src/HOL/Analysis/Finite_Cartesian_Product.thy
author wenzelm
Wed, 28 Feb 2018 13:37:33 +0100
changeset 67731 184c293f0a33
parent 67686 2c58505bf151
child 67732 39d80006fc29
permissions -rw-r--r--
clarified use of vec type syntax;
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
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(*  Title:      HOL/Analysis/Finite_Cartesian_Product.thy
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    Author:     Amine Chaieb, University of Cambridge
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*)
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section \<open>Definition of finite Cartesian product types.\<close>
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theory Finite_Cartesian_Product
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imports
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  Euclidean_Space
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  L2_Norm
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  "HOL-Library.Numeral_Type"
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  "HOL-Library.Countable_Set"
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  "HOL-Library.FuncSet"
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begin
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subsection \<open>Finite Cartesian products, with indexing and lambdas.\<close>
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typedef ('a, 'b) vec = "UNIV :: ('b::finite \<Rightarrow> 'a) set"
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  morphisms vec_nth vec_lambda ..
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declare vec_lambda_inject [simplified, simp]
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bundle vec_syntax begin
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notation
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  vec_nth (infixl "$" 90) and
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  vec_lambda (binder "\<chi>" 10)
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end
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bundle no_vec_syntax begin
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no_notation
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  vec_nth (infixl "$" 90) and
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  vec_lambda (binder "\<chi>" 10)
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end
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unbundle vec_syntax
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text \<open>
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  Concrete syntax for \<open>('a, 'b) vec\<close>:
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    \<^item> \<open>'a^'b\<close> becomes \<open>('a, 'b::finite) vec\<close>
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    \<^item> \<open>'a^'b::_\<close> becomes \<open>('a, 'b) vec\<close> without extra sort-constraint
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\<close>
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syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
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parse_translation \<open>
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  let
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    fun vec t u = Syntax.const @{type_syntax vec} $ t $ u;
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    fun finite_vec_tr [t, u] =
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      (case Term_Position.strip_positions u of
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        v as Free (x, _) =>
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          if Lexicon.is_tid x then
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            vec t (Syntax.const @{syntax_const "_ofsort"} $ v $
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              Syntax.const @{class_syntax finite})
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          else vec t u
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      | _ => vec t u)
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  in
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    [(@{syntax_const "_finite_vec"}, K finite_vec_tr)]
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  end
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\<close>
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lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
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  by (simp add: vec_nth_inject [symmetric] fun_eq_iff)
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lemma vec_lambda_beta [simp]: "vec_lambda g $ i = g i"
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  by (simp add: vec_lambda_inverse)
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lemma vec_lambda_unique: "(\<forall>i. f$i = g i) \<longleftrightarrow> vec_lambda g = f"
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  by (auto simp add: vec_eq_iff)
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lemma vec_lambda_eta [simp]: "(\<chi> i. (g$i)) = g"
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  by (simp add: vec_eq_iff)
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subsection \<open>Cardinality of vectors\<close>
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instance vec :: (finite, finite) finite
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eberlm <eberlm@in.tum.de>
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    74
proof
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
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    75
  show "finite (UNIV :: ('a, 'b) vec set)"
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eberlm <eberlm@in.tum.de>
parents: 64267
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    76
  proof (subst bij_betw_finite)
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eberlm <eberlm@in.tum.de>
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    77
    show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
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    78
      by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
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eberlm <eberlm@in.tum.de>
parents: 64267
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    79
    have "finite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
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    80
      by (intro finite_PiE) auto
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eberlm <eberlm@in.tum.de>
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    81
    also have "(PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set)) = Pi UNIV (\<lambda>_. UNIV)"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
    82
      by auto
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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    83
    finally show "finite \<dots>" .
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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    84
  qed
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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    85
qed
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eberlm <eberlm@in.tum.de>
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    86
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eberlm <eberlm@in.tum.de>
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lemma countable_PiE:
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  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
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    89
  by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
    90
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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    91
instance vec :: (countable, finite) countable
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eberlm <eberlm@in.tum.de>
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    92
proof
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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    93
  have "countable (UNIV :: ('a, 'b) vec set)"
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eberlm <eberlm@in.tum.de>
parents: 64267
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    94
  proof (rule countableI_bij2)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
    95
    show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
    96
      by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
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eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
    97
    have "countable (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
    98
      by (intro countable_PiE) auto
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
    99
    also have "(PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set)) = Pi UNIV (\<lambda>_. UNIV)"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   100
      by auto
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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diff changeset
   101
    finally show "countable \<dots>" .
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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diff changeset
   102
  qed
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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diff changeset
   103
  thus "\<exists>t::('a, 'b) vec \<Rightarrow> nat. inj t"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   104
    by (auto elim!: countableE)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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diff changeset
   105
qed
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   106
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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   107
lemma infinite_UNIV_vec:
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eberlm <eberlm@in.tum.de>
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   108
  assumes "infinite (UNIV :: 'a set)"
67731
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   109
  shows   "infinite (UNIV :: ('a^'b) set)"
66281
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   110
proof (subst bij_betw_finite)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   111
  show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   112
    by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   113
  have "infinite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" (is "infinite ?A")
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   114
  proof
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   115
    assume "finite ?A"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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diff changeset
   116
    hence "finite ((\<lambda>f. f undefined) ` ?A)"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   117
      by (rule finite_imageI)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   118
    also have "(\<lambda>f. f undefined) ` ?A = UNIV"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   119
      by auto
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   120
    finally show False 
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   121
      using \<open>infinite (UNIV :: 'a set)\<close> by contradiction
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   122
  qed
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   123
  also have "?A = Pi UNIV (\<lambda>_. UNIV)" 
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   124
    by auto
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   125
  finally show "infinite (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" .
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   126
qed
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   127
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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   128
lemma CARD_vec [simp]:
67731
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   129
  "CARD('a^'b) = CARD('a) ^ CARD('b)"
66281
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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   130
proof (cases "finite (UNIV :: 'a set)")
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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   131
  case True
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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   132
  show ?thesis
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
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   133
  proof (subst bij_betw_same_card)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   134
    show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   135
      by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   136
    have "CARD('a) ^ CARD('b) = card (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   137
      (is "_ = card ?A")
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   138
      by (subst card_PiE) (auto simp: prod_constant)
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   139
    
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   140
    also have "?A = Pi UNIV (\<lambda>_. UNIV)" 
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   141
      by auto
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   142
    finally show "card \<dots> = CARD('a) ^ CARD('b)" ..
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   143
  qed
6ad54b84ca5d facts about cardinality of vector type
eberlm <eberlm@in.tum.de>
parents: 64267
diff changeset
   144
qed (simp_all add: infinite_UNIV_vec)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   145
67683
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   146
lemma countable_vector:
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   147
  fixes B:: "'n::finite \<Rightarrow> 'a set"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   148
  assumes "\<And>i. countable (B i)"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   149
  shows "countable {V. \<forall>i::'n::finite. V $ i \<in> B i}"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   150
proof -
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   151
  have "f \<in> ($) ` {V. \<forall>i. V $ i \<in> B i}" if "f \<in> Pi\<^sub>E UNIV B" for f
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   152
  proof -
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   153
    have "\<exists>W. (\<forall>i. W $ i \<in> B i) \<and> ($) W = f"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   154
      by (metis that PiE_iff UNIV_I vec_lambda_inverse)
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   155
    then show "f \<in> ($) ` {v. \<forall>i. v $ i \<in> B i}"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   156
      by blast
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   157
  qed
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   158
  then have "Pi\<^sub>E UNIV B = vec_nth ` {V. \<forall>i::'n. V $ i \<in> B i}"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   159
    by blast
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   160
  then have "countable (vec_nth ` {V. \<forall>i. V $ i \<in> B i})"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   161
    by (metis finite_class.finite_UNIV countable_PiE assms)
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   162
  then have "countable (vec_lambda ` vec_nth ` {V. \<forall>i. V $ i \<in> B i})"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   163
    by auto
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   164
  then show ?thesis
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   165
    by (simp add: image_comp o_def vec_nth_inverse)
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   166
qed
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   167
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   168
subsection \<open>Group operations and class instances\<close>
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   169
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   170
instantiation vec :: (zero, finite) zero
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   171
begin
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   172
  definition "0 \<equiv> (\<chi> i. 0)"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   173
  instance ..
33175
2083bde13ce1 distinguished session for multivariate analysis
himmelma
parents:
diff changeset
   174
end
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   175
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   176
instantiation vec :: (plus, finite) plus
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   177
begin
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67155
diff changeset
   178
  definition "(+) \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   179
  instance ..
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   180
end
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   181
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   182
instantiation vec :: (minus, finite) minus
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   183
begin
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67155
diff changeset
   184
  definition "(-) \<equiv> (\<lambda> x y. (\<chi> i. x$i - y$i))"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   185
  instance ..
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   186
end
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   187
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   188
instantiation vec :: (uminus, finite) uminus
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   189
begin
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   190
  definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x$i)))"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   191
  instance ..
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   192
end
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   193
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   194
lemma zero_index [simp]: "0 $ i = 0"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   195
  unfolding zero_vec_def by simp
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   196
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   197
lemma vector_add_component [simp]: "(x + y)$i = x$i + y$i"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   198
  unfolding plus_vec_def by simp
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   199
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   200
lemma vector_minus_component [simp]: "(x - y)$i = x$i - y$i"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   201
  unfolding minus_vec_def by simp
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   202
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   203
lemma vector_uminus_component [simp]: "(- x)$i = - (x$i)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   204
  unfolding uminus_vec_def by simp
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   205
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   206
instance vec :: (semigroup_add, finite) semigroup_add
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   207
  by standard (simp add: vec_eq_iff add.assoc)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   208
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   209
instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   210
  by standard (simp add: vec_eq_iff add.commute)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   211
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   212
instance vec :: (monoid_add, finite) monoid_add
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   213
  by standard (simp_all add: vec_eq_iff)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   214
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   215
instance vec :: (comm_monoid_add, finite) comm_monoid_add
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   216
  by standard (simp add: vec_eq_iff)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   217
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   218
instance vec :: (cancel_semigroup_add, finite) cancel_semigroup_add
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   219
  by standard (simp_all add: vec_eq_iff)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   220
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   221
instance vec :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   222
  by standard (simp_all add: vec_eq_iff diff_diff_eq)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   223
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   224
instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   225
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   226
instance vec :: (group_add, finite) group_add
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   227
  by standard (simp_all add: vec_eq_iff)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   228
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   229
instance vec :: (ab_group_add, finite) ab_group_add
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   230
  by standard (simp_all add: vec_eq_iff)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   231
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   232
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   233
subsection \<open>Real vector space\<close>
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   234
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   235
instantiation vec :: (real_vector, finite) real_vector
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   236
begin
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   237
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   238
definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   239
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   240
lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   241
  unfolding scaleR_vec_def by simp
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   242
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   243
instance
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   244
  by standard (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   245
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   246
end
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   247
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   248
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   249
subsection \<open>Topological space\<close>
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   250
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   251
instantiation vec :: (topological_space, finite) topological_space
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   252
begin
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   253
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   254
definition [code del]:
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   255
  "open (S :: ('a ^ 'b) set) \<longleftrightarrow>
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   256
    (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   257
      (\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> S))"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   258
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   259
instance proof
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   260
  show "open (UNIV :: ('a ^ 'b) set)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   261
    unfolding open_vec_def by auto
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   262
next
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   263
  fix S T :: "('a ^ 'b) set"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   264
  assume "open S" "open T" thus "open (S \<inter> T)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   265
    unfolding open_vec_def
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   266
    apply clarify
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   267
    apply (drule (1) bspec)+
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   268
    apply (clarify, rename_tac Sa Ta)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   269
    apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   270
    apply (simp add: open_Int)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   271
    done
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   272
next
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   273
  fix K :: "('a ^ 'b) set set"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   274
  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   275
    unfolding open_vec_def
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   276
    apply clarify
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   277
    apply (drule (1) bspec)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   278
    apply (drule (1) bspec)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   279
    apply clarify
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   280
    apply (rule_tac x=A in exI)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   281
