5298
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5299
moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5300
{ fix z assume "z:?rhs" hence "z:?lhs" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5301
  using rel_interior_convex_cone_aux[of S "fst z" "snd z"] assms by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5302
}
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5303
ultimately show ?thesis by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5304
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5305
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5306
lemma convex_hull_finite_union:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5307
assumes "finite I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5308
assumes "!i:I. (convex (S i) & (S i) ~= {})"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5309
shows "convex hull (Union (S ` I)) = 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5310
       {setsum (%i. c i *\<^sub>R s i) I |c s. (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. s i : S i)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5311
  (is "?lhs = ?rhs")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5312
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5313
{ fix x assume "x : ?rhs" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5314
  from this obtain c s 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5315
    where *: "setsum (%i. c i *\<^sub>R s i) I=x" "(setsum c I = 1)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5316
     "(!i:I. c i >= 0) & (!i:I. s i : S i)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5317
  hence "!i:I. s i : convex hull (Union (S ` I))" using hull_subset[of "Union (S ` I)" convex] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5318
  hence "x : ?lhs" unfolding *(1)[THEN sym]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5319
     apply (subst convex_setsum[of I "convex hull Union (S ` I)" c s])
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5320
     using * assms convex_convex_hull by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5321
} hence "?lhs >= ?rhs" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5322
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5323
{ fix i assume "i:I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5324
    from this assms have "EX p. p : S i" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5325
} 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5326
from this obtain p where p_def: "!i:I. p i : S i" by metis
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5327
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5328
{ fix i assume "i:I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5329
  { fix x assume "x : S i"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5330
    def c == "(%j. if (j=i) then (1::real) else 0)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5331
    hence *: "setsum c I = 1" using `finite I` `i:I` setsum_delta[of I i "(%(j::'a). (1::real))"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5332
    def s == "(%j. if (j=i) then x else p j)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5333
    hence "!j. c j *\<^sub>R s j = (if (j=i) then x else 0)" using c_def by (auto simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5334
    hence "x = setsum (%i. c i *\<^sub>R s i) I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5335
       using s_def c_def `finite I` `i:I` setsum_delta[of I i "(%(j::'a). x)"] by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5336
    hence "x : ?rhs" apply auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5337
      apply (rule_tac x="c" in exI) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5338
      apply (rule_tac x="s" in exI) using * c_def s_def p_def `x : S i` by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5339
  } hence "?rhs >= S i" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5340
} hence *: "?rhs >= Union (S ` I)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5341
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5342
{ fix u v assume uv: "(u :: real)>=0 & v>=0 & u+v=1"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5343
  fix x y assume xy: "(x : ?rhs) & (y : ?rhs)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5344
  from xy obtain c s where xc: "x=setsum (%i. c i *\<^sub>R s i) I &
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5345
     (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. s i : S i)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5346
  from xy obtain d t where yc: "y=setsum (%i. d i *\<^sub>R t i) I &
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5347
     (!i:I. d i >= 0) & (setsum d I = 1) & (!i:I. t i : S i)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5348
  def e == "(%i. u * (c i)+v * (d i))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5349
  have ge0: "!i:I. e i >= 0"  using e_def xc yc uv by (simp add: mult_nonneg_nonneg)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5350
  have "setsum (%i. u * c i) I = u * setsum c I" by (simp add: setsum_right_distrib)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5351
  moreover have "setsum (%i. v * d i) I = v * setsum d I" by (simp add: setsum_right_distrib)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5352
  ultimately have sum1: "setsum e I = 1" using e_def xc yc uv by (simp add: setsum_addf)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5353
  def q == "(%i. if (e i = 0) then (p i) 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5354
                 else (u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5355
  { fix i assume "i:I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5356
    { assume "e i = 0" hence "q i : S i" using `i:I` p_def q_def by auto }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5357
    moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5358
    { assume "e i ~= 0" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5359
      hence "q i : S i" using mem_convex_alt[of "S i" "s i" "t i" "u * (c i)" "v * (d i)"] 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5360
         mult_nonneg_nonneg[of u "c i"] mult_nonneg_nonneg[of v "d i"]
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5361
         assms q_def e_def `i:I` `e i ~= 0` xc yc uv by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5362
    } ultimately have "q i : S i" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5363
  } hence qs: "!i:I. q i : S i" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5364
  { fix i assume "i:I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5365
    { assume "e i = 0" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5366
      have ge: "u * (c i) >= 0 & v * (d i) >= 0" using xc yc uv `i:I` by (simp add: mult_nonneg_nonneg)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5367
      moreover hence "u * (c i) <= 0 & v * (d i) <= 0" using `e i = 0` e_def `i:I` by simp 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5368
      ultimately have "u * (c i) = 0 & v * (d i) = 0" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5369
      hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5370
         using `e i = 0` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5371
    }
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5372
    moreover
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5373
    { assume "e i ~= 0"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5374
      hence "(u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i) = q i"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5375
         using q_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5376
      hence "e i *\<^sub>R ((u * (c i)/(e i))*\<^sub>R (s i)+(v * (d i)/(e i))*\<^sub>R (t i))
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5377
             = (e i) *\<^sub>R (q i)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5378
      hence "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5379
         using `e i ~= 0` by (simp add: algebra_simps)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5380
    } ultimately have 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5381
      "(u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5382
  } hence *: "!i:I.
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5383
    (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i) = (e i) *\<^sub>R (q i)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5384
  have "u *\<^sub>R x + v *\<^sub>R y =
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5385
       setsum (%i. (u * (c i))*\<^sub>R (s i)+(v * (d i))*\<^sub>R (t i)) I"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5386
          using xc yc by (simp add: algebra_simps scaleR_right.setsum setsum_addf)
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5387
  also have "...=setsum (%i. (e i) *\<^sub>R (q i)) I" using * by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5388
  finally have "u *\<^sub>R x + v *\<^sub>R y = setsum (%i. (e i) *\<^sub>R (q i)) I" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5389
  hence "u *\<^sub>R x + v *\<^sub>R y : ?rhs" using ge0 sum1 qs by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5390
} hence "convex ?rhs" unfolding convex_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5391
hence "?rhs : convex" unfolding mem_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5392
from this show ?thesis using `?lhs >= ?rhs` * 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5393
   hull_minimal[of "Union (S ` I)" "?rhs" "convex"] by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5394
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5395
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5396
lemma convex_hull_union_two:
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5397
fixes S T :: "('m::euclidean_space) set"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5398
assumes "convex S" "S ~= {}" "convex T" "T ~= {}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5399
shows "convex hull (S Un T) = {u *\<^sub>R s + v *\<^sub>R t |u v s t. u>=0 & v>=0 & u+v=1 & s:S & t:T}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5400
  (is "?lhs = ?rhs")
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5401
proof-
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5402
def I == "{(1::nat),2}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5403
def s == "(%i. (if i=(1::nat) then S else T))"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5404
have "Union (s ` I) = S Un T" using s_def I_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5405
hence "convex hull (Union (s ` I)) = convex hull (S Un T)" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5406
moreover have "convex hull Union (s ` I) =
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5407
    {SUM i:I. c i *\<^sub>R sa i |c sa. (ALL i:I. 0 <= c i) & setsum c I = 1 & (ALL i:I. sa i : s i)}"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5408
    apply (subst convex_hull_finite_union[of I s]) using assms s_def I_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5409
moreover have 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5410
  "{SUM i:I. c i *\<^sub>R sa i |c sa. (ALL i:I. 0 <= c i) & setsum c I = 1 & (ALL i:I. sa i : s i)} <=
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5411
  ?rhs"
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5412
  using s_def I_def by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5413
ultimately have "?lhs<=?rhs" by auto 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5414
{ fix x assume "x : ?rhs" 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5415
  from this obtain u v s t 
