(* Title: Library/Multivariate_Analysis/Euclidean_Space.thy
Author: Amine Chaieb, University of Cambridge
*)
header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
theory Euclidean_Space
imports
Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
Finite_Cartesian_Product Infinite_Set Numeral_Type
Inner_Product L2_Norm
uses "positivstellensatz.ML" ("normarith.ML")
begin
subsection{* Basic componentwise operations on vectors. *}
instantiation cart :: (plus,finite) plus
begin
definition vector_add_def : "op + \<equiv> (\<lambda> x y. (\<chi> i. (x$i) + (y$i)))"
instance ..
end
instantiation cart :: (times,finite) times
begin
definition vector_mult_def : "op * \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))"
instance ..
end
instantiation cart :: (minus,finite) minus
begin
definition vector_minus_def : "op - \<equiv> (\<lambda> x y. (\<chi> i. (x$i) - (y$i)))"
instance ..
end
instantiation cart :: (uminus,finite) uminus
begin
definition vector_uminus_def : "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x$i)))"
instance ..
end
instantiation cart :: (zero,finite) zero
begin
definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
instance ..
end
instantiation cart :: (one,finite) one
begin
definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
instance ..
end
instantiation cart :: (scaleR, finite) scaleR
begin
definition vector_scaleR_def: "scaleR = (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
instance ..
end
instantiation cart :: (ord,finite) ord
begin
definition vector_le_def:
"less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"
definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"
instance by (intro_classes)
end
text{* The ordering on one-dimensional vectors is linear. *}
class cart_one = assumes UNIV_one: "card (UNIV \<Colon> 'a set) = Suc 0"
begin
subclass finite
proof from UNIV_one show "finite (UNIV :: 'a set)"
by (auto intro!: card_ge_0_finite) qed
end
instantiation cart :: (linorder,cart_one) linorder begin
instance proof
guess a B using UNIV_one[where 'a='b] unfolding card_Suc_eq apply- by(erule exE)+
hence *:"UNIV = {a}" by auto
have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P a" unfolding * by auto hence all:"\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a" by auto
fix x y z::"'a^'b::cart_one" note * = vector_le_def vector_less_def all Cart_eq
show "x\<le>x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x\<le>y \<or> y\<le>x" unfolding * by(auto simp only:field_simps)
{ assume "x\<le>y" "y\<le>z" thus "x\<le>z" unfolding * by(auto simp only:field_simps) }
{ assume "x\<le>y" "y\<le>x" thus "x=y" unfolding * by(auto simp only:field_simps) }
qed end
text{* Also the scalar-vector multiplication. *}
definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
where "c *s x = (\<chi> i. c * (x$i))"
text{* Constant Vectors *}
definition "vec x = (\<chi> i. x)"
subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
method_setup vector = {*
let
val ss1 = HOL_basic_ss addsimps [@{thm setsum_addf} RS sym,
@{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
@{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
val ss2 = @{simpset} addsimps
[@{thm vector_add_def}, @{thm vector_mult_def},
@{thm vector_minus_def}, @{thm vector_uminus_def},
@{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
@{thm vector_scaleR_def},
@{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}]
fun vector_arith_tac ths =
simp_tac ss1
THEN' (fn i => rtac @{thm setsum_cong2} i
ORELSE rtac @{thm setsum_0'} i
ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
(* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *)
THEN' asm_full_simp_tac (ss2 addsimps ths)
in
Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
end
*} "Lifts trivial vector statements to real arith statements"
lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
text{* Obvious "component-pushing". *}
lemma vec_component [simp]: "vec x $ i = x"
by (vector vec_def)
lemma vector_add_component [simp]: "(x + y)$i = x$i + y$i"
by vector
lemma vector_minus_component [simp]: "(x - y)$i = x$i - y$i"