    apply fast
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   282
    done
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   283
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   284
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   285
end
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   286
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   287
lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   288
  unfolding open_vec_def by auto
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   289
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   290
lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   291
  unfolding open_vec_def
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   292
  apply clarify
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   293
  apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   294
  done
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   295
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   296
lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   297
  unfolding closed_open vimage_Compl [symmetric]
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   298
  by (rule open_vimage_vec_nth)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   299
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   300
lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   301
proof -
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   302
  have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   303
  thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   304
    by (simp add: closed_INT closed_vimage_vec_nth)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   305
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   306
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   307
lemma tendsto_vec_nth [tendsto_intros]:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   308
  assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   309
  shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   310
proof (rule topological_tendstoI)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   311
  fix S assume "open S" "a $ i \<in> S"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   312
  then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   313
    by (simp_all add: open_vimage_vec_nth)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   314
  with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y $ i) -` S) net"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   315
    by (rule topological_tendstoD)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   316
  then show "eventually (\<lambda>x. f x $ i \<in> S) net"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   317
    by simp
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   318
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   319
44631
6820684c7a58 generalize lemma isCont_vec_nth
huffman
parents: 44630
diff changeset
   320
lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
6820684c7a58 generalize lemma isCont_vec_nth
huffman
parents: 44630
diff changeset
   321
  unfolding isCont_def by (rule tendsto_vec_nth)
6820684c7a58 generalize lemma isCont_vec_nth
huffman
parents: 44630
diff changeset
   322
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   323
lemma vec_tendstoI:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   324
  assumes "\<And>i. ((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   325
  shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   326
proof (rule topological_tendstoI)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   327
  fix S assume "open S" and "a \<in> S"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   328
  then obtain A where A: "\<And>i. open (A i)" "\<And>i. a $ i \<in> A i"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   329
    and S: "\<And>y. \<forall>i. y $ i \<in> A i \<Longrightarrow> y \<in> S"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   330
    unfolding open_vec_def by metis
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   331
  have "\<And>i. eventually (\<lambda>x. f x $ i \<in> A i) net"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   332
    using assms A by (rule topological_tendstoD)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   333
  hence "eventually (\<lambda>x. \<forall>i. f x $ i \<in> A i) net"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   334
    by (rule eventually_all_finite)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   335
  thus "eventually (\<lambda>x. f x \<in> S) net"
61810
3c5040d5694a sorted out eventually_mono
paulson <lp15@cam.ac.uk>
parents: 61169
diff changeset
   336
    by (rule eventually_mono, simp add: S)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   337
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   338
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   339
lemma tendsto_vec_lambda [tendsto_intros]:
61973
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   340
  assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
0c7e865fa7cb more symbols;
wenzelm
parents: 61969
diff changeset
   341
  shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   342
  using assms by (simp add: vec_tendstoI)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   343
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   344
lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   345
proof (rule openI)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   346
  fix a assume "a \<in> (\<lambda>x. x $ i) ` S"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   347
  then obtain z where "a = z $ i" and "z \<in> S" ..
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   348
  then obtain A where A: "\<forall>i. open (A i) \<and> z $ i \<in> A i"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   349
    and S: "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   350
    using \<open>open S\<close> unfolding open_vec_def by auto
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   351
  hence "A i \<subseteq> (\<lambda>x. x $ i) ` S"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   352
    by (clarsimp, rule_tac x="\<chi> j. if j = i then x else z $ j" in image_eqI,
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   353
      simp_all)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   354
  hence "open (A i) \<and> a \<in> A i \<and> A i \<subseteq> (\<lambda>x. x $ i) ` S"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   355
    using A \<open>a = z $ i\<close> by simp
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   356
  then show "\<exists>T. open T \<and> a \<in> T \<and> T \<subseteq> (\<lambda>x. x $ i) ` S" by - (rule exI)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   357
qed
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   358
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   359
instance vec :: (perfect_space, finite) perfect_space
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   360
proof
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   361
  fix x :: "'a ^ 'b" show "\<not> open {x}"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   362
  proof
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   363
    assume "open {x}"
62102
877463945ce9 fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents: 62101
diff changeset
   364
    hence "\<forall>i. open ((\<lambda>x. x $ i) ` {x})" by (fast intro: open_image_vec_nth)
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   365
    hence "\<forall>i. open {x $ i}" by simp
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   366
    thus "False" by (simp add: not_open_singleton)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   367
  qed
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   368
qed
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   369
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44282
diff changeset
   370
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   371
subsection \<open>Metric space\<close>
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   372
(* TODO: Product of uniform spaces and compatibility with metric_spaces! *)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   373
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   374
instantiation vec :: (metric_space, finite) dist
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   375
begin
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   376
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   377
definition
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   378
  "dist x y = L2_set (\<lambda>i. dist (x$i) (y$i)) UNIV"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   379
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   380
instance ..
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   381
end
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   382
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   383
instantiation vec :: (metric_space, finite) uniformity_dist
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   384
begin
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   385
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   386
definition [code del]:
67731
184c293f0a33 clarified use of vec type syntax;
wenzelm
parents: 67686
diff changeset
   387
  "(uniformity :: (('a^'b::_) \<times> ('a^'b::_)) filter) =
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   388
    (INF e:{0 <..}. principal {(x, y). dist x y < e})"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   389
62102
877463945ce9 fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents: 62101
diff changeset
   390
instance
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   391
  by standard (rule uniformity_vec_def)
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   392
end
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   393
62102
877463945ce9 fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents: 62101
diff changeset
   394
declare uniformity_Abort[where 'a="'a :: metric_space ^ 'b :: finite", code]
877463945ce9 fix code generation for uniformity: uniformity is a non-computable pure data.
hoelzl
parents: 62101
diff changeset
   395
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   396
instantiation vec :: (metric_space, finite) metric_space
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   397
begin
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   398
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   399
lemma dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   400
  unfolding dist_vec_def by (rule member_le_L2_set) simp_all
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   401
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   402
instance proof
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   403
  fix x y :: "'a ^ 'b"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   404
  show "dist x y = 0 \<longleftrightarrow> x = y"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   405
    unfolding dist_vec_def
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   406
    by (simp add: L2_set_eq_0_iff vec_eq_iff)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   407
next
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   408
  fix x y z :: "'a ^ 'b"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   409
  show "dist x y \<le> dist x z + dist y z"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   410
    unfolding dist_vec_def
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   411
    apply (rule order_trans [OF _ L2_set_triangle_ineq])
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   412
    apply (simp add: L2_set_mono dist_triangle2)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   413
    done
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   414
next
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   415
  fix S :: "('a ^ 'b) set"
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   416
  have *: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
44630
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   417
  proof
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   418
    assume "open S" show "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   419
    proof
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   420
      fix x assume "x \<in> S"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   421
      obtain A where A: "\<forall>i. open (A i)" "\<forall>i. x $ i \<in> A i"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   422
        and S: "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   423
        using \<open>open S\<close> and \<open>x \<in> S\<close> unfolding open_vec_def by metis
44630
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   424
      have "\<forall>i\<in>UNIV. \<exists>r>0. \<forall>y. dist y (x $ i) < r \<longrightarrow> y \<in> A i"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   425
        using A unfolding open_dist by simp
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   426
      hence "\<exists>r. \<forall>i\<in>UNIV. 0 < r i \<and> (\<forall>y. dist y (x $ i) < r i \<longrightarrow> y \<in> A i)"
44681
49ef76b4a634 remove duplicate lemma finite_choice in favor of finite_set_choice
huffman
parents: 44631
diff changeset
   427
        by (rule finite_set_choice [OF finite])
44630
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   428
      then obtain r where r1: "\<forall>i. 0 < r i"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   429
        and r2: "\<forall>i y. dist y (x $ i) < r i \<longrightarrow> y \<in> A i" by fast
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   430
      have "0 < Min (range r) \<and> (\<forall>y. dist y x < Min (range r) \<longrightarrow> y \<in> S)"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   431
        by (simp add: r1 r2 S le_less_trans [OF dist_vec_nth_le])
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   432
      thus "\<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" ..
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   433
    qed
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   434
  next
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   435
    assume *: "\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S" show "open S"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   436
    proof (unfold open_vec_def, rule)
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   437
      fix x assume "x \<in> S"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   438
      then obtain e where "0 < e" and S: "\<forall>y. dist y x < e \<longrightarrow> y \<in> S"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   439
        using * by fast
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62397
diff changeset
   440
      define r where [abs_def]: "r i = e / sqrt (of_nat CARD('b))" for i :: 'b
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   441
      from \<open>0 < e\<close> have r: "\<forall>i. 0 < r i"
56541
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 54230
diff changeset
   442
        unfolding r_def by simp_all
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   443
      from \<open>0 < e\<close> have e: "e = L2_set r UNIV"
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   444
        unfolding r_def by (simp add: L2_set_constant)
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62397
diff changeset
   445
      define A where "A i = {y. dist (x $ i) y < r i}" for i
44630
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   446
      have "\<forall>i. open (A i) \<and> x $ i \<in> A i"
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   447
        unfolding A_def by (simp add: open_ball r)
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   448
      moreover have "\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S"
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   449
        by (simp add: A_def S dist_vec_def e L2_set_strict_mono dist_commute)
44630
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   450
      ultimately show "\<exists>A. (\<forall>i. open (A i) \<and> x $ i \<in> A i) \<and>
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   451
        (\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S)" by metis
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   452
    qed
d08cb39b628a convert proof to Isar-style
huffman
parents: 44571
diff changeset
   453
  qed
62101
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   454
  show "open S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   455
    unfolding * eventually_uniformity_metric
26c0a70f78a3 add uniform spaces
hoelzl
parents: 61973
diff changeset
   456
    by (simp del: split_paired_All add: dist_vec_def dist_commute)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   457
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   458
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   459
end
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   460
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   461
lemma Cauchy_vec_nth:
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   462
  "Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   463
  unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   464
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   465
lemma vec_CauchyI:
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   466
  fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   467
  assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   468
  shows "Cauchy (\<lambda>n. X n)"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   469
proof (rule metric_CauchyI)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   470
  fix r :: real assume "0 < r"
56541
0e3abadbef39 made divide_pos_pos a simp rule
nipkow
parents: 54230
diff changeset
   471
  hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
63040
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62397
diff changeset
   472
  define N where "N i = (LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s)" for i
eb4ddd18d635 eliminated old 'def';
wenzelm
parents: 62397
diff changeset
   473
  define M where "M = Max (range N)"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   474
  have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   475
    using X \<open>0 < ?s\<close> by (rule metric_CauchyD)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   476
  hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m $ i) (X n $ i) < ?s"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   477
    unfolding N_def by (rule LeastI_ex)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   478
  hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < ?s"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   479
    unfolding M_def by simp
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   480
  {
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   481
    fix m n :: nat
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   482
    assume "M \<le> m" "M \<le> n"
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   483
    have "dist (X m) (X n) = L2_set (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   484
      unfolding dist_vec_def ..
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   485
    also have "\<dots> \<le> sum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   486
      by (rule L2_set_le_sum [OF zero_le_dist])
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   487
    also have "\<dots> < sum (\<lambda>i::'n. ?s) UNIV"
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   488
      by (rule sum_strict_mono, simp_all add: M \<open>M \<le> m\<close> \<open>M \<le> n\<close>)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   489
    also have "\<dots> = r"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   490
      by simp
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   491
    finally have "dist (X m) (X n) < r" .
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   492
  }
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   493
  hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   494
    by simp
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   495
  then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   496
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   497
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   498
instance vec :: (complete_space, finite) complete_space
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   499
proof
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   500
  fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
61969
e01015e49041 more symbols;
wenzelm
parents: 61810
diff changeset
   501
  have "\<And>i. (\<lambda>n. X n $ i) \<longlonglongrightarrow> lim (\<lambda>n. X n $ i)"
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   502
    using Cauchy_vec_nth [OF \<open>Cauchy X\<close>]
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   503
    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
61969
e01015e49041 more symbols;
wenzelm
parents: 61810
diff changeset
   504
  hence "X \<longlonglongrightarrow> vec_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   505
    by (simp add: vec_tendstoI)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   506
  then show "convergent X"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   507
    by (rule convergentI)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   508
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   509
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   510
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   511
subsection \<open>Normed vector space\<close>
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   512
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   513
instantiation vec :: (real_normed_vector, finite) real_normed_vector
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   514
begin
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   515
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   516
definition "norm x = L2_set (\<lambda>i. norm (x$i)) UNIV"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   517
44141
0697c01ff3ea follow standard naming scheme for sgn_vec_def
huffman
parents: 44136
diff changeset
   518
definition "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   519
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   520
instance proof
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   521
  fix a :: real and x y :: "'a ^ 'b"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   522
  show "norm x = 0 \<longleftrightarrow> x = 0"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   523
    unfolding norm_vec_def
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   524
    by (simp add: L2_set_eq_0_iff vec_eq_iff)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   525
  show "norm (x + y) \<le> norm x + norm y"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   526
    unfolding norm_vec_def
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   527
    apply (rule order_trans [OF _ L2_set_triangle_ineq])
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   528
    apply (simp add: L2_set_mono norm_triangle_ineq)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   529
    done
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   530
  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   531
    unfolding norm_vec_def
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   532
    by (simp add: L2_set_right_distrib)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   533
  show "sgn x = scaleR (inverse (norm x)) x"
44141
0697c01ff3ea follow standard naming scheme for sgn_vec_def
huffman
parents: 44136
diff changeset
   534
    by (rule sgn_vec_def)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   535
  show "dist x y = norm (x - y)"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   536
    unfolding dist_vec_def norm_vec_def
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   537
    by (simp add: dist_norm)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   538
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   539
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   540
end
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   541
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   542
lemma norm_nth_le: "norm (x $ i) \<le> norm x"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   543
unfolding norm_vec_def
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   544
by (rule member_le_L2_set) simp_all
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   545
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44233
diff changeset
   546
lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x $ i)"
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 60420
diff changeset
   547
apply standard
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   548
apply (rule vector_add_component)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   549
apply (rule vector_scaleR_component)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   550
apply (rule_tac x="1" in exI, simp add: norm_nth_le)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   551
done
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   552
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   553
instance vec :: (banach, finite) banach ..
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   554
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   555
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   556
subsection \<open>Inner product space\<close>
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   557
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   558
instantiation vec :: (real_inner, finite) real_inner
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   559
begin
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   560
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   561
definition "inner x y = sum (\<lambda>i. inner (x$i) (y$i)) UNIV"
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   562
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   563
instance proof
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   564
  fix r :: real and x y z :: "'a ^ 'b"
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   565
  show "inner x y = inner y x"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   566
    unfolding inner_vec_def
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   567
    by (simp add: inner_commute)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   568
  show "inner (x + y) z = inner x z + inner y z"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   569
    unfolding inner_vec_def
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   570
    by (simp add: inner_add_left sum.distrib)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   571
  show "inner (scaleR r x) y = r * inner x y"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   572
    unfolding inner_vec_def
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   573
    by (simp add: sum_distrib_left)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   574
  show "0 \<le> inner x x"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   575
    unfolding inner_vec_def
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   576
    by (simp add: sum_nonneg)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   577
  show "inner x x = 0 \<longleftrightarrow> x = 0"
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   578
    unfolding inner_vec_def
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   579
    by (simp add: vec_eq_iff sum_nonneg_eq_0_iff)
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   580
  show "norm x = sqrt (inner x x)"
67155
9e5b05d54f9d canonical name
nipkow
parents: 66453
diff changeset
   581
    unfolding inner_vec_def norm_vec_def L2_set_def
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   582
    by (simp add: power2_norm_eq_inner)
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   583
qed
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   584
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   585
end
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   586
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   587
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   588
subsection \<open>Euclidean space\<close>
44135
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   589
60420
884f54e01427 isabelle update_cartouches;
wenzelm
parents: 59815
diff changeset
   590
text \<open>Vectors pointing along a single axis.\<close>
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   591
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   592
definition "axis k x = (\<chi> i. if i = k then x else 0)"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   593
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   594
lemma axis_nth [simp]: "axis i x $ i = x"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   595
  unfolding axis_def by simp
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   596
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   597
lemma axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   598
  unfolding axis_def vec_eq_iff by auto
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   599
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   600
lemma inner_axis_axis:
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   601
  "inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   602
  unfolding inner_vec_def
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   603
  apply (cases "i = j")
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   604
  apply clarsimp
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   605
  apply (subst sum.remove [of _ j], simp_all)
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   606
  apply (rule sum.neutral, simp add: axis_def)
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   607
  apply (rule sum.neutral, simp add: axis_def)
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   608
  done
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   609
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   610
lemma sum_single:
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   611
  assumes "finite A" and "k \<in> A" and "f k = y"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   612
  assumes "\<And>i. i \<in> A \<Longrightarrow> i \<noteq> k \<Longrightarrow> f i = 0"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   613
  shows "(\<Sum>i\<in>A. f i) = y"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   614
  apply (subst sum.remove [OF assms(1,2)])
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   615
  apply (simp add: sum.neutral assms(3,4))
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   616
  done
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   617
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   618
lemma inner_axis: "inner x (axis i y) = inner (x $ i) y"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   619
  unfolding inner_vec_def
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63918
diff changeset
   620
  apply (rule_tac k=i in sum_single)
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   621
  apply simp_all
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   622
  apply (simp add: axis_def)
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   623
  done
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   624
67683
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   625
lemma inner_axis': "inner(axis i y) x = inner y (x $ i)"
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   626
  by (simp add: inner_axis inner_commute)
817944aeac3f Lots of new material about matrices, etc.
paulson <lp15@cam.ac.uk>
parents: 67673
diff changeset
   627
44136
e63ad7d5158d more uniform naming scheme for finite cartesian product type and related theorems
huffman
parents: 44135
diff changeset
   628
instantiation vec :: (euclidean_space, finite) euclidean_space
44135
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   629
begin
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   630
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   631
definition "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   632
44135
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   633
instance proof
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   634
  show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   635
    unfolding Basis_vec_def by simp
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   636
next
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   637
  show "finite (Basis :: ('a ^ 'b) set)"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   638
    unfolding Basis_vec_def by simp
44135
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   639
next
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   640
  fix u v :: "'a ^ 'b"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   641
  assume "u \<in> Basis" and "v \<in> Basis"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   642
  thus "inner u v = (if u = v then 1 else 0)"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   643
    unfolding Basis_vec_def
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   644
    by (auto simp add: inner_axis_axis axis_eq_axis inner_Basis)
44135
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   645
next
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   646
  fix x :: "'a ^ 'b"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   647
  show "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> x = 0"
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   648
    unfolding Basis_vec_def
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   649
    by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   650
qed
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   651
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   652
lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   653
  apply (simp add: Basis_vec_def)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   654
  apply (subst card_UN_disjoint)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   655
     apply simp
44166
d12d89a66742 modify euclidean_space class to include basis set
huffman
parents: 44165
diff changeset
   656
    apply simp
50526
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   657
   apply (auto simp: axis_eq_axis) [1]
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   658
  apply (subst card_UN_disjoint)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   659
     apply (auto simp: axis_eq_axis)
899c9c4e4a4c Remove the indexed basis from the definition of euclidean spaces and only use the set of Basis vectors
hoelzl
parents: 50252
diff changeset
   660
  done
44135
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   661
36591
df38e0c5c123 move class instantiations from Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 36590
diff changeset
   662
end
44135
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   663
62397
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62102
diff changeset
   664
lemma cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62102
diff changeset
   665
  by (simp add: inner_axis)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62102
diff changeset
   666
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62102
diff changeset
   667
lemma axis_in_Basis: "a \<in> Basis \<Longrightarrow> axis i a \<in> Basis"
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62102
diff changeset
   668
  by (auto simp add: Basis_vec_def axis_eq_axis)
5ae24f33d343 Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents: 62102
diff changeset
   669
44135
18b4ab6854f1 move euclidean_space instance from Cartesian_Euclidean_Space.thy to Finite_Cartesian_Product.thy
huffman
parents: 42290
diff changeset
   670
end