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5416
    where *: "x=u *\<^sub>R s + v *\<^sub>R t & u>=0 & v>=0 & u+v=1 & s:S & t:T" by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5417
  hence "x : convex hull {s,t}" using convex_hull_2[of s t] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5418
  hence "x : convex hull (S Un T)" using * hull_mono[of "{s, t}" "S Un T"] by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5419
} hence "?lhs >= ?rhs" by blast
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5420
from this show ?thesis using `?lhs<=?rhs` by auto
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5421
qed
0e5d48096f58 Extend convex analysis by Bogdan Grechuk
hoelzl
parents: 39302
diff changeset
  5422
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5423
subsection {* Convexity on direct sums *}
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5424
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5425
lemma closure_sum:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5426
  fixes S T :: "('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5427
  shows "closure S \<oplus> closure T \<subseteq> closure (S \<oplus> T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5428
proof-
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5429
  have "(closure S) \<oplus> (closure T) = (\<lambda>(x,y). x + y) ` (closure S \<times> closure T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5430
    by (simp add: set_plus_image)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5431
  also have "... = (\<lambda>(x,y). x + y) ` closure (S \<times> T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5432
    using closure_direct_sum by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5433
  also have "... \<subseteq> closure (S \<oplus> T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5434
    using fst_snd_linear closure_linear_image[of "(\<lambda>(x,y). x + y)" "S \<times> T"]
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5435
    by (auto simp: set_plus_image)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5436
  finally show ?thesis
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5437
    by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5438
qed
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5439
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5440
lemma convex_oplus:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5441
fixes S T :: "('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5442
assumes "convex S" "convex T"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5443
shows "convex (S \<oplus> T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5444
proof-
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5445
have "{x + y |x y. x : S & y : T} = {c. EX a:S. EX b:T. c = a + b}" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5446
thus ?thesis unfolding set_plus_def using convex_sums[of S T] assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5447
qed
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5448
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5449
lemma convex_hull_sum:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5450
fixes S T :: "('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5451
shows "convex hull (S \<oplus> T) = (convex hull S) \<oplus> (convex hull T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5452
proof-
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5453
have "(convex hull S) \<oplus> (convex hull T) =
40897
1eb1b2f9d062 adapt proofs to changed set_plus_image (cf. ee8d0548c148);
hoelzl
parents: 40887
diff changeset
  5454
      (%(x,y). x + y) ` ((convex hull S) <*> (convex hull T))"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5455
   by (simp add: set_plus_image)
40897
1eb1b2f9d062 adapt proofs to changed set_plus_image (cf. ee8d0548c148);
hoelzl
parents: 40887
diff changeset
  5456
also have "... = (%(x,y). x + y) ` (convex hull (S <*> T))" using convex_hull_direct_sum by auto
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5457
also have "...= convex hull (S \<oplus> T)" using fst_snd_linear linear_conv_bounded_linear
40897
1eb1b2f9d062 adapt proofs to changed set_plus_image (cf. ee8d0548c148);
hoelzl
parents: 40887
diff changeset
  5458
   convex_hull_linear_image[of "(%(x,y). x + y)" "S <*> T"] by (auto simp add: set_plus_image)
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5459
finally show ?thesis by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5460
qed
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5461
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5462
lemma rel_interior_sum:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5463
fixes S T :: "('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5464
assumes "convex S" "convex T"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5465
shows "rel_interior (S \<oplus> T) = (rel_interior S) \<oplus> (rel_interior T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5466
proof-
40897
1eb1b2f9d062 adapt proofs to changed set_plus_image (cf. ee8d0548c148);
hoelzl
parents: 40887
diff changeset
  5467
have "(rel_interior S) \<oplus> (rel_interior T) = (%(x,y). x + y) ` (rel_interior S <*> rel_interior T)"
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5468
   by (simp add: set_plus_image)
40897
1eb1b2f9d062 adapt proofs to changed set_plus_image (cf. ee8d0548c148);
hoelzl
parents: 40887
diff changeset
  5469
also have "... = (%(x,y). x + y) ` rel_interior (S <*> T)" using rel_interior_direct_sum assms by auto
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5470
also have "...= rel_interior (S \<oplus> T)" using fst_snd_linear convex_direct_sum assms
40897
1eb1b2f9d062 adapt proofs to changed set_plus_image (cf. ee8d0548c148);
hoelzl
parents: 40887
diff changeset
  5471
   rel_interior_convex_linear_image[of "(%(x,y). x + y)" "S <*> T"] by (auto simp add: set_plus_image)
40887
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5472
finally show ?thesis by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5473
qed
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5474
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5475
lemma convex_sum_gen:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5476
  fixes S :: "'a \<Rightarrow> 'n::euclidean_space set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5477
  assumes "\<And>i. i \<in> I \<Longrightarrow> (convex (S i))"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5478
  shows "convex (setsum_set S I)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5479
proof cases
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5480
  assume "finite I" from this assms show ?thesis
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5481
    by induct (auto simp: convex_oplus)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5482
qed auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5483
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5484
lemma convex_hull_sum_gen:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5485
fixes S :: "'a => ('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5486
shows "convex hull (setsum_set S I) = setsum_set (%i. (convex hull (S i))) I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5487
apply (subst setsum_set_linear) using convex_hull_sum convex_hull_singleton by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5488
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5489
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5490
lemma rel_interior_sum_gen:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5491
fixes S :: "'a => ('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5492
assumes "!i:I. (convex (S i))"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5493
shows "rel_interior (setsum_set S I) = setsum_set (%i. (rel_interior (S i))) I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5494
apply (subst setsum_set_cond_linear[of convex])
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5495
  using rel_interior_sum rel_interior_sing[of "0"] assms by (auto simp add: convex_oplus)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5496
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5497
lemma convex_rel_open_direct_sum:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5498
fixes S T :: "('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5499
assumes "convex S" "rel_open S" "convex T" "rel_open T"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5500
shows "convex (S <*> T) & rel_open (S <*> T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5501
by (metis assms convex_direct_sum rel_interior_direct_sum rel_open_def)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5502
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5503
lemma convex_rel_open_sum:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5504
fixes S T :: "('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5505
assumes "convex S" "rel_open S" "convex T" "rel_open T"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5506
shows "convex (S \<oplus> T) & rel_open (S \<oplus> T)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5507
by (metis assms convex_oplus rel_interior_sum rel_open_def)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5508
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5509
lemma convex_hull_finite_union_cones:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5510
assumes "finite I" "I ~= {}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5511
assumes "!i:I. (convex (S i) & cone (S i) & (S i) ~= {})"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5512
shows "convex hull (Union (S ` I)) = setsum_set S I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5513
  (is "?lhs = ?rhs")
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5514
proof-
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5515
{ fix x assume "x : ?lhs"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5516
  from this obtain c xs where x_def: "x=setsum (%i. c i *\<^sub>R xs i) I &
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5517
     (!i:I. c i >= 0) & (setsum c I = 1) & (!i:I. xs i : S i)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5518
     using convex_hull_finite_union[of I S] assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5519
  def s == "(%i. c i *\<^sub>R xs i)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5520
  { fix i assume "i:I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5521
    hence "s i : S i" using s_def x_def assms mem_cone[of "S i" "xs i" "c i"] by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5522
  } hence "!i:I. s i : S i" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5523
  moreover have "x = setsum s I" using x_def s_def by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5524
  ultimately have "x : ?rhs" using set_setsum_alt[of I S] assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5525
}
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5526
moreover
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5527
{ fix x assume "x : ?rhs"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5528
  from this obtain s where x_def: "x=setsum s I & (!i:I. s i : S i)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5529
     using set_setsum_alt[of I S] assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5530
  def xs == "(%i. of_nat(card I) *\<^sub>R s i)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5531
  hence "x=setsum (%i. ((1 :: real)/of_nat(card I)) *\<^sub>R xs i) I" using x_def assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5532
  moreover have "!i:I. xs i : S i" using x_def xs_def assms by (simp add: cone_def)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5533
  moreover have "(!i:I. (1 :: real)/of_nat(card I) >= 0)" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5534
  moreover have "setsum (%i. (1 :: real)/of_nat(card I)) I = 1" using assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5535
  ultimately have "x : ?lhs" apply (subst convex_hull_finite_union[of I S])
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5536
    using assms apply blast
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5537
    using assms apply blast
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5538
    apply rule apply (rule_tac x="(%i. (1 :: real)/of_nat(card I))" in exI) by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5539
} ultimately show ?thesis by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5540
qed
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5541
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5542
lemma convex_hull_union_cones_two:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5543
fixes S T :: "('m::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5544
assumes "convex S" "cone S" "S ~= {}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5545
assumes "convex T" "cone T" "T ~= {}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5546
shows "convex hull (S Un T) = S \<oplus> T"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5547
proof-
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5548
def I == "{(1::nat),2}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5549
def A == "(%i. (if i=(1::nat) then S else T))"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5550
have "Union (A ` I) = S Un T" using A_def I_def by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5551
hence "convex hull (Union (A ` I)) = convex hull (S Un T)" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5552
moreover have "convex hull Union (A ` I) = setsum_set A I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5553
    apply (subst convex_hull_finite_union_cones[of I A]) using assms A_def I_def by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5554
moreover have
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5555
  "setsum_set A I = S \<oplus> T" using A_def I_def
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5556
     unfolding set_plus_def apply auto unfolding set_plus_def by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5557
ultimately show ?thesis by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5558
qed
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5559
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5560
lemma rel_interior_convex_hull_union:
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5561
fixes S :: "'a => ('n::euclidean_space) set"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5562
assumes "finite I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5563
assumes "!i:I. convex (S i) & (S i) ~= {}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5564
shows "rel_interior (convex hull (Union (S ` I))) =  {setsum (%i. c i *\<^sub>R s i) I
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5565
       |c s. (!i:I. c i > 0) & (setsum c I = 1) & (!i:I. s i : rel_interior(S i))}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5566
(is "?lhs=?rhs")
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5567
proof-
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5568
{ assume "I={}" hence ?thesis using convex_hull_empty rel_interior_empty by auto }
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5569
moreover
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5570
{ assume "I ~= {}"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5571
  def C0 == "convex hull (Union (S ` I))"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5572
  have "!i:I. C0 >= S i" unfolding C0_def using hull_subset[of "Union (S ` I)"] by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5573
  def K0 == "cone hull ({(1 :: real)} <*> C0)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5574
  def K == "(%i. cone hull ({(1 :: real)} <*> (S i)))"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5575
  have "!i:I. K i ~= {}" unfolding K_def using assms by (simp add: cone_hull_empty_iff[symmetric])
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5576
  { fix i assume "i:I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5577
    hence "convex (K i)" unfolding K_def apply (subst convex_cone_hull) apply (subst convex_direct_sum)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5578
    using assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5579
  }
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5580
  hence convK: "!i:I. convex (K i)" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5581
  { fix i assume "i:I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5582
    hence "K0 >= K i" unfolding K0_def K_def apply (subst hull_mono) using `!i:I. C0 >= S i` by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5583
  }
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5584
  hence "K0 >= Union (K ` I)" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5585
  moreover have "K0 : convex" unfolding mem_def K0_def
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5586
     apply (subst convex_cone_hull) apply (subst convex_direct_sum)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5587
     unfolding C0_def using convex_convex_hull by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5588
  ultimately have geq: "K0 >= convex hull (Union (K ` I))" using hull_minimal[of _ "K0" "convex"] by blast
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5589
  have "!i:I. K i >= {(1 :: real)} <*> (S i)" using K_def by (simp add: hull_subset)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5590
  hence "Union (K ` I) >= {(1 :: real)} <*> Union (S ` I)" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5591
  hence "convex hull Union (K ` I) >= convex hull ({(1 :: real)} <*> Union (S ` I))" by (simp add: hull_mono)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5592
  hence "convex hull Union (K ` I) >= {(1 :: real)} <*> C0" unfolding C0_def
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5593
     using convex_hull_direct_sum[of "{(1 :: real)}" "Union (S ` I)"] convex_hull_singleton by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5594
  moreover have "convex hull(Union (K ` I)) : cone" unfolding mem_def apply (subst cone_convex_hull)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5595
     using cone_Union[of "K ` I"] apply auto unfolding K_def using cone_cone_hull by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5596
  ultimately have "convex hull (Union (K ` I)) >= K0"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5597
     unfolding K0_def using hull_minimal[of _ "convex hull (Union (K ` I))" "cone"] by blast
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5598
  hence "K0 = convex hull (Union (K ` I))" using geq by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5599
  also have "...=setsum_set K I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5600
     apply (subst convex_hull_finite_union_cones[of I K])
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5601
     using assms apply blast
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5602
     using `I ~= {}` apply blast
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5603
     unfolding K_def apply rule
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5604
     apply (subst convex_cone_hull) apply (subst convex_direct_sum)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5605
     using assms cone_cone_hull `!i:I. K i ~= {}` K_def by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5606
  finally have "K0 = setsum_set K I" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5607
  hence *: "rel_interior K0 = setsum_set (%i. (rel_interior (K i))) I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5608
     using rel_interior_sum_gen[of I K] convK by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5609
  { fix x assume "x : ?lhs"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5610
    hence "((1::real),x) : rel_interior K0" using K0_def C0_def
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5611
       rel_interior_convex_cone_aux[of C0 "(1::real)" x] convex_convex_hull by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5612
    from this obtain k where k_def: "((1::real),x) = setsum k I & (!i:I. k i : rel_interior (K i))"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5613
      using `finite I` * set_setsum_alt[of I "(%i. rel_interior (K i))"] by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5614
    { fix i assume "i:I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5615
      hence "(convex (S i)) & k i : rel_interior (cone hull {1} <*> S i)" using k_def K_def assms by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5616
      hence "EX ci si. k i = (ci, ci *\<^sub>R si) & 0 < ci & si : rel_interior (S i)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5617
         using rel_interior_convex_cone[of "S i"] by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5618
    }
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5619
    from this obtain c s where cs_def: "!i:I. (k i = (c i, c i *\<^sub>R s i) & 0 < c i
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5620
          & s i : rel_interior (S i))" by metis
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5621
    hence "x = (SUM i:I. c i *\<^sub>R s i) & setsum c I = 1" using k_def by (simp add: setsum_prod)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5622
    hence "x : ?rhs" using k_def apply auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5623
       apply (rule_tac x="c" in exI) apply (rule_tac x="s" in exI) using cs_def by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5624
  }
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5625
  moreover
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5626
  { fix x assume "x : ?rhs"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5627
    from this obtain c s where cs_def: "x=setsum (%i. c i *\<^sub>R s i) I &
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5628
       (!i:I. c i > 0) & (setsum c I = 1) & (!i:I. s i : rel_interior(S i))" by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5629
    def k == "(%i. (c i, c i *\<^sub>R s i))"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5630
    { fix i assume "i:I"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5631
      hence "k i : rel_interior (K i)"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5632
         using k_def K_def assms cs_def rel_interior_convex_cone[of "S i"] by auto
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5633
    }
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5634
    hence "((1::real),x) : rel_interior K0"
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5635
       using K0_def * set_setsum_alt[of I "(%i. rel_interior (K i))"] assms k_def cs_def
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5636
       apply auto apply (rule_tac x="k" in exI) by (simp add: setsum_prod)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5637
    hence "x : ?lhs" using K0_def C0_def
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5638
       rel_interior_convex_cone_aux[of C0 "(1::real)" x] by (auto simp add: convex_convex_hull)
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5639
  }
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5640
  ultimately have ?thesis by blast
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5641
} ultimately show ?thesis by blast
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5642
qed
ee8d0548c148 Prove rel_interior_convex_hull_union (by Grechuck Bogdan).
hoelzl
parents: 40719
diff changeset
  5643
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
  5644
end