src/HOL/Multivariate_Analysis/Linear_Algebra.thy
author huffman
Wed, 10 Aug 2011 09:23:42 -0700
changeset 44133 691c52e900ca
child 44142 8e27e0177518
permissions -rw-r--r--
split Linear_Algebra.thy from Euclidean_Space.thy
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
44133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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(*  Title:      HOL/Multivariate_Analysis/Linear_Algebra.thy
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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    Author:     Amine Chaieb, University of Cambridge
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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*)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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header {* Elementary linear algebra on Euclidean spaces *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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theory Linear_Algebra
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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imports
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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     9
  Euclidean_Space
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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    10
  "~~/src/HOL/Library/Infinite_Set"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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    11
  L2_Norm
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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  "~~/src/HOL/Library/Convex"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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uses
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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  "~~/src/HOL/Library/positivstellensatz.ML"  (* FIXME duplicate use!? *)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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  ("normarith.ML")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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begin
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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  by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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    21
notation inner (infix "\<bullet>" 70)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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    23
subsection {* A connectedness or intermediate value lemma with several applications. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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    24
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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lemma connected_real_lemma:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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  fixes f :: "real \<Rightarrow> 'a::metric_space"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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    27
  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
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parents:
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  and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    29
  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    30
  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    31
  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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    32
  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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    33
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    34
  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    35
  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    36
  have Sub: "\<exists>y. isUb UNIV ?S y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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    37
    apply (rule exI[where x= b])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    38
    using ab fb e12 by (auto simp add: isUb_def setle_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    39
  from reals_complete[OF Se Sub] obtain l where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
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    40
    l: "isLub UNIV ?S l"by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    41
  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    42
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    43
    by (metis linorder_linear)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    44
  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    45
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    46
    by (metis linorder_linear not_le)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    47
    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    48
    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    49
    have "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    50
    then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    51
    {assume le2: "f l \<in> e2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    52
      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    53
      hence lap: "l - a > 0" using alb by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    54
      from e2[rule_format, OF le2] obtain e where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    55
        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    56
      from dst[OF alb e(1)] obtain d where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    57
        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    58
      let ?d' = "min (d/2) ((l - a)/2)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    59
      have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    60
        by (simp add: min_max.less_infI2)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    61
      then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    62
      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    63
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    64
      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    65
      moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    66
      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    67
      ultimately have False using e12 alb d' by auto}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    68
    moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    69
    {assume le1: "f l \<in> e1"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    70
    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    71
      hence blp: "b - l > 0" using alb by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    72
      from e1[rule_format, OF le1] obtain e where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    73
        e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    74
      from dst[OF alb e(1)] obtain d where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    75
        d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    76
      have "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    77
      then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    78
      then obtain d' where d': "d' > 0" "d' < d" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    79
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    80
      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    81
      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    82
      with l d' have False
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    83
        by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    84
    ultimately show ?thesis using alb by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    85
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    86
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    87
text{* One immediately useful corollary is the existence of square roots! --- Should help to get rid of all the development of square-root for reals as a special case *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    88
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    89
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    90
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    91
  have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    92
  thus ?thesis by (simp add: field_simps power2_eq_square)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    93
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    94
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    95
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    96
  using isCont_power[OF isCont_ident, of 2, unfolded isCont_def LIM_eq, rule_format, of e x] apply (auto simp add: power2_eq_square)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    97
  apply (rule_tac x="s" in exI)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    98
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
    99
  apply (erule_tac x=y in allE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   100
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   101
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   102
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   103
lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   104
  using real_sqrt_le_iff[of x "y^2"] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   105
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   106
lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   107
  using real_sqrt_le_mono[of "x^2" y] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   108
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   109
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   110
  using real_sqrt_less_mono[of "x^2" y] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   111
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   112
lemma sqrt_even_pow2: assumes n: "even n"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   113
  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   114
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   115
  from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   116
  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   117
    by (simp only: power_mult[symmetric] mult_commute)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   118
  then show ?thesis  using m by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   119
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   120
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   121
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   122
  apply (cases "x = 0", simp_all)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   123
  using sqrt_divide_self_eq[of x]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   124
  apply (simp add: inverse_eq_divide field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   125
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   126
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   127
text{* Hence derive more interesting properties of the norm. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   128
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   129
(* FIXME: same as norm_scaleR
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   130
lemma norm_mul[simp]: "norm(a *\<^sub>R x) = abs(a) * norm x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   131
  by (simp add: norm_vector_def setL2_right_distrib abs_mult)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   132
*)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   133
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   134
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   135
  by (simp add: setL2_def power2_eq_square)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   136
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   137
lemma norm_cauchy_schwarz:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   138
  shows "inner x y <= norm x * norm y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   139
  using Cauchy_Schwarz_ineq2[of x y] by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   140
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   141
lemma norm_cauchy_schwarz_abs:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   142
  shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   143
  by (rule Cauchy_Schwarz_ineq2)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   144
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   145
lemma norm_triangle_sub:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   146
  fixes x y :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   147
  shows "norm x \<le> norm y  + norm (x - y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   148
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   149
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   150
lemma real_abs_norm: "\<bar>norm x\<bar> = norm x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   151
  by (rule abs_norm_cancel)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   152
lemma real_abs_sub_norm: "\<bar>norm x - norm y\<bar> <= norm(x - y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   153
  by (rule norm_triangle_ineq3)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   154
lemma norm_le: "norm(x) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   155
  by (simp add: norm_eq_sqrt_inner) 
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   156
lemma norm_lt: "norm(x) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   157
  by (simp add: norm_eq_sqrt_inner)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   158
lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   159
  apply(subst order_eq_iff) unfolding norm_le by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   160
lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   161
  unfolding norm_eq_sqrt_inner by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   162
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   163
text{* Squaring equations and inequalities involving norms.  *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   164
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   165
lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   166
  by (simp add: norm_eq_sqrt_inner)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   167
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   168
lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   169
  by (auto simp add: norm_eq_sqrt_inner)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   170
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   171
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   172
proof
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   173
  assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   174
  then have "\<bar>x\<bar>\<twosuperior> \<le> \<bar>y\<bar>\<twosuperior>" by (rule power_mono, simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   175
  then show "x\<twosuperior> \<le> y\<twosuperior>" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   176
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   177
  assume "x\<twosuperior> \<le> y\<twosuperior>"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   178
  then have "sqrt (x\<twosuperior>) \<le> sqrt (y\<twosuperior>)" by (rule real_sqrt_le_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   179
  then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   180
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   181
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   182
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   183
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   184
  using norm_ge_zero[of x]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   185
  apply arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   186
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   187
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   188
lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   189
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   190
  using norm_ge_zero[of x]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   191
  apply arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   192
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   193
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   194
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   195
  by (metis not_le norm_ge_square)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   196
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   197
  by (metis norm_le_square not_less)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   198
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   199
text{* Dot product in terms of the norm rather than conversely. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   200
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   201
lemmas inner_simps = inner.add_left inner.add_right inner.diff_right inner.diff_left 
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   202
inner.scaleR_left inner.scaleR_right
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   203
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   204
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   205
  unfolding power2_norm_eq_inner inner_simps inner_commute by auto 
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   206
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   207
lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   208
  unfolding power2_norm_eq_inner inner_simps inner_commute by(auto simp add:algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   209
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   210
text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   211
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   212
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   213
proof
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   214
  assume ?lhs then show ?rhs by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   215
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   216
  assume ?rhs
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   217
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   218
  hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" by (simp add: inner_simps inner_commute)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   219
  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: field_simps inner_simps inner_commute)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   220
  then show "x = y" by (simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   221
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   222
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   223
subsection{* General linear decision procedure for normed spaces. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   224
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   225
lemma norm_cmul_rule_thm:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   226
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   227
  shows "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(scaleR c x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   228
  unfolding norm_scaleR
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   229
  apply (erule mult_left_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   230
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   231
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   232
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   233
  (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   234
lemma norm_add_rule_thm:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   235
  fixes x1 x2 :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   236
  shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   237
  by (rule order_trans [OF norm_triangle_ineq add_mono])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   238
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   239
lemma ge_iff_diff_ge_0: "(a::'a::linordered_ring) \<ge> b == a - b \<ge> 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   240
  by (simp add: field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   241
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   242
lemma pth_1:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   243
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   244
  shows "x == scaleR 1 x" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   245
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   246
lemma pth_2:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   247
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   248
  shows "x - y == x + -y" by (atomize (full)) simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   249
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   250
lemma pth_3:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   251
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   252
  shows "- x == scaleR (-1) x" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   253
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   254
lemma pth_4:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   255
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   256
  shows "scaleR 0 x == 0" and "scaleR c 0 = (0::'a)" by simp_all
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   257
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   258
lemma pth_5:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   259
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   260
  shows "scaleR c (scaleR d x) == scaleR (c * d) x" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   261
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   262
lemma pth_6:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   263
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   264
  shows "scaleR c (x + y) == scaleR c x + scaleR c y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   265
  by (simp add: scaleR_right_distrib)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   266
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   267
lemma pth_7:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   268
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   269
  shows "0 + x == x" and "x + 0 == x" by simp_all
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   270
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   271
lemma pth_8:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   272
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   273
  shows "scaleR c x + scaleR d x == scaleR (c + d) x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   274
  by (simp add: scaleR_left_distrib)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   275
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   276
lemma pth_9:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   277
  fixes x :: "'a::real_normed_vector" shows
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   278
  "(scaleR c x + z) + scaleR d x == scaleR (c + d) x + z"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   279
  "scaleR c x + (scaleR d x + z) == scaleR (c + d) x + z"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   280
  "(scaleR c x + w) + (scaleR d x + z) == scaleR (c + d) x + (w + z)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   281
  by (simp_all add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   282
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   283
lemma pth_a:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   284
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   285
  shows "scaleR 0 x + y == y" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   286
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   287
lemma pth_b:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   288
  fixes x :: "'a::real_normed_vector" shows
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   289
  "scaleR c x + scaleR d y == scaleR c x + scaleR d y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   290
  "(scaleR c x + z) + scaleR d y == scaleR c x + (z + scaleR d y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   291
  "scaleR c x + (scaleR d y + z) == scaleR c x + (scaleR d y + z)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   292
  "(scaleR c x + w) + (scaleR d y + z) == scaleR c x + (w + (scaleR d y + z))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   293
  by (simp_all add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   294
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   295
lemma pth_c:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   296
  fixes x :: "'a::real_normed_vector" shows
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   297
  "scaleR c x + scaleR d y == scaleR d y + scaleR c x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   298
  "(scaleR c x + z) + scaleR d y == scaleR d y + (scaleR c x + z)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   299
  "scaleR c x + (scaleR d y + z) == scaleR d y + (scaleR c x + z)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   300
  "(scaleR c x + w) + (scaleR d y + z) == scaleR d y + ((scaleR c x + w) + z)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   301
  by (simp_all add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   302
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   303
lemma pth_d:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   304
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   305
  shows "x + 0 == x" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   306
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   307
lemma norm_imp_pos_and_ge:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   308
  fixes x :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   309
  shows "norm x == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   310
  by atomize auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   311
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   312
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   313
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   314
lemma norm_pths:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   315
  fixes x :: "'a::real_normed_vector" shows
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   316
  "x = y \<longleftrightarrow> norm (x - y) \<le> 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   317
  "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   318
  using norm_ge_zero[of "x - y"] by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   319
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   320
use "normarith.ML"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   321
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   322
method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   323
*} "prove simple linear statements about vector norms"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   324
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   325
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   326
text{* Hence more metric properties. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   327
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   328
lemma norm_triangle_half_r:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   329
  shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   330
  using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   331
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   332
lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2" 
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   333
  shows "norm (x - x') < e"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   334
  using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   335
  unfolding dist_norm[THEN sym] .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   336
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   337
lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   338
  by (metis order_trans norm_triangle_ineq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   339
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   340
lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   341
  by (metis basic_trans_rules(21) norm_triangle_ineq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   342
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   343
lemma dist_triangle_add:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   344
  fixes x y x' y' :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   345
  shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   346
  by norm
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   347
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   348
lemma dist_triangle_add_half:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   349
  fixes x x' y y' :: "'a::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   350
  shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   351
  by norm
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   352
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   353
lemma setsum_clauses:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   354
  shows "setsum f {} = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   355
  and "finite S \<Longrightarrow> setsum f (insert x S) =
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   356
                 (if x \<in> S then setsum f S else f x + setsum f S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   357
  by (auto simp add: insert_absorb)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   358
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   359
lemma setsum_norm:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   360
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   361
  assumes fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   362
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   363
proof(induct rule: finite_induct[OF fS])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   364
  case 1 thus ?case by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   365
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   366
  case (2 x S)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   367
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   368
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   369
    using "2.hyps" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   370
  finally  show ?case  using "2.hyps" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   371
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   372
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   373
lemma setsum_norm_le:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   374
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   375
  assumes fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   376
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   377
  shows "norm (setsum f S) \<le> setsum g S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   378
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   379
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   380
    by - (rule setsum_mono, simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   381
  then show ?thesis using setsum_norm[OF fS, of f] fg
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   382
    by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   383
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   384
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   385
lemma setsum_norm_bound:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   386
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   387
  assumes fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   388
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   389
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   390
  using setsum_norm_le[OF fS K] setsum_constant[symmetric]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   391
  by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   392
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   393
lemma setsum_group:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   394
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   395
  shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   396
  apply (subst setsum_image_gen[OF fS, of g f])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   397
  apply (rule setsum_mono_zero_right[OF fT fST])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   398
  by (auto intro: setsum_0')
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   399
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   400
lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> y = setsum (\<lambda>x. f x \<bullet> y) S "
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   401
  apply(induct rule: finite_induct) by(auto simp add: inner_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   402
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   403
lemma dot_rsum: "finite S \<Longrightarrow> y \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   404
  apply(induct rule: finite_induct) by(auto simp add: inner_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   405
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   406
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   407
proof
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   408
  assume "\<forall>x. x \<bullet> y = x \<bullet> z"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   409
  hence "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   410
  hence "(y - z) \<bullet> (y - z) = 0" ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   411
  thus "y = z" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   412
qed simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   413
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   414
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   415
proof
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   416
  assume "\<forall>z. x \<bullet> z = y \<bullet> z"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   417
  hence "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   418
  hence "(x - y) \<bullet> (x - y) = 0" ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   419
  thus "x = y" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   420
qed simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   421
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   422
subsection{* Orthogonality. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   423
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   424
context real_inner
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   425
begin
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   426
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   427
definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   428
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   429
lemma orthogonal_clauses:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   430
  "orthogonal a 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   431
  "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   432
  "orthogonal a x \<Longrightarrow> orthogonal a (-x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   433
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   434
  "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   435
  "orthogonal 0 a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   436
  "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   437
  "orthogonal x a \<Longrightarrow> orthogonal (-x) a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   438
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   439
  "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   440
  unfolding orthogonal_def inner_simps inner_add_left inner_add_right inner_diff_left inner_diff_right (*FIXME*) by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   441
 
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   442
end
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   443
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   444
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   445
  by (simp add: orthogonal_def inner_commute)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   446
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   447
subsection{* Linear functions. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   448
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   449
definition
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   450
  linear :: "('a::real_vector \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   451
  "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   452
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   453
lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   454
  shows "linear f" using assms unfolding linear_def by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   455
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   456
lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   457
  by (simp add: linear_def algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   458
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   459
lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   460
  by (simp add: linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   461
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   462
lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   463
  by (simp add: linear_def algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   464
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   465
lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   466
  by (simp add: linear_def algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   467
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   468
lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   469
  by (simp add: linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   470
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   471
lemma linear_id: "linear id" by (simp add: linear_def id_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   472
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   473
lemma linear_zero: "linear (\<lambda>x. 0)" by (simp add: linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   474
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   475
lemma linear_compose_setsum:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   476
  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   477
  shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   478
  using lS
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   479
  apply (induct rule: finite_induct[OF fS])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   480
  by (auto simp add: linear_zero intro: linear_compose_add)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   481
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   482
lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   483
  unfolding linear_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   484
  apply clarsimp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   485
  apply (erule allE[where x="0::'a"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   486
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   487
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   488
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   489
lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x" by (simp add: linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   490
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   491
lemma linear_neg: "linear f ==> f (-x) = - f x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   492
  using linear_cmul [where c="-1"] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   493
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   494
lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   495
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   496
lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   497
  by (simp add: diff_minus linear_add linear_neg)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   498
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   499
lemma linear_setsum:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   500
  assumes lf: "linear f" and fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   501
  shows "f (setsum g S) = setsum (f o g) S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   502
proof (induct rule: finite_induct[OF fS])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   503
  case 1 thus ?case by (simp add: linear_0[OF lf])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   504
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   505
  case (2 x F)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   506
  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   507
    by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   508
  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   509
  also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   510
  finally show ?case .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   511
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   512
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   513
lemma linear_setsum_mul:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   514
  assumes lf: "linear f" and fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   515
  shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   516
  using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   517
  linear_cmul[OF lf] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   518
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   519
lemma linear_injective_0:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   520
  assumes lf: "linear f"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   521
  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   522
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   523
  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   524
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   525
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   526
    by (simp add: linear_sub[OF lf])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   527
  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   528
  finally show ?thesis .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   529
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   530
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   531
subsection{* Bilinear functions. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   532
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   533
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   534
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   535
lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   536
  by (simp add: bilinear_def linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   537
lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   538
  by (simp add: bilinear_def linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   539
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   540
lemma bilinear_lmul: "bilinear h ==> h (c *\<^sub>R x) y = c *\<^sub>R (h x y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   541
  by (simp add: bilinear_def linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   542
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   543
lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   544
  by (simp add: bilinear_def linear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   545
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   546
lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   547
  by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   548
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   549
lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   550
  by (simp only: scaleR_minus1_left [symmetric] bilinear_rmul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   551
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   552
lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   553
  using add_imp_eq[of x y 0] by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   554
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   555
lemma bilinear_lzero:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   556
  assumes bh: "bilinear h" shows "h 0 x = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   557
  using bilinear_ladd[OF bh, of 0 0 x]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   558
    by (simp add: eq_add_iff field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   559
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   560
lemma bilinear_rzero:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   561
  assumes bh: "bilinear h" shows "h x 0 = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   562
  using bilinear_radd[OF bh, of x 0 0 ]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   563
    by (simp add: eq_add_iff field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   564
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   565
lemma bilinear_lsub: "bilinear h ==> h (x - y) z = h x z - h y z"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   566
  by (simp  add: diff_minus bilinear_ladd bilinear_lneg)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   567
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   568
lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   569
  by (simp  add: diff_minus bilinear_radd bilinear_rneg)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   570
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   571
lemma bilinear_setsum:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   572
  assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   573
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   574
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   575
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   576
    apply (rule linear_setsum[unfolded o_def])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   577
    using bh fS by (auto simp add: bilinear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   578
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   579
    apply (rule setsum_cong, simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   580
    apply (rule linear_setsum[unfolded o_def])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   581
    using bh fT by (auto simp add: bilinear_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   582
  finally show ?thesis unfolding setsum_cartesian_product .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   583
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   584
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   585
subsection{* Adjoints. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   586
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   587
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   588
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   589
lemma adjoint_unique:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   590
  assumes "\<forall>x y. inner (f x) y = inner x (g y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   591
  shows "adjoint f = g"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   592
unfolding adjoint_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   593
proof (rule some_equality)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   594
  show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   595
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   596
  fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   597
  hence "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   598
  hence "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   599
  hence "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   600
  hence "\<forall>y. h y = g y" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   601
  thus "h = g" by (simp add: ext)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   602
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   603
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   604
lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   605
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   606
subsection{* Interlude: Some properties of real sets *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   607
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   608
lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   609
  shows "\<forall>n \<ge> m. d n < e m"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   610
  using assms apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   611
  apply (erule_tac x="n" in allE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   612
  apply (erule_tac x="n" in allE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   613
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   614
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   615
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   616
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   617
lemma infinite_enumerate: assumes fS: "infinite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   618
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   619
unfolding subseq_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   620
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   621
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   622
lemma approachable_lt_le: "(\<exists>(d::real)>0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   623
apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   624
apply (rule_tac x="d/2" in exI)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   625
apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   626
done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   627
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   628
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   629
lemma triangle_lemma:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   630
  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   631
  shows "x <= y + z"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   632
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   633
  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: mult_nonneg_nonneg)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   634
  with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   635
  from y z have yz: "y + z \<ge> 0" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   636
  from power2_le_imp_le[OF th yz] show ?thesis .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   637
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   638
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   639
text {* TODO: move to NthRoot *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   640
lemma sqrt_add_le_add_sqrt:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   641
  assumes x: "0 \<le> x" and y: "0 \<le> y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   642
  shows "sqrt (x + y) \<le> sqrt x + sqrt y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   643
apply (rule power2_le_imp_le)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   644
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   645
apply (simp add: mult_nonneg_nonneg x y)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   646
apply (simp add: add_nonneg_nonneg x y)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   647
done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   648
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   649
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   650
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   651
definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   652
  "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   653
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   654
lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   655
  unfolding hull_def by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   656
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   657
lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   658
unfolding hull_def subset_iff by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   659
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   660
lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   661
using hull_same[of s S] hull_in[of S s] by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   662
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   663
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   664
lemma hull_hull: "S hull (S hull s) = S hull s"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   665
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   666
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   667
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   668
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   669
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   670
lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   671
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   672
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   673
lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   674
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   675
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   676
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   677
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   678
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   679
lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   680
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   681
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   682
lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   683
           ==> (S hull s = t)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   684
unfolding hull_def by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   685
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   686
lemma hull_induct: "(\<And>x. x\<in> S \<Longrightarrow> P x) \<Longrightarrow> Q {x. P x} \<Longrightarrow> \<forall>x\<in> Q hull S. P x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   687
  using hull_minimal[of S "{x. P x}" Q]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   688
  by (auto simp add: subset_eq Collect_def mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   689
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   690
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   691
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   692
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   693
unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   694
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   695
lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   696
  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   697
apply rule
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   698
apply (rule hull_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   699
unfolding Un_subset_iff
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   700
apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   701
apply (rule hull_minimal)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   702
apply (metis hull_union_subset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   703
apply (metis hull_in T)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   704
done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   705
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   706
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   707
  unfolding hull_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   708
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   709
lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   710
by (metis hull_redundant_eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   711
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   712
text{* Archimedian properties and useful consequences. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   713
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   714
lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   715
  using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   716
lemmas real_arch_lt = reals_Archimedean2
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   717
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   718
lemmas real_arch = reals_Archimedean3
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   719
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   720
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   721
  using reals_Archimedean
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   722
  apply (auto simp add: field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   723
  apply (subgoal_tac "inverse (real n) > 0")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   724
  apply arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   725
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   726
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   727
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   728
lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   729
proof(induct n)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   730
  case 0 thus ?case by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   731
next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   732
  case (Suc n)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   733
  hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   734
  from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   735
  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   736
  also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   737
    apply (simp add: field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   738
    using mult_left_mono[OF p Suc.prems] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   739
  finally show ?case  by (simp add: real_of_nat_Suc field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   740
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   741
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   742
lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   743
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   744
  from x have x0: "x - 1 > 0" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   745
  from real_arch[OF x0, rule_format, of y]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   746
  obtain n::nat where n:"y < real n * (x - 1)" by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   747
  from x0 have x00: "x- 1 \<ge> 0" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   748
  from real_pow_lbound[OF x00, of n] n
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   749
  have "y < x^n" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   750
  then show ?thesis by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   751
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   752
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   753
lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   754
  using real_arch_pow[of 2 x] by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   755
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   756
lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   757
  shows "\<exists>n. x^n < y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   758
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   759
  {assume x0: "x > 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   760
    from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   761
    from real_arch_pow[OF ix, of "1/y"]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   762
    obtain n where n: "1/y < (1/x)^n" by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   763
    then
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   764
    have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   765
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   766
  {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   767
  ultimately show ?thesis by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   768
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   769
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   770
lemma forall_pos_mono: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n::nat. n \<noteq> 0 ==> P(inverse(real n))) \<Longrightarrow> (\<And>e. 0 < e ==> P e)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   771
  by (metis real_arch_inv)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   772
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   773
lemma forall_pos_mono_1: "(\<And>d e::real. d < e \<Longrightarrow> P d ==> P e) \<Longrightarrow> (\<And>n. P(inverse(real (Suc n)))) ==> 0 < e ==> P e"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   774
  apply (rule forall_pos_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   775
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   776
  apply (atomize)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   777
  apply (erule_tac x="n - 1" in allE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   778
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   779
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   780
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   781
lemma real_archimedian_rdiv_eq_0: assumes x0: "x \<ge> 0" and c: "c \<ge> 0" and xc: "\<forall>(m::nat)>0. real m * x \<le> c"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   782
  shows "x = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   783
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   784
  {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   785
    from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   786
    with xc[rule_format, of n] have "n = 0" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   787
    with n c have False by simp}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   788
  then show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   789
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   790
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   791
subsection {* Geometric progression *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   792
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   793
lemma sum_gp_basic: "((1::'a::{field}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   794
  (is "?lhs = ?rhs")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   795
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   796
  {assume x1: "x = 1" hence ?thesis by simp}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   797
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   798
  {assume x1: "x\<noteq>1"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   799
    hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   800
    from geometric_sum[OF x1, of "Suc n", unfolded x1']
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   801
    have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   802
      unfolding atLeastLessThanSuc_atLeastAtMost
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   803
      using x1' apply (auto simp only: field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   804
      apply (simp add: field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   805
      done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   806
    then have ?thesis by (simp add: field_simps) }
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   807
  ultimately show ?thesis by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   808
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   809
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   810
lemma sum_gp_multiplied: assumes mn: "m <= n"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   811
  shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   812
  (is "?lhs = ?rhs")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   813
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   814
  let ?S = "{0..(n - m)}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   815
  from mn have mn': "n - m \<ge> 0" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   816
  let ?f = "op + m"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   817
  have i: "inj_on ?f ?S" unfolding inj_on_def by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   818
  have f: "?f ` ?S = {m..n}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   819
    using mn apply (auto simp add: image_iff Bex_def) by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   820
  have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   821
    by (rule ext, simp add: power_add power_mult)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   822
  from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   823
  have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   824
  then show ?thesis unfolding sum_gp_basic using mn
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   825
    by (simp add: field_simps power_add[symmetric])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   826
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   827
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   828
lemma sum_gp: "setsum (op ^ (x::'a::{field})) {m .. n} =
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   829
   (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   830
                    else (x^ m - x^ (Suc n)) / (1 - x))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   831
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   832
  {assume nm: "n < m" hence ?thesis by simp}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   833
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   834
  {assume "\<not> n < m" hence nm: "m \<le> n" by arith
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   835
    {assume x: "x = 1"  hence ?thesis by simp}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   836
    moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   837
    {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   838
      from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   839
    ultimately have ?thesis by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   840
  }
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   841
  ultimately show ?thesis by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   842
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   843
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   844
lemma sum_gp_offset: "setsum (op ^ (x::'a::{field})) {m .. m+n} =
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   845
  (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   846
  unfolding sum_gp[of x m "m + n"] power_Suc
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   847
  by (simp add: field_simps power_add)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   848
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   849
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   850
subsection{* A bit of linear algebra. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   851
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   852
definition (in real_vector)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   853
  subspace :: "'a set \<Rightarrow> bool" where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   854
  "subspace S \<longleftrightarrow> 0 \<in> S \<and> (\<forall>x\<in> S. \<forall>y \<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x \<in>S. c *\<^sub>R x \<in>S )"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   855
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   856
definition (in real_vector) "span S = (subspace hull S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   857
definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   858
abbreviation (in real_vector) "independent s == ~(dependent s)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   859
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   860
text {* Closure properties of subspaces. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   861
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   862
lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   863
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   864
lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   865
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   866
lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   867
  by (metis subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   868
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   869
lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   870
  by (metis subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   871
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   872
lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   873
  by (metis scaleR_minus1_left subspace_mul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   874
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   875
lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   876
  by (metis diff_minus subspace_add subspace_neg)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   877
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   878
lemma (in real_vector) subspace_setsum:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   879
  assumes sA: "subspace A" and fB: "finite B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   880
  and f: "\<forall>x\<in> B. f x \<in> A"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   881
  shows "setsum f B \<in> A"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   882
  using  fB f sA
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   883
  apply(induct rule: finite_induct[OF fB])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   884
  by (simp add: subspace_def sA, auto simp add: sA subspace_add)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   885
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   886
lemma subspace_linear_image:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   887
  assumes lf: "linear f" and sS: "subspace S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   888
  shows "subspace(f ` S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   889
  using lf sS linear_0[OF lf]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   890
  unfolding linear_def subspace_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   891
  apply (auto simp add: image_iff)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   892
  apply (rule_tac x="x + y" in bexI, auto)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   893
  apply (rule_tac x="c *\<^sub>R x" in bexI, auto)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   894
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   895
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   896
lemma subspace_linear_preimage: "linear f ==> subspace S ==> subspace {x. f x \<in> S}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   897
  by (auto simp add: subspace_def linear_def linear_0[of f])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   898
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   899
lemma subspace_trivial: "subspace {0}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   900
  by (simp add: subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   901
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   902
lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   903
  by (simp add: subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   904
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   905
lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   906
  by (metis span_def hull_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   907
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   908
lemma (in real_vector) subspace_span: "subspace(span S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   909
  unfolding span_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   910
  apply (rule hull_in[unfolded mem_def])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   911
  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   912
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   913
  apply (erule_tac x="X" in ballE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   914
  apply (simp add: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   915
  apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   916
  apply (erule_tac x="X" in ballE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   917
  apply (erule_tac x="X" in ballE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   918
  apply (erule_tac x="X" in ballE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   919
  apply (clarsimp simp add: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   920
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   921
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   922
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   923
  apply (erule_tac x="X" in ballE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   924
  apply (erule_tac x="X" in ballE)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   925
  apply (simp add: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   926
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   927
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   928
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   929
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   930
lemma (in real_vector) span_clauses:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   931
  "a \<in> S ==> a \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   932
  "0 \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   933
  "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   934
  "x \<in> span S \<Longrightarrow> c *\<^sub>R x \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   935
  by (metis span_def hull_subset subset_eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   936
     (metis subspace_span subspace_def)+
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   937
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   938
lemma (in real_vector) span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   939
  and P: "subspace P" and x: "x \<in> span S" shows "P x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   940
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   941
  from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   942
  from P have P': "P \<in> subspace" by (simp add: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   943
  from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   944
  show "P x" by (metis mem_def subset_eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   945
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   946
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   947
lemma span_empty[simp]: "span {} = {0}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   948
  apply (simp add: span_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   949
  apply (rule hull_unique)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   950
  apply (auto simp add: mem_def subspace_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   951
  unfolding mem_def[of "0::'a", symmetric]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   952
  apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   953
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   954
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   955
lemma (in real_vector) independent_empty[intro]: "independent {}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   956
  by (simp add: dependent_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   957
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   958
lemma dependent_single[simp]:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   959
  "dependent {x} \<longleftrightarrow> x = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   960
  unfolding dependent_def by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   961
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   962
lemma (in real_vector) independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   963
  apply (clarsimp simp add: dependent_def span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   964
  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   965
  apply force
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   966
  apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   967
  apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   968
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   969
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   970
lemma (in real_vector) span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   971
  by (metis order_antisym span_def hull_minimal mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   972
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   973
lemma (in real_vector) span_induct': assumes SP: "\<forall>x \<in> S. P x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   974
  and P: "subspace P" shows "\<forall>x \<in> span S. P x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   975
  using span_induct SP P by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   976
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   977
inductive (in real_vector) span_induct_alt_help for S:: "'a \<Rightarrow> bool"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   978
  where
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   979
  span_induct_alt_help_0: "span_induct_alt_help S 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   980
  | span_induct_alt_help_S: "x \<in> S \<Longrightarrow> span_induct_alt_help S z \<Longrightarrow> span_induct_alt_help S (c *\<^sub>R x + z)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   981
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   982
lemma span_induct_alt':
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   983
  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" shows "\<forall>x \<in> span S. h x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   984
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   985
  {fix x:: "'a" assume x: "span_induct_alt_help S x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   986
    have "h x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   987
      apply (rule span_induct_alt_help.induct[OF x])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   988
      apply (rule h0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   989
      apply (rule hS, assumption, assumption)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   990
      done}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   991
  note th0 = this
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   992
  {fix x assume x: "x \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   993
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   994
    have "span_induct_alt_help S x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   995
      proof(rule span_induct[where x=x and S=S])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   996
        show "x \<in> span S" using x .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   997
      next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   998
        fix x assume xS : "x \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
   999
          from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1000
          show "span_induct_alt_help S x" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1001
        next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1002
        have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1003
        moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1004
        {fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1005
          from h
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1006
          have "span_induct_alt_help S (x + y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1007
            apply (induct rule: span_induct_alt_help.induct)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1008
            apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1009
            unfolding add_assoc
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1010
            apply (rule span_induct_alt_help_S)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1011
            apply assumption
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1012
            apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1013
            done}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1014
        moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1015
        {fix c x assume xt: "span_induct_alt_help S x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1016
          then have "span_induct_alt_help S (c *\<^sub>R x)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1017
            apply (induct rule: span_induct_alt_help.induct)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1018
            apply (simp add: span_induct_alt_help_0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1019
            apply (simp add: scaleR_right_distrib)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1020
            apply (rule span_induct_alt_help_S)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1021
            apply assumption
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1022
            apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1023
            done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1024
        }
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1025
        ultimately show "subspace (span_induct_alt_help S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1026
          unfolding subspace_def mem_def Ball_def by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1027
      qed}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1028
  with th0 show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1029
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1030
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1031
lemma span_induct_alt:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1032
  assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1033
  shows "h x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1034
using span_induct_alt'[of h S] h0 hS x by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1035
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1036
text {* Individual closure properties. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1037
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1038
lemma span_span: "span (span A) = span A"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1039
  unfolding span_def hull_hull ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1040
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1041
lemma (in real_vector) span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1042
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1043
lemma (in real_vector) span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1044
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1045
lemma span_inc: "S \<subseteq> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1046
  by (metis subset_eq span_superset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1047
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1048
lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1049
  unfolding dependent_def apply(rule_tac x=0 in bexI)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1050
  using assms span_0 by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1051
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1052
lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1053
  by (metis subspace_add subspace_span)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1054
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1055
lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1056
  by (metis subspace_span subspace_mul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1057
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1058
lemma span_neg: "x \<in> span S ==> - x \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1059
  by (metis subspace_neg subspace_span)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1060
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1061
lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1062
  by (metis subspace_span subspace_sub)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1063
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1064
lemma (in real_vector) span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1065
  by (rule subspace_setsum, rule subspace_span)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1066
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1067
lemma span_add_eq: "x \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1068
  apply (auto simp only: span_add span_sub)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1069
  apply (subgoal_tac "(x + y) - x \<in> span S", simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1070
  by (simp only: span_add span_sub)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1071
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1072
text {* Mapping under linear image. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1073
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1074
lemma span_linear_image: assumes lf: "linear f"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1075
  shows "span (f ` S) = f ` (span S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1076
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1077
  {fix x
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1078
    assume x: "x \<in> span (f ` S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1079
    have "x \<in> f ` span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1080
      apply (rule span_induct[where x=x and S = "f ` S"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1081
      apply (clarsimp simp add: image_iff)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1082
      apply (frule span_superset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1083
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1084
      apply (simp only: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1085
      apply (rule subspace_linear_image[OF lf])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1086
      apply (rule subspace_span)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1087
      apply (rule x)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1088
      done}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1089
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1090
  {fix x assume x: "x \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1091
    have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_eqI)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1092
      unfolding mem_def Collect_def ..
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1093
    have "f x \<in> span (f ` S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1094
      apply (rule span_induct[where S=S])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1095
      apply (rule span_superset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1096
      apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1097
      apply (subst th0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1098
      apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1099
      apply (rule x)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1100
      done}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1101
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1102
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1103
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1104
text {* The key breakdown property. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1105
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1106
lemma span_breakdown:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1107
  assumes bS: "b \<in> S" and aS: "a \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1108
  shows "\<exists>k. a - k *\<^sub>R b \<in> span (S - {b})" (is "?P a")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1109
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1110
  {fix x assume xS: "x \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1111
    {assume ab: "x = b"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1112
      then have "?P x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1113
        apply simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1114
        apply (rule exI[where x="1"], simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1115
        by (rule span_0)}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1116
    moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1117
    {assume ab: "x \<noteq> b"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1118
      then have "?P x"  using xS
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1119
        apply -
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1120
        apply (rule exI[where x=0])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1121
        apply (rule span_superset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1122
        by simp}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1123
    ultimately have "?P x" by blast}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1124
  moreover have "subspace ?P"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1125
    unfolding subspace_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1126
    apply auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1127
    apply (simp add: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1128
    apply (rule exI[where x=0])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1129
    using span_0[of "S - {b}"]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1130
    apply (simp add: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1131
    apply (clarsimp simp add: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1132
    apply (rule_tac x="k + ka" in exI)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1133
    apply (subgoal_tac "x + y - (k + ka) *\<^sub>R b = (x - k*\<^sub>R b) + (y - ka *\<^sub>R b)")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1134
    apply (simp only: )
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1135
    apply (rule span_add[unfolded mem_def])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1136
    apply assumption+
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1137
    apply (simp add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1138
    apply (clarsimp simp add: mem_def)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1139
    apply (rule_tac x= "c*k" in exI)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1140
    apply (subgoal_tac "c *\<^sub>R x - (c * k) *\<^sub>R b = c*\<^sub>R (x - k*\<^sub>R b)")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1141
    apply (simp only: )
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1142
    apply (rule span_mul[unfolded mem_def])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1143
    apply assumption
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1144
    by (simp add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1145
  ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1146
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1147
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1148
lemma span_breakdown_eq:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1149
  "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1150
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1151
  {assume x: "x \<in> span (insert a S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1152
    from x span_breakdown[of "a" "insert a S" "x"]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1153
    have ?rhs apply clarsimp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1154
      apply (rule_tac x= "k" in exI)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1155
      apply (rule set_rev_mp[of _ "span (S - {a})" _])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1156
      apply assumption
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1157
      apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1158
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1159
      done}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1160
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1161
  { fix k assume k: "x - k *\<^sub>R a \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1162
    have eq: "x = (x - k *\<^sub>R a) + k *\<^sub>R a" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1163
    have "(x - k *\<^sub>R a) + k *\<^sub>R a \<in> span (insert a S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1164
      apply (rule span_add)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1165
      apply (rule set_rev_mp[of _ "span S" _])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1166
      apply (rule k)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1167
      apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1168
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1169
      apply (rule span_mul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1170
      apply (rule span_superset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1171
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1172
      done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1173
    then have ?lhs using eq by metis}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1174
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1175
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1176
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1177
text {* Hence some "reversal" results. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1178
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1179
lemma in_span_insert:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1180
  assumes a: "a \<in> span (insert b S)" and na: "a \<notin> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1181
  shows "b \<in> span (insert a S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1182
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1183
  from span_breakdown[of b "insert b S" a, OF insertI1 a]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1184
  obtain k where k: "a - k*\<^sub>R b \<in> span (S - {b})" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1185
  {assume k0: "k = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1186
    with k have "a \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1187
      apply (simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1188
      apply (rule set_rev_mp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1189
      apply assumption
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1190
      apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1191
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1192
      done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1193
    with na  have ?thesis by blast}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1194
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1195
  {assume k0: "k \<noteq> 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1196
    have eq: "b = (1/k) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1197
    from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1198
      by (simp add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1199
    from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1200
      by (rule span_mul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1201
    hence th: "(1/k) *\<^sub>R a - b \<in> span (S - {b})"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1202
      unfolding eq' .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1203
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1204
    from k
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1205
    have ?thesis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1206
      apply (subst eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1207
      apply (rule span_sub)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1208
      apply (rule span_mul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1209
      apply (rule span_superset)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1210
      apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1211
      apply (rule set_rev_mp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1212
      apply (rule th)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1213
      apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1214
      using na by blast}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1215
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1216
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1217
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1218
lemma in_span_delete:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1219
  assumes a: "a \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1220
  and na: "a \<notin> span (S-{b})"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1221
  shows "b \<in> span (insert a (S - {b}))"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1222
  apply (rule in_span_insert)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1223
  apply (rule set_rev_mp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1224
  apply (rule a)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1225
  apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1226
  apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1227
  apply (rule na)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1228
  done
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1229
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1230
text {* Transitivity property. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1231
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1232
lemma span_trans:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1233
  assumes x: "x \<in> span S" and y: "y \<in> span (insert x S)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1234
  shows "y \<in> span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1235
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1236
  from span_breakdown[of x "insert x S" y, OF insertI1 y]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1237
  obtain k where k: "y -k*\<^sub>R x \<in> span (S - {x})" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1238
  have eq: "y = (y - k *\<^sub>R x) + k *\<^sub>R x" by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1239
  show ?thesis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1240
    apply (subst eq)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1241
    apply (rule span_add)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1242
    apply (rule set_rev_mp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1243
    apply (rule k)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1244
    apply (rule span_mono)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1245
    apply blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1246
    apply (rule span_mul)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1247
    by (rule x)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1248
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1249
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1250
lemma span_insert_0[simp]: "span (insert 0 S) = span S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1251
  using span_mono[of S "insert 0 S"] by (auto intro: span_trans span_0)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1252
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1253
text {* An explicit expansion is sometimes needed. *}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1254
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1255
lemma span_explicit:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1256
  "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = y}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1257
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1258
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1259
  {fix x assume x: "x \<in> ?E"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1260
    then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1261
      by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1262
    have "x \<in> span P"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1263
      unfolding u[symmetric]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1264
      apply (rule span_setsum[OF fS])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1265
      using span_mono[OF SP]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1266
      by (auto intro: span_superset span_mul)}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1267
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1268
  have "\<forall>x \<in> span P. x \<in> ?E"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1269
    unfolding mem_def Collect_def
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1270
  proof(rule span_induct_alt')
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1271
    show "?h 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1272
      apply (rule exI[where x="{}"]) by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1273
  next
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1274
    fix c x y
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1275
    assume x: "x \<in> P" and hy: "?h y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1276
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1277
      and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y" by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1278
    let ?S = "insert x S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1279
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1280
                  else u y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1281
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1282
    {assume xS: "x \<in> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1283
      have S1: "S = (S - {x}) \<union> {x}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1284
        and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1285
      have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1286
        using xS
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1287
        by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1288
          setsum_clauses(2)[OF fS] cong del: if_weak_cong)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1289
      also have "\<dots> = (\<Sum>v\<in>S. u v *\<^sub>R v) + c *\<^sub>R x"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1290
        apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1291
        by (simp add: algebra_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1292
      also have "\<dots> = c*\<^sub>R x + y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1293
        by (simp add: add_commute u)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1294
      finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1295
    then have "?Q ?S ?u (c*\<^sub>R x + y)" using th0 by blast}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1296
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1297
  {assume xS: "x \<notin> S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1298
    have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *\<^sub>R v) = y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1299
      unfolding u[symmetric]
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1300
      apply (rule setsum_cong2)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1301
      using xS by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1302
    have "?Q ?S ?u (c*\<^sub>R x + y)" using fS xS th0
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1303
      by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1304
  ultimately have "?Q ?S ?u (c*\<^sub>R x + y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1305
    by (cases "x \<in> S", simp, simp)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1306
    then show "?h (c*\<^sub>R x + y)"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1307
      apply -
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1308
      apply (rule exI[where x="?S"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1309
      apply (rule exI[where x="?u"]) by metis
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1310
  qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1311
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1312
qed
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1313
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1314
lemma dependent_explicit:
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1315
  "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>v\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R v) S = 0))" (is "?lhs = ?rhs")
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1316
proof-
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1317
  {assume dP: "dependent P"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1318
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1319
      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *\<^sub>R v) S = a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1320
      unfolding dependent_def span_explicit by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1321
    let ?S = "insert a S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1322
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1323
    let ?v = a
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1324
    from aP SP have aS: "a \<notin> S" by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1325
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1326
    have s0: "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1327
      using fS aS
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1328
      apply (simp add: setsum_clauses field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1329
      apply (subst (2) ua[symmetric])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1330
      apply (rule setsum_cong2)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1331
      by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1332
    with th0 have ?rhs
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1333
      apply -
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1334
      apply (rule exI[where x= "?S"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1335
      apply (rule exI[where x= "?u"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1336
      by clarsimp}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1337
  moreover
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1338
  {fix S u v assume fS: "finite S"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1339
      and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1340
    and u: "setsum (\<lambda>v. u v *\<^sub>R v) S = 0"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1341
    let ?a = v
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1342
    let ?S = "S - {v}"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1343
    let ?u = "\<lambda>i. (- u i) / u v"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1344
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1345
    have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1346
      using fS vS uv
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1347
      by (simp add: setsum_diff1 divide_inverse field_simps)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1348
    also have "\<dots> = ?a"
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1349
      unfolding scaleR_right.setsum [symmetric] u
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1350
      using uv by simp
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1351
    finally  have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1352
    with th0 have ?lhs
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1353
      unfolding dependent_def span_explicit
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1354
      apply -
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1355
      apply (rule bexI[where x= "?a"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1356
      apply (simp_all del: scaleR_minus_left)
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1357
      apply (rule exI[where x= "?S"])
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1358
      by (auto simp del: scaleR_minus_left)}
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy
huffman
parents:
diff changeset
  1359
  ultimately show ?thesis by blast
691c52e900ca split Linear_Algebra.thy from Euclidean_Space.thy