by vector
lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
by vector
lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
by vector
lemma vector_uminus_component [simp]: "(- x)$i = - (x$i)"
by vector
lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
by vector
lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
lemmas vector_component =
vec_component vector_add_component vector_mult_component
vector_smult_component vector_minus_component vector_uminus_component
vector_scaleR_component cond_component
subsection {* Some frequently useful arithmetic lemmas over vectors. *}
instance cart :: (semigroup_add,finite) semigroup_add
apply (intro_classes) by (vector add_assoc)
instance cart :: (monoid_add,finite) monoid_add
apply (intro_classes) by vector+
instance cart :: (group_add,finite) group_add
apply (intro_classes) by (vector algebra_simps)+
instance cart :: (ab_semigroup_add,finite) ab_semigroup_add
apply (intro_classes) by (vector add_commute)
instance cart :: (comm_monoid_add,finite) comm_monoid_add
apply (intro_classes) by vector
instance cart :: (ab_group_add,finite) ab_group_add
apply (intro_classes) by vector+
instance cart :: (cancel_semigroup_add,finite) cancel_semigroup_add
apply (intro_classes)
by (vector Cart_eq)+
instance cart :: (cancel_ab_semigroup_add,finite) cancel_ab_semigroup_add
apply (intro_classes)
by (vector Cart_eq)
instance cart :: (real_vector, finite) real_vector
by default (vector scaleR_left_distrib scaleR_right_distrib)+
instance cart :: (semigroup_mult,finite) semigroup_mult
apply (intro_classes) by (vector mult_assoc)
instance cart :: (monoid_mult,finite) monoid_mult
apply (intro_classes) by vector+
instance cart :: (ab_semigroup_mult,finite) ab_semigroup_mult
apply (intro_classes) by (vector mult_commute)
instance cart :: (ab_semigroup_idem_mult,finite) ab_semigroup_idem_mult
apply (intro_classes) by (vector mult_idem)
instance cart :: (comm_monoid_mult,finite) comm_monoid_mult
apply (intro_classes) by vector
fun vector_power where
"vector_power x 0 = 1"
| "vector_power x (Suc n) = x * vector_power x n"
instance cart :: (semiring,finite) semiring
apply (intro_classes) by (vector field_simps)+
instance cart :: (semiring_0,finite) semiring_0
apply (intro_classes) by (vector field_simps)+
instance cart :: (semiring_1,finite) semiring_1
apply (intro_classes) by vector
instance cart :: (comm_semiring,finite) comm_semiring
apply (intro_classes) by (vector field_simps)+
instance cart :: (comm_semiring_0,finite) comm_semiring_0 by (intro_classes)
instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
instance cart :: (semiring_0_cancel,finite) semiring_0_cancel by (intro_classes)
instance cart :: (comm_semiring_0_cancel,finite) comm_semiring_0_cancel by (intro_classes)
instance cart :: (ring,finite) ring by (intro_classes)
instance cart :: (semiring_1_cancel,finite) semiring_1_cancel by (intro_classes)
instance cart :: (comm_semiring_1,finite) comm_semiring_1 by (intro_classes)
instance cart :: (ring_1,finite) ring_1 ..
instance cart :: (real_algebra,finite) real_algebra
apply intro_classes
apply (simp_all add: vector_scaleR_def field_simps)
apply vector
apply vector
done
instance cart :: (real_algebra_1,finite) real_algebra_1 ..
lemma of_nat_index:
"(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
apply (induct n)
apply vector
apply vector
done
lemma zero_index[simp]:
"(0 :: 'a::zero ^'n)$i = 0" by vector
lemma one_index[simp]:
"(1 :: 'a::one ^'n)$i = 1" by vector
lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
proof-
have "(1::'a) + of_nat n = 0 \<longleftrightarrow> of_nat 1 + of_nat n = (of_nat 0 :: 'a)" by simp
also have "\<dots> \<longleftrightarrow> 1 + n = 0" by (simp only: of_nat_add[symmetric] of_nat_eq_iff)
finally show ?thesis by simp
qed
instance cart :: (semiring_char_0,finite) semiring_char_0
proof (intro_classes)
fix m n ::nat
show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
by (simp add: Cart_eq of_nat_index)
qed
instance cart :: (comm_ring_1,finite) comm_ring_1 by intro_classes
instance cart :: (ring_char_0,finite) ring_char_0 by intro_classes
