src/HOL/Library/Euclidean_Space.thy
author huffman
Fri, 29 May 2009 15:32:33 -0700
changeset 31344 fc09ec06b89b
parent 31340 5cddd98abe14
child 31389 3affcbc60c6d
permissions -rw-r--r--
instance ^ :: (metric_space, finite) metric_space
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
     1
(* Title:      Library/Euclidean_Space
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
     2
   Author:     Amine Chaieb, University of Cambridge
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
     3
*)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
     4
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
     5
header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
     6
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
     7
theory Euclidean_Space
30661
54858c8ad226 tuned header
haftmann
parents: 30582
diff changeset
     8
imports
30665
haftmann
parents: 30655 30661
diff changeset
     9
  Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    10
  Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
30045
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
    11
  Inner_Product
31118
541d43bee678 Isolated decision procedure for noms and the general arithmetic solver
chaieb
parents: 31034
diff changeset
    12
uses "positivstellensatz.ML" ("normarith.ML")
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    13
begin
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    14
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    15
text{* Some common special cases.*}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    16
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    17
lemma forall_1: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    18
  by (metis num1_eq_iff)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    19
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    20
lemma exhaust_2:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    21
  fixes x :: 2 shows "x = 1 \<or> x = 2"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    22
proof (induct x)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    23
  case (of_int z)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    24
  then have "0 <= z" and "z < 2" by simp_all
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    25
  then have "z = 0 | z = 1" by arith
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    26
  then show ?case by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    27
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    28
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    29
lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    30
  by (metis exhaust_2)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    31
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    32
lemma exhaust_3:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    33
  fixes x :: 3 shows "x = 1 \<or> x = 2 \<or> x = 3"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    34
proof (induct x)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    35
  case (of_int z)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    36
  then have "0 <= z" and "z < 3" by simp_all
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    37
  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    38
  then show ?case by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    39
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    40
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    41
lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    42
  by (metis exhaust_3)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    43
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    44
lemma UNIV_1: "UNIV = {1::1}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    45
  by (auto simp add: num1_eq_iff)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    46
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    47
lemma UNIV_2: "UNIV = {1::2, 2::2}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    48
  using exhaust_2 by auto
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    49
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    50
lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    51
  using exhaust_3 by auto
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    52
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    53
lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    54
  unfolding UNIV_1 by simp
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    55
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    56
lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    57
  unfolding UNIV_2 by simp
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    58
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    59
lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    60
  unfolding UNIV_3 by (simp add: add_ac)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    61
29906
80369da39838 section -> subsection
huffman
parents: 29905
diff changeset
    62
subsection{* Basic componentwise operations on vectors. *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    63
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    64
instantiation "^" :: (plus,type) plus
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    65
begin
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
    66
definition  vector_add_def : "op + \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) + (y$i)))"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    67
instance ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    68
end
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    69
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    70
instantiation "^" :: (times,type) times
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    71
begin
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
    72
  definition vector_mult_def : "op * \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) * (y$i)))"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    73
  instance ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    74
end
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    75
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    76
instantiation "^" :: (minus,type) minus begin
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    77
  definition vector_minus_def : "op - \<equiv> (\<lambda> x y.  (\<chi> i. (x$i) - (y$i)))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    78
instance ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    79
end
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    80
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    81
instantiation "^" :: (uminus,type) uminus begin
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    82
  definition vector_uminus_def : "uminus \<equiv> (\<lambda> x.  (\<chi> i. - (x$i)))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    83
instance ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    84
end
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    85
instantiation "^" :: (zero,type) zero begin
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
    86
  definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    87
instance ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    88
end
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    89
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    90
instantiation "^" :: (one,type) one begin
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
    91
  definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    92
instance ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    93
end
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    94
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    95
instantiation "^" :: (ord,type) ord
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    96
 begin
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
    97
definition vector_less_eq_def:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    98
  "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
    99
definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   100
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   101
instance by (intro_classes)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   102
end
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   103
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   104
instantiation "^" :: (scaleR, type) scaleR
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   105
begin
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   106
definition vector_scaleR_def: "scaleR = (\<lambda> r x.  (\<chi> i. scaleR r (x$i)))"
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   107
instance ..
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   108
end
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   109
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   110
text{* Also the scalar-vector multiplication. *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   111
31275
1ba01cdd9a9a Changed prioriy of vector_scalar_mult
himmelma
parents: 31118
diff changeset
   112
definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   113
  where "c *s x = (\<chi> i. c * (x$i))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   114
31275
1ba01cdd9a9a Changed prioriy of vector_scalar_mult
himmelma
parents: 31118
diff changeset
   115
text{* Constant Vectors *} 
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   116
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   117
definition "vec x = (\<chi> i. x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   118
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   119
text{* Dot products. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   120
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   121
definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   122
  "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) UNIV"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   123
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   124
lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \<bullet> y = (x$1) * (y$1)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   125
  by (simp add: dot_def setsum_1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   126
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   127
lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   128
  by (simp add: dot_def setsum_2)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   129
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   130
lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   131
  by (simp add: dot_def setsum_3)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   132
29906
80369da39838 section -> subsection
huffman
parents: 29905
diff changeset
   133
subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   134
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   135
method_setup vector = {*
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   136
let
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   137
  val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym,
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   138
  @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   139
  @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   140
  val ss2 = @{simpset} addsimps
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   141
             [@{thm vector_add_def}, @{thm vector_mult_def},
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   142
              @{thm vector_minus_def}, @{thm vector_uminus_def},
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   143
              @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   144
              @{thm vector_scaleR_def},
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   145
              @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   146
 fun vector_arith_tac ths =
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   147
   simp_tac ss1
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   148
   THEN' (fn i => rtac @{thm setsum_cong2} i
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   149
         ORELSE rtac @{thm setsum_0'} i
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   150
         ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   151
   (* THEN' TRY o clarify_tac HOL_cs  THEN' (TRY o rtac @{thm iffI}) *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   152
   THEN' asm_full_simp_tac (ss2 addsimps ths)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   153
 in
30549
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   154
  Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   155
 end
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   156
*} "Lifts trivial vector statements to real arith statements"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   157
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   158
lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   159
lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   160
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   161
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   162
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   163
text{* Obvious "component-pushing". *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   164
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   165
lemma vec_component [simp]: "(vec x :: 'a ^ 'n)$i = x"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   166
  by (vector vec_def)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   167
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   168
lemma vector_add_component [simp]:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   169
  fixes x y :: "'a::{plus} ^ 'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   170
  shows "(x + y)$i = x$i + y$i"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   171
  by vector
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   172
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   173
lemma vector_minus_component [simp]:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   174
  fixes x y :: "'a::{minus} ^ 'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   175
  shows "(x - y)$i = x$i - y$i"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   176
  by vector
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   177
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   178
lemma vector_mult_component [simp]:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   179
  fixes x y :: "'a::{times} ^ 'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   180
  shows "(x * y)$i = x$i * y$i"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   181
  by vector
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   182
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   183
lemma vector_smult_component [simp]:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   184
  fixes y :: "'a::{times} ^ 'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   185
  shows "(c *s y)$i = c * (y$i)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   186
  by vector
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   187
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   188
lemma vector_uminus_component [simp]:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   189
  fixes x :: "'a::{uminus} ^ 'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   190
  shows "(- x)$i = - (x$i)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   191
  by vector
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   192
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   193
lemma vector_scaleR_component [simp]:
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   194
  fixes x :: "'a::scaleR ^ 'n"
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   195
  shows "(scaleR r x)$i = scaleR r (x$i)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   196
  by vector
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   197
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   198
lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   199
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   200
lemmas vector_component =
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   201
  vec_component vector_add_component vector_mult_component
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   202
  vector_smult_component vector_minus_component vector_uminus_component
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   203
  vector_scaleR_component cond_component
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   204
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   205
subsection {* Some frequently useful arithmetic lemmas over vectors. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   206
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   207
instance "^" :: (semigroup_add,type) semigroup_add
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   208
  apply (intro_classes) by (vector add_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   209
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   210
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   211
instance "^" :: (monoid_add,type) monoid_add
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   212
  apply (intro_classes) by vector+
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   213
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   214
instance "^" :: (group_add,type) group_add
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   215
  apply (intro_classes) by (vector algebra_simps)+
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   216
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   217
instance "^" :: (ab_semigroup_add,type) ab_semigroup_add
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   218
  apply (intro_classes) by (vector add_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   219
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   220
instance "^" :: (comm_monoid_add,type) comm_monoid_add
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   221
  apply (intro_classes) by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   222
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   223
instance "^" :: (ab_group_add,type) ab_group_add
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   224
  apply (intro_classes) by vector+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   225
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   226
instance "^" :: (cancel_semigroup_add,type) cancel_semigroup_add
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   227
  apply (intro_classes)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   228
  by (vector Cart_eq)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   229
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   230
instance "^" :: (cancel_ab_semigroup_add,type) cancel_ab_semigroup_add
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   231
  apply (intro_classes)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   232
  by (vector Cart_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   233
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   234
instance "^" :: (real_vector, type) real_vector
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   235
  by default (vector scaleR_left_distrib scaleR_right_distrib)+
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   236
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   237
instance "^" :: (semigroup_mult,type) semigroup_mult
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   238
  apply (intro_classes) by (vector mult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   239
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   240
instance "^" :: (monoid_mult,type) monoid_mult
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   241
  apply (intro_classes) by vector+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   242
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   243
instance "^" :: (ab_semigroup_mult,type) ab_semigroup_mult
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   244
  apply (intro_classes) by (vector mult_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   245
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   246
instance "^" :: (ab_semigroup_idem_mult,type) ab_semigroup_idem_mult
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   247
  apply (intro_classes) by (vector mult_idem)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   248
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   249
instance "^" :: (comm_monoid_mult,type) comm_monoid_mult
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   250
  apply (intro_classes) by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   251
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   252
fun vector_power :: "('a::{one,times} ^'n) \<Rightarrow> nat \<Rightarrow> 'a^'n" where
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   253
  "vector_power x 0 = 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   254
  | "vector_power x (Suc n) = x * vector_power x n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   255
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   256
instance "^" :: (semiring,type) semiring
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   257
  apply (intro_classes) by (vector ring_simps)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   258
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   259
instance "^" :: (semiring_0,type) semiring_0
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   260
  apply (intro_classes) by (vector ring_simps)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   261
instance "^" :: (semiring_1,type) semiring_1
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   262
  apply (intro_classes) by vector
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   263
instance "^" :: (comm_semiring,type) comm_semiring
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   264
  apply (intro_classes) by (vector ring_simps)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   265
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   266
instance "^" :: (comm_semiring_0,type) comm_semiring_0 by (intro_classes)
29905
26ee9656872a add instance for cancel_comm_monoid_add
huffman
parents: 29881
diff changeset
   267
instance "^" :: (cancel_comm_monoid_add, type) cancel_comm_monoid_add ..
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   268
instance "^" :: (semiring_0_cancel,type) semiring_0_cancel by (intro_classes)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   269
instance "^" :: (comm_semiring_0_cancel,type) comm_semiring_0_cancel by (intro_classes)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   270
instance "^" :: (ring,type) ring by (intro_classes)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   271
instance "^" :: (semiring_1_cancel,type) semiring_1_cancel by (intro_classes)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   272
instance "^" :: (comm_semiring_1,type) comm_semiring_1 by (intro_classes)
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   273
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   274
instance "^" :: (ring_1,type) ring_1 ..
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   275
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   276
instance "^" :: (real_algebra,type) real_algebra
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   277
  apply intro_classes
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   278
  apply (simp_all add: vector_scaleR_def ring_simps)
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   279
  apply vector
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   280
  apply vector
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   281
  done
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   282
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   283
instance "^" :: (real_algebra_1,type) real_algebra_1 ..
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   284
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   285
lemma of_nat_index:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   286
  "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   287
  apply (induct n)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   288
  apply vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   289
  apply vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   290
  done
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   291
lemma zero_index[simp]:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   292
  "(0 :: 'a::zero ^'n)$i = 0" by vector
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   293
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   294
lemma one_index[simp]:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   295
  "(1 :: 'a::one ^'n)$i = 1" by vector
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   296
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   297
lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   298
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   299
  have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   300
  also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   301
  finally show ?thesis by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   302
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   303
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   304
instance "^" :: (semiring_char_0,type) semiring_char_0
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   305
proof (intro_classes)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   306
  fix m n ::nat
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   307
  show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   308
    by (simp add: Cart_eq of_nat_index)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   309
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   310
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   311
instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
   312
instance "^" :: (ring_char_0,type) ring_char_0 by intro_classes
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   313
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   314
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   315
  by (vector mult_assoc)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   316
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   317
  by (vector ring_simps)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   318
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   319
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   320
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   321
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   322
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   323
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   324
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   325
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   326
lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   327
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   328
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   329
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   330
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   331
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   332
  by (simp add: Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   333
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   334
subsection {* Square root of sum of squares *}
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   335
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   336
definition
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   337
  "setL2 f A = sqrt (\<Sum>i\<in>A. (f i)\<twosuperior>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   338
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   339
lemma setL2_cong:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   340
  "\<lbrakk>A = B; \<And>x. x \<in> B \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   341
  unfolding setL2_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   342
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   343
lemma strong_setL2_cong:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   344
  "\<lbrakk>A = B; \<And>x. x \<in> B =simp=> f x = g x\<rbrakk> \<Longrightarrow> setL2 f A = setL2 g B"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   345
  unfolding setL2_def simp_implies_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   346
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   347
lemma setL2_infinite [simp]: "\<not> finite A \<Longrightarrow> setL2 f A = 0"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   348
  unfolding setL2_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   349
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   350
lemma setL2_empty [simp]: "setL2 f {} = 0"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   351
  unfolding setL2_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   352
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   353
lemma setL2_insert [simp]:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   354
  "\<lbrakk>finite F; a \<notin> F\<rbrakk> \<Longrightarrow>
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   355
    setL2 f (insert a F) = sqrt ((f a)\<twosuperior> + (setL2 f F)\<twosuperior>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   356
  unfolding setL2_def by (simp add: setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   357
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   358
lemma setL2_nonneg [simp]: "0 \<le> setL2 f A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   359
  unfolding setL2_def by (simp add: setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   360
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   361
lemma setL2_0': "\<forall>a\<in>A. f a = 0 \<Longrightarrow> setL2 f A = 0"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   362
  unfolding setL2_def by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   363
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   364
lemma setL2_mono:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   365
  assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<le> g i"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   366
  assumes "\<And>i. i \<in> K \<Longrightarrow> 0 \<le> f i"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   367
  shows "setL2 f K \<le> setL2 g K"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   368
  unfolding setL2_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   369
  by (simp add: setsum_nonneg setsum_mono power_mono prems)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   370
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   371
lemma setL2_right_distrib:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   372
  "0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   373
  unfolding setL2_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   374
  apply (simp add: power_mult_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   375
  apply (simp add: setsum_right_distrib [symmetric])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   376
  apply (simp add: real_sqrt_mult setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   377
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   378
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   379
lemma setL2_left_distrib:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   380
  "0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   381
  unfolding setL2_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   382
  apply (simp add: power_mult_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   383
  apply (simp add: setsum_left_distrib [symmetric])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   384
  apply (simp add: real_sqrt_mult setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   385
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   386
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   387
lemma setsum_nonneg_eq_0_iff:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   388
  fixes f :: "'a \<Rightarrow> 'b::pordered_ab_group_add"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   389
  shows "\<lbrakk>finite A; \<forall>x\<in>A. 0 \<le> f x\<rbrakk> \<Longrightarrow> setsum f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   390
  apply (induct set: finite, simp)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   391
  apply (simp add: add_nonneg_eq_0_iff setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   392
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   393
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   394
lemma setL2_eq_0_iff: "finite A \<Longrightarrow> setL2 f A = 0 \<longleftrightarrow> (\<forall>x\<in>A. f x = 0)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   395
  unfolding setL2_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   396
  by (simp add: setsum_nonneg setsum_nonneg_eq_0_iff)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   397
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   398
lemma setL2_triangle_ineq:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   399
  shows "setL2 (\<lambda>i. f i + g i) A \<le> setL2 f A + setL2 g A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   400
proof (cases "finite A")
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   401
  case False
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   402
  thus ?thesis by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   403
next
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   404
  case True
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   405
  thus ?thesis
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   406
  proof (induct set: finite)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   407
    case empty
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   408
    show ?case by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   409
  next
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   410
    case (insert x F)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   411
    hence "sqrt ((f x + g x)\<twosuperior> + (setL2 (\<lambda>i. f i + g i) F)\<twosuperior>) \<le>
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   412
           sqrt ((f x + g x)\<twosuperior> + (setL2 f F + setL2 g F)\<twosuperior>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   413
      by (intro real_sqrt_le_mono add_left_mono power_mono insert
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   414
                setL2_nonneg add_increasing zero_le_power2)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   415
    also have
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   416
      "\<dots> \<le> sqrt ((f x)\<twosuperior> + (setL2 f F)\<twosuperior>) + sqrt ((g x)\<twosuperior> + (setL2 g F)\<twosuperior>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   417
      by (rule real_sqrt_sum_squares_triangle_ineq)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   418
    finally show ?case
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   419
      using insert by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   420
  qed
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   421
qed
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   422
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   423
lemma sqrt_sum_squares_le_sum:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   424
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt (x\<twosuperior> + y\<twosuperior>) \<le> x + y"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   425
  apply (rule power2_le_imp_le)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   426
  apply (simp add: power2_sum)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   427
  apply (simp add: mult_nonneg_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   428
  apply (simp add: add_nonneg_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   429
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   430
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   431
lemma setL2_le_setsum [rule_format]:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   432
  "(\<forall>i\<in>A. 0 \<le> f i) \<longrightarrow> setL2 f A \<le> setsum f A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   433
  apply (cases "finite A")
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   434
  apply (induct set: finite)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   435
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   436
  apply clarsimp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   437
  apply (erule order_trans [OF sqrt_sum_squares_le_sum])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   438
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   439
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   440
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   441
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   442
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   443
lemma sqrt_sum_squares_le_sum_abs: "sqrt (x\<twosuperior> + y\<twosuperior>) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   444
  apply (rule power2_le_imp_le)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   445
  apply (simp add: power2_sum)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   446
  apply (simp add: mult_nonneg_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   447
  apply (simp add: add_nonneg_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   448
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   449
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   450
lemma setL2_le_setsum_abs: "setL2 f A \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   451
  apply (cases "finite A")
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   452
  apply (induct set: finite)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   453
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   454
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   455
  apply (rule order_trans [OF sqrt_sum_squares_le_sum_abs])
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   456
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   457
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   458
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   459
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   460
lemma setL2_mult_ineq_lemma:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   461
  fixes a b c d :: real
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   462
  shows "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   463
proof -
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   464
  have "0 \<le> (a * d - b * c)\<twosuperior>" by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   465
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * d) * (b * c)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   466
    by (simp only: power2_diff power_mult_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   467
  also have "\<dots> = a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior> - 2 * (a * c) * (b * d)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   468
    by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   469
  finally show "2 * (a * c) * (b * d) \<le> a\<twosuperior> * d\<twosuperior> + b\<twosuperior> * c\<twosuperior>"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   470
    by simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   471
qed
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   472
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   473
lemma setL2_mult_ineq: "(\<Sum>i\<in>A. \<bar>f i\<bar> * \<bar>g i\<bar>) \<le> setL2 f A * setL2 g A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   474
  apply (cases "finite A")
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   475
  apply (induct set: finite)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   476
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   477
  apply (rule power2_le_imp_le, simp)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   478
  apply (rule order_trans)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   479
  apply (rule power_mono)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   480
  apply (erule add_left_mono)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   481
  apply (simp add: add_nonneg_nonneg mult_nonneg_nonneg setsum_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   482
  apply (simp add: power2_sum)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   483
  apply (simp add: power_mult_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   484
  apply (simp add: right_distrib left_distrib)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   485
  apply (rule ord_le_eq_trans)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   486
  apply (rule setL2_mult_ineq_lemma)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   487
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   488
  apply (intro mult_nonneg_nonneg setL2_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   489
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   490
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   491
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   492
lemma member_le_setL2: "\<lbrakk>finite A; i \<in> A\<rbrakk> \<Longrightarrow> f i \<le> setL2 f A"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   493
  apply (rule_tac s="insert i (A - {i})" and t="A" in subst)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   494
  apply fast
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   495
  apply (subst setL2_insert)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   496
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   497
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   498
  apply simp
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   499
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   500
31344
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   501
subsection {* Metric *}
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   502
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   503
instantiation "^" :: (metric_space, finite) metric_space
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   504
begin
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   505
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   506
definition dist_vector_def:
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   507
  "dist (x::'a^'b) (y::'a^'b) = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   508
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   509
instance proof
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   510
  fix x y :: "'a ^ 'b"
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   511
  show "dist x y = 0 \<longleftrightarrow> x = y"
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   512
    unfolding dist_vector_def
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   513
    by (simp add: setL2_eq_0_iff Cart_eq)
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   514
next
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   515
  fix x y z :: "'a ^ 'b"
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   516
  show "dist x y \<le> dist x z + dist y z"
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   517
    unfolding dist_vector_def
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   518
    apply (rule order_trans [OF _ setL2_triangle_ineq])
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   519
    apply (simp add: setL2_mono dist_triangle2)
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   520
    done
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   521
qed
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   522
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   523
end
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   524
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   525
subsection {* Norms *}
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   526
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   527
instantiation "^" :: (real_normed_vector, finite) real_normed_vector
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   528
begin
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   529
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   530
definition vector_norm_def:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   531
  "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   532
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   533
definition vector_sgn_def:
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   534
  "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   535
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   536
instance proof
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   537
  fix a :: real and x y :: "'a ^ 'b"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   538
  show "0 \<le> norm x"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   539
    unfolding vector_norm_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   540
    by (rule setL2_nonneg)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   541
  show "norm x = 0 \<longleftrightarrow> x = 0"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   542
    unfolding vector_norm_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   543
    by (simp add: setL2_eq_0_iff Cart_eq)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   544
  show "norm (x + y) \<le> norm x + norm y"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   545
    unfolding vector_norm_def
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   546
    apply (rule order_trans [OF _ setL2_triangle_ineq])
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   547
    apply (simp add: setL2_mono norm_triangle_ineq)
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   548
    done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   549
  show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   550
    unfolding vector_norm_def
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   551
    by (simp add: norm_scaleR setL2_right_distrib)
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   552
  show "sgn x = scaleR (inverse (norm x)) x"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   553
    by (rule vector_sgn_def)
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   554
  show "dist x y = norm (x - y)"
31344
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   555
    unfolding dist_vector_def vector_norm_def
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   556
    by (simp add: dist_norm)
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   557
qed
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   558
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   559
end
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   560
30045
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   561
subsection {* Inner products *}
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   562
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   563
instantiation "^" :: (real_inner, finite) real_inner
30045
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   564
begin
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   565
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   566
definition vector_inner_def:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   567
  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
30045
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   568
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   569
instance proof
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   570
  fix r :: real and x y z :: "'a ^ 'b"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   571
  show "inner x y = inner y x"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   572
    unfolding vector_inner_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   573
    by (simp add: inner_commute)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   574
  show "inner (x + y) z = inner x z + inner y z"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   575
    unfolding vector_inner_def
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   576
    by (simp add: inner_left_distrib setsum_addf)
30045
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   577
  show "inner (scaleR r x) y = r * inner x y"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   578
    unfolding vector_inner_def
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   579
    by (simp add: inner_scaleR_left setsum_right_distrib)
30045
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   580
  show "0 \<le> inner x x"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   581
    unfolding vector_inner_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   582
    by (simp add: setsum_nonneg)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   583
  show "inner x x = 0 \<longleftrightarrow> x = 0"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   584
    unfolding vector_inner_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   585
    by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   586
  show "norm x = sqrt (inner x x)"
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   587
    unfolding vector_inner_def vector_norm_def setL2_def
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   588
    by (simp add: power2_norm_eq_inner)
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   589
qed
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   590
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   591
end
b8ddd7667eed real_inner class instance for vectors
huffman
parents: 30041
diff changeset
   592
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   593
subsection{* Properties of the dot product.  *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   594
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   595
lemma dot_sym: "(x::'a:: {comm_monoid_add, ab_semigroup_mult} ^ 'n) \<bullet> y = y \<bullet> x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   596
  by (vector mult_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   597
lemma dot_ladd: "((x::'a::ring ^ 'n) + y) \<bullet> z = (x \<bullet> z) + (y \<bullet> z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   598
  by (vector ring_simps)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   599
lemma dot_radd: "x \<bullet> (y + (z::'a::ring ^ 'n)) = (x \<bullet> y) + (x \<bullet> z)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   600
  by (vector ring_simps)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   601
lemma dot_lsub: "((x::'a::ring ^ 'n) - y) \<bullet> z = (x \<bullet> z) - (y \<bullet> z)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   602
  by (vector ring_simps)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   603
lemma dot_rsub: "(x::'a::ring ^ 'n) \<bullet> (y - z) = (x \<bullet> y) - (x \<bullet> z)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   604
  by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   605
lemma dot_lmult: "(c *s x) \<bullet> y = (c::'a::ring) * (x \<bullet> y)" by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   606
lemma dot_rmult: "x \<bullet> (c *s y) = (c::'a::comm_ring) * (x \<bullet> y)" by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   607
lemma dot_lneg: "(-x) \<bullet> (y::'a::ring ^ 'n) = -(x \<bullet> y)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   608
lemma dot_rneg: "(x::'a::ring ^ 'n) \<bullet> (-y) = -(x \<bullet> y)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   609
lemma dot_lzero[simp]: "0 \<bullet> x = (0::'a::{comm_monoid_add, mult_zero})" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   610
lemma dot_rzero[simp]: "x \<bullet> 0 = (0::'a::{comm_monoid_add, mult_zero})" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   611
lemma dot_pos_le[simp]: "(0::'a\<Colon>ordered_ring_strict) <= x \<bullet> x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   612
  by (simp add: dot_def setsum_nonneg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   613
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   614
lemma setsum_squares_eq_0_iff: assumes fS: "finite F" and fp: "\<forall>x \<in> F. f x \<ge> (0 ::'a::pordered_ab_group_add)" shows "setsum f F = 0 \<longleftrightarrow> (ALL x:F. f x = 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   615
using fS fp setsum_nonneg[OF fp]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   616
proof (induct set: finite)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   617
  case empty thus ?case by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   618
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   619
  case (insert x F)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   620
  from insert.prems have Fx: "f x \<ge> 0" and Fp: "\<forall> a \<in> F. f a \<ge> 0" by simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   621
  from insert.hyps Fp setsum_nonneg[OF Fp]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   622
  have h: "setsum f F = 0 \<longleftrightarrow> (\<forall>a \<in>F. f a = 0)" by metis
31034
736f521ad036 dropped duplicate lemma sum_nonneg_eq_zero_iff
haftmann
parents: 31021
diff changeset
   623
  from add_nonneg_eq_0_iff[OF Fx  setsum_nonneg[OF Fp]] insert.hyps(1,2)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   624
  show ?case by (simp add: h)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   625
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   626
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   627
lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) = 0"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   628
  by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   629
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   630
lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   631
  by (auto simp add: le_less)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   632
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   633
subsection{* The collapse of the general concepts to dimension one. *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   634
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   635
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   636
  by (simp add: Cart_eq forall_1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   637
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   638
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   639
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   640
  apply (erule_tac x= "x$1" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   641
  apply (simp only: vector_one[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   642
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   643
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   644
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   645
  by (simp add: vector_norm_def UNIV_1)
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   646
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   647
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   648
  by (simp add: norm_vector_1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   649
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   650
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   651
  by (auto simp add: norm_real dist_norm)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   652
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   653
subsection {* A connectedness or intermediate value lemma with several applications. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   654
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   655
lemma connected_real_lemma:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   656
  fixes f :: "real \<Rightarrow> real ^ 'n::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   657
  assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   658
  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"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   659
  and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   660
  and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   661
  and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   662
  shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   663
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   664
  let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   665
  have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   666
  have Sub: "\<exists>y. isUb UNIV ?S y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   667
    apply (rule exI[where x= b])
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   668
    using ab fb e12 by (auto simp add: isUb_def setle_def)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   669
  from reals_complete[OF Se Sub] obtain l where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   670
    l: "isLub UNIV ?S l"by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   671
  have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   672
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   673
    by (metis linorder_linear)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   674
  have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   675
    apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   676
    by (metis linorder_linear not_le)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   677
    have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   678
    have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   679
    have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   680
    {assume le2: "f l \<in> e2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   681
      from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   682
      hence lap: "l - a > 0" using alb by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   683
      from e2[rule_format, OF le2] obtain e where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   684
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   685
      from dst[OF alb e(1)] obtain d where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   686
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   687
      have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   688
	apply ferrack by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   689
      then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   690
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   691
      from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   692
      moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   693
      have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   694
      ultimately have False using e12 alb d' by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   695
    moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   696
    {assume le1: "f l \<in> e1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   697
    from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   698
      hence blp: "b - l > 0" using alb by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   699
      from e1[rule_format, OF le1] obtain e where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   700
	e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   701
      from dst[OF alb e(1)] obtain d where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   702
	d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   703
      have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   704
      then obtain d' where d': "d' > 0" "d' < d" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   705
      from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   706
      hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   707
      with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   708
      with l d' have False
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   709
	by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   710
    ultimately show ?thesis using alb by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   711
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   712
29881
58f3c48dbbb7 fix document generation
huffman
parents: 29844
diff changeset
   713
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 @{typ "real^1"} *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   714
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   715
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   716
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   717
  have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   718
  thus ?thesis by (simp add: ring_simps power2_eq_square)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   719
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   720
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   721
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
31340
5cddd98abe14 fix reference to LIM_def
huffman
parents: 31289
diff changeset
   722
  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)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   723
  apply (rule_tac x="s" in exI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   724
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   725
  apply (erule_tac x=y in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   726
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   727
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   728
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   729
lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   730
  using real_sqrt_le_iff[of x "y^2"] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   731
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   732
lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   733
  using real_sqrt_le_mono[of "x^2" y] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   734
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   735
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   736
  using real_sqrt_less_mono[of "x^2" y] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   737
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   738
lemma sqrt_even_pow2: assumes n: "even n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   739
  shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   740
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   741
  from n obtain m where m: "n = 2*m" unfolding even_nat_equiv_def2
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   742
    by (auto simp add: nat_number)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   743
  from m  have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   744
    by (simp only: power_mult[symmetric] mult_commute)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   745
  then show ?thesis  using m by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   746
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   747
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   748
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   749
  apply (cases "x = 0", simp_all)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   750
  using sqrt_divide_self_eq[of x]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   751
  apply (simp add: inverse_eq_divide real_sqrt_ge_0_iff field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   752
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   753
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   754
text{* Hence derive more interesting properties of the norm. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   755
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   756
text {*
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   757
  This type-specific version is only here
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   758
  to make @{text normarith.ML} happy.
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   759
*}
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   760
lemma norm_0: "norm (0::real ^ _) = 0"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   761
  by (rule norm_zero)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   762
30263
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
   763
lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   764
  by (simp add: vector_norm_def vector_component setL2_right_distrib
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   765
           abs_mult cong: strong_setL2_cong)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   766
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (x \<bullet> x = (0::real))"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   767
  by (simp add: vector_norm_def dot_def setL2_def power2_eq_square)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   768
lemma real_vector_norm_def: "norm x = sqrt (x \<bullet> x)"
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   769
  by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   770
lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   771
  by (simp add: real_vector_norm_def)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   772
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n::finite)" by (metis norm_eq_zero)
30263
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
   773
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   774
  by vector
30263
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
   775
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   776
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
30263
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
   777
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   778
  by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   779
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==>  a *s x = a *s y ==> (x = y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   780
  by (metis vector_mul_lcancel)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   781
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   782
  by (metis vector_mul_rcancel)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   783
lemma norm_cauchy_schwarz:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   784
  fixes x y :: "real ^ 'n::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   785
  shows "x \<bullet> y <= norm x * norm y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   786
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   787
  {assume "norm x = 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   788
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   789
  moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   790
  {assume "norm y = 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   791
    hence ?thesis by (simp add: dot_lzero dot_rzero)}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   792
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   793
  {assume h: "norm x \<noteq> 0" "norm y \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   794
    let ?z = "norm y *s x - norm x *s y"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   795
    from h have p: "norm x * norm y > 0" by (metis norm_ge_zero le_less zero_compare_simps)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   796
    from dot_pos_le[of ?z]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   797
    have "(norm x * norm y) * (x \<bullet> y) \<le> norm x ^2 * norm y ^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   798
      apply (simp add: dot_rsub dot_lsub dot_lmult dot_rmult ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   799
      by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   800
    hence "x\<bullet>y \<le> (norm x ^2 * norm y ^2) / (norm x * norm y)" using p
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   801
      by (simp add: field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   802
    hence ?thesis using h by (simp add: power2_eq_square)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   803
  ultimately show ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   804
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   805
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   806
lemma norm_cauchy_schwarz_abs:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   807
  fixes x y :: "real ^ 'n::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   808
  shows "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   809
  using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   810
  by (simp add: real_abs_def dot_rneg)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   811
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   812
lemma norm_triangle_sub: "norm (x::real ^'n::finite) <= norm(y) + norm(x - y)"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   813
  using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   814
lemma norm_triangle_le: "norm(x::real ^'n::finite) + norm y <= e ==> norm(x + y) <= e"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   815
  by (metis order_trans norm_triangle_ineq)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   816
lemma norm_triangle_lt: "norm(x::real ^'n::finite) + norm(y) < e ==> norm(x + y) < e"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   817
  by (metis basic_trans_rules(21) norm_triangle_ineq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   818
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   819
lemma setsum_delta:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   820
  assumes fS: "finite S"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   821
  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   822
proof-
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   823
  let ?f = "(\<lambda>k. if k=a then b k else 0)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   824
  {assume a: "a \<notin> S"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   825
    hence "\<forall> k\<in> S. ?f k = 0" by simp
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   826
    hence ?thesis  using a by simp}
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   827
  moreover
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   828
  {assume a: "a \<in> S"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   829
    let ?A = "S - {a}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   830
    let ?B = "{a}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   831
    have eq: "S = ?A \<union> ?B" using a by blast
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   832
    have dj: "?A \<inter> ?B = {}" by simp
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   833
    from fS have fAB: "finite ?A" "finite ?B" by auto
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   834
    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   835
      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   836
      by simp
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   837
    then have ?thesis  using a by simp}
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   838
  ultimately show ?thesis by blast
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   839
qed
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   840
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   841
lemma component_le_norm: "\<bar>x$i\<bar> <= norm (x::real ^ 'n::finite)"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   842
  apply (simp add: vector_norm_def)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   843
  apply (rule member_le_setL2, simp_all)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   844
  done
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   845
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   846
lemma norm_bound_component_le: "norm(x::real ^ 'n::finite) <= e
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   847
                ==> \<bar>x$i\<bar> <= e"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   848
  by (metis component_le_norm order_trans)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   849
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   850
lemma norm_bound_component_lt: "norm(x::real ^ 'n::finite) < e
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   851
                ==> \<bar>x$i\<bar> < e"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   852
  by (metis component_le_norm basic_trans_rules(21))
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   853
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   854
lemma norm_le_l1: "norm (x:: real ^'n::finite) <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   855
  by (simp add: vector_norm_def setL2_le_setsum)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   856
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   857
lemma real_abs_norm: "\<bar>norm x\<bar> = norm (x :: real ^ _)"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   858
  by (rule abs_norm_cancel)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   859
lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n::finite) - norm y\<bar> <= norm(x - y)"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   860
  by (rule norm_triangle_ineq3)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   861
lemma norm_le: "norm(x::real ^ _) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   862
  by (simp add: real_vector_norm_def)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   863
lemma norm_lt: "norm(x::real ^ _) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   864
  by (simp add: real_vector_norm_def)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   865
lemma norm_eq: "norm (x::real ^ _) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   866
  by (simp add: order_eq_iff norm_le)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   867
lemma norm_eq_1: "norm(x::real ^ _) = 1 \<longleftrightarrow> x \<bullet> x = 1"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   868
  by (simp add: real_vector_norm_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   869
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   870
text{* Squaring equations and inequalities involving norms.  *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   871
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   872
lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   873
  by (simp add: real_vector_norm_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   874
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   875
lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   876
  by (auto simp add: real_vector_norm_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   877
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   878
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   879
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   880
  have "x^2 \<le> y^2 \<longleftrightarrow> (x -y) * (y + x) \<le> 0" by (simp add: ring_simps power2_eq_square)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   881
  also have "\<dots> \<longleftrightarrow> \<bar>x\<bar> \<le> \<bar>y\<bar>" apply (simp add: zero_compare_simps real_abs_def not_less) by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   882
finally show ?thesis ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   883
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   884
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   885
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   886
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   887
  using norm_ge_zero[of x]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   888
  apply arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   889
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   890
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   891
lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
   892
  apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   893
  using norm_ge_zero[of x]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   894
  apply arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   895
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   896
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   897
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   898
  by (metis not_le norm_ge_square)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   899
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   900
  by (metis norm_le_square not_less)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   901
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   902
text{* Dot product in terms of the norm rather than conversely. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   903
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   904
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   905
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   906
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   907
lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   908
  by (simp add: norm_pow_2 dot_ladd dot_radd dot_lsub dot_rsub dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   909
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   910
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   911
text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   912
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   913
lemma vector_eq: "(x:: real ^ 'n::finite) = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y\<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   914
proof
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   915
  assume "?lhs" then show ?rhs by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   916
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   917
  assume ?rhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   918
  then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y\<bullet> y = 0" by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   919
  hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   920
    by (simp add: dot_rsub dot_lsub dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   921
  then have "(x - y) \<bullet> (x - y) = 0" by (simp add: ring_simps dot_lsub dot_rsub)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   922
  then show "x = y" by (simp add: dot_eq_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   923
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   924
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   925
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   926
subsection{* General linear decision procedure for normed spaces. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   927
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   928
lemma norm_cmul_rule_thm: "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(c *s x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   929
  apply (clarsimp simp add: norm_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   930
  apply (rule mult_mono1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   931
  apply simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   932
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   933
30263
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
   934
  (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   935
lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n::finite) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   936
  apply (rule norm_triangle_le) by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   937
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   938
lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   939
  by (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   940
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   941
lemma pth_1: "(x::real^'n) == 1 *s x" by (simp only: vector_smult_lid)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   942
lemma pth_2: "x - (y::real^'n) == x + -y" by (atomize (full)) simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   943
lemma pth_3: "(-x::real^'n) == -1 *s x" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   944
lemma pth_4: "0 *s (x::real^'n) == 0" "c *s 0 = (0::real ^ 'n)" by vector+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   945
lemma pth_5: "c *s (d *s x) == (c * d) *s (x::real ^ 'n)" by (atomize (full)) vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   946
lemma pth_6: "(c::real) *s (x + y) == c *s x + c *s y" by (atomize (full)) (vector ring_simps)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   947
lemma pth_7: "0 + x == (x::real^'n)" "x + 0 == x" by simp_all
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   948
lemma pth_8: "(c::real) *s x + d *s x == (c + d) *s x" by (atomize (full)) (vector ring_simps)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   949
lemma pth_9: "((c::real) *s x + z) + d *s x == (c + d) *s x + z"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   950
  "c *s x + (d *s x + z) == (c + d) *s x + z"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   951
  "(c *s x + w) + (d *s x + z) == (c + d) *s x + (w + z)" by ((atomize (full)), vector ring_simps)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   952
lemma pth_a: "(0::real) *s x + y == y" by (atomize (full)) vector
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   953
lemma pth_b: "(c::real) *s x + d *s y == c *s x + d *s y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   954
  "(c *s x + z) + d *s y == c *s x + (z + d *s y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   955
  "c *s x + (d *s y + z) == c *s x + (d *s y + z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   956
  "(c *s x + w) + (d *s y + z) == c *s x + (w + (d *s y + z))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   957
  by ((atomize (full)), vector)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   958
lemma pth_c: "(c::real) *s x + d *s y == d *s y + c *s x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   959
  "(c *s x + z) + d *s y == d *s y + (c *s x + z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   960
  "c *s x + (d *s y + z) == d *s y + (c *s x + z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   961
  "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   962
lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   963
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   964
lemma norm_imp_pos_and_ge: "norm (x::real ^ _) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   965
  by (atomize) (auto simp add: norm_ge_zero)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   966
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   967
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   968
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
   969
lemma norm_pths:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   970
  "(x::real ^'n::finite) = y \<longleftrightarrow> norm (x - y) \<le> 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   971
  "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
   972
  using norm_ge_zero[of "x - y"] by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   973
31344
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   974
lemma vector_dist_norm:
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   975
  fixes x y :: "real ^ _"
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   976
  shows "dist x y = norm (x - y)"
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   977
  by (rule dist_norm)
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
   978
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   979
use "normarith.ML"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   980
30549
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   981
method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   982
*} "Proves simple linear statements about vector norms"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   983
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   984
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   985
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   986
text{* Hence more metric properties. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
   987
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   988
lemma dist_triangle_alt:
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   989
  fixes x y z :: "'a::metric_space"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   990
  shows "dist y z <= dist x y + dist x z"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   991
using dist_triangle [of y z x] by (simp add: dist_commute)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   992
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   993
lemma dist_pos_lt:
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   994
  fixes x y :: "'a::metric_space"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   995
  shows "x \<noteq> y ==> 0 < dist x y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   996
by (simp add: zero_less_dist_iff)
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   997
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   998
lemma dist_nz:
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
   999
  fixes x y :: "'a::metric_space"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1000
  shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1001
by (simp add: zero_less_dist_iff)
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1002
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1003
lemma dist_triangle_le:
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1004
  fixes x y z :: "'a::metric_space"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1005
  shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1006
by (rule order_trans [OF dist_triangle2])
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1007
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1008
lemma dist_triangle_lt:
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1009
  fixes x y z :: "'a::metric_space"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1010
  shows "dist x z + dist y z < e ==> dist x y < e"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1011
by (rule le_less_trans [OF dist_triangle2])
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1012
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1013
lemma dist_triangle_half_l:
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1014
  fixes x1 x2 y :: "'a::metric_space"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1015
  shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1016
by (rule dist_triangle_lt [where z=y], simp)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1017
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1018
lemma dist_triangle_half_r:
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1019
  fixes x1 x2 y :: "'a::metric_space"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1020
  shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1021
by (rule dist_triangle_half_l, simp_all add: dist_commute)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1022
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1023
lemma dist_triangle_add:
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1024
  fixes x y x' y' :: "'a::real_normed_vector"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1025
  shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1026
unfolding dist_norm by (rule norm_diff_triangle_ineq)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1027
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1028
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1029
  unfolding dist_norm vector_ssub_ldistrib[symmetric] norm_mul ..
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1030
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1031
lemma dist_triangle_add_half:
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1032
  fixes x x' y y' :: "'a::real_normed_vector"
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1033
  shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1034
by (rule le_less_trans [OF dist_triangle_add], simp)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1035
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1036
lemma setsum_component [simp]:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1037
  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1038
  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1039
  by (cases "finite S", induct S set: finite, simp_all)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1040
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1041
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1042
  by (simp add: Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1043
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1044
lemma setsum_clauses:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1045
  shows "setsum f {} = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1046
  and "finite S \<Longrightarrow> setsum f (insert x S) =
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1047
                 (if x \<in> S then setsum f S else f x + setsum f S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1048
  by (auto simp add: insert_absorb)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1049
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1050
lemma setsum_cmul:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1051
  fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1052
  shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1053
  by (simp add: Cart_eq setsum_right_distrib)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1054
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1055
lemma setsum_norm:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1056
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1057
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1058
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1059
proof(induct rule: finite_induct[OF fS])
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1060
  case 1 thus ?case by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1061
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1062
  case (2 x S)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1063
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1064
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1065
    using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1066
  finally  show ?case  using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1067
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1068
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1069
lemma real_setsum_norm:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1070
  fixes f :: "'a \<Rightarrow> real ^'n::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1071
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1072
  shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1073
proof(induct rule: finite_induct[OF fS])
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  1074
  case 1 thus ?case by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1075
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1076
  case (2 x S)
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  1077
  from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1078
  also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1079
    using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1080
  finally  show ?case  using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1081
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1082
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1083
lemma setsum_norm_le:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1084
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1085
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1086
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1087
  shows "norm (setsum f S) \<le> setsum g S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1088
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1089
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1090
    by - (rule setsum_mono, simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1091
  then show ?thesis using setsum_norm[OF fS, of f] fg
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1092
    by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1093
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1094
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1095
lemma real_setsum_norm_le:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1096
  fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1097
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1098
  and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1099
  shows "norm (setsum f S) \<le> setsum g S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1100
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1101
  from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1102
    by - (rule setsum_mono, simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1103
  then show ?thesis using real_setsum_norm[OF fS, of f] fg
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1104
    by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1105
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1106
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1107
lemma setsum_norm_bound:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1108
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1109
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1110
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1111
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1112
  using setsum_norm_le[OF fS K] setsum_constant[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1113
  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1114
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1115
lemma real_setsum_norm_bound:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1116
  fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1117
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1118
  and K: "\<forall>x \<in> S. norm (f x) \<le> K"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1119
  shows "norm (setsum f S) \<le> of_nat (card S) * K"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1120
  using real_setsum_norm_le[OF fS K] setsum_constant[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1121
  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1122
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1123
lemma setsum_vmul:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1124
  fixes f :: "'a \<Rightarrow> 'b::{real_normed_vector,semiring, mult_zero}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1125
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1126
  shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1127
proof(induct rule: finite_induct[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1128
  case 1 then show ?case by (simp add: vector_smult_lzero)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1129
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1130
  case (2 x F)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1131
  from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1132
    by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1133
  also have "\<dots> = f x *s v + setsum f F *s v"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1134
    by (simp add: vector_sadd_rdistrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1135
  also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1136
  finally show ?case .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1137
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1138
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1139
(* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"]  ---
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1140
 Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1141
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1142
lemma setsum_add_split: assumes mn: "(m::nat) \<le> n + 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1143
  shows "setsum f {m..n + p} = setsum f {m..n} + setsum f {n + 1..n + p}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1144
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1145
  let ?A = "{m .. n}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1146
  let ?B = "{n + 1 .. n + p}"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1147
  have eq: "{m .. n+p} = ?A \<union> ?B" using mn by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1148
  have d: "?A \<inter> ?B = {}" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1149
  from setsum_Un_disjoint[of "?A" "?B" f] eq d show ?thesis by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1150
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1151
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1152
lemma setsum_natinterval_left:
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1153
  assumes mn: "(m::nat) <= n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1154
  shows "setsum f {m..n} = f m + setsum f {m + 1..n}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1155
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1156
  from mn have "{m .. n} = insert m {m+1 .. n}" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1157
  then show ?thesis by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1158
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1159
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1160
lemma setsum_natinterval_difff:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1161
  fixes f:: "nat \<Rightarrow> ('a::ab_group_add)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1162
  shows  "setsum (\<lambda>k. f k - f(k + 1)) {(m::nat) .. n} =
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1163
          (if m <= n then f m - f(n + 1) else 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1164
by (induct n, auto simp add: ring_simps not_le le_Suc_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1165
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1166
lemmas setsum_restrict_set' = setsum_restrict_set[unfolded Int_def]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1167
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1168
lemma setsum_setsum_restrict:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1169
  "finite S \<Longrightarrow> finite T \<Longrightarrow> setsum (\<lambda>x. setsum (\<lambda>y. f x y) {y. y\<in> T \<and> R x y}) S = setsum (\<lambda>y. setsum (\<lambda>x. f x y) {x. x \<in> S \<and> R x y}) T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1170
  apply (simp add: setsum_restrict_set'[unfolded mem_def] mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1171
  by (rule setsum_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1172
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1173
lemma setsum_image_gen: assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1174
  shows "setsum g S = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1175
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1176
  {fix x assume "x \<in> S" then have "{y. y\<in> f`S \<and> f x = y} = {f x}" by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1177
  note th0 = this
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1178
  have "setsum g S = setsum (\<lambda>x. setsum (\<lambda>y. g x) {y. y\<in> f`S \<and> f x = y}) S"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1179
    apply (rule setsum_cong2)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1180
    by (simp add: th0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1181
  also have "\<dots> = setsum (\<lambda>y. setsum g {x. x \<in> S \<and> f x = y}) (f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1182
    apply (rule setsum_setsum_restrict[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1183
    by (rule finite_imageI[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1184
  finally show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1185
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1186
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1187
    (* FIXME: Here too need stupid finiteness assumption on T!!! *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1188
lemma setsum_group:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1189
  assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1190
  shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1191
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1192
apply (subst setsum_image_gen[OF fS, of g f])
30263
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
  1193
apply (rule setsum_mono_zero_right[OF fT fST])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1194
by (auto intro: setsum_0')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1195
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1196
lemma vsum_norm_allsubsets_bound:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1197
  fixes f:: "'a \<Rightarrow> real ^'n::finite"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1198
  assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1199
  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1200
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1201
  let ?d = "real CARD('n)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1202
  let ?nf = "\<lambda>x. norm (f x)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1203
  let ?U = "UNIV :: 'n set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1204
  have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1205
    by (rule setsum_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1206
  have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1207
  have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1208
    apply (rule setsum_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1209
    by (rule norm_le_l1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1210
  also have "\<dots> \<le> 2 * ?d * e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1211
    unfolding th0 th1
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1212
  proof(rule setsum_bounded)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1213
    fix i assume i: "i \<in> ?U"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1214
    let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1215
    let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1216
    have thp: "P = ?Pp \<union> ?Pn" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1217
    have thp0: "?Pp \<inter> ?Pn ={}" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1218
    have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1219
    have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1220
      using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1221
      by (auto intro: abs_le_D1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1222
    have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1223
      using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1224
      by (auto simp add: setsum_negf intro: abs_le_D1)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1225
    have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1226
      apply (subst thp)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1227
      apply (rule setsum_Un_zero)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1228
      using fP thp0 by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1229
    also have "\<dots> \<le> 2*e" using Pne Ppe by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1230
    finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1231
  qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1232
  finally show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1233
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1234
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1235
lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> (y::'a::{comm_ring}^'n) = setsum (\<lambda>x. f x \<bullet> y) S "
30263
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
  1236
  by (induct rule: finite_induct, auto simp add: dot_lzero dot_ladd dot_radd)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1237
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1238
lemma dot_rsum: "finite S \<Longrightarrow> (y::'a::{comm_ring}^'n) \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1239
  by (induct rule: finite_induct, auto simp add: dot_rzero dot_radd)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1240
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1241
subsection{* Basis vectors in coordinate directions. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1242
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1243
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1244
definition "basis k = (\<chi> i. if i = k then 1 else 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1245
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1246
lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1247
  unfolding basis_def by simp
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1248
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1249
lemma delta_mult_idempotent:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1250
  "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1251
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1252
lemma norm_basis:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1253
  shows "norm (basis k :: real ^'n::finite) = 1"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1254
  apply (simp add: basis_def real_vector_norm_def dot_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1255
  apply (vector delta_mult_idempotent)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1256
  using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1257
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1258
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1259
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1260
lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1261
  by (rule norm_basis)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1262
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1263
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n::finite). norm x = c"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1264
  apply (rule exI[where x="c *s basis arbitrary"])
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1265
  by (simp only: norm_mul norm_basis)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1266
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1267
lemma vector_choose_dist: assumes e: "0 <= e"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1268
  shows "\<exists>(y::real^'n::finite). dist x y = e"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1269
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1270
  from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1271
    by blast
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  1272
  then have "dist x (x - c) = e" by (simp add: dist_norm)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1273
  then show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1274
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1275
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1276
lemma basis_inj: "inj (basis :: 'n \<Rightarrow> real ^'n::finite)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1277
  by (simp add: inj_on_def Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1278
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1279
lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1280
  by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1281
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1282
lemma basis_expansion:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1283
  "setsum (\<lambda>i. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n::finite)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1284
  by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1285
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1286
lemma basis_expansion_unique:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1287
  "setsum (\<lambda>i. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n::finite) \<longleftrightarrow> (\<forall>i. f i = x$i)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1288
  by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1289
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1290
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1291
  by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1292
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1293
lemma dot_basis:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1294
  shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n::finite) = (x$i :: 'a::semiring_1)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1295
  by (auto simp add: dot_def basis_def cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1296
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1297
lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1298
  by (auto simp add: Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1299
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1300
lemma basis_nonzero:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1301
  shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1302
  by (simp add: basis_eq_0)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1303
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1304
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1305
  apply (auto simp add: Cart_eq dot_basis)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1306
  apply (erule_tac x="basis i" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1307
  apply (simp add: dot_basis)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1308
  apply (subgoal_tac "y = z")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1309
  apply simp
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1310
  apply (simp add: Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1311
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1312
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1313
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1314
  apply (auto simp add: Cart_eq dot_basis)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1315
  apply (erule_tac x="basis i" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1316
  apply (simp add: dot_basis)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1317
  apply (subgoal_tac "x = y")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1318
  apply simp
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1319
  apply (simp add: Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1320
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1321
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1322
subsection{* Orthogonality. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1323
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1324
definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1325
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1326
lemma orthogonal_basis:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1327
  shows "orthogonal (basis i :: 'a^'n::finite) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1328
  by (auto simp add: orthogonal_def dot_def basis_def cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1329
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1330
lemma orthogonal_basis_basis:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1331
  shows "orthogonal (basis i :: 'a::ring_1^'n::finite) (basis j) \<longleftrightarrow> i \<noteq> j"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1332
  unfolding orthogonal_basis[of i] basis_component[of j] by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1333
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1334
  (* FIXME : Maybe some of these require less than comm_ring, but not all*)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1335
lemma orthogonal_clauses:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1336
  "orthogonal a (0::'a::comm_ring ^'n)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1337
  "orthogonal a x ==> orthogonal a (c *s x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1338
  "orthogonal a x ==> orthogonal a (-x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1339
  "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1340
  "orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1341
  "orthogonal 0 a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1342
  "orthogonal x a ==> orthogonal (c *s x) a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1343
  "orthogonal x a ==> orthogonal (-x) a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1344
  "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1345
  "orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1346
  unfolding orthogonal_def dot_rneg dot_rmult dot_radd dot_rsub
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1347
  dot_lzero dot_rzero dot_lneg dot_lmult dot_ladd dot_lsub
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1348
  by simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1349
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1350
lemma orthogonal_commute: "orthogonal (x::'a::{ab_semigroup_mult,comm_monoid_add} ^'n)y \<longleftrightarrow> orthogonal y x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1351
  by (simp add: orthogonal_def dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1352
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1353
subsection{* Explicit vector construction from lists. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1354
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1355
primrec from_nat :: "nat \<Rightarrow> 'a::{monoid_add,one}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1356
where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1357
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1358
lemma from_nat [simp]: "from_nat = of_nat"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1359
by (rule ext, induct_tac x, simp_all)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1360
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1361
primrec
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1362
  list_fun :: "nat \<Rightarrow> _ list \<Rightarrow> _ \<Rightarrow> _"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1363
where
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1364
  "list_fun n [] = (\<lambda>x. 0)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1365
| "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1366
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1367
definition "vector l = (\<chi> i. list_fun 1 l i)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1368
(*definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"*)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1369
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1370
lemma vector_1: "(vector[x]) $1 = x"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1371
  unfolding vector_def by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1372
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1373
lemma vector_2:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1374
 "(vector[x,y]) $1 = x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1375
 "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1376
  unfolding vector_def by simp_all
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1377
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1378
lemma vector_3:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1379
 "(vector [x,y,z] ::('a::zero)^3)$1 = x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1380
 "(vector [x,y,z] ::('a::zero)^3)$2 = y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1381
 "(vector [x,y,z] ::('a::zero)^3)$3 = z"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1382
  unfolding vector_def by simp_all
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1383
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1384
lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1385
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1386
  apply (erule_tac x="v$1" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1387
  apply (subgoal_tac "vector [v$1] = v")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1388
  apply simp
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1389
  apply (vector vector_def)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1390
  apply (simp add: forall_1)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1391
  done
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1392
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1393
lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1394
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1395
  apply (erule_tac x="v$1" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1396
  apply (erule_tac x="v$2" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1397
  apply (subgoal_tac "vector [v$1, v$2] = v")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1398
  apply simp
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1399
  apply (vector vector_def)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1400
  apply (simp add: forall_2)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1401
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1402
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1403
lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1404
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1405
  apply (erule_tac x="v$1" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1406
  apply (erule_tac x="v$2" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1407
  apply (erule_tac x="v$3" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1408
  apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1409
  apply simp
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1410
  apply (vector vector_def)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1411
  apply (simp add: forall_3)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1412
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1413
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1414
subsection{* Linear functions. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1415
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1416
definition "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *s x) = c *s f x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1417
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1418
lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1419
  by (vector linear_def Cart_eq ring_simps)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1420
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1421
lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1422
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1423
lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1424
  by (vector linear_def Cart_eq ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1425
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1426
lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1427
  by (vector linear_def Cart_eq ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1428
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1429
lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1430
  by (simp add: linear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1431
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1432
lemma linear_id: "linear id" by (simp add: linear_def id_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1433
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1434
lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1435
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1436
lemma linear_compose_setsum:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1437
  assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^ 'm)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1438
  shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1439
  using lS
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1440
  apply (induct rule: finite_induct[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1441
  by (auto simp add: linear_zero intro: linear_compose_add)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1442
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1443
lemma linear_vmul_component:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1444
  fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1445
  assumes lf: "linear f"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1446
  shows "linear (\<lambda>x. f x $ k *s v)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1447
  using lf
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1448
  apply (auto simp add: linear_def )
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1449
  by (vector ring_simps)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1450
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1451
lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1452
  unfolding linear_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1453
  apply clarsimp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1454
  apply (erule allE[where x="0::'a"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1455
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1456
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1457
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1458
lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1459
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1460
lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1461
  unfolding vector_sneg_minus1
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1462
  using linear_cmul[of f] by auto
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1463
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1464
lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1465
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1466
lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1467
  by (simp add: diff_def linear_add linear_neg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1468
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1469
lemma linear_setsum:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1470
  fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1471
  assumes lf: "linear f" and fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1472
  shows "f (setsum g S) = setsum (f o g) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1473
proof (induct rule: finite_induct[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1474
  case 1 thus ?case by (simp add: linear_0[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1475
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1476
  case (2 x F)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1477
  have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1478
    by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1479
  also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1480
  also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1481
  finally show ?case .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1482
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1483
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1484
lemma linear_setsum_mul:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1485
  fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1486
  assumes lf: "linear f" and fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1487
  shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1488
  using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1489
  linear_cmul[OF lf] by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1490
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1491
lemma linear_injective_0:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1492
  assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1493
  shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1494
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1495
  have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1496
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1497
  also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1498
    by (simp add: linear_sub[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1499
  also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1500
  finally show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1501
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1502
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1503
lemma linear_bounded:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1504
  fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1505
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1506
  shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1507
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1508
  let ?S = "UNIV:: 'm set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1509
  let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1510
  have fS: "finite ?S" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1511
  {fix x:: "real ^ 'm"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1512
    let ?g = "(\<lambda>i. (x$i) *s (basis i) :: real ^ 'm)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1513
    have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1514
      by (simp only:  basis_expansion)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1515
    also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1516
      using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1517
      by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1518
    finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1519
    {fix i assume i: "i \<in> ?S"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1520
      from component_le_norm[of x i]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1521
      have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1522
      unfolding norm_mul
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1523
      apply (simp only: mult_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1524
      apply (rule mult_mono)
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1525
      by (auto simp add: ring_simps norm_ge_zero) }
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1526
    then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1527
    from real_setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1528
    have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1529
  then show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1530
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1531
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1532
lemma linear_bounded_pos:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1533
  fixes f:: "real ^'n::finite \<Rightarrow> real ^ 'm::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1534
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1535
  shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1536
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1537
  from linear_bounded[OF lf] obtain B where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1538
    B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1539
  let ?K = "\<bar>B\<bar> + 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1540
  have Kp: "?K > 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1541
    {assume C: "B < 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1542
      have "norm (1::real ^ 'n) > 0" by (simp add: zero_less_norm_iff)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1543
      with C have "B * norm (1:: real ^ 'n) < 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1544
	by (simp add: zero_compare_simps)
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1545
      with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1546
    }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1547
    then have Bp: "B \<ge> 0" by ferrack
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1548
    {fix x::"real ^ 'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1549
      have "norm (f x) \<le> ?K *  norm x"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1550
      using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  1551
      apply (auto simp add: ring_simps split add: abs_split)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  1552
      apply (erule order_trans, simp)
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  1553
      done
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1554
  }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1555
  then show ?thesis using Kp by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1556
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1557
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1558
subsection{* Bilinear functions. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1559
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1560
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1561
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1562
lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1563
  by (simp add: bilinear_def linear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1564
lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1565
  by (simp add: bilinear_def linear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1566
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1567
lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1568
  by (simp add: bilinear_def linear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1569
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1570
lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1571
  by (simp add: bilinear_def linear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1572
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1573
lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1574
  by (simp only: vector_sneg_minus1 bilinear_lmul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1575
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1576
lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1577
  by (simp only: vector_sneg_minus1 bilinear_rmul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1578
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1579
lemma  (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1580
  using add_imp_eq[of x y 0] by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1581
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1582
lemma bilinear_lzero:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1583
  fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1584
  using bilinear_ladd[OF bh, of 0 0 x]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1585
    by (simp add: eq_add_iff ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1586
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1587
lemma bilinear_rzero:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1588
  fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1589
  using bilinear_radd[OF bh, of x 0 0 ]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1590
    by (simp add: eq_add_iff ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1591
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1592
lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ 'n)) z = h x z - h y z"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1593
  by (simp  add: diff_def bilinear_ladd bilinear_lneg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1594
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1595
lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ 'n)) = h z x - h z y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1596
  by (simp  add: diff_def bilinear_radd bilinear_rneg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1597
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1598
lemma bilinear_setsum:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1599
  fixes h:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m \<Rightarrow> 'a ^ 'k"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1600
  assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1601
  shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1602
proof-
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1603
  have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1604
    apply (rule linear_setsum[unfolded o_def])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1605
    using bh fS by (auto simp add: bilinear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1606
  also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1607
    apply (rule setsum_cong, simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1608
    apply (rule linear_setsum[unfolded o_def])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1609
    using bh fT by (auto simp add: bilinear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1610
  finally show ?thesis unfolding setsum_cartesian_product .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1611
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1612
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1613
lemma bilinear_bounded:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1614
  fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1615
  assumes bh: "bilinear h"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1616
  shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1617
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1618
  let ?M = "UNIV :: 'm set"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1619
  let ?N = "UNIV :: 'n set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1620
  let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1621
  have fM: "finite ?M" and fN: "finite ?N" by simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1622
  {fix x:: "real ^ 'm" and  y :: "real^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1623
    have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1624
    also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))"  unfolding bilinear_setsum[OF bh fM fN] ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1625
    finally have th: "norm (h x y) = \<dots>" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1626
    have "norm (h x y) \<le> ?B * norm x * norm y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1627
      apply (simp add: setsum_left_distrib th)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1628
      apply (rule real_setsum_norm_le)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1629
      using fN fM
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1630
      apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1631
      apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] norm_mul ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1632
      apply (rule mult_mono)
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1633
      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1634
      apply (rule mult_mono)
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1635
      apply (auto simp add: norm_ge_zero zero_le_mult_iff component_le_norm)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1636
      done}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1637
  then show ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1638
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1639
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1640
lemma bilinear_bounded_pos:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1641
  fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1642
  assumes bh: "bilinear h"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1643
  shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1644
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1645
  from bilinear_bounded[OF bh] obtain B where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1646
    B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1647
  let ?K = "\<bar>B\<bar> + 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1648
  have Kp: "?K > 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1649
  have KB: "B < ?K" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1650
  {fix x::"real ^'m" and y :: "real ^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1651
    from KB Kp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1652
    have "B * norm x * norm y \<le> ?K * norm x * norm y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1653
      apply -
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1654
      apply (rule mult_right_mono, rule mult_right_mono)
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  1655
      by (auto simp add: norm_ge_zero)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1656
    then have "norm (h x y) \<le> ?K * norm x * norm y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1657
      using B[rule_format, of x y] by simp}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1658
  with Kp show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1659
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1660
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1661
subsection{* Adjoints. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1662
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1663
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1664
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1665
lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1666
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1667
lemma adjoint_works_lemma:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1668
  fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1669
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1670
  shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1671
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1672
  let ?N = "UNIV :: 'n set"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1673
  let ?M = "UNIV :: 'm set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1674
  have fN: "finite ?N" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1675
  have fM: "finite ?M" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1676
  {fix y:: "'a ^ 'm"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1677
    let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: 'a ^ 'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1678
    {fix x
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1679
      have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1680
	by (simp only: basis_expansion)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1681
      also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1682
	unfolding linear_setsum[OF lf fN]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1683
	by (simp add: linear_cmul[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1684
      finally have "f x \<bullet> y = x \<bullet> ?w"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1685
	apply (simp only: )
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1686
	apply (simp add: dot_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] ring_simps)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1687
	done}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1688
  }
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1689
  then show ?thesis unfolding adjoint_def
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1690
    some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1691
    using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1692
    by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1693
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1694
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1695
lemma adjoint_works:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1696
  fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1697
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1698
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1699
  using adjoint_works_lemma[OF lf] by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1700
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1701
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1702
lemma adjoint_linear:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1703
  fixes f :: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1704
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1705
  shows "linear (adjoint f)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1706
  by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1707
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1708
lemma adjoint_clauses:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1709
  fixes f:: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1710
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1711
  shows "x \<bullet> adjoint f y = f x \<bullet> y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1712
  and "adjoint f y \<bullet> x = y \<bullet> f x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1713
  by (simp_all add: adjoint_works[OF lf] dot_sym )
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1714
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1715
lemma adjoint_adjoint:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1716
  fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1717
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1718
  shows "adjoint (adjoint f) = f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1719
  apply (rule ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1720
  by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1721
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1722
lemma adjoint_unique:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1723
  fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1724
  assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1725
  shows "f' = adjoint f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1726
  apply (rule ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1727
  using u
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1728
  by (simp add: vector_eq_rdot[symmetric] adjoint_clauses[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1729
29881
58f3c48dbbb7 fix document generation
huffman
parents: 29844
diff changeset
  1730
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1731
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1732
consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1733
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1734
defs (overloaded)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1735
matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1736
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1737
abbreviation
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1738
  matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1739
  where "m ** m' == m\<star> m'"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1740
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1741
defs (overloaded)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1742
  matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1743
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1744
abbreviation
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1745
  matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1746
  where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1747
  "m *v v == m \<star> v"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1748
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1749
defs (overloaded)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1750
  vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) (UNIV :: 'm set)) :: 'a^'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1751
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1752
abbreviation
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1753
  vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1754
  where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1755
  "v v* m == v \<star> m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1756
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1757
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1758
definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1759
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1760
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1761
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1762
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1763
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1764
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1765
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1766
  by (vector matrix_matrix_mult_def setsum_addf[symmetric] ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1767
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1768
lemma setsum_delta':
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1769
  assumes fS: "finite S" shows
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1770
  "setsum (\<lambda>k. if a = k then b k else 0) S =
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1771
     (if a\<in> S then b a else 0)"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1772
  using setsum_delta[OF fS, of a b, symmetric]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1773
  by (auto intro: setsum_cong)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1774
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1775
lemma matrix_mul_lid:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1776
  fixes A :: "'a::semiring_1 ^ 'm ^ 'n::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1777
  shows "mat 1 ** A = A"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1778
  apply (simp add: matrix_matrix_mult_def mat_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1779
  apply vector
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1780
  by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite]  mult_1_left mult_zero_left if_True UNIV_I)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1781
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1782
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1783
lemma matrix_mul_rid:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1784
  fixes A :: "'a::semiring_1 ^ 'm::finite ^ 'n"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1785
  shows "A ** mat 1 = A"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1786
  apply (simp add: matrix_matrix_mult_def mat_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1787
  apply vector
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1788
  by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite]  mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1789
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1790
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1791
  apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1792
  apply (subst setsum_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1793
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1794
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1795
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1796
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1797
  apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1798
  apply (subst setsum_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1799
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1800
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1801
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1802
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1803
  apply (vector matrix_vector_mult_def mat_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1804
  by (simp add: cond_value_iff cond_application_beta
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1805
    setsum_delta' cong del: if_weak_cong)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1806
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1807
lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1808
  by (simp add: matrix_matrix_mult_def transp_def Cart_eq mult_commute)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1809
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1810
lemma matrix_eq:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1811
  fixes A B :: "'a::semiring_1 ^ 'n::finite ^ 'm"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1812
  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1813
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1814
  apply (subst Cart_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1815
  apply clarify
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1816
  apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq cong del: if_weak_cong)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1817
  apply (erule_tac x="basis ia" in allE)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1818
  apply (erule_tac x="i" in allE)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1819
  by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1820
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1821
lemma matrix_vector_mul_component:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1822
  shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1823
  by (simp add: matrix_vector_mult_def dot_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1824
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1825
lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1826
  apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1827
  apply (subst setsum_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1828
  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1829
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1830
lemma transp_mat: "transp (mat n) = mat n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1831
  by (vector transp_def mat_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1832
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1833
lemma transp_transp: "transp(transp A) = A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1834
  by (vector transp_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1835
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1836
lemma row_transp:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1837
  fixes A:: "'a::semiring_1^'n^'m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1838
  shows "row i (transp A) = column i A"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1839
  by (simp add: row_def column_def transp_def Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1840
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1841
lemma column_transp:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1842
  fixes A:: "'a::semiring_1^'n^'m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1843
  shows "column i (transp A) = row i A"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1844
  by (simp add: row_def column_def transp_def Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1845
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1846
lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1847
by (auto simp add: rows_def columns_def row_transp intro: set_ext)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1848
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1849
lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1850
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1851
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1852
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1853
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1854
  by (simp add: matrix_vector_mult_def dot_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1855
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1856
lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1857
  by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1858
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1859
lemma vector_componentwise:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1860
  "(x::'a::ring_1^'n::finite) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1861
  apply (subst basis_expansion[symmetric])
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1862
  by (vector Cart_eq setsum_component)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1863
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1864
lemma linear_componentwise:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1865
  fixes f:: "'a::ring_1 ^ 'm::finite \<Rightarrow> 'a ^ 'n"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1866
  assumes lf: "linear f"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1867
  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1868
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1869
  let ?M = "(UNIV :: 'm set)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1870
  let ?N = "(UNIV :: 'n set)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1871
  have fM: "finite ?M" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1872
  have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1873
    unfolding vector_smult_component[symmetric]
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1874
    unfolding setsum_component[of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1875
    ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1876
  then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1877
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1878
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1879
text{* Inverse matrices  (not necessarily square) *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1880
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1881
definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1882
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1883
definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1884
        (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1885
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1886
text{* Correspondence between matrices and linear operators. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1887
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1888
definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1889
where "matrix f = (\<chi> i j. (f(basis j))$i)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1890
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1891
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1892
  by (simp add: linear_def matrix_vector_mult_def Cart_eq ring_simps setsum_right_distrib setsum_addf)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1893
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1894
lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n::finite)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1895
apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1896
apply clarify
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1897
apply (rule linear_componentwise[OF lf, symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1898
done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1899
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1900
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n::finite))" by (simp add: ext matrix_works)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1901
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1902
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n::finite)) = A"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1903
  by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1904
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1905
lemma matrix_compose:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1906
  assumes lf: "linear (f::'a::comm_ring_1^'n::finite \<Rightarrow> 'a^'m::finite)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1907
  and lg: "linear (g::'a::comm_ring_1^'m::finite \<Rightarrow> 'a^'k)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1908
  shows "matrix (g o f) = matrix g ** matrix f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1909
  using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1910
  by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1911
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1912
lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s ((transp A)$i)) (UNIV:: 'n set)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1913
  by (simp add: matrix_vector_mult_def transp_def Cart_eq mult_commute)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1914
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1915
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n::finite^'m::finite) *v x) = (\<lambda>x. transp A *v x)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1916
  apply (rule adjoint_unique[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1917
  apply (rule matrix_vector_mul_linear)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1918
  apply (simp add: transp_def dot_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1919
  apply (subst setsum_commute)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1920
  apply (auto simp add: mult_ac)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1921
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1922
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1923
lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n::finite \<Rightarrow> 'a ^ 'm::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1924
  shows "matrix(adjoint f) = transp(matrix f)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1925
  apply (subst matrix_vector_mul[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1926
  unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1927
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1928
subsection{* Interlude: Some properties of real sets *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1929
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1930
lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1931
  shows "\<forall>n \<ge> m. d n < e m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1932
  using prems apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1933
  apply (erule_tac x="n" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1934
  apply (erule_tac x="n" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1935
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1936
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1937
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1938
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1939
lemma real_convex_bound_lt:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1940
  assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1941
  and uv: "u + v = 1"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1942
  shows "u * x + v * y < a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1943
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1944
  have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1945
  have "a = a * (u + v)" unfolding uv  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1946
  hence th: "u * a + v * a = a" by (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1947
  from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_compare_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1948
  from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_compare_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1949
  from xa ya u v have "u * x + v * y < u * a + v * a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1950
    apply (cases "u = 0", simp_all add: uv')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1951
    apply(rule mult_strict_left_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1952
    using uv' apply simp_all
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1953
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1954
    apply (rule add_less_le_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1955
    apply(rule mult_strict_left_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1956
    apply simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1957
    apply (rule mult_left_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1958
    apply simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1959
    done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1960
  thus ?thesis unfolding th .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1961
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1962
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1963
lemma real_convex_bound_le:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1964
  assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1965
  and uv: "u + v = 1"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1966
  shows "u * x + v * y \<le> a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1967
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1968
  from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1969
  also have "\<dots> \<le> (u + v) * a" by (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1970
  finally show ?thesis unfolding uv by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1971
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1972
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1973
lemma infinite_enumerate: assumes fS: "infinite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1974
  shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1975
unfolding subseq_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1976
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1977
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1978
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)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1979
apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1980
apply (rule_tac x="d/2" in exI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1981
apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1982
done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1983
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1984
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  1985
lemma triangle_lemma:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1986
  assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1987
  shows "x <= y + z"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1988
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1989
  have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y  by (simp add: zero_compare_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1990
  with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1991
  from y z have yz: "y + z \<ge> 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1992
  from power2_le_imp_le[OF th yz] show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1993
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1994
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1995
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1996
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1997
   (\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  1998
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1999
  let ?S = "(UNIV :: 'n set)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2000
  {assume H: "?rhs"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2001
    then have ?lhs by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2002
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2003
  {assume H: "?lhs"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2004
    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2005
    let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2006
    {fix i
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2007
      from f have "P i (f i)" by metis
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2008
      then have "P i (?x$i)" by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2009
    }
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2010
    hence "\<forall>i. P i (?x$i)" by metis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2011
    hence ?rhs by metis }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2012
  ultimately show ?thesis by metis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2013
qed
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2014
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2015
(* Supremum and infimum of real sets *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2016
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2017
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2018
definition rsup:: "real set \<Rightarrow> real" where
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2019
  "rsup S = (SOME a. isLub UNIV S a)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2020
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2021
lemma rsup_alt: "rsup S = (SOME a. (\<forall>x \<in> S. x \<le> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<le> b) \<longrightarrow> a \<le> b))"  by (auto simp  add: isLub_def rsup_def leastP_def isUb_def setle_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2022
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2023
lemma rsup: assumes Se: "S \<noteq> {}" and b: "\<exists>b. S *<= b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2024
  shows "isLub UNIV S (rsup S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2025
using Se b
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2026
unfolding rsup_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2027
apply clarify
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2028
apply (rule someI_ex)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2029
apply (rule reals_complete)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2030
by (auto simp add: isUb_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2031
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2032
lemma rsup_le: assumes Se: "S \<noteq> {}" and Sb: "S *<= b" shows "rsup S \<le> b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2033
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2034
  from Sb have bu: "isUb UNIV S b" by (simp add: isUb_def setle_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2035
  from rsup[OF Se] Sb have "isLub UNIV S (rsup S)"  by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2036
  then show ?thesis using bu by (auto simp add: isLub_def leastP_def setle_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2037
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2038
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2039
lemma rsup_finite_Max: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2040
  shows "rsup S = Max S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2041
using fS Se
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2042
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2043
  let ?m = "Max S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2044
  from Max_ge[OF fS] have Sm: "\<forall> x\<in> S. x \<le> ?m" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2045
  with rsup[OF Se] have lub: "isLub UNIV S (rsup S)" by (metis setle_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2046
  from Max_in[OF fS Se] lub have mrS: "?m \<le> rsup S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2047
    by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2048
  moreover
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2049
  have "rsup S \<le> ?m" using Sm lub
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2050
    by (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2051
  ultimately  show ?thesis by arith
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2052
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2053
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2054
lemma rsup_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2055
  shows "rsup S \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2056
  using rsup_finite_Max[OF fS Se] Max_in[OF fS Se] by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2057
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2058
lemma rsup_finite_Ub: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2059
  shows "isUb S S (rsup S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2060
  using rsup_finite_Max[OF fS Se] rsup_finite_in[OF fS Se] Max_ge[OF fS]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2061
  unfolding isUb_def setle_def by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2062
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2063
lemma rsup_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2064
  shows "a \<le> rsup S \<longleftrightarrow> (\<exists> x \<in> S. a \<le> x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2065
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2066
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2067
lemma rsup_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2068
  shows "a \<ge> rsup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2069
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2070
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2071
lemma rsup_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2072
  shows "a < rsup S \<longleftrightarrow> (\<exists> x \<in> S. a < x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2073
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2074
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2075
lemma rsup_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2076
  shows "a > rsup S \<longleftrightarrow> (\<forall> x \<in> S. a > x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2077
using rsup_finite_Ub[OF fS Se] by (auto simp add: isUb_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2078
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2079
lemma rsup_unique: assumes b: "S *<= b" and S: "\<forall>b' < b. \<exists>x \<in> S. b' < x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2080
  shows "rsup S = b"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2081
using b S
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2082
unfolding setle_def rsup_alt
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2083
apply -
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2084
apply (rule some_equality)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2085
apply (metis  linorder_not_le order_eq_iff[symmetric])+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2086
done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2087
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2088
lemma rsup_le_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. T *<= b) \<Longrightarrow> rsup S \<le> rsup T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2089
  apply (rule rsup_le)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2090
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2091
  using rsup[of T] by (auto simp add: isLub_def leastP_def setge_def setle_def isUb_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2092
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2093
lemma isUb_def': "isUb R S = (\<lambda>x. S *<= x \<and> x \<in> R)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2094
  apply (rule ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2095
  by (metis isUb_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2096
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2097
lemma UNIV_trivial: "UNIV x" using UNIV_I[of x] by (metis mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2098
lemma rsup_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2099
  shows "a \<le> rsup S \<and> rsup S \<le> b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2100
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2101
  from rsup[OF Se] u have lub: "isLub UNIV S (rsup S)" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2102
  hence b: "rsup S \<le> b" using u by (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2103
  from Se obtain y where y: "y \<in> S" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2104
  from lub l have "a \<le> rsup S" apply (auto simp add: isLub_def leastP_def setle_def setge_def isUb_def')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2105
    apply (erule ballE[where x=y])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2106
    apply (erule ballE[where x=y])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2107
    apply arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2108
    using y apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2109
    done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2110
  with b show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2111
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2112
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2113
lemma rsup_abs_le: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rsup S\<bar> \<le> a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2114
  unfolding abs_le_interval_iff  using rsup_bounds[of S "-a" a]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2115
  by (auto simp add: setge_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2116
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2117
lemma rsup_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rsup S - l\<bar> \<le> e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2118
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2119
  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2120
  show ?thesis using S b rsup_bounds[of S "l - e" "l+e"] unfolding th
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2121
    by  (auto simp add: setge_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2122
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2123
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2124
definition rinf:: "real set \<Rightarrow> real" where
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2125
  "rinf S = (SOME a. isGlb UNIV S a)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2126
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2127
lemma rinf_alt: "rinf S = (SOME a. (\<forall>x \<in> S. x \<ge> a) \<and> (\<forall>b. (\<forall>x \<in> S. x \<ge> b) \<longrightarrow> a \<ge> b))"  by (auto simp  add: isGlb_def rinf_def greatestP_def isLb_def setle_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2128
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2129
lemma reals_complete_Glb: assumes Se: "\<exists>x. x \<in> S" and lb: "\<exists> y. isLb UNIV S y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2130
  shows "\<exists>(t::real). isGlb UNIV S t"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2131
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2132
  let ?M = "uminus ` S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2133
  from lb have th: "\<exists>y. isUb UNIV ?M y" apply (auto simp add: isUb_def isLb_def setle_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2134
    by (rule_tac x="-y" in exI, auto)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2135
  from Se have Me: "\<exists>x. x \<in> ?M" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2136
  from reals_complete[OF Me th] obtain t where t: "isLub UNIV ?M t" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2137
  have "isGlb UNIV S (- t)" using t
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2138
    apply (auto simp add: isLub_def isGlb_def leastP_def greatestP_def setle_def setge_def isUb_def isLb_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2139
    apply (erule_tac x="-y" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2140
    apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2141
    done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2142
  then show ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2143
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2144
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2145
lemma rinf: assumes Se: "S \<noteq> {}" and b: "\<exists>b. b <=* S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2146
  shows "isGlb UNIV S (rinf S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2147
using Se b
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2148
unfolding rinf_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2149
apply clarify
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2150
apply (rule someI_ex)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2151
apply (rule reals_complete_Glb)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2152
apply (auto simp add: isLb_def setle_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2153
done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2154
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2155
lemma rinf_ge: assumes Se: "S \<noteq> {}" and Sb: "b <=* S" shows "rinf S \<ge> b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2156
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2157
  from Sb have bu: "isLb UNIV S b" by (simp add: isLb_def setge_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2158
  from rinf[OF Se] Sb have "isGlb UNIV S (rinf S)"  by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2159
  then show ?thesis using bu by (auto simp add: isGlb_def greatestP_def setle_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2160
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2161
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2162
lemma rinf_finite_Min: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2163
  shows "rinf S = Min S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2164
using fS Se
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2165
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2166
  let ?m = "Min S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2167
  from Min_le[OF fS] have Sm: "\<forall> x\<in> S. x \<ge> ?m" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2168
  with rinf[OF Se] have glb: "isGlb UNIV S (rinf S)" by (metis setge_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2169
  from Min_in[OF fS Se] glb have mrS: "?m \<ge> rinf S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2170
    by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2171
  moreover
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2172
  have "rinf S \<ge> ?m" using Sm glb
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2173
    by (auto simp add: isGlb_def greatestP_def isLb_def setle_def setge_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2174
  ultimately  show ?thesis by arith
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2175
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2176
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2177
lemma rinf_finite_in: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2178
  shows "rinf S \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2179
  using rinf_finite_Min[OF fS Se] Min_in[OF fS Se] by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2180
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2181
lemma rinf_finite_Lb: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2182
  shows "isLb S S (rinf S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2183
  using rinf_finite_Min[OF fS Se] rinf_finite_in[OF fS Se] Min_le[OF fS]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2184
  unfolding isLb_def setge_def by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2185
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2186
lemma rinf_finite_ge_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2187
  shows "a \<le> rinf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2188
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2189
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2190
lemma rinf_finite_le_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2191
  shows "a \<ge> rinf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2192
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2193
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2194
lemma rinf_finite_gt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2195
  shows "a < rinf S \<longleftrightarrow> (\<forall> x \<in> S. a < x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2196
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2197
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2198
lemma rinf_finite_lt_iff: assumes fS: "finite S" and Se: "S \<noteq> {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2199
  shows "a > rinf S \<longleftrightarrow> (\<exists> x \<in> S. a > x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2200
using rinf_finite_Lb[OF fS Se] by (auto simp add: isLb_def setge_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2201
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2202
lemma rinf_unique: assumes b: "b <=* S" and S: "\<forall>b' > b. \<exists>x \<in> S. b' > x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2203
  shows "rinf S = b"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2204
using b S
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2205
unfolding setge_def rinf_alt
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2206
apply -
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2207
apply (rule some_equality)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2208
apply (metis  linorder_not_le order_eq_iff[symmetric])+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2209
done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2210
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2211
lemma rinf_ge_subset: "S\<noteq>{} \<Longrightarrow> S \<subseteq> T \<Longrightarrow> (\<exists>b. b <=* T) \<Longrightarrow> rinf S >= rinf T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2212
  apply (rule rinf_ge)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2213
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2214
  using rinf[of T] by (auto simp add: isGlb_def greatestP_def setge_def setle_def isLb_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2215
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2216
lemma isLb_def': "isLb R S = (\<lambda>x. x <=* S \<and> x \<in> R)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2217
  apply (rule ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2218
  by (metis isLb_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2219
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2220
lemma rinf_bounds: assumes Se: "S \<noteq> {}" and l: "a <=* S" and u: "S *<= b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2221
  shows "a \<le> rinf S \<and> rinf S \<le> b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2222
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2223
  from rinf[OF Se] l have lub: "isGlb UNIV S (rinf S)" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2224
  hence b: "a \<le> rinf S" using l by (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2225
  from Se obtain y where y: "y \<in> S" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2226
  from lub u have "b \<ge> rinf S" apply (auto simp add: isGlb_def greatestP_def setle_def setge_def isLb_def')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2227
    apply (erule ballE[where x=y])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2228
    apply (erule ballE[where x=y])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2229
    apply arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2230
    using y apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2231
    done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2232
  with b show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2233
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2234
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2235
lemma rinf_abs_ge: "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>rinf S\<bar> \<le> a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2236
  unfolding abs_le_interval_iff  using rinf_bounds[of S "-a" a]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2237
  by (auto simp add: setge_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2238
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2239
lemma rinf_asclose: assumes S:"S \<noteq> {}" and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e" shows "\<bar>rinf S - l\<bar> \<le> e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2240
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2241
  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e" by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2242
  show ?thesis using S b rinf_bounds[of S "l - e" "l+e"] unfolding th
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2243
    by  (auto simp add: setge_def setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2244
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2245
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2246
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2247
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2248
subsection{* Operator norm. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2249
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2250
definition "onorm f = rsup {norm (f x)| x. norm x = 1}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2251
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2252
lemma norm_bound_generalize:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2253
  fixes f:: "real ^'n::finite \<Rightarrow> real^'m::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2254
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2255
  shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2256
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2257
  {assume H: ?rhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2258
    {fix x :: "real^'n" assume x: "norm x = 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2259
      from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2260
    then have ?lhs by blast }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2261
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2262
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2263
  {assume H: ?lhs
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2264
    from H[rule_format, of "basis arbitrary"]
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2265
    have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"]
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  2266
      by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2267
    {fix x :: "real ^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2268
      {assume "x = 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  2269
	then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2270
      moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2271
      {assume x0: "x \<noteq> 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  2272
	hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2273
	let ?c = "1/ norm x"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  2274
	have "norm (?c*s x) = 1" using x0 by (simp add: n0 norm_mul)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2275
	with H have "norm (f(?c*s x)) \<le> b" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2276
	hence "?c * norm (f x) \<le> b"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2277
	  by (simp add: linear_cmul[OF lf] norm_mul)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2278
	hence "norm (f x) \<le> b * norm x"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  2279
	  using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2280
      ultimately have "norm (f x) \<le> b * norm x" by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2281
    then have ?rhs by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2282
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2283
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2284
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2285
lemma onorm:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2286
  fixes f:: "real ^'n::finite \<Rightarrow> real ^'m::finite"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2287
  assumes lf: "linear f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2288
  shows "norm (f x) <= onorm f * norm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2289
  and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2290
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2291
  {
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2292
    let ?S = "{norm (f x) |x. norm x = 1}"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2293
    have Se: "?S \<noteq> {}" using  norm_basis by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2294
    from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2295
      unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2296
    {from rsup[OF Se b, unfolded onorm_def[symmetric]]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2297
      show "norm (f x) <= onorm f * norm x"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2298
	apply -
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2299
	apply (rule spec[where x = x])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2300
	unfolding norm_bound_generalize[OF lf, symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2301
	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2302
    {
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2303
      show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2304
	using rsup[OF Se b, unfolded onorm_def[symmetric]]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2305
	unfolding norm_bound_generalize[OF lf, symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2306
	by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2307
  }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2308
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2309
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2310
lemma onorm_pos_le: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" shows "0 <= onorm f"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2311
  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2312
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2313
lemma onorm_eq_0: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2314
  shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2315
  using onorm[OF lf]
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  2316
  apply (auto simp add: onorm_pos_le)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2317
  apply atomize
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2318
  apply (erule allE[where x="0::real"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2319
  using onorm_pos_le[OF lf]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2320
  apply arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2321
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2322
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2323
lemma onorm_const: "onorm(\<lambda>x::real^'n::finite. (y::real ^ 'm::finite)) = norm y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2324
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2325
  let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2326
  have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2327
    by(auto intro: vector_choose_size set_ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2328
  show ?thesis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2329
    unfolding onorm_def th
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2330
    apply (rule rsup_unique) by (simp_all  add: setle_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2331
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2332
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2333
lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n::finite \<Rightarrow> real ^'m::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2334
  shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2335
  unfolding onorm_eq_0[OF lf, symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2336
  using onorm_pos_le[OF lf] by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2337
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2338
lemma onorm_compose:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2339
  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2340
  and lg: "linear (g::real^'k::finite \<Rightarrow> real^'n::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2341
  shows "onorm (f o g) <= onorm f * onorm g"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2342
  apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2343
  unfolding o_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2344
  apply (subst mult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2345
  apply (rule order_trans)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2346
  apply (rule onorm(1)[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2347
  apply (rule mult_mono1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2348
  apply (rule onorm(1)[OF lg])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2349
  apply (rule onorm_pos_le[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2350
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2351
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2352
lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2353
  shows "onorm (\<lambda>x. - f x) \<le> onorm f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2354
  using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  2355
  unfolding norm_minus_cancel by metis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2356
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2357
lemma onorm_neg: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2358
  shows "onorm (\<lambda>x. - f x) = onorm f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2359
  using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2360
  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2361
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2362
lemma onorm_triangle:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2363
  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and lg: "linear g"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2364
  shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2365
  apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2366
  apply (rule order_trans)
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  2367
  apply (rule norm_triangle_ineq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2368
  apply (simp add: distrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2369
  apply (rule add_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2370
  apply (rule onorm(1)[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2371
  apply (rule onorm(1)[OF lg])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2372
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2373
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2374
lemma onorm_triangle_le: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2375
  \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2376
  apply (rule order_trans)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2377
  apply (rule onorm_triangle)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2378
  apply assumption+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2379
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2380
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2381
lemma onorm_triangle_lt: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2382
  ==> onorm(\<lambda>x. f x + g x) < e"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2383
  apply (rule order_le_less_trans)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2384
  apply (rule onorm_triangle)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2385
  by assumption+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2386
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2387
(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2388
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2389
definition vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x = (\<chi> i. x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2390
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2391
definition dest_vec1:: "'a ^1 \<Rightarrow> 'a"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2392
  where "dest_vec1 x = (x$1)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2393
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2394
lemma vec1_component[simp]: "(vec1 x)$1 = x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2395
  by (simp add: vec1_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2396
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2397
lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2398
  by (simp_all add: vec1_def dest_vec1_def Cart_eq forall_1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2399
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2400
lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2401
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2402
lemma exists_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(vec1 x))" by (metis vec1_dest_vec1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2403
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2404
lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"  by (metis vec1_dest_vec1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2405
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2406
lemma exists_dest_vec1: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P(dest_vec1 x))"by (metis vec1_dest_vec1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2407
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2408
lemma vec1_eq[simp]:  "vec1 x = vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2409
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2410
lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y" by (metis vec1_dest_vec1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2411
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2412
lemma vec1_in_image_vec1: "vec1 x \<in> (vec1 ` S) \<longleftrightarrow> x \<in> S" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2413
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2414
lemma vec1_vec: "vec1 x = vec x" by (vector vec1_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2415
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2416
lemma vec1_add: "vec1(x + y) = vec1 x + vec1 y" by (vector vec1_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2417
lemma vec1_sub: "vec1(x - y) = vec1 x - vec1 y" by (vector vec1_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2418
lemma vec1_cmul: "vec1(c* x) = c *s vec1 x " by (vector vec1_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2419
lemma vec1_neg: "vec1(- x) = - vec1 x " by (vector vec1_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2420
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2421
lemma vec1_setsum: assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2422
  shows "vec1(setsum f S) = setsum (vec1 o f) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2423
  apply (induct rule: finite_induct[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2424
  apply (simp add: vec1_vec)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2425
  apply (auto simp add: vec1_add)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2426
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2427
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2428
lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2429
  by (simp add: dest_vec1_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2430
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2431
lemma dest_vec1_vec: "dest_vec1(vec x) = x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2432
  by (simp add: vec1_vec[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2433
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2434
lemma dest_vec1_add: "dest_vec1(x + y) = dest_vec1 x + dest_vec1 y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2435
 by (metis vec1_dest_vec1 vec1_add)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2436
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2437
lemma dest_vec1_sub: "dest_vec1(x - y) = dest_vec1 x - dest_vec1 y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2438
 by (metis vec1_dest_vec1 vec1_sub)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2439
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2440
lemma dest_vec1_cmul: "dest_vec1(c*sx) = c * dest_vec1 x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2441
 by (metis vec1_dest_vec1 vec1_cmul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2442
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2443
lemma dest_vec1_neg: "dest_vec1(- x) = - dest_vec1 x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2444
 by (metis vec1_dest_vec1 vec1_neg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2445
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2446
lemma dest_vec1_0[simp]: "dest_vec1 0 = 0" by (metis vec_0 dest_vec1_vec)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2447
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2448
lemma dest_vec1_sum: assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2449
  shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2450
  apply (induct rule: finite_induct[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2451
  apply (simp add: dest_vec1_vec)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2452
  apply (auto simp add: dest_vec1_add)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2453
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2454
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2455
lemma norm_vec1: "norm(vec1 x) = abs(x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2456
  by (simp add: vec1_def norm_real)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2457
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2458
lemma dist_vec1: "dist(vec1 x) (vec1 y) = abs(x - y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2459
  by (simp only: dist_real vec1_component)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2460
lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2461
  by (metis vec1_dest_vec1 norm_vec1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2462
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2463
lemma linear_vmul_dest_vec1:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2464
  fixes f:: "'a::semiring_1^'n \<Rightarrow> 'a^1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2465
  shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2466
  unfolding dest_vec1_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2467
  apply (rule linear_vmul_component)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2468
  by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2469
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2470
lemma linear_from_scalars:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2471
  assumes lf: "linear (f::'a::comm_ring_1 ^1 \<Rightarrow> 'a^'n)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2472
  shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2473
  apply (rule ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2474
  apply (subst matrix_works[OF lf, symmetric])
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2475
  apply (auto simp add: Cart_eq matrix_vector_mult_def dest_vec1_def column_def  mult_commute UNIV_1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2476
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2477
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2478
lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a^1)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2479
  shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2480
  apply (rule ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2481
  apply (subst matrix_works[OF lf, symmetric])
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2482
  apply (simp add: Cart_eq matrix_vector_mult_def vec1_def row_def dot_def mult_commute forall_1)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2483
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2484
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2485
lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2486
  by (simp add: dest_vec1_eq[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2487
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2488
lemma setsum_scalars: assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2489
  shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2490
  unfolding vec1_setsum[OF fS] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2491
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2492
lemma dest_vec1_wlog_le: "(\<And>(x::'a::linorder ^ 1) y. P x y \<longleftrightarrow> P y x)  \<Longrightarrow> (\<And>x y. dest_vec1 x <= dest_vec1 y ==> P x y) \<Longrightarrow> P x y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2493
  apply (cases "dest_vec1 x \<le> dest_vec1 y")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2494
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2495
  apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2496
  apply (auto)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2497
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2498
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2499
text{* Pasting vectors. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2500
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2501
lemma linear_fstcart: "linear fstcart"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2502
  by (auto simp add: linear_def Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2503
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2504
lemma linear_sndcart: "linear sndcart"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2505
  by (auto simp add: linear_def Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2506
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2507
lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2508
  by (simp add: Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2509
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2510
lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b + 'c)) + fstcart y"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2511
  by (simp add: Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2512
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2513
lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b + 'c))"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2514
  by (simp add: Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2515
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2516
lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b + 'c))"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2517
  by (simp add: Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2518
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2519
lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b + 'c)) - fstcart y"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2520
  by (simp add: Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2521
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2522
lemma fstcart_setsum:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2523
  fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2524
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2525
  shows "fstcart (setsum f S) = setsum (\<lambda>i. fstcart (f i)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2526
  by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2527
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2528
lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2529
  by (simp add: Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2530
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2531
lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b + 'c)) + sndcart y"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2532
  by (simp add: Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2533
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2534
lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b + 'c))"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2535
  by (simp add: Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2536
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2537
lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b + 'c))"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2538
  by (simp add: Cart_eq)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2539
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2540
lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b + 'c)) - sndcart y"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2541
  by (simp add: Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2542
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2543
lemma sndcart_setsum:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2544
  fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2545
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2546
  shows "sndcart (setsum f S) = setsum (\<lambda>i. sndcart (f i)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2547
  by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2548
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2549
lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2550
  by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2551
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2552
lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2553
  by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2554
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2555
lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2556
  by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2557
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2558
lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2559
  unfolding vector_sneg_minus1 pastecart_cmul ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2560
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2561
lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2562
  by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2563
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2564
lemma pastecart_setsum:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2565
  fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2566
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2567
  shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2568
  by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2569
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2570
lemma setsum_Plus:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2571
  "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2572
    (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2573
  unfolding Plus_def
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2574
  by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2575
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2576
lemma setsum_UNIV_sum:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2577
  fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2578
  shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2579
  apply (subst UNIV_Plus_UNIV [symmetric])
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2580
  apply (rule setsum_Plus [OF finite finite])
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2581
  done
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2583
lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2584
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2585
  have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2586
    by (simp add: pastecart_fst_snd)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2587
  have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2588
    by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2589
  then show ?thesis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2590
    unfolding th0
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  2591
    unfolding real_vector_norm_def real_sqrt_le_iff id_def
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2592
    by (simp add: dot_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2593
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2594
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2595
lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
31344
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
  2596
  unfolding dist_norm by (metis fstcart_sub[symmetric] norm_fstcart)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2597
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2598
lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2599
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2600
  have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2601
    by (simp add: pastecart_fst_snd)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2602
  have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2603
    by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2604
  then show ?thesis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2605
    unfolding th0
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  2606
    unfolding real_vector_norm_def real_sqrt_le_iff id_def
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2607
    by (simp add: dot_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2608
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2609
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2610
lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
31344
fc09ec06b89b instance ^ :: (metric_space, finite) metric_space
huffman
parents: 31340
diff changeset
  2611
  unfolding dist_norm by (metis sndcart_sub[symmetric] norm_sndcart)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2612
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2613
lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n::finite) (x2::'a::{times,comm_monoid_add}^'m::finite)) \<bullet> (pastecart y1 y2) =  x1 \<bullet> y1 + x2 \<bullet> y2"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2614
  by (simp add: dot_def setsum_UNIV_sum pastecart_def)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2615
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  2616
lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ 'm::finite) + norm(y::real^'n::finite)"
30040
e2cd1acda1ab real_normed_vector instance
huffman
parents: 30039
diff changeset
  2617
  unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff id_def
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2618
  apply (rule power2_le_imp_le)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2619
  apply (simp add: real_sqrt_pow2[OF add_nonneg_nonneg[OF dot_pos_le[of x] dot_pos_le[of y]]])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2620
  apply (auto simp add: power2_eq_square ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2621
  apply (simp add: power2_eq_square[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2622
  apply (rule mult_nonneg_nonneg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2623
  apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2624
  apply (rule add_nonneg_nonneg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2625
  apply (simp_all add: real_sqrt_pow2[OF dot_pos_le])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2626
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2627
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2628
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2629
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2630
definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2631
  "S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2632
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2633
lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2634
  unfolding hull_def by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2635
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2636
lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2637
unfolding hull_def subset_iff by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2638
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2639
lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2640
using hull_same[of s S] hull_in[of S s] by metis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2641
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2642
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2643
lemma hull_hull: "S hull (S hull s) = S hull s"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2644
  unfolding hull_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2645
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2646
lemma hull_subset: "s \<subseteq> (S hull s)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2647
  unfolding hull_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2648
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2649
lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2650
  unfolding hull_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2651
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2652
lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2653
  unfolding hull_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2654
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2655
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2656
  unfolding hull_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2657
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2658
lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow>  s \<subseteq> t"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2659
  unfolding hull_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2660
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2661
lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2662
           ==> (S hull s = t)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2663
unfolding hull_def by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2664
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2665
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"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2666
  using hull_minimal[of S "{x. P x}" Q]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2667
  by (auto simp add: subset_eq Collect_def mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2668
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2669
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2670
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2671
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2672
unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2673
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2674
lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2675
  shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2676
apply rule
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2677
apply (rule hull_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2678
unfolding Un_subset_iff
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2679
apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2680
apply (rule hull_minimal)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2681
apply (metis hull_union_subset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2682
apply (metis hull_in T)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2683
done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2684
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2685
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2686
  unfolding hull_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2687
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2688
lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2689
by (metis hull_redundant_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2690
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2691
text{* Archimedian properties and useful consequences. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2692
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2693
lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2694
  using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2695
lemmas real_arch_lt = reals_Archimedean2
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2696
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2697
lemmas real_arch = reals_Archimedean3
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2698
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2699
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2700
  using reals_Archimedean
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2701
  apply (auto simp add: field_simps inverse_positive_iff_positive)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2702
  apply (subgoal_tac "inverse (real n) > 0")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2703
  apply arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2704
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2705
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2706
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2707
lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2708
proof(induct n)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2709
  case 0 thus ?case by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2710
next
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2711
  case (Suc n)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2712
  hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2713
  from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2714
  from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2715
  also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2716
    apply (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2717
    using mult_left_mono[OF p Suc.prems] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2718
  finally show ?case  by (simp add: real_of_nat_Suc ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2719
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2720
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2721
lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2722
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2723
  from x have x0: "x - 1 > 0" by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2724
  from real_arch[OF x0, rule_format, of y]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2725
  obtain n::nat where n:"y < real n * (x - 1)" by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2726
  from x0 have x00: "x- 1 \<ge> 0" by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2727
  from real_pow_lbound[OF x00, of n] n
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2728
  have "y < x^n" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2729
  then show ?thesis by metis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2730
qed
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2731
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2732
lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2733
  using real_arch_pow[of 2 x] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2734
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2735
lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2736
  shows "\<exists>n. x^n < y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2737
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2738
  {assume x0: "x > 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2739
    from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2740
    from real_arch_pow[OF ix, of "1/y"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2741
    obtain n where n: "1/y < (1/x)^n" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2742
    then
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2743
    have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2744
  moreover
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2745
  {assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2746
  ultimately show ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2747
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2748
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2749
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)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2750
  by (metis real_arch_inv)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2751
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2752
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"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2753
  apply (rule forall_pos_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2754
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2755
  apply (atomize)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2756
  apply (erule_tac x="n - 1" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2757
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2758
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2759
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2760
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"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2761
  shows "x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2762
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2763
  {assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2764
    from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x"  by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2765
    with xc[rule_format, of n] have "n = 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2766
    with n c have False by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2767
  then show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2768
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2769
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2770
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2771
(* Relate max and min to sup and inf.                                        *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2772
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2773
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2774
lemma real_max_rsup: "max x y = rsup {x,y}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2775
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2776
  have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2777
  from rsup_finite_le_iff[OF f, of "max x y"] have "rsup {x,y} \<le> max x y" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2778
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2779
  have "max x y \<le> rsup {x,y}" using rsup_finite_ge_iff[OF f, of "max x y"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2780
    by (simp add: linorder_linear)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2781
  ultimately show ?thesis by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2782
qed
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2783
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2784
lemma real_min_rinf: "min x y = rinf {x,y}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2785
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2786
  have f: "finite {x, y}" "{x,y} \<noteq> {}"  by simp_all
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2787
  from rinf_finite_le_iff[OF f, of "min x y"] have "rinf {x,y} \<le> min x y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2788
    by (simp add: linorder_linear)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2789
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2790
  have "min x y \<le> rinf {x,y}" using rinf_finite_ge_iff[OF f, of "min x y"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2791
    by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2792
  ultimately show ?thesis by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2793
qed
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2794
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2795
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2796
(* Geometric progression.                                                    *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2797
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2798
31021
53642251a04f farewell to class recpower
haftmann
parents: 30960
diff changeset
  2799
lemma sum_gp_basic: "((1::'a::{field}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2800
  (is "?lhs = ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2801
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2802
  {assume x1: "x = 1" hence ?thesis by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2803
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2804
  {assume x1: "x\<noteq>1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2805
    hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2806
    from geometric_sum[OF x1, of "Suc n", unfolded x1']
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2807
    have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2808
      unfolding atLeastLessThanSuc_atLeastAtMost
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2809
      using x1' apply (auto simp only: field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2810
      apply (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2811
      done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2812
    then have ?thesis by (simp add: ring_simps) }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2813
  ultimately show ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2814
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2815
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2816
lemma sum_gp_multiplied: assumes mn: "m <= n"
31021
53642251a04f farewell to class recpower
haftmann
parents: 30960
diff changeset
  2817
  shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2818
  (is "?lhs = ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2819
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2820
  let ?S = "{0..(n - m)}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2821
  from mn have mn': "n - m \<ge> 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2822
  let ?f = "op + m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2823
  have i: "inj_on ?f ?S" unfolding inj_on_def by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2824
  have f: "?f ` ?S = {m..n}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2825
    using mn apply (auto simp add: image_iff Bex_def) by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2826
  have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2827
    by (rule ext, simp add: power_add power_mult)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2828
  from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2829
  have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2830
  then show ?thesis unfolding sum_gp_basic using mn
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2831
    by (simp add: ring_simps power_add[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2832
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2833
31021
53642251a04f farewell to class recpower
haftmann
parents: 30960
diff changeset
  2834
lemma sum_gp: "setsum (op ^ (x::'a::{field})) {m .. n} =
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2835
   (if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2836
                    else (x^ m - x^ (Suc n)) / (1 - x))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2837
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2838
  {assume nm: "n < m" hence ?thesis by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2839
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2840
  {assume "\<not> n < m" hence nm: "m \<le> n" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2841
    {assume x: "x = 1"  hence ?thesis by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2842
    moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2843
    {assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2844
      from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2845
    ultimately have ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2846
  }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2847
  ultimately show ?thesis by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2848
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2849
31021
53642251a04f farewell to class recpower
haftmann
parents: 30960
diff changeset
  2850
lemma sum_gp_offset: "setsum (op ^ (x::'a::{field})) {m .. m+n} =
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2851
  (if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2852
  unfolding sum_gp[of x m "m + n"] power_Suc
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2853
  by (simp add: ring_simps power_add)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2854
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2855
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2856
subsection{* A bit of linear algebra. *}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2857
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2858
definition "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 *s x \<in>S )"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2859
definition "span S = (subspace hull S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2860
definition "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2861
abbreviation "independent s == ~(dependent s)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2862
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2863
(* Closure properties of subspaces.                                          *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2864
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2865
lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2866
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2867
lemma subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2868
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2869
lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2870
  by (metis subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2871
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2872
lemma subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2873
  by (metis subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2874
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2875
lemma subspace_neg: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> - x \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2876
  by (metis vector_sneg_minus1 subspace_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2877
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2878
lemma subspace_sub: "subspace S \<Longrightarrow> (x::'a::ring_1^'n) \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2879
  by (metis diff_def subspace_add subspace_neg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2880
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2881
lemma subspace_setsum:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2882
  assumes sA: "subspace A" and fB: "finite B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2883
  and f: "\<forall>x\<in> B. f x \<in> A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2884
  shows "setsum f B \<in> A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2885
  using  fB f sA
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2886
  apply(induct rule: finite_induct[OF fB])
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2887
  by (simp add: subspace_def sA, auto simp add: sA subspace_add)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2888
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2889
lemma subspace_linear_image:
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2890
  assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and sS: "subspace S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2891
  shows "subspace(f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2892
  using lf sS linear_0[OF lf]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2893
  unfolding linear_def subspace_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2894
  apply (auto simp add: image_iff)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2895
  apply (rule_tac x="x + y" in bexI, auto)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2896
  apply (rule_tac x="c*s x" in bexI, auto)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2897
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2898
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2899
lemma subspace_linear_preimage: "linear (f::'a::semiring_1^'n \<Rightarrow> _) ==> subspace S ==> subspace {x. f x \<in> S}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2900
  by (auto simp add: subspace_def linear_def linear_0[of f])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2901
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2902
lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2903
  by (simp add: subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2904
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2905
lemma subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2906
  by (simp add: subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2907
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2908
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2909
lemma span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2910
  by (metis span_def hull_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2911
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2912
lemma subspace_span: "subspace(span S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2913
  unfolding span_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2914
  apply (rule hull_in[unfolded mem_def])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2915
  apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2916
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2917
  apply (erule_tac x="X" in ballE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2918
  apply (simp add: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2919
  apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2920
  apply (erule_tac x="X" in ballE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2921
  apply (erule_tac x="X" in ballE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2922
  apply (erule_tac x="X" in ballE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2923
  apply (clarsimp simp add: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2924
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2925
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2926
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2927
  apply (erule_tac x="X" in ballE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2928
  apply (erule_tac x="X" in ballE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2929
  apply (simp add: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2930
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2931
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2932
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2933
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2934
lemma span_clauses:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2935
  "a \<in> S ==> a \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2936
  "0 \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2937
  "x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2938
  "x \<in> span S \<Longrightarrow> c *s x \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2939
  by (metis span_def hull_subset subset_eq subspace_span subspace_def)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2940
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2941
lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2942
  and P: "subspace P" and x: "x \<in> span S" shows "P x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2943
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2944
  from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2945
  from P have P': "P \<in> subspace" by (simp add: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2946
  from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2947
  show "P x" by (metis mem_def subset_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2948
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2949
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2950
lemma span_empty: "span {} = {(0::'a::semiring_0 ^ 'n)}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2951
  apply (simp add: span_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2952
  apply (rule hull_unique)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2953
  apply (auto simp add: mem_def subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2954
  unfolding mem_def[of "0::'a^'n", symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2955
  apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2956
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2957
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2958
lemma independent_empty: "independent {}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2959
  by (simp add: dependent_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2960
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2961
lemma independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2962
  apply (clarsimp simp add: dependent_def span_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2963
  apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2964
  apply force
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2965
  apply (rule span_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2966
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2967
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2968
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2969
lemma span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow>  subspace B \<Longrightarrow> span A = B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2970
  by (metis order_antisym span_def hull_minimal mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2971
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2972
lemma span_induct': assumes SP: "\<forall>x \<in> S. P x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2973
  and P: "subspace P" shows "\<forall>x \<in> span S. P x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2974
  using span_induct SP P by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2975
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2976
inductive span_induct_alt_help for S:: "'a::semiring_1^'n \<Rightarrow> bool"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2977
  where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2978
  span_induct_alt_help_0: "span_induct_alt_help S 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2979
  | span_induct_alt_help_S: "x \<in> S \<Longrightarrow> span_induct_alt_help S z \<Longrightarrow> span_induct_alt_help S (c *s x + z)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2980
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2981
lemma span_induct_alt':
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2982
  assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c*s x + y)" shows "\<forall>x \<in> span S. h x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2983
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2984
  {fix x:: "'a^'n" assume x: "span_induct_alt_help S x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2985
    have "h x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2986
      apply (rule span_induct_alt_help.induct[OF x])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2987
      apply (rule h0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2988
      apply (rule hS, assumption, assumption)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2989
      done}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2990
  note th0 = this
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2991
  {fix x assume x: "x \<in> span S"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  2992
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2993
    have "span_induct_alt_help S x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2994
      proof(rule span_induct[where x=x and S=S])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2995
	show "x \<in> span S" using x .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2996
      next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2997
	fix x assume xS : "x \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2998
	  from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  2999
	  show "span_induct_alt_help S x" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3000
	next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3001
	have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3002
	moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3003
	{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3004
	  from h
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3005
	  have "span_induct_alt_help S (x + y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3006
	    apply (induct rule: span_induct_alt_help.induct)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3007
	    apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3008
	    unfolding add_assoc
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3009
	    apply (rule span_induct_alt_help_S)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3010
	    apply assumption
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3011
	    apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3012
	    done}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3013
	moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3014
	{fix c x assume xt: "span_induct_alt_help S x"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3015
	  then have "span_induct_alt_help S (c*s x)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3016
	    apply (induct rule: span_induct_alt_help.induct)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3017
	    apply (simp add: span_induct_alt_help_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3018
	    apply (simp add: vector_smult_assoc vector_add_ldistrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3019
	    apply (rule span_induct_alt_help_S)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3020
	    apply assumption
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3021
	    apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3022
	    done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3023
	}
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3024
	ultimately show "subspace (span_induct_alt_help S)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3025
	  unfolding subspace_def mem_def Ball_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3026
      qed}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3027
  with th0 show ?thesis by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3028
qed
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3029
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3030
lemma span_induct_alt:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3031
  assumes h0: "h (0::'a::semiring_1^'n)" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c*s x + y)" and x: "x \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3032
  shows "h x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3033
using span_induct_alt'[of h S] h0 hS x by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3034
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3035
(* Individual closure properties. *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3036
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3037
lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3038
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3039
lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3040
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3041
lemma span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3042
  by (metis subspace_add subspace_span)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3043
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3044
lemma span_mul: "x \<in> span S ==> (c *s x) \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3045
  by (metis subspace_span subspace_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3046
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3047
lemma span_neg: "x \<in> span S ==> - (x::'a::ring_1^'n) \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3048
  by (metis subspace_neg subspace_span)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3049
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3050
lemma span_sub: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3051
  by (metis subspace_span subspace_sub)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3052
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3053
lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3054
  apply (rule subspace_setsum)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3055
  by (metis subspace_span subspace_setsum)+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3056
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3057
lemma span_add_eq: "(x::'a::ring_1^'n) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3058
  apply (auto simp only: span_add span_sub)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3059
  apply (subgoal_tac "(x + y) - x \<in> span S", simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3060
  by (simp only: span_add span_sub)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3061
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3062
(* Mapping under linear image. *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3063
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3064
lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ 'n => _)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3065
  shows "span (f ` S) = f ` (span S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3066
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3067
  {fix x
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3068
    assume x: "x \<in> span (f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3069
    have "x \<in> f ` span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3070
      apply (rule span_induct[where x=x and S = "f ` S"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3071
      apply (clarsimp simp add: image_iff)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3072
      apply (frule span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3073
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3074
      apply (simp only: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3075
      apply (rule subspace_linear_image[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3076
      apply (rule subspace_span)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3077
      apply (rule x)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3078
      done}
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3079
  moreover
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3080
  {fix x assume x: "x \<in> span S"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3081
    have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3082
      unfolding mem_def Collect_def ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3083
    have "f x \<in> span (f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3084
      apply (rule span_induct[where S=S])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3085
      apply (rule span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3086
      apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3087
      apply (subst th0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3088
      apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3089
      apply (rule x)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3090
      done}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3091
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3092
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3093
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3094
(* The key breakdown property. *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3095
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3096
lemma span_breakdown:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3097
  assumes bS: "(b::'a::ring_1 ^ 'n) \<in> S" and aS: "a \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3098
  shows "\<exists>k. a - k*s b \<in> span (S - {b})" (is "?P a")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3099
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3100
  {fix x assume xS: "x \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3101
    {assume ab: "x = b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3102
      then have "?P x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3103
	apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3104
	apply (rule exI[where x="1"], simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3105
	by (rule span_0)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3106
    moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3107
    {assume ab: "x \<noteq> b"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3108
      then have "?P x"  using xS
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3109
	apply -
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3110
	apply (rule exI[where x=0])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3111
	apply (rule span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3112
	by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3113
    ultimately have "?P x" by blast}
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3114
  moreover have "subspace ?P"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3115
    unfolding subspace_def
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3116
    apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3117
    apply (simp add: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3118
    apply (rule exI[where x=0])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3119
    using span_0[of "S - {b}"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3120
    apply (simp add: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3121
    apply (clarsimp simp add: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3122
    apply (rule_tac x="k + ka" in exI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3123
    apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3124
    apply (simp only: )
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3125
    apply (rule span_add[unfolded mem_def])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3126
    apply assumption+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3127
    apply (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3128
    apply (clarsimp simp add: mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3129
    apply (rule_tac x= "c*k" in exI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3130
    apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3131
    apply (simp only: )
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3132
    apply (rule span_mul[unfolded mem_def])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3133
    apply assumption
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3134
    by (vector ring_simps)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3135
  ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3136
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3137
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3138
lemma span_breakdown_eq:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3139
  "(x::'a::ring_1^'n) \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *s a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3140
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3141
  {assume x: "x \<in> span (insert a S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3142
    from x span_breakdown[of "a" "insert a S" "x"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3143
    have ?rhs apply clarsimp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3144
      apply (rule_tac x= "k" in exI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3145
      apply (rule set_rev_mp[of _ "span (S - {a})" _])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3146
      apply assumption
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3147
      apply (rule span_mono)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3148
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3149
      done}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3150
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3151
  { fix k assume k: "x - k *s a \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3152
    have eq: "x = (x - k *s a) + k *s a" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3153
    have "(x - k *s a) + k *s a \<in> span (insert a S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3154
      apply (rule span_add)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3155
      apply (rule set_rev_mp[of _ "span S" _])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3156
      apply (rule k)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3157
      apply (rule span_mono)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3158
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3159
      apply (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3160
      apply (rule span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3161
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3162
      done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3163
    then have ?lhs using eq by metis}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3164
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3165
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3166
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3167
(* Hence some "reversal" results.*)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3168
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3169
lemma in_span_insert:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3170
  assumes a: "(a::'a::field^'n) \<in> span (insert b S)" and na: "a \<notin> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3171
  shows "b \<in> span (insert a S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3172
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3173
  from span_breakdown[of b "insert b S" a, OF insertI1 a]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3174
  obtain k where k: "a - k*s b \<in> span (S - {b})" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3175
  {assume k0: "k = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3176
    with k have "a \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3177
      apply (simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3178
      apply (rule set_rev_mp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3179
      apply assumption
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3180
      apply (rule span_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3181
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3182
      done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3183
    with na  have ?thesis by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3184
  moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3185
  {assume k0: "k \<noteq> 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3186
    have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3187
    from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3188
      by (vector field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3189
    from k have "(1/k) *s (a - k*s b) \<in> span (S - {b})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3190
      by (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3191
    hence th: "(1/k) *s a - b \<in> span (S - {b})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3192
      unfolding eq' .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3193
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3194
    from k
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3195
    have ?thesis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3196
      apply (subst eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3197
      apply (rule span_sub)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3198
      apply (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3199
      apply (rule span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3200
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3201
      apply (rule set_rev_mp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3202
      apply (rule th)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3203
      apply (rule span_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3204
      using na by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3205
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3206
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3207
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3208
lemma in_span_delete:
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3209
  assumes a: "(a::'a::field^'n) \<in> span S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3210
  and na: "a \<notin> span (S-{b})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3211
  shows "b \<in> span (insert a (S - {b}))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3212
  apply (rule in_span_insert)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3213
  apply (rule set_rev_mp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3214
  apply (rule a)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3215
  apply (rule span_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3216
  apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3217
  apply (rule na)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3218
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3219
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3220
(* Transitivity property. *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3221
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3222
lemma span_trans:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3223
  assumes x: "(x::'a::ring_1^'n) \<in> span S" and y: "y \<in> span (insert x S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3224
  shows "y \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3225
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3226
  from span_breakdown[of x "insert x S" y, OF insertI1 y]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3227
  obtain k where k: "y -k*s x \<in> span (S - {x})" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3228
  have eq: "y = (y - k *s x) + k *s x" by vector
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3229
  show ?thesis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3230
    apply (subst eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3231
    apply (rule span_add)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3232
    apply (rule set_rev_mp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3233
    apply (rule k)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3234
    apply (rule span_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3235
    apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3236
    apply (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3237
    by (rule x)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3238
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3239
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3240
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3241
(* An explicit expansion is sometimes needed.                                *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3242
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3243
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3244
lemma span_explicit:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3245
  "span P = {y::'a::semiring_1^'n. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *s v) S = y}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3246
  (is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3247
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3248
  {fix x assume x: "x \<in> ?E"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3249
    then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *s v) S = x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3250
      by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3251
    have "x \<in> span P"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3252
      unfolding u[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3253
      apply (rule span_setsum[OF fS])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3254
      using span_mono[OF SP]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3255
      by (auto intro: span_superset span_mul)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3256
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3257
  have "\<forall>x \<in> span P. x \<in> ?E"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3258
    unfolding mem_def Collect_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3259
  proof(rule span_induct_alt')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3260
    show "?h 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3261
      apply (rule exI[where x="{}"]) by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3262
  next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3263
    fix c x y
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3264
    assume x: "x \<in> P" and hy: "?h y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3265
    from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3266
      and u: "setsum (\<lambda>v. u v *s v) S = y" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3267
    let ?S = "insert x S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3268
    let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3269
                  else u y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3270
    from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3271
    {assume xS: "x \<in> S"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3272
      have S1: "S = (S - {x}) \<union> {x}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3273
	and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3274
      have "setsum (\<lambda>v. ?u v *s v) ?S =(\<Sum>v\<in>S - {x}. u v *s v) + (u x + c) *s x"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3275
	using xS
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3276
	by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3277
	  setsum_clauses(2)[OF fS] cong del: if_weak_cong)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3278
      also have "\<dots> = (\<Sum>v\<in>S. u v *s v) + c *s x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3279
	apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3280
	by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3281
      also have "\<dots> = c*s x + y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3282
	by (simp add: add_commute u)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3283
      finally have "setsum (\<lambda>v. ?u v *s v) ?S = c*s x + y" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3284
    then have "?Q ?S ?u (c*s x + y)" using th0 by blast}
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3285
  moreover
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3286
  {assume xS: "x \<notin> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3287
    have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *s v) = y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3288
      unfolding u[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3289
      apply (rule setsum_cong2)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3290
      using xS by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3291
    have "?Q ?S ?u (c*s x + y)" using fS xS th0
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3292
      by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3293
  ultimately have "?Q ?S ?u (c*s x + y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3294
    by (cases "x \<in> S", simp, simp)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3295
    then show "?h (c*s x + y)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3296
      apply -
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3297
      apply (rule exI[where x="?S"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3298
      apply (rule exI[where x="?u"]) by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3299
  qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3300
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3301
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3302
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3303
lemma dependent_explicit:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3304
  "dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>(v::'a::{idom,field}^'n) \<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *s v) S = 0))" (is "?lhs = ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3305
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3306
  {assume dP: "dependent P"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3307
    then obtain a S u where aP: "a \<in> P" and fS: "finite S"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3308
      and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3309
      unfolding dependent_def span_explicit by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3310
    let ?S = "insert a S"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3311
    let ?u = "\<lambda>y. if y = a then - 1 else u y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3312
    let ?v = a
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3313
    from aP SP have aS: "a \<notin> S" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3314
    from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3315
    have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3316
      using fS aS
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3317
      apply (simp add: vector_smult_lneg vector_smult_lid setsum_clauses ring_simps )
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3318
      apply (subst (2) ua[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3319
      apply (rule setsum_cong2)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3320
      by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3321
    with th0 have ?rhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3322
      apply -
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3323
      apply (rule exI[where x= "?S"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3324
      apply (rule exI[where x= "?u"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3325
      by clarsimp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3326
  moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3327
  {fix S u v assume fS: "finite S"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3328
      and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3329
    and u: "setsum (\<lambda>v. u v *s v) S = 0"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3330
    let ?a = v
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3331
    let ?S = "S - {v}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3332
    let ?u = "\<lambda>i. (- u i) / u v"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3333
    have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P"       using fS SP vS by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3334
    have "setsum (\<lambda>v. ?u v *s v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *s (u v *s v)) S - ?u v *s v"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3335
      using fS vS uv
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3336
      by (simp add: setsum_diff1 vector_smult_lneg divide_inverse
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3337
	vector_smult_assoc field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3338
    also have "\<dots> = ?a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3339
      unfolding setsum_cmul u
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3340
      using uv by (simp add: vector_smult_lneg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3341
    finally  have "setsum (\<lambda>v. ?u v *s v) ?S = ?a" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3342
    with th0 have ?lhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3343
      unfolding dependent_def span_explicit
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3344
      apply -
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3345
      apply (rule bexI[where x= "?a"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3346
      apply simp_all
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3347
      apply (rule exI[where x= "?S"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3348
      by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3349
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3350
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3351
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3352
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3353
lemma span_finite:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3354
  assumes fS: "finite S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3355
  shows "span S = {(y::'a::semiring_1^'n). \<exists>u. setsum (\<lambda>v. u v *s v) S = y}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3356
  (is "_ = ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3357
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3358
  {fix y assume y: "y \<in> span S"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3359
    from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3360
      u: "setsum (\<lambda>v. u v *s v) S' = y" unfolding span_explicit by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3361
    let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3362
    from setsum_restrict_set[OF fS, of "\<lambda>v. u v *s v" S', symmetric] SS'
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3363
    have "setsum (\<lambda>v. ?u v *s v) S = setsum (\<lambda>v. u v *s v) S'"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3364
      unfolding cond_value_iff cond_application_beta
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3365
      apply (simp add: cond_value_iff cong del: if_weak_cong)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3366
      apply (rule setsum_cong)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3367
      apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3368
      done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3369
    hence "setsum (\<lambda>v. ?u v *s v) S = y" by (metis u)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3370
    hence "y \<in> ?rhs" by auto}
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3371
  moreover
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3372
  {fix y u assume u: "setsum (\<lambda>v. u v *s v) S = y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3373
    then have "y \<in> span S" using fS unfolding span_explicit by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3374
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3375
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3376
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3377
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3378
(* Standard bases are a spanning set, and obviously finite.                  *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3379
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3380
lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n::finite | i. i \<in> (UNIV :: 'n set)} = UNIV"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3381
apply (rule set_ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3382
apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3383
apply (subst basis_expansion[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3384
apply (rule span_setsum)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3385
apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3386
apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3387
apply (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3388
apply (rule span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3389
apply (auto simp add: Collect_def mem_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3390
done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3391
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3392
lemma has_size_stdbasis: "{basis i ::real ^'n::finite | i. i \<in> (UNIV :: 'n set)} hassize CARD('n)" (is "?S hassize ?n")
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3393
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3394
  have eq: "?S = basis ` UNIV" by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3395
  show ?thesis unfolding eq
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3396
    apply (rule hassize_image_inj[OF basis_inj])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3397
    by (simp add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3398
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3399
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3400
lemma finite_stdbasis: "finite {basis i ::real^'n::finite |i. i\<in> (UNIV:: 'n set)}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3401
  using has_size_stdbasis[unfolded hassize_def]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3402
  ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3403
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3404
lemma card_stdbasis: "card {basis i ::real^'n::finite |i. i\<in> (UNIV :: 'n set)} = CARD('n)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3405
  using has_size_stdbasis[unfolded hassize_def]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3406
  ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3407
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3408
lemma independent_stdbasis_lemma:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3409
  assumes x: "(x::'a::semiring_1 ^ 'n) \<in> span (basis ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3410
  and iS: "i \<notin> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3411
  shows "(x$i) = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3412
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3413
  let ?U = "UNIV :: 'n set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3414
  let ?B = "basis ` S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3415
  let ?P = "\<lambda>(x::'a^'n). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3416
 {fix x::"'a^'n" assume xS: "x\<in> ?B"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3417
   from xS have "?P x" by auto}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3418
 moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3419
 have "subspace ?P"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3420
   by (auto simp add: subspace_def Collect_def mem_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3421
 ultimately show ?thesis
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3422
   using x span_induct[of ?B ?P x] iS by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3423
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3424
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3425
lemma independent_stdbasis: "independent {basis i ::real^'n::finite |i. i\<in> (UNIV :: 'n set)}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3426
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3427
  let ?I = "UNIV :: 'n set"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3428
  let ?b = "basis :: _ \<Rightarrow> real ^'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3429
  let ?B = "?b ` ?I"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3430
  have eq: "{?b i|i. i \<in> ?I} = ?B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3431
    by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3432
  {assume d: "dependent ?B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3433
    then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3434
      unfolding dependent_def by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3435
    have eq1: "?B - {?b k} = ?B - ?b ` {k}"  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3436
    have eq2: "?B - {?b k} = ?b ` (?I - {k})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3437
      unfolding eq1
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3438
      apply (rule inj_on_image_set_diff[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3439
      apply (rule basis_inj) using k(1) by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3440
    from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3441
    from independent_stdbasis_lemma[OF th0, of k, simplified]
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3442
    have False by simp}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3443
  then show ?thesis unfolding eq dependent_def ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3444
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3445
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3446
(* This is useful for building a basis step-by-step.                         *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3447
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3448
lemma independent_insert:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3449
  "independent(insert (a::'a::field ^'n) S) \<longleftrightarrow>
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3450
      (if a \<in> S then independent S
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3451
                else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3452
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3453
  {assume aS: "a \<in> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3454
    hence ?thesis using insert_absorb[OF aS] by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3455
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3456
  {assume aS: "a \<notin> S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3457
    {assume i: ?lhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3458
      then have ?rhs using aS
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3459
	apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3460
	apply (rule conjI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3461
	apply (rule independent_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3462
	apply assumption
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3463
	apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3464
	by (simp add: dependent_def)}
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3465
    moreover
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3466
    {assume i: ?rhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3467
      have ?lhs using i aS
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3468
	apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3469
	apply (auto simp add: dependent_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3470
	apply (case_tac "aa = a", auto)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3471
	apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3472
	apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3473
	apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3474
	apply (subgoal_tac "insert aa (S - {aa}) = S")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3475
	apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3476
	apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3477
	apply (rule in_span_insert)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3478
	apply assumption
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3479
	apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3480
	apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3481
	done}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3482
    ultimately have ?thesis by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3483
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3484
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3485
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3486
(* The degenerate case of the Exchange Lemma.  *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3487
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3488
lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3489
  by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3490
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3491
lemma span_span: "span (span A) = span A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3492
  unfolding span_def hull_hull ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3493
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3494
lemma span_inc: "S \<subseteq> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3495
  by (metis subset_eq span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3496
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3497
lemma spanning_subset_independent:
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3498
  assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^'n) set)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3499
  and AsB: "A \<subseteq> span B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3500
  shows "A = B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3501
proof
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3502
  from BA show "B \<subseteq> A" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3503
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3504
  from span_mono[OF BA] span_mono[OF AsB]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3505
  have sAB: "span A = span B" unfolding span_span by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3506
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3507
  {fix x assume x: "x \<in> A"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3508
    from iA have th0: "x \<notin> span (A - {x})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3509
      unfolding dependent_def using x by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3510
    from x have xsA: "x \<in> span A" by (blast intro: span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3511
    have "A - {x} \<subseteq> A" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3512
    hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3513
    {assume xB: "x \<notin> B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3514
      from xB BA have "B \<subseteq> A -{x}" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3515
      hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3516
      with th1 th0 sAB have "x \<notin> span A" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3517
      with x have False by (metis span_superset)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3518
    then have "x \<in> B" by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3519
  then show "A \<subseteq> B" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3520
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3521
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3522
(* The general case of the Exchange Lemma, the key to what follows.  *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3523
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3524
lemma exchange_lemma:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3525
  assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3526
  and sp:"s \<subseteq> span t"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3527
  shows "\<exists>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3528
using f i sp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3529
proof(induct c\<equiv>"card(t - s)" arbitrary: s t rule: nat_less_induct)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3530
  fix n:: nat and s t :: "('a ^'n) set"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3531
  assume H: " \<forall>m<n. \<forall>(x:: ('a ^'n) set) xa.
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3532
                finite xa \<longrightarrow>
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3533
                independent x \<longrightarrow>
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3534
                x \<subseteq> span xa \<longrightarrow>
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3535
                m = card (xa - x) \<longrightarrow>
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3536
                (\<exists>t'. (t' hassize card xa) \<and>
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3537
                      x \<subseteq> t' \<and> t' \<subseteq> x \<union> xa \<and> x \<subseteq> span t')"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3538
    and ft: "finite t" and s: "independent s" and sp: "s \<subseteq> span t"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3539
    and n: "n = card (t - s)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3540
  let ?P = "\<lambda>t'. (t' hassize card t) \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3541
  let ?ths = "\<exists>t'. ?P t'"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3542
  {assume st: "s \<subseteq> t"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3543
    from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3544
      by (auto simp add: hassize_def intro: span_superset)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3545
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3546
  {assume st: "t \<subseteq> s"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3547
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3548
    from spanning_subset_independent[OF st s sp]
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3549
      st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3550
      by (auto simp add: hassize_def intro: span_superset)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3551
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3552
  {assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3553
    from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3554
      from b have "t - {b} - s \<subset> t - s" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3555
      then have cardlt: "card (t - {b} - s) < n" using n ft
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3556
 	by (auto intro: psubset_card_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3557
      from b ft have ct0: "card t \<noteq> 0" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3558
    {assume stb: "s \<subseteq> span(t -{b})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3559
      from ft have ftb: "finite (t -{b})" by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3560
      from H[rule_format, OF cardlt ftb s stb]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3561
      obtain u where u: "u hassize card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3562
      let ?w = "insert b u"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3563
      have th0: "s \<subseteq> insert b u" using u by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3564
      from u(3) b have "u \<subseteq> s \<union> t" by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3565
      then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3566
      have bu: "b \<notin> u" using b u by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3567
      from u(1) have fu: "finite u" by (simp add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3568
      from u(1) ft b have "u hassize (card t - 1)" by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3569
      then
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3570
      have th2: "insert b u hassize card t"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3571
	using  card_insert_disjoint[OF fu bu] ct0 by (auto simp add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3572
      from u(4) have "s \<subseteq> span u" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3573
      also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3574
      finally have th3: "s \<subseteq> span (insert b u)" .      from th0 th1 th2 th3 have th: "?P ?w"  by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3575
      from th have ?ths by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3576
    moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3577
    {assume stb: "\<not> s \<subseteq> span(t -{b})"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3578
      from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3579
      have ab: "a \<noteq> b" using a b by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3580
      have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3581
      have mlt: "card ((insert a (t - {b})) - s) < n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3582
	using cardlt ft n  a b by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3583
      have ft': "finite (insert a (t - {b}))" using ft by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3584
      {fix x assume xs: "x \<in> s"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3585
	have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3586
	from b(1) have "b \<in> span t" by (simp add: span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3587
	have bs: "b \<in> span (insert a (t - {b}))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3588
	  by (metis in_span_delete a sp mem_def subset_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3589
	from xs sp have "x \<in> span t" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3590
	with span_mono[OF t]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3591
	have x: "x \<in> span (insert b (insert a (t - {b})))" ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3592
	from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3593
      then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3594
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3595
      from H[rule_format, OF mlt ft' s sp' refl] obtain u where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3596
	u: "u hassize card (insert a (t -{b}))" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3597
	"s \<subseteq> span u" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3598
      from u a b ft at ct0 have "?P u" by (auto simp add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3599
      then have ?ths by blast }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3600
    ultimately have ?ths by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3601
  }
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3602
  ultimately
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3603
  show ?ths  by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3604
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3605
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3606
(* This implies corresponding size bounds.                                   *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3607
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3608
lemma independent_span_bound:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3609
  assumes f: "finite t" and i: "independent (s::('a::field^'n) set)" and sp:"s \<subseteq> span t"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3610
  shows "finite s \<and> card s \<le> card t"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3611
  by (metis exchange_lemma[OF f i sp] hassize_def finite_subset card_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3612
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3613
lemma finite_Atleast_Atmost[simp]: "finite {f x |x. x\<in> {(i::'a::finite_intvl_succ) .. j}}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3614
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3615
  have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3616
  show ?thesis unfolding eq
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3617
    apply (rule finite_imageI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3618
    apply (rule finite_intvl)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3619
    done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3620
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3621
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3622
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3623
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3624
  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3625
  show ?thesis unfolding eq
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3626
    apply (rule finite_imageI)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3627
    apply (rule finite)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3628
    done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3629
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3630
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3631
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3632
lemma independent_bound:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3633
  fixes S:: "(real^'n::finite) set"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3634
  shows "independent S \<Longrightarrow> finite S \<and> card S <= CARD('n)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3635
  apply (subst card_stdbasis[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3636
  apply (rule independent_span_bound)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3637
  apply (rule finite_Atleast_Atmost_nat)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3638
  apply assumption
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3639
  unfolding span_stdbasis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3640
  apply (rule subset_UNIV)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3641
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3642
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3643
lemma dependent_biggerset: "(finite (S::(real ^'n::finite) set) ==> card S > CARD('n)) ==> dependent S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3644
  by (metis independent_bound not_less)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3645
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3646
(* Hence we can create a maximal independent subset.                         *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3647
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3648
lemma maximal_independent_subset_extend:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3649
  assumes sv: "(S::(real^'n::finite) set) \<subseteq> V" and iS: "independent S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3650
  shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3651
  using sv iS
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3652
proof(induct d\<equiv> "CARD('n) - card S" arbitrary: S rule: nat_less_induct)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3653
  fix n and S:: "(real^'n) set"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3654
  assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = CARD('n) - card S \<longrightarrow>
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3655
              (\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3656
    and sv: "S \<subseteq> V" and i: "independent S" and n: "n = CARD('n) - card S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3657
  let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3658
  let ?ths = "\<exists>x. ?P x"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3659
  let ?d = "CARD('n)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3660
  {assume "V \<subseteq> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3661
    then have ?ths  using sv i by blast }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3662
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3663
  {assume VS: "\<not> V \<subseteq> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3664
    from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3665
    from a have aS: "a \<notin> S" by (auto simp add: span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3666
    have th0: "insert a S \<subseteq> V" using a sv by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3667
    from independent_insert[of a S]  i a
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3668
    have th1: "independent (insert a S)" by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3669
    have mlt: "?d - card (insert a S) < n"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3670
      using aS a n independent_bound[OF th1]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3671
      by auto
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3672
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3673
    from H[rule_format, OF mlt th0 th1 refl]
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3674
    obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3675
      by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3676
    from B have "?P B" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3677
    then have ?ths by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3678
  ultimately show ?ths by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3679
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3680
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3681
lemma maximal_independent_subset:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3682
  "\<exists>(B:: (real ^'n::finite) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3683
  by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3684
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3685
(* Notion of dimension.                                                      *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3686
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3687
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3688
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3689
lemma basis_exists:  "\<exists>B. (B :: (real ^'n::finite) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3690
unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n)"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3691
unfolding hassize_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3692
using maximal_independent_subset[of V] independent_bound
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3693
by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3694
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3695
(* Consequences of independence or spanning for cardinality.                 *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3696
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3697
lemma independent_card_le_dim: "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B \<le> dim V"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3698
by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3699
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3700
lemma span_card_ge_dim:  "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3701
  by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3702
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3703
lemma basis_card_eq_dim:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3704
  "B \<subseteq> (V:: (real ^'n::finite) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3705
  by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3706
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3707
lemma dim_unique: "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> B hassize n \<Longrightarrow> dim V = n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3708
  by (metis basis_card_eq_dim hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3709
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3710
(* More lemmas about dimension.                                              *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3711
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3712
lemma dim_univ: "dim (UNIV :: (real^'n::finite) set) = CARD('n)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3713
  apply (rule dim_unique[of "{basis i |i. i\<in> (UNIV :: 'n set)}"])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3714
  by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3715
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3716
lemma dim_subset:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3717
  "(S:: (real ^'n::finite) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3718
  using basis_exists[of T] basis_exists[of S]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3719
  by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3720
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3721
lemma dim_subset_univ: "dim (S:: (real^'n::finite) set) \<le> CARD('n)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3722
  by (metis dim_subset subset_UNIV dim_univ)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3723
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3724
(* Converses to those.                                                       *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3725
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3726
lemma card_ge_dim_independent:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3727
  assumes BV:"(B::(real ^'n::finite) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3728
  shows "V \<subseteq> span B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3729
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3730
  {fix a assume aV: "a \<in> V"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3731
    {assume aB: "a \<notin> span B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3732
      then have iaB: "independent (insert a B)" using iB aV  BV by (simp add: independent_insert)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3733
      from aV BV have th0: "insert a B \<subseteq> V" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3734
      from aB have "a \<notin>B" by (auto simp add: span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3735
      with independent_card_le_dim[OF th0 iaB] dVB  have False by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3736
    then have "a \<in> span B"  by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3737
  then show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3738
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3739
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3740
lemma card_le_dim_spanning:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3741
  assumes BV: "(B:: (real ^'n::finite) set) \<subseteq> V" and VB: "V \<subseteq> span B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3742
  and fB: "finite B" and dVB: "dim V \<ge> card B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3743
  shows "independent B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3744
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3745
  {fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3746
    from a fB have c0: "card B \<noteq> 0" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3747
    from a fB have cb: "card (B -{a}) = card B - 1" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3748
    from BV a have th0: "B -{a} \<subseteq> V" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3749
    {fix x assume x: "x \<in> V"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3750
      from a have eq: "insert a (B -{a}) = B" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3751
      from x VB have x': "x \<in> span B" by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3752
      from span_trans[OF a(2), unfolded eq, OF x']
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3753
      have "x \<in> span (B -{a})" . }
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3754
    then have th1: "V \<subseteq> span (B -{a})" by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3755
    have th2: "finite (B -{a})" using fB by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3756
    from span_card_ge_dim[OF th0 th1 th2]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3757
    have c: "dim V \<le> card (B -{a})" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3758
    from c c0 dVB cb have False by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3759
  then show ?thesis unfolding dependent_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3760
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3761
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3762
lemma card_eq_dim: "(B:: (real ^'n::finite) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3763
  by (metis hassize_def order_eq_iff card_le_dim_spanning
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3764
    card_ge_dim_independent)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3765
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3766
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3767
(* More general size bound lemmas.                                           *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3768
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3769
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3770
lemma independent_bound_general:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3771
  "independent (S:: (real^'n::finite) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3772
  by (metis independent_card_le_dim independent_bound subset_refl)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3773
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3774
lemma dependent_biggerset_general: "(finite (S:: (real^'n::finite) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3775
  using independent_bound_general[of S] by (metis linorder_not_le)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3776
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3777
lemma dim_span: "dim (span (S:: (real ^'n::finite) set)) = dim S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3778
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3779
  have th0: "dim S \<le> dim (span S)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3780
    by (auto simp add: subset_eq intro: dim_subset span_superset)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3781
  from basis_exists[of S]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3782
  obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3783
  from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3784
  have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3785
  have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3786
  from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3787
    using fB(2)  by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3788
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3789
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3790
lemma subset_le_dim: "(S:: (real ^'n::finite) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3791
  by (metis dim_span dim_subset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3792
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3793
lemma span_eq_dim: "span (S:: (real ^'n::finite) set) = span T ==> dim S = dim T"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3794
  by (metis dim_span)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3795
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3796
lemma spans_image:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3797
  assumes lf: "linear (f::'a::semiring_1^'n \<Rightarrow> _)" and VB: "V \<subseteq> span B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3798
  shows "f ` V \<subseteq> span (f ` B)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3799
  unfolding span_linear_image[OF lf]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3800
  by (metis VB image_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3801
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3802
lemma dim_image_le:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3803
  fixes f :: "real^'n::finite \<Rightarrow> real^'m::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3804
  assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n::finite) set)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3805
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3806
  from basis_exists[of S] obtain B where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3807
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3808
  from B have fB: "finite B" "card B = dim S" unfolding hassize_def by blast+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3809
  have "dim (f ` S) \<le> card (f ` B)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3810
    apply (rule span_card_ge_dim)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3811
    using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3812
  also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3813
  finally show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3814
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3815
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3816
(* Relation between bases and injectivity/surjectivity of map.               *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3817
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3818
lemma spanning_surjective_image:
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3819
  assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^'n) set)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3820
  and lf: "linear f" and sf: "surj f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3821
  shows "UNIV \<subseteq> span (f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3822
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3823
  have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3824
  also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3825
finally show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3826
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3827
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3828
lemma independent_injective_image:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3829
  assumes iS: "independent (S::('a::semiring_1^'n) set)" and lf: "linear f" and fi: "inj f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3830
  shows "independent (f ` S)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3831
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3832
  {fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3833
    have eq: "f ` S - {f a} = f ` (S - {a})" using fi
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3834
      by (auto simp add: inj_on_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3835
    from a have "f a \<in> f ` span (S -{a})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3836
      unfolding eq span_linear_image[OF lf, of "S - {a}"]  by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3837
    hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3838
    with a(1) iS  have False by (simp add: dependent_def) }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3839
  then show ?thesis unfolding dependent_def by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3840
qed
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3841
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3842
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3843
(* Picking an orthogonal replacement for a spanning set.                     *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3844
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3845
    (* FIXME : Move to some general theory ?*)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3846
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3847
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3848
lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n::finite) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3849
  apply (cases "b = 0", simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3850
  apply (simp add: dot_rsub dot_rmult)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3851
  unfolding times_divide_eq_right[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3852
  by (simp add: field_simps dot_eq_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3853
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3854
lemma basis_orthogonal:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3855
  fixes B :: "(real ^'n::finite) set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3856
  assumes fB: "finite B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3857
  shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3858
  (is " \<exists>C. ?P B C")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3859
proof(induct rule: finite_induct[OF fB])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3860
  case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3861
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3862
  case (2 a B)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3863
  note fB = `finite B` and aB = `a \<notin> B`
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3864
  from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3865
  obtain C where C: "finite C" "card C \<le> card B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3866
    "span C = span B" "pairwise orthogonal C" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3867
  let ?a = "a - setsum (\<lambda>x. (x\<bullet>a / (x\<bullet>x)) *s x) C"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3868
  let ?C = "insert ?a C"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3869
  from C(1) have fC: "finite ?C" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3870
  from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3871
  {fix x k
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3872
    have th0: "\<And>(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3873
    have "x - k *s (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *s x)) \<in> span C \<longleftrightarrow> x - k *s a \<in> span C"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3874
      apply (simp only: vector_ssub_ldistrib th0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3875
      apply (rule span_add_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3876
      apply (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3877
      apply (rule span_setsum[OF C(1)])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3878
      apply clarify
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3879
      apply (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3880
      by (rule span_superset)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3881
  then have SC: "span ?C = span (insert a B)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3882
    unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3883
  thm pairwise_def
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3884
  {fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3885
    {assume xa: "x = ?a" and ya: "y = ?a"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3886
      have "orthogonal x y" using xa ya xy by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3887
    moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3888
    {assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3889
      from ya have Cy: "C = insert y (C - {y})" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3890
      have fth: "finite (C - {y})" using C by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3891
      have "orthogonal x y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3892
	using xa ya
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3893
	unfolding orthogonal_def xa dot_lsub dot_rsub diff_eq_0_iff_eq
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3894
	apply simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3895
	apply (subst Cy)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3896
	using C(1) fth
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3897
	apply (simp only: setsum_clauses)
30263
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
  3898
	thm dot_ladd
c88af4619a73 fixed proofs; added rules as default simp-rules
chaieb
parents: 30242
diff changeset
  3899
	apply (auto simp add: dot_ladd dot_radd dot_lmult dot_rmult dot_eq_0 dot_sym[of y a] dot_lsum[OF fth])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3900
	apply (rule setsum_0')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3901
	apply clarsimp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3902
	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3903
	by auto}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3904
    moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3905
    {assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3906
      from xa have Cx: "C = insert x (C - {x})" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3907
      have fth: "finite (C - {x})" using C by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3908
      have "orthogonal x y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3909
	using xa ya
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3910
	unfolding orthogonal_def ya dot_rsub dot_lsub diff_eq_0_iff_eq
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3911
	apply simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3912
	apply (subst Cx)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3913
	using C(1) fth
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3914
	apply (simp only: setsum_clauses)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3915
	apply (subst dot_sym[of x])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3916
	apply (auto simp add: dot_radd dot_rmult dot_eq_0 dot_sym[of x a] dot_rsum[OF fth])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3917
	apply (rule setsum_0')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3918
	apply clarsimp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3919
	apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
29844
4ac95212efcc fixed proof -- removed unnecessary sorry
chaieb
parents: 29842
diff changeset
  3920
	by auto}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3921
    moreover
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3922
    {assume xa: "x \<in> C" and ya: "y \<in> C"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3923
      have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3924
    ultimately have "orthogonal x y" using xC yC by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3925
  then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3926
  from fC cC SC CPO have "?P (insert a B) ?C" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3927
  then show ?case by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3928
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3929
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3930
lemma orthogonal_basis_exists:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3931
  fixes V :: "(real ^'n::finite) set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3932
  shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (B hassize dim V) \<and> pairwise orthogonal B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3933
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3934
  from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "B hassize dim V" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3935
  from B have fB: "finite B" "card B = dim V" by (simp_all add: hassize_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3936
  from basis_orthogonal[OF fB(1)] obtain C where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3937
    C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3938
  from C B
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3939
  have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3940
  from span_mono[OF B(3)]  C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3941
  from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3942
  have iC: "independent C" by (simp add: dim_span)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3943
  from C fB have "card C \<le> dim V" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3944
  moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3945
    by (simp add: dim_span)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3946
  ultimately have CdV: "C hassize dim V" unfolding hassize_def using C(1) by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3947
  from C B CSV CdV iC show ?thesis by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3948
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3949
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3950
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31285
diff changeset
  3951
  by (metis set_eq_subset span_mono span_span span_inc) (* FIXME: slow *)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3952
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3953
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3954
(* Low-dimensional subset is in a hyperplane (weak orthogonal complement).   *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3955
(* ------------------------------------------------------------------------- *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3956
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3957
lemma span_not_univ_orthogonal:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3958
  assumes sU: "span S \<noteq> UNIV"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3959
  shows "\<exists>(a:: real ^'n::finite). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3960
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3961
  from sU obtain a where a: "a \<notin> span S" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3962
  from orthogonal_basis_exists obtain B where
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3963
    B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "B hassize dim S" "pairwise orthogonal B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3964
    by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3965
  from B have fB: "finite B" "card B = dim S" by (simp_all add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3966
  from span_mono[OF B(2)] span_mono[OF B(3)]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3967
  have sSB: "span S = span B" by (simp add: span_span)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3968
  let ?a = "a - setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3969
  have "setsum (\<lambda>b. (a\<bullet>b / (b\<bullet>b)) *s b) B \<in> span S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3970
    unfolding sSB
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3971
    apply (rule span_setsum[OF fB(1)])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3972
    apply clarsimp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3973
    apply (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3974
    by (rule span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3975
  with a have a0:"?a  \<noteq> 0" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3976
  have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3977
  proof(rule span_induct')
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3978
    show "subspace (\<lambda>x. ?a \<bullet> x = 0)"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3979
      by (auto simp add: subspace_def mem_def dot_radd dot_rmult)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3980
  next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3981
    {fix x assume x: "x \<in> B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3982
      from x have B': "B = insert x (B - {x})" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3983
      have fth: "finite (B - {x})" using fB by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3984
      have "?a \<bullet> x = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3985
	apply (subst B') using fB fth
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3986
	unfolding setsum_clauses(2)[OF fth]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3987
	apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3988
	apply (clarsimp simp add: dot_lsub dot_ladd dot_lmult dot_lsum dot_eq_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3989
	apply (rule setsum_0', rule ballI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3990
	unfolding dot_sym
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3991
	by (auto simp add: x field_simps dot_eq_0 intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3992
    then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3993
  qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3994
  with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3995
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3996
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  3997
lemma span_not_univ_subset_hyperplane:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  3998
  assumes SU: "span S \<noteq> (UNIV ::(real^'n::finite) set)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  3999
  shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4000
  using span_not_univ_orthogonal[OF SU] by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4001
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4002
lemma lowdim_subset_hyperplane:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4003
  assumes d: "dim S < CARD('n::finite)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4004
  shows "\<exists>(a::real ^'n::finite). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4005
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4006
  {assume "span S = UNIV"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4007
    hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4008
    hence "dim S = CARD('n)" by (simp add: dim_span dim_univ)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4009
    with d have False by arith}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4010
  hence th: "span S \<noteq> UNIV" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4011
  from span_not_univ_subset_hyperplane[OF th] show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4012
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4013
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4014
(* We can extend a linear basis-basis injection to the whole set.            *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4015
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4016
lemma linear_indep_image_lemma:
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4017
  assumes lf: "linear f" and fB: "finite B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4018
  and ifB: "independent (f ` B)"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4019
  and fi: "inj_on f B" and xsB: "x \<in> span B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4020
  and fx: "f (x::'a::field^'n) = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4021
  shows "x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4022
  using fB ifB fi xsB fx
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4023
proof(induct arbitrary: x rule: finite_induct[OF fB])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4024
  case 1 thus ?case by (auto simp add:  span_empty)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4025
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4026
  case (2 a b x)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4027
  have fb: "finite b" using "2.prems" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4028
  have th0: "f ` b \<subseteq> f ` (insert a b)"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4029
    apply (rule image_mono) by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4030
  from independent_mono[ OF "2.prems"(2) th0]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4031
  have ifb: "independent (f ` b)"  .
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4032
  have fib: "inj_on f b"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4033
    apply (rule subset_inj_on [OF "2.prems"(3)])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4034
    by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4035
  from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4036
  obtain k where k: "x - k*s a \<in> span (b -{a})" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4037
  have "f (x - k*s a) \<in> span (f ` b)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4038
    unfolding span_linear_image[OF lf]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4039
    apply (rule imageI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4040
    using k span_mono[of "b-{a}" b] by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4041
  hence "f x - k*s f a \<in> span (f ` b)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4042
    by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4043
  hence th: "-k *s f a \<in> span (f ` b)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4044
    using "2.prems"(5) by (simp add: vector_smult_lneg)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4045
  {assume k0: "k = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4046
    from k0 k have "x \<in> span (b -{a})" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4047
    then have "x \<in> span b" using span_mono[of "b-{a}" b]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4048
      by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4049
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4050
  {assume k0: "k \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4051
    from span_mul[OF th, of "- 1/ k"] k0
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4052
    have th1: "f a \<in> span (f ` b)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4053
      by (auto simp add: vector_smult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4054
    from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4055
    have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4056
    from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4057
    have "f a \<notin> span (f ` b)" using tha
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4058
      using "2.hyps"(2)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4059
      "2.prems"(3) by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4060
    with th1 have False by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4061
    then have "x \<in> span b" by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4062
  ultimately have xsb: "x \<in> span b" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4063
  from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4064
  show "x = 0" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4065
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4066
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4067
(* We can extend a linear mapping from basis.                                *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4068
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4069
lemma linear_independent_extend_lemma:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4070
  assumes fi: "finite B" and ib: "independent B"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4071
  shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4072
           \<and> (\<forall>x\<in> span B. \<forall>c. g (c*s x) = c *s g x)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4073
           \<and> (\<forall>x\<in> B. g x = f x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4074
using ib fi
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4075
proof(induct rule: finite_induct[OF fi])
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4076
  case 1 thus ?case by (auto simp add: span_empty)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4077
next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4078
  case (2 a b)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4079
  from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4080
    by (simp_all add: independent_insert)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4081
  from "2.hyps"(3)[OF ibf] obtain g where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4082
    g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4083
    "\<forall>x\<in>span b. \<forall>c. g (c *s x) = c *s g x" "\<forall>x\<in>b. g x = f x" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4084
  let ?h = "\<lambda>z. SOME k. (z - k *s a) \<in> span b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4085
  {fix z assume z: "z \<in> span (insert a b)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4086
    have th0: "z - ?h z *s a \<in> span b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4087
      apply (rule someI_ex)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4088
      unfolding span_breakdown_eq[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4089
      using z .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4090
    {fix k assume k: "z - k *s a \<in> span b"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4091
      have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4092
	by (simp add: ring_simps vector_sadd_rdistrib[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4093
      from span_sub[OF th0 k]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4094
      have khz: "(k - ?h z) *s a \<in> span b" by (simp add: eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4095
      {assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4096
	from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4097
	have "a \<in> span b" by (simp add: vector_smult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4098
	with "2.prems"(1) "2.hyps"(2) have False
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4099
	  by (auto simp add: dependent_def)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4100
      then have "k = ?h z" by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4101
    with th0 have "z - ?h z *s a \<in> span b \<and> (\<forall>k. z - k *s a \<in> span b \<longrightarrow> k = ?h z)" by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4102
  note h = this
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4103
  let ?g = "\<lambda>z. ?h z *s f a + g (z - ?h z *s a)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4104
  {fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4105
    have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4106
      by (vector ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4107
    have addh: "?h (x + y) = ?h x + ?h y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4108
      apply (rule conjunct2[OF h, rule_format, symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4109
      apply (rule span_add[OF x y])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4110
      unfolding tha
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4111
      by (metis span_add x y conjunct1[OF h, rule_format])
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4112
    have "?g (x + y) = ?g x + ?g y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4113
      unfolding addh tha
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4114
      g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4115
      by (simp add: vector_sadd_rdistrib)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4116
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4117
  {fix x:: "'a^'n" and c:: 'a  assume x: "x \<in> span (insert a b)"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4118
    have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4119
      by (vector ring_simps)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4120
    have hc: "?h (c *s x) = c * ?h x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4121
      apply (rule conjunct2[OF h, rule_format, symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4122
      apply (metis span_mul x)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4123
      by (metis tha span_mul x conjunct1[OF h])
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4124
    have "?g (c *s x) = c*s ?g x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4125
      unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4126
      by (vector ring_simps)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4127
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4128
  {fix x assume x: "x \<in> (insert a b)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4129
    {assume xa: "x = a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4130
      have ha1: "1 = ?h a"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4131
	apply (rule conjunct2[OF h, rule_format])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4132
	apply (metis span_superset insertI1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4133
	using conjunct1[OF h, OF span_superset, OF insertI1]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4134
	by (auto simp add: span_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4135
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4136
      from xa ha1[symmetric] have "?g x = f x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4137
	apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4138
	using g(2)[rule_format, OF span_0, of 0]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4139
	by simp}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4140
    moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4141
    {assume xb: "x \<in> b"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4142
      have h0: "0 = ?h x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4143
	apply (rule conjunct2[OF h, rule_format])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4144
	apply (metis  span_superset insertI1 xb x)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4145
	apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4146
	apply (metis span_superset xb)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4147
	done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4148
      have "?g x = f x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4149
	by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4150
    ultimately have "?g x = f x" using x by blast }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4151
  ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4152
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4153
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4154
lemma linear_independent_extend:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4155
  assumes iB: "independent (B:: (real ^'n::finite) set)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4156
  shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4157
proof-
30303
haftmann
parents: 30066
diff changeset
  4158
  from maximal_independent_subset_extend[of B UNIV] iB
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4159
  obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4160
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4161
  from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4162
  obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4163
           \<and> (\<forall>x\<in> span C. \<forall>c. g (c*s x) = c *s g x)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4164
           \<and> (\<forall>x\<in> C. g x = f x)" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4165
  from g show ?thesis unfolding linear_def using C
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4166
    apply clarsimp by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4167
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4168
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4169
(* Can construct an isomorphism between spaces of same dimension.            *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4170
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4171
lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4172
  and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4173
using fB c
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4174
proof(induct arbitrary: B rule: finite_induct[OF fA])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4175
  case 1 thus ?case by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4176
next
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4177
  case (2 x s t)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4178
  thus ?case
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4179
  proof(induct rule: finite_induct[OF "2.prems"(1)])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4180
    case 1    then show ?case by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4181
  next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4182
    case (2 y t)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4183
    from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4184
    from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4185
      f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4186
    from f "2.prems"(2) "2.hyps"(2) show ?case
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4187
      apply -
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4188
      apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4189
      by (auto simp add: inj_on_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4190
  qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4191
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4192
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4193
lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4194
  c: "card A = card B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4195
  shows "A = B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4196
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4197
  from fB AB have fA: "finite A" by (auto intro: finite_subset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4198
  from fA fB have fBA: "finite (B - A)" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4199
  have e: "A \<inter> (B - A) = {}" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4200
  have eq: "A \<union> (B - A) = B" using AB by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4201
  from card_Un_disjoint[OF fA fBA e, unfolded eq c]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4202
  have "card (B - A) = 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4203
  hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4204
  with AB show "A = B" by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4205
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4206
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4207
lemma subspace_isomorphism:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4208
  assumes s: "subspace (S:: (real ^'n::finite) set)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4209
  and t: "subspace (T :: (real ^ 'm::finite) set)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4210
  and d: "dim S = dim T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4211
  shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4212
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4213
  from basis_exists[of S] obtain B where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4214
    B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4215
  from basis_exists[of T] obtain C where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4216
    C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "C hassize dim T" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4217
  from B(4) C(4) card_le_inj[of B C] d obtain f where
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4218
    f: "f ` B \<subseteq> C" "inj_on f B" unfolding hassize_def by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4219
  from linear_independent_extend[OF B(2)] obtain g where
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4220
    g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4221
  from B(4) have fB: "finite B" by (simp add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4222
  from C(4) have fC: "finite C" by (simp add: hassize_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4223
  from inj_on_iff_eq_card[OF fB, of f] f(2)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4224
  have "card (f ` B) = card B" by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4225
  with B(4) C(4) have ceq: "card (f ` B) = card C" using d
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4226
    by (simp add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4227
  have "g ` B = f ` B" using g(2)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4228
    by (auto simp add: image_iff)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4229
  also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4230
  finally have gBC: "g ` B = C" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4231
  have gi: "inj_on g B" using f(2) g(2)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4232
    by (auto simp add: inj_on_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4233
  note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4234
  {fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4235
    from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4236
    from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4237
    have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4238
    have "x=y" using g0[OF th1 th0] by simp }
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4239
  then have giS: "inj_on g S"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4240
    unfolding inj_on_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4241
  from span_subspace[OF B(1,3) s]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4242
  have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4243
  also have "\<dots> = span C" unfolding gBC ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4244
  also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4245
  finally have gS: "g ` S = T" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4246
  from g(1) gS giS show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4247
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4248
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4249
(* linear functions are equal on a subspace if they are on a spanning set.   *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4250
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4251
lemma subspace_kernel:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4252
  assumes lf: "linear (f::'a::semiring_1 ^'n \<Rightarrow> _)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4253
  shows "subspace {x. f x = 0}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4254
apply (simp add: subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4255
by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4256
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4257
lemma linear_eq_0_span:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4258
  assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4259
  shows "\<forall>x \<in> span B. f x = (0::'a::semiring_1 ^'n)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4260
proof
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4261
  fix x assume x: "x \<in> span B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4262
  let ?P = "\<lambda>x. f x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4263
  from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4264
  with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4265
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4266
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4267
lemma linear_eq_0:
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4268
  assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4269
  shows "\<forall>x \<in> S. f x = (0::'a::semiring_1^'n)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4270
  by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4271
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4272
lemma linear_eq:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4273
  assumes lf: "linear (f::'a::ring_1^'n \<Rightarrow> _)" and lg: "linear g" and S: "S \<subseteq> span B"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4274
  and fg: "\<forall> x\<in> B. f x = g x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4275
  shows "\<forall>x\<in> S. f x = g x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4276
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4277
  let ?h = "\<lambda>x. f x - g x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4278
  from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4279
  from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4280
  show ?thesis by simp
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4281
qed
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4282
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4283
lemma linear_eq_stdbasis:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4284
  assumes lf: "linear (f::'a::ring_1^'m::finite \<Rightarrow> 'a^'n::finite)" and lg: "linear g"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4285
  and fg: "\<forall>i. f (basis i) = g(basis i)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4286
  shows "f = g"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4287
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4288
  let ?U = "UNIV :: 'm set"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4289
  let ?I = "{basis i:: 'a^'m|i. i \<in> ?U}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4290
  {fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4291
    from equalityD2[OF span_stdbasis]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4292
    have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4293
    from linear_eq[OF lf lg IU] fg x
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4294
    have "f x = g x" unfolding Collect_def  Ball_def mem_def by metis}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4295
  then show ?thesis by (auto intro: ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4296
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4297
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4298
(* Similar results for bilinear functions.                                   *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4299
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4300
lemma bilinear_eq:
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4301
  assumes bf: "bilinear (f:: 'a::ring^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4302
  and bg: "bilinear g"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4303
  and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4304
  and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4305
  shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4306
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4307
  let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4308
  from bf bg have sp: "subspace ?P"
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4309
    unfolding bilinear_def linear_def subspace_def bf bg
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4310
    by(auto simp add: span_0 mem_def bilinear_lzero[OF bf] bilinear_lzero[OF bg] span_add Ball_def intro:  bilinear_ladd[OF bf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4311
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4312
  have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4313
    apply -
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4314
    apply (rule ballI)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4315
    apply (rule span_induct[of B ?P])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4316
    defer
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4317
    apply (rule sp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4318
    apply assumption
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4319
    apply (clarsimp simp add: Ball_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4320
    apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4321
    using fg
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4322
    apply (auto simp add: subspace_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4323
    using bf bg unfolding bilinear_def linear_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4324
    by(auto simp add: span_0 mem_def bilinear_rzero[OF bf] bilinear_rzero[OF bg] span_add Ball_def intro:  bilinear_ladd[OF bf])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4325
  then show ?thesis using SB TC by (auto intro: ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4326
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4327
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4328
lemma bilinear_eq_stdbasis:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4329
  assumes bf: "bilinear (f:: 'a::ring_1^'m::finite \<Rightarrow> 'a^'n::finite \<Rightarrow> 'a^'p)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4330
  and bg: "bilinear g"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4331
  and fg: "\<forall>i j. f (basis i) (basis j) = g (basis i) (basis j)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4332
  shows "f = g"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4333
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4334
  from fg have th: "\<forall>x \<in> {basis i| i. i\<in> (UNIV :: 'm set)}. \<forall>y\<in>  {basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y" by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4335
  from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4336
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4337
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4338
(* Detailed theorems about left and right invertibility in general case.     *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4339
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4340
lemma left_invertible_transp:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4341
  "(\<exists>(B::'a^'n^'m). B ** transp (A::'a^'n^'m) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B::'a^'m^'n). A ** B = mat 1)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4342
  by (metis matrix_transp_mul transp_mat transp_transp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4343
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4344
lemma right_invertible_transp:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4345
  "(\<exists>(B::'a^'n^'m). transp (A::'a^'n^'m) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B::'a^'m^'n). B ** A = mat 1)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4346
  by (metis matrix_transp_mul transp_mat transp_transp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4347
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4348
lemma linear_injective_left_inverse:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4349
  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and fi: "inj f"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4350
  shows "\<exists>g. linear g \<and> g o f = id"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4351
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4352
  from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4353
  obtain h:: "real ^'m \<Rightarrow> real ^'n" where h: "linear h" " \<forall>x \<in> f ` {basis i|i. i \<in> (UNIV::'n set)}. h x = inv f x" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4354
  from h(2)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4355
  have th: "\<forall>i. (h \<circ> f) (basis i) = id (basis i)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4356
    using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4357
    by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4358
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4359
  from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4360
  have "h o f = id" .
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4361
  then show ?thesis using h(1) by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4362
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4363
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4364
lemma linear_surjective_right_inverse:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4365
  assumes lf: "linear (f:: real ^'m::finite \<Rightarrow> real ^'n::finite)" and sf: "surj f"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4366
  shows "\<exists>g. linear g \<and> f o g = id"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4367
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4368
  from linear_independent_extend[OF independent_stdbasis]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4369
  obtain h:: "real ^'n \<Rightarrow> real ^'m" where
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4370
    h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> (UNIV :: 'n set)}. h x = inv f x" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4371
  from h(2)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4372
  have th: "\<forall>i. (f o h) (basis i) = id (basis i)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4373
    using sf
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4374
    apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4375
    apply (erule_tac x="basis i" in allE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4376
    by auto
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4377
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4378
  from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4379
  have "f o h = id" .
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4380
  then show ?thesis using h(1) by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4381
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4382
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4383
lemma matrix_left_invertible_injective:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4384
"(\<exists>B. (B::real^'m^'n) ** (A::real^'n::finite^'m::finite) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4385
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4386
  {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4387
    from xy have "B*v (A *v x) = B *v (A*v y)" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4388
    hence "x = y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4389
      unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4390
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4391
  {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4392
    hence i: "inj (op *v A)" unfolding inj_on_def by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4393
    from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4394
    obtain g where g: "linear g" "g o op *v A = id" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4395
    have "matrix g ** A = mat 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4396
      unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4397
      using g(2) by (simp add: o_def id_def stupid_ext)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4398
    then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4399
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4400
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4401
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4402
lemma matrix_left_invertible_ker:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4403
  "(\<exists>B. (B::real ^'m::finite^'n::finite) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4404
  unfolding matrix_left_invertible_injective
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4405
  using linear_injective_0[OF matrix_vector_mul_linear, of A]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4406
  by (simp add: inj_on_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4407
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4408
lemma matrix_right_invertible_surjective:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4409
"(\<exists>B. (A::real^'n::finite^'m::finite) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4410
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4411
  {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4412
    {fix x :: "real ^ 'm"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4413
      have "A *v (B *v x) = x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4414
	by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4415
    hence "surj (op *v A)" unfolding surj_def by metis }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4416
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4417
  {assume sf: "surj (op *v A)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4418
    from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4419
    obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4420
      by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4421
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4422
    have "A ** (matrix g) = mat 1"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4423
      unfolding matrix_eq  matrix_vector_mul_lid
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4424
	matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4425
      using g(2) unfolding o_def stupid_ext[symmetric] id_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4426
      .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4427
    hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4428
  }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4429
  ultimately show ?thesis unfolding surj_def by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4430
qed
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4431
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4432
lemma matrix_left_invertible_independent_columns:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4433
  fixes A :: "real^'n::finite^'m::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4434
  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4435
   (is "?lhs \<longleftrightarrow> ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4436
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4437
  let ?U = "UNIV :: 'n set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4438
  {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4439
    {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4440
      and i: "i \<in> ?U"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4441
      let ?x = "\<chi> i. c i"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4442
      have th0:"A *v ?x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4443
	using c
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4444
	unfolding matrix_mult_vsum Cart_eq
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4445
	by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4446
      from k[rule_format, OF th0] i
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4447
      have "c i = 0" by (vector Cart_eq)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4448
    hence ?rhs by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4449
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4450
  {assume H: ?rhs
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4451
    {fix x assume x: "A *v x = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4452
      let ?c = "\<lambda>i. ((x$i ):: real)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4453
      from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4454
      have "x = 0" by vector}}
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4455
  ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4456
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4457
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4458
lemma matrix_right_invertible_independent_rows:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4459
  fixes A :: "real^'n::finite^'m::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4460
  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4461
  unfolding left_invertible_transp[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4462
    matrix_left_invertible_independent_columns
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4463
  by (simp add: column_transp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4464
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4465
lemma matrix_right_invertible_span_columns:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4466
  "(\<exists>(B::real ^'n::finite^'m::finite). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4467
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4468
  let ?U = "UNIV :: 'm set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4469
  have fU: "finite ?U" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4470
  have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4471
    unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4472
    apply (subst eq_commute) ..
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4473
  have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4474
  {assume h: ?lhs
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4475
    {fix x:: "real ^'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4476
	from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4477
	  where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4478
	have "x \<in> span (columns A)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4479
	  unfolding y[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4480
	  apply (rule span_setsum[OF fU])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4481
	  apply clarify
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4482
	  apply (rule span_mul)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4483
	  apply (rule span_superset)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4484
	  unfolding columns_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4485
	  by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4486
    then have ?rhs unfolding rhseq by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4487
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4488
  {assume h:?rhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4489
    let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4490
    {fix y have "?P y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4491
      proof(rule span_induct_alt[of ?P "columns A"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4492
	show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4493
	  apply (rule exI[where x=0])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4494
	  by (simp add: zero_index vector_smult_lzero)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4495
      next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4496
	fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4497
	from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4498
	  unfolding columns_def by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4499
	from y2 obtain x:: "real ^'m" where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4500
	  x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4501
	let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4502
	show "?P (c*s y1 + y2)"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4503
	  proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] cond_value_iff right_distrib cond_application_beta cong del: if_weak_cong)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4504
	    fix j
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4505
	    have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4506
           else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" using i(1)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4507
	      by (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4508
	    have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4509
           else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4510
	      apply (rule setsum_cong[OF refl])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4511
	      using th by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4512
	    also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4513
	      by (simp add: setsum_addf)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4514
	    also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4515
	      unfolding setsum_delta[OF fU]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4516
	      using i(1) by simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4517
	    finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4518
           else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4519
	  qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4520
	next
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4521
	  show "y \<in> span (columns A)" unfolding h by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4522
	qed}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4523
    then have ?lhs unfolding lhseq ..}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4524
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4525
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4526
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4527
lemma matrix_left_invertible_span_rows:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4528
  "(\<exists>(B::real^'m::finite^'n::finite). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4529
  unfolding right_invertible_transp[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4530
  unfolding columns_transp[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4531
  unfolding matrix_right_invertible_span_columns
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4532
 ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4533
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4534
(* An injective map real^'n->real^'n is also surjective.                       *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4535
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4536
lemma linear_injective_imp_surjective:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4537
  assumes lf: "linear (f:: real ^'n::finite \<Rightarrow> real ^'n)" and fi: "inj f"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4538
  shows "surj f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4539
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4540
  let ?U = "UNIV :: (real ^'n) set"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4541
  from basis_exists[of ?U] obtain B
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4542
    where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4543
    by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4544
  from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4545
  have th: "?U \<subseteq> span (f ` B)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4546
    apply (rule card_ge_dim_independent)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4547
    apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4548
    apply (rule independent_injective_image[OF B(2) lf fi])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4549
    apply (rule order_eq_refl)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4550
    apply (rule sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4551
    unfolding d
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4552
    apply (rule card_image)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4553
    apply (rule subset_inj_on[OF fi])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4554
    by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4555
  from th show ?thesis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4556
    unfolding span_linear_image[OF lf] surj_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4557
    using B(3) by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4558
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4559
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4560
(* And vice versa.                                                           *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4561
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4562
lemma surjective_iff_injective_gen:
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4563
  assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4564
  and ST: "f ` S \<subseteq> T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4565
  shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4566
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4567
  {assume h: "?lhs"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4568
    {fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4569
      from x fS have S0: "card S \<noteq> 0" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4570
      {assume xy: "x \<noteq> y"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4571
	have th: "card S \<le> card (f ` (S - {y}))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4572
	  unfolding c
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4573
	  apply (rule card_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4574
	  apply (rule finite_imageI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4575
	  using fS apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4576
	  using h xy x y f unfolding subset_eq image_iff
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4577
	  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4578
	  apply (case_tac "xa = f x")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4579
	  apply (rule bexI[where x=x])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4580
	  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4581
	  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4582
	also have " \<dots> \<le> card (S -{y})"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4583
	  apply (rule card_image_le)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4584
	  using fS by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4585
	also have "\<dots> \<le> card S - 1" using y fS by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4586
	finally have False  using S0 by arith }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4587
      then have "x = y" by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4588
    then have ?rhs unfolding inj_on_def by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4589
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4590
  {assume h: ?rhs
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4591
    have "f ` S = T"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4592
      apply (rule card_subset_eq[OF fT ST])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4593
      unfolding card_image[OF h] using c .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4594
    then have ?lhs by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4595
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4596
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4597
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4598
lemma linear_surjective_imp_injective:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4599
  assumes lf: "linear (f::real ^'n::finite => real ^'n)" and sf: "surj f"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4600
  shows "inj f"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4601
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4602
  let ?U = "UNIV :: (real ^'n) set"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4603
  from basis_exists[of ?U] obtain B
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4604
    where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "B hassize dim ?U"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4605
    by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4606
  {fix x assume x: "x \<in> span B" and fx: "f x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4607
    from B(4) have fB: "finite B" by (simp add: hassize_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4608
    from B(4) have d: "dim ?U = card B" by (simp add: hassize_def)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4609
    have fBi: "independent (f ` B)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4610
      apply (rule card_le_dim_spanning[of "f ` B" ?U])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4611
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4612
      using sf B(3)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4613
      unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4614
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4615
      using fB apply (blast intro: finite_imageI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4616
      unfolding d
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4617
      apply (rule card_image_le)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4618
      apply (rule fB)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4619
      done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4620
    have th0: "dim ?U \<le> card (f ` B)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4621
      apply (rule span_card_ge_dim)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4622
      apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4623
      unfolding span_linear_image[OF lf]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4624
      apply (rule subset_trans[where B = "f ` UNIV"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4625
      using sf unfolding surj_def apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4626
      apply (rule image_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4627
      apply (rule B(3))
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4628
      apply (metis finite_imageI fB)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4629
      done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4630
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4631
    moreover have "card (f ` B) \<le> card B"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4632
      by (rule card_image_le, rule fB)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4633
    ultimately have th1: "card B = card (f ` B)" unfolding d by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4634
    have fiB: "inj_on f B"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4635
      unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4636
    from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4637
    have "x = 0" by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4638
  note th = this
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4639
  from th show ?thesis unfolding linear_injective_0[OF lf]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4640
    using B(3) by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4641
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4642
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4643
(* Hence either is enough for isomorphism.                                   *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4644
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4645
lemma left_right_inverse_eq:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4646
  assumes fg: "f o g = id" and gh: "g o h = id"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4647
  shows "f = h"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4648
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4649
  have "f = f o (g o h)" unfolding gh by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4650
  also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4651
  finally show "f = h" unfolding fg by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4652
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4653
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4654
lemma isomorphism_expand:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4655
  "f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4656
  by (simp add: expand_fun_eq o_def id_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4657
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4658
lemma linear_injective_isomorphism:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4659
  assumes lf: "linear (f :: real^'n::finite \<Rightarrow> real ^'n)" and fi: "inj f"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4660
  shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4661
unfolding isomorphism_expand[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4662
using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4663
by (metis left_right_inverse_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4664
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4665
lemma linear_surjective_isomorphism:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4666
  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and sf: "surj f"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4667
  shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4668
unfolding isomorphism_expand[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4669
using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4670
by (metis left_right_inverse_eq)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4671
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4672
(* Left and right inverses are the same for R^N->R^N.                        *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4673
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4674
lemma linear_inverse_left:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4675
  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and lf': "linear f'"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4676
  shows "f o f' = id \<longleftrightarrow> f' o f = id"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4677
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4678
  {fix f f':: "real ^'n \<Rightarrow> real ^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4679
    assume lf: "linear f" "linear f'" and f: "f o f' = id"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4680
    from f have sf: "surj f"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4681
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4682
      apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4683
      by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4684
    from linear_surjective_isomorphism[OF lf(1) sf] lf f
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4685
    have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4686
      by metis}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4687
  then show ?thesis using lf lf' by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4688
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4689
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4690
(* Moreover, a one-sided inverse is automatically linear.                    *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4691
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4692
lemma left_inverse_linear:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4693
  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and gf: "g o f = id"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4694
  shows "linear g"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4695
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4696
  from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4697
    by metis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4698
  from linear_injective_isomorphism[OF lf fi]
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4699
  obtain h:: "real ^'n \<Rightarrow> real ^'n" where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4700
    h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4701
  have "h = g" apply (rule ext) using gf h(2,3)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4702
    apply (simp add: o_def id_def stupid_ext[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4703
    by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4704
  with h(1) show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4705
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4706
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4707
lemma right_inverse_linear:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4708
  assumes lf: "linear (f:: real ^'n::finite \<Rightarrow> real ^'n)" and gf: "f o g = id"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4709
  shows "linear g"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4710
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4711
  from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4712
    by metis
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4713
  from linear_surjective_isomorphism[OF lf fi]
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4714
  obtain h:: "real ^'n \<Rightarrow> real ^'n" where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4715
    h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4716
  have "h = g" apply (rule ext) using gf h(2,3)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4717
    apply (simp add: o_def id_def stupid_ext[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4718
    by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4719
  with h(1) show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4720
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4721
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4722
(* The same result in terms of square matrices.                              *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4723
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4724
lemma matrix_left_right_inverse:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4725
  fixes A A' :: "real ^'n::finite^'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4726
  shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4727
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4728
  {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4729
    have sA: "surj (op *v A)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4730
      unfolding surj_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4731
      apply clarify
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4732
      apply (rule_tac x="(A' *v y)" in exI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4733
      by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4734
    from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4735
    obtain f' :: "real ^'n \<Rightarrow> real ^'n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4736
      where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4737
    have th: "matrix f' ** A = mat 1"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4738
      by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4739
    hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4740
    hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4741
    hence "matrix f' ** A = A' ** A" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4742
    hence "A' ** A = mat 1" by (simp add: th)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4743
  then show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4744
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4745
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4746
(* Considering an n-element vector as an n-by-1 or 1-by-n matrix.            *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4747
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4748
definition "rowvector v = (\<chi> i j. (v$j))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4749
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4750
definition "columnvector v = (\<chi> i j. (v$i))"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4751
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4752
lemma transp_columnvector:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4753
 "transp(columnvector v) = rowvector v"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4754
  by (simp add: transp_def rowvector_def columnvector_def Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4755
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4756
lemma transp_rowvector: "transp(rowvector v) = columnvector v"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4757
  by (simp add: transp_def columnvector_def rowvector_def Cart_eq)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4758
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4759
lemma dot_rowvector_columnvector:
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4760
  "columnvector (A *v v) = A ** columnvector v"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4761
  by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4762
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4763
lemma dot_matrix_product: "(x::'a::semiring_1^'n::finite) \<bullet> y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4764
  by (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4765
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4766
lemma dot_matrix_vector_mul:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4767
  fixes A B :: "real ^'n::finite ^'n" and x y :: "real ^'n"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4768
  shows "(A *v x) \<bullet> (B *v y) =
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4769
      (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4770
unfolding dot_matrix_product transp_columnvector[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4771
  dot_rowvector_columnvector matrix_transp_mul matrix_mul_assoc ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4772
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4773
(* Infinity norm.                                                            *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4774
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4775
definition "infnorm (x::real^'n::finite) = rsup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4776
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4777
lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4778
  by auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4779
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4780
lemma infnorm_set_image:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4781
  "{abs(x$i) |i. i\<in> (UNIV :: 'n set)} =
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4782
  (\<lambda>i. abs(x$i)) ` (UNIV :: 'n set)" by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4783
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4784
lemma infnorm_set_lemma:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4785
  shows "finite {abs((x::'a::abs ^'n::finite)$i) |i. i\<in> (UNIV :: 'n set)}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4786
  and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4787
  unfolding infnorm_set_image
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4788
  by (auto intro: finite_imageI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4789
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4790
lemma infnorm_pos_le: "0 \<le> infnorm (x::real^'n::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4791
  unfolding infnorm_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4792
  unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4793
  unfolding infnorm_set_image
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4794
  by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4795
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4796
lemma infnorm_triangle: "infnorm ((x::real^'n::finite) + y) \<le> infnorm x + infnorm y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4797
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4798
  have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4799
  have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4800
  have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4801
  show ?thesis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4802
  unfolding infnorm_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4803
  unfolding rsup_finite_le_iff[ OF infnorm_set_lemma]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4804
  apply (subst diff_le_eq[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4805
  unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4806
  unfolding infnorm_set_image bex_simps
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4807
  apply (subst th)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4808
  unfolding th1
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4809
  unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4810
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4811
  unfolding infnorm_set_image ball_simps bex_simps
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4812
  apply simp
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4813
  apply (metis th2)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4814
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4815
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4816
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4817
lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n::finite) = 0"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4818
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4819
  have "infnorm x <= 0 \<longleftrightarrow> x = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4820
    unfolding infnorm_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4821
    unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4822
    unfolding infnorm_set_image ball_simps
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4823
    by vector
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4824
  then show ?thesis using infnorm_pos_le[of x] by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4825
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4826
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4827
lemma infnorm_0: "infnorm 0 = 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4828
  by (simp add: infnorm_eq_0)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4829
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4830
lemma infnorm_neg: "infnorm (- x) = infnorm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4831
  unfolding infnorm_def
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4832
  apply (rule cong[of "rsup" "rsup"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4833
  apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4834
  apply (rule set_ext)
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4835
  apply auto
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4836
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4837
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4838
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4839
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4840
  have "y - x = - (x - y)" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4841
  then show ?thesis  by (metis infnorm_neg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4842
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4843
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4844
lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4845
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4846
  have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4847
    by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4848
  from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4849
  have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4850
    "infnorm y \<le> infnorm (x - y) + infnorm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4851
    by (simp_all add: ring_simps infnorm_neg diff_def[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4852
  from th[OF ths]  show ?thesis .
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4853
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4854
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4855
lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4856
  using infnorm_pos_le[of x] by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4857
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4858
lemma component_le_infnorm:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4859
  shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4860
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4861
  let ?U = "UNIV :: 'n set"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4862
  let ?S = "{\<bar>x$i\<bar> |i. i\<in> ?U}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4863
  have fS: "finite ?S" unfolding image_Collect[symmetric]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4864
    apply (rule finite_imageI) unfolding Collect_def mem_def by simp
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4865
  have S0: "?S \<noteq> {}" by blast
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4866
  have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4867
  from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4868
  show ?thesis unfolding infnorm_def isUb_def setle_def
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4869
    unfolding infnorm_set_image ball_simps by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4870
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4871
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4872
lemma infnorm_mul_lemma: "infnorm(a *s x) <= \<bar>a\<bar> * infnorm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4873
  apply (subst infnorm_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4874
  unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4875
  unfolding infnorm_set_image ball_simps
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4876
  apply (simp add: abs_mult)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4877
  apply (rule allI)
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4878
  apply (cut_tac component_le_infnorm[of x])
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4879
  apply (rule mult_mono)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4880
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4881
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4882
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4883
lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4884
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4885
  {assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4886
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4887
  {assume a0: "a \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4888
    from a0 have th: "(1/a) *s (a *s x) = x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4889
      by (simp add: vector_smult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4890
    from a0 have ap: "\<bar>a\<bar> > 0" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4891
    from infnorm_mul_lemma[of "1/a" "a *s x"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4892
    have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*s x)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4893
      unfolding th by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4894
    with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *s x))" by (simp add: field_simps)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4895
    then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4896
      using ap by (simp add: field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4897
    with infnorm_mul_lemma[of a x] have ?thesis by arith }
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4898
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4899
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4900
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4901
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4902
  using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4903
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4904
(* Prove that it differs only up to a bound from Euclidean norm.             *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4905
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4906
lemma infnorm_le_norm: "infnorm x \<le> norm x"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4907
  unfolding infnorm_def rsup_finite_le_iff[OF infnorm_set_lemma]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4908
  unfolding infnorm_set_image  ball_simps
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4909
  by (metis component_le_norm)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4910
lemma card_enum: "card {1 .. n} = n" by auto
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4911
lemma norm_le_infnorm: "norm(x) <= sqrt(real CARD('n)) * infnorm(x::real ^'n::finite)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4912
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4913
  let ?d = "CARD('n)"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4914
  have "real ?d \<ge> 0" by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4915
  hence d2: "(sqrt (real ?d))^2 = real ?d"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4916
    by (auto intro: real_sqrt_pow2)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4917
  have th: "sqrt (real ?d) * infnorm x \<ge> 0"
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4918
    by (simp add: zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4919
  have th1: "x\<bullet>x \<le> (sqrt (real ?d) * infnorm x)^2"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4920
    unfolding power_mult_distrib d2
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4921
    apply (subst power2_abs[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4922
    unfolding real_of_nat_def dot_def power2_eq_square[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4923
    apply (subst power2_abs[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4924
    apply (rule setsum_bounded)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4925
    apply (rule power_mono)
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4926
    unfolding abs_of_nonneg[OF infnorm_pos_le]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4927
    unfolding infnorm_def  rsup_finite_ge_iff[OF infnorm_set_lemma]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4928
    unfolding infnorm_set_image bex_simps
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4929
    apply blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4930
    by (rule abs_ge_zero)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4931
  from real_le_lsqrt[OF dot_pos_le th th1]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4932
  show ?thesis unfolding real_vector_norm_def id_def .
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4933
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4934
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4935
(* Equality in Cauchy-Schwarz and triangle inequalities.                     *)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4936
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4937
lemma norm_cauchy_schwarz_eq: "(x::real ^'n::finite) \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *s y = norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4938
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4939
  {assume h: "x = 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4940
    hence ?thesis by simp}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4941
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4942
  {assume h: "y = 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4943
    hence ?thesis by simp}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4944
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4945
  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4946
    from dot_eq_0[of "norm y *s x - norm x *s y"]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4947
    have "?rhs \<longleftrightarrow> (norm y * (norm y * norm x * norm x - norm x * (x \<bullet> y)) - norm x * (norm y * (y \<bullet> x) - norm x * norm y * norm y) =  0)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4948
      using x y
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4949
      unfolding dot_rsub dot_lsub dot_lmult dot_rmult
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4950
      unfolding norm_pow_2[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4951
      apply (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4952
      apply metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4953
      done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4954
    also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4955
      by (simp add: ring_simps dot_sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4956
    also have "\<dots> \<longleftrightarrow> ?lhs" using x y
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4957
      apply simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4958
      by metis
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4959
    finally have ?thesis by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4960
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4961
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4962
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4963
lemma norm_cauchy_schwarz_abs_eq:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4964
  fixes x y :: "real ^ 'n::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4965
  shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4966
                norm x *s y = norm y *s x \<or> norm(x) *s y = - norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4967
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4968
  have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4969
  have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4970
    apply simp by vector
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4971
  also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4972
     (-x) \<bullet> y = norm x * norm y)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4973
    unfolding norm_cauchy_schwarz_eq[symmetric]
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4974
    unfolding norm_minus_cancel
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4975
      norm_mul by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4976
  also have "\<dots> \<longleftrightarrow> ?lhs"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4977
    unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] dot_lneg
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4978
    by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4979
  finally show ?thesis ..
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4980
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4981
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4982
lemma norm_triangle_eq:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4983
  fixes x y :: "real ^ 'n::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  4984
  shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4985
proof-
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4986
  {assume x: "x =0 \<or> y =0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4987
    hence ?thesis by (cases "x=0", simp_all)}
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4988
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4989
  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4990
    hence "norm x \<noteq> 0" "norm y \<noteq> 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4991
      by simp_all
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  4992
    hence n: "norm x > 0" "norm y > 0"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4993
      using norm_ge_zero[of x] norm_ge_zero[of y]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4994
      by arith+
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4995
    have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4996
    have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  4997
      apply (rule th) using n norm_ge_zero[of "x + y"]
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4998
      by arith
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  4999
    also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5000
      unfolding norm_cauchy_schwarz_eq[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5001
      unfolding norm_pow_2 dot_ladd dot_radd
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5002
      by (simp add: norm_pow_2[symmetric] power2_eq_square dot_sym ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5003
    finally have ?thesis .}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5004
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5005
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5006
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5007
(* Collinearity.*)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5008
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5009
definition "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *s u)"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5010
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5011
lemma collinear_empty:  "collinear {}" by (simp add: collinear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5012
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  5013
lemma collinear_sing: "collinear {(x::'a::ring_1^'n)}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5014
  apply (simp add: collinear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5015
  apply (rule exI[where x=0])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5016
  by simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5017
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5018
lemma collinear_2: "collinear {(x::'a::ring_1^'n),y}"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5019
  apply (simp add: collinear_def)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5020
  apply (rule exI[where x="x - y"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5021
  apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5022
  apply (rule exI[where x=0], simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5023
  apply (rule exI[where x=1], simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5024
  apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5025
  apply (rule exI[where x=0], simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5026
  done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5027
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5028
lemma collinear_lemma: "collinear {(0::real^'n),x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *s x)" (is "?lhs \<longleftrightarrow> ?rhs")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5029
proof-
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  5030
  {assume "x=0 \<or> y = 0" hence ?thesis
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5031
      by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5032
  moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5033
  {assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5034
    {assume h: "?lhs"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5035
      then obtain u where u: "\<forall> x\<in> {0,x,y}. \<forall>y\<in> {0,x,y}. \<exists>c. x - y = c *s u" unfolding collinear_def by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5036
      from u[rule_format, of x 0] u[rule_format, of y 0]
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  5037
      obtain cx and cy where
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5038
	cx: "x = cx*s u" and cy: "y = cy*s u"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5039
	by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5040
      from cx x have cx0: "cx \<noteq> 0" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5041
      from cy y have cy0: "cy \<noteq> 0" by auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5042
      let ?d = "cy / cx"
30489
5d7d0add1741 remove trailing spaces
huffman
parents: 30305
diff changeset
  5043
      from cx cy cx0 have "y = ?d *s x"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5044
	by (simp add: vector_smult_assoc)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5045
      hence ?rhs using x y by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5046
    moreover
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5047
    {assume h: "?rhs"
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5048
      then obtain c where c: "y = c*s x" using x y by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5049
      have ?lhs unfolding collinear_def c
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5050
	apply (rule exI[where x=x])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5051
	apply auto
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5052
	apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5053
	apply (rule exI[where x= "-c"], simp only: vector_smult_lneg)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5054
	apply (rule exI[where x=1], simp)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5055
	apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5056
	apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5057
	done}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5058
    ultimately have ?thesis by blast}
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5059
  ultimately show ?thesis by blast
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5060
qed
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5061
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  5062
lemma norm_cauchy_schwarz_equal:
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  5063
  fixes x y :: "real ^ 'n::finite"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  5064
  shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5065
unfolding norm_cauchy_schwarz_abs_eq
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  5066
apply (cases "x=0", simp_all add: collinear_2)
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  5067
apply (cases "y=0", simp_all add: collinear_2 insert_commute)
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5068
unfolding collinear_lemma
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5069
apply simp
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5070
apply (subgoal_tac "norm x \<noteq> 0")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5071
apply (subgoal_tac "norm y \<noteq> 0")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5072
apply (rule iffI)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5073
apply (cases "norm x *s y = norm y *s x")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5074
apply (rule exI[where x="(1/norm x) * norm y"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5075
apply (drule sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5076
unfolding vector_smult_assoc[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5077
apply (simp add: vector_smult_assoc field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5078
apply (rule exI[where x="(1/norm x) * - norm y"])
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5079
apply clarify
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5080
apply (drule sym)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5081
unfolding vector_smult_assoc[symmetric]
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5082
apply (simp add: vector_smult_assoc field_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5083
apply (erule exE)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5084
apply (erule ssubst)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5085
unfolding vector_smult_assoc
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5086
unfolding norm_mul
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5087
apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5088
apply (case_tac "c <= 0", simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5089
apply (simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5090
apply (case_tac "c <= 0", simp add: ring_simps)
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5091
apply (simp add: ring_simps)
30041
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  5092
apply simp
9becd197a40e remove duplicated lemmas about norm
huffman
parents: 30040
diff changeset
  5093
apply simp
29842
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5094
done
4ac60c7d9b78 (Real) Vectors in Euclidean space, and elementary linear algebra.
chaieb
parents:
diff changeset
  5095
30039
7208c88df507 fix real_vector, real_algebra instances
huffman
parents: 29906
diff changeset
  5096
end