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
by (vector mult_assoc)
lemma vector_sadd_rdistrib: "((a::'a::semiring) + b) *s x = a *s x + b *s x"
by (vector field_simps)
lemma vector_add_ldistrib: "(c::'a::semiring) *s (x + y) = c *s x + c *s y"
by (vector field_simps)
lemma vector_smult_lzero[simp]: "(0::'a::mult_zero) *s x = 0" by vector
lemma vector_smult_lid[simp]: "(1::'a::monoid_mult) *s x = x" by vector
lemma vector_ssub_ldistrib: "(c::'a::ring) *s (x - y) = c *s x - c *s y"
by (vector field_simps)
lemma vector_smult_rneg: "(c::'a::ring) *s -x = -(c *s x)" by vector
lemma vector_smult_lneg: "- (c::'a::ring) *s x = -(c *s x)" by vector
lemma vector_sneg_minus1: "-x = (- (1::'a::ring_1)) *s x" by vector
lemma vector_smult_rzero[simp]: "c *s 0 = (0::'a::mult_zero ^ 'n)" by vector
lemma vector_sub_rdistrib: "((a::'a::ring) - b) *s x = a *s x - b *s x"
by (vector field_simps)
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
by (simp add: Cart_eq)
subsection {* Topological space *}
instantiation cart :: (topological_space, finite) topological_space
begin
definition open_vector_def:
"open (S :: ('a ^ 'b) set) \<longleftrightarrow>
(\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>
(\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> S))"
instance proof
show "open (UNIV :: ('a ^ 'b) set)"
unfolding open_vector_def by auto
next
fix S T :: "('a ^ 'b) set"
assume "open S" "open T" thus "open (S \<inter> T)"
unfolding open_vector_def
apply clarify
apply (drule (1) bspec)+
apply (clarify, rename_tac Sa Ta)
apply (rule_tac x="\<lambda>i. Sa i \<inter> Ta i" in exI)
apply (simp add: open_Int)
done
next
fix K :: "('a ^ 'b) set set"
assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
unfolding open_vector_def
apply clarify
apply (drule (1) bspec)
apply (drule (1) bspec)
apply clarify
apply (rule_tac x=A in exI)
apply fast
done
qed
end
lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
unfolding open_vector_def by auto
lemma open_vimage_Cart_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
unfolding open_vector_def
apply clarify
apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
done
lemma closed_vimage_Cart_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_Cart_nth)
lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
proof -
have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
by (simp add: closed_INT closed_vimage_Cart_nth)
qed
lemma tendsto_Cart_nth [tendsto_intros]:
assumes "((\<lambda>x. f x) ---> a) net"
shows "((\<lambda>x. f x $ i) ---> a $ i) net"
proof (rule topological_tendstoI)
fix S assume "open S" "a $ i \<in> S"
then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
by (simp_all add: open_vimage_Cart_nth)
with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y $ i) -` S) net"
by (rule topological_tendstoD)
then show "eventually (\<lambda>x. f x $ i \<in> S) net"
by simp
qed
subsection {* Metric *}
(* TODO: move somewhere else *)
lemma finite_choice: "finite A \<Longrightarrow> \<forall>x\<in>A. \<exists>y. P x y \<Longrightarrow> \<exists>f. \<forall>x\<in>A. P x (f x)"
apply (induct set: finite, simp_all)
apply (clarify, rename_tac y)
apply (rule_tac x="f(x:=y)" in exI, simp)
done
instantiation cart :: (metric_space, finite) metric_space
begin
definition dist_vector_def:
"dist (x::'a^'b) (y::'a^'b) = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
lemma dist_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
unfolding dist_vector_def
by (rule member_le_setL2) simp_all
instance proof
fix x y :: "'a ^ 'b"
show "dist x y = 0 \<longleftrightarrow> x = y"
unfolding dist_vector_def
by (simp add: setL2_eq_0_iff Cart_eq)
next
fix x y z :: "'a ^ 'b"
show "dist x y \<le> dist x z + dist y z"
unfolding dist_vector_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
apply (simp add: setL2_mono dist_triangle2)
done
next
(* FIXME: long proof! *)
fix S :: "('a ^ 'b) set"
show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
unfolding open_vector_def open_dist
apply safe
apply (drule (1) bspec)
apply clarify
apply (subgoal_tac "\<exists>e>0. \<forall>i y. dist y (x$i) < e \<longrightarrow> y \<in> A i")
apply clarify
apply (rule_tac x=e in exI, clarify)
apply (drule spec, erule mp, clarify)
apply (drule spec, drule spec, erule mp)
apply (erule le_less_trans [OF dist_nth_le])
apply (subgoal_tac "\<forall>i\<in>UNIV. \<exists>e>0. \<forall>y. dist y (x$i) < e \<longrightarrow> y \<in> A i")
apply (drule finite_choice [OF finite], clarify)
apply (rule_tac x="Min (range f)" in exI, simp)
apply clarify
apply (drule_tac x=i in spec, clarify)
apply (erule (1) bspec)
apply (drule (1) bspec, clarify)
apply (subgoal_tac "\<exists>r. (\<forall>i::'b. 0 < r i) \<and> e = setL2 r UNIV")
apply clarify
apply (rule_tac x="\<lambda>i. {y. dist y (x$i) < r i}" in exI)
apply (rule conjI)
apply clarify
apply (rule conjI)
apply (clarify, rename_tac y)
apply (rule_tac x="r i - dist y (x$i)" in exI, rule conjI, simp)
apply clarify
apply (simp only: less_diff_eq)
apply (erule le_less_trans [OF dist_triangle])
apply simp
apply clarify
apply (drule spec, erule mp)
apply (simp add: dist_vector_def setL2_strict_mono)
apply (rule_tac x="\<lambda>i. e / sqrt (of_nat CARD('b))" in exI)
apply (simp add: divide_pos_pos setL2_constant)
done
qed
end
lemma LIMSEQ_Cart_nth:
"(X ----> a) \<Longrightarrow> (\<lambda>n. X n $ i) ----> a $ i"
unfolding LIMSEQ_conv_tendsto by (rule tendsto_Cart_nth)
lemma LIM_Cart_nth:
"(f -- x --> y) \<Longrightarrow> (\<lambda>x. f x $ i) -- x --> y $ i"
unfolding LIM_conv_tendsto by (rule tendsto_Cart_nth)
lemma Cauchy_Cart_nth:
"Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_nth_le])
lemma LIMSEQ_vector:
fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
assumes X: "\<And>i. (\<lambda>n. X n $ i) ----> (a $ i)"
shows "X ----> a"
proof (rule metric_LIMSEQ_I)
fix r :: real assume "0 < r"
then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
by (simp add: divide_pos_pos)
def N \<equiv> "\<lambda>i. LEAST N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"
def M \<equiv> "Max (range N)"
have "\<And>i. \<exists>N. \<forall>n\<ge>N. dist (X n $ i) (a $ i) < ?s"
using X `0 < ?s` by (rule metric_LIMSEQ_D)
hence "\<And>i. \<forall>n\<ge>N i. dist (X n $ i) (a $ i) < ?s"
unfolding N_def by (rule LeastI_ex)
hence M: "\<And>i. \<forall>n\<ge>M. dist (X n $ i) (a $ i) < ?s"
unfolding M_def by simp
{
fix n :: nat assume "M \<le> n"
have "dist (X n) a = setL2 (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"
unfolding dist_vector_def ..
also have "\<dots> \<le> setsum (\<lambda>i. dist (X n $ i) (a $ i)) UNIV"
by (rule setL2_le_setsum [OF zero_le_dist])
also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
by (rule setsum_strict_mono, simp_all add: M `M \<le> n`)
also have "\<dots> = r"
by simp
finally have "dist (X n) a < r" .
}
hence "\<forall>n\<ge>M. dist (X n) a < r"
by simp
then show "\<exists>M. \<forall>n\<ge>M. dist (X n) a < r" ..
qed
lemma Cauchy_vector:
fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
shows "Cauchy (\<lambda>n. X n)"
proof (rule metric_CauchyI)
fix r :: real assume "0 < r"
then have "0 < r / of_nat CARD('n)" (is "0 < ?s")
by (simp add: divide_pos_pos)
def N \<equiv> "\<lambda>i. LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
def M \<equiv> "Max (range N)"
have "\<And>i. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s"
using X `0 < ?s` by (rule metric_CauchyD)
hence "\<And>i. \<forall>m\<ge>N i. \<forall>n\<ge>N i. dist (X m $ i) (X n $ i) < ?s"
unfolding N_def by (rule LeastI_ex)
hence M: "\<And>i. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m $ i) (X n $ i) < ?s"
unfolding M_def by simp
{
fix m n :: nat
assume "M \<le> m" "M \<le> n"
have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
unfolding dist_vector_def ..
also have "\<dots> \<le> setsum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
by (rule setL2_le_setsum [OF zero_le_dist])
also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
by (rule setsum_strict_mono, simp_all add: M `M \<le> m` `M \<le> n`)
also have "\<dots> = r"
by simp
finally have "dist (X m) (X n) < r" .
}
hence "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r"
by simp
then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
qed
instance cart :: (complete_space, finite) complete_space
proof
fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
have "\<And>i. (\<lambda>n. X n $ i) ----> lim (\<lambda>n. X n $ i)"
using Cauchy_Cart_nth [OF `Cauchy X`]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
hence "X ----> Cart_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
by (simp add: LIMSEQ_vector)
then show "convergent X"
by (rule convergentI)
qed
subsection {* Norms *}
instantiation cart :: (real_normed_vector, finite) real_normed_vector
begin
definition norm_vector_def:
"norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"
definition vector_sgn_def:
"sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
instance proof
fix a :: real and x y :: "'a ^ 'b"
show "0 \<le> norm x"
unfolding norm_vector_def
by (rule setL2_nonneg)
show "norm x = 0 \<longleftrightarrow> x = 0"
unfolding norm_vector_def
by (simp add: setL2_eq_0_iff Cart_eq)
show "norm (x + y) \<le> norm x + norm y"
unfolding norm_vector_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
apply (simp add: setL2_mono norm_triangle_ineq)
done
show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
unfolding norm_vector_def
by (simp add: setL2_right_distrib)
show "sgn x = scaleR (inverse (norm x)) x"
by (rule vector_sgn_def)
show "dist x y = norm (x - y)"
unfolding dist_vector_def norm_vector_def
by (simp add: dist_norm)
qed
end
lemma norm_nth_le: "norm (x $ i) \<le> norm x"
unfolding norm_vector_def
by (rule member_le_setL2) simp_all
interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"
apply default
apply (rule vector_add_component)
apply (rule vector_scaleR_component)
apply (rule_tac x="1" in exI, simp add: norm_nth_le)
done
instance cart :: (banach, finite) banach ..
subsection {* Inner products *}
abbreviation inner_bullet (infix "\<bullet>" 70) where "x \<bullet> y \<equiv> inner x y"
instantiation cart :: (real_inner, finite) real_inner
begin
definition inner_vector_def:
"inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
instance proof
fix r :: real and x y z :: "'a ^ 'b"
show "inner x y = inner y x"
unfolding inner_vector_def
by (simp add: inner_commute)
show "inner (x + y) z = inner x z + inner y z"
unfolding inner_vector_def
by (simp add: inner_add_left setsum_addf)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_vector_def
by (simp add: setsum_right_distrib)
show "0 \<le> inner x x"
unfolding inner_vector_def
by (simp add: setsum_nonneg)
show "inner x x = 0 \<longleftrightarrow> x = 0"
unfolding inner_vector_def
by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
show "norm x = sqrt (inner x x)"
unfolding inner_vector_def norm_vector_def setL2_def
by (simp add: power2_norm_eq_inner)
qed
end
subsection {* A connectedness or intermediate value lemma with several applications. *}
lemma connected_real_lemma:
fixes f :: "real \<Rightarrow> 'a::metric_space"
assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
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"
and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
and e2: "\<forall>y \<in> e2. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e2"
and e12: "~(\<exists>x \<ge> a. x <= b \<and> f x \<in> e1 \<and> f x \<in> e2)"
shows "\<exists>x \<ge> a. x <= b \<and> f x \<notin> e1 \<and> f x \<notin> e2" (is "\<exists> x. ?P x")
proof-
let ?S = "{c. \<forall>x \<ge> a. x <= c \<longrightarrow> f x \<in> e1}"
have Se: " \<exists>x. x \<in> ?S" apply (rule exI[where x=a]) by (auto simp add: fa)
have Sub: "\<exists>y. isUb UNIV ?S y"
apply (rule exI[where x= b])
using ab fb e12 by (auto simp add: isUb_def setle_def)
from reals_complete[OF Se Sub] obtain l where
l: "isLub UNIV ?S l"by blast
have alb: "a \<le> l" "l \<le> b" using l ab fa fb e12
apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
by (metis linorder_linear)
have ale1: "\<forall>z \<ge> a. z < l \<longrightarrow> f z \<in> e1" using l
apply (auto simp add: isLub_def leastP_def isUb_def setle_def setge_def)
by (metis linorder_linear not_le)
have th1: "\<And>z x e d :: real. z <= x + e \<Longrightarrow> e < d ==> z < x \<or> abs(z - x) < d" by arith
have th2: "\<And>e x:: real. 0 < e ==> ~(x + e <= x)" by arith
have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by dlo
{assume le2: "f l \<in> e2"
from le2 fa fb e12 alb have la: "l \<noteq> a" by metis
hence lap: "l - a > 0" using alb by arith
from e2[rule_format, OF le2] obtain e where
e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e2" by metis
from dst[OF alb e(1)] obtain d where
d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" using lap d(1)
apply ferrack by arith
then obtain d' where d': "d' > 0" "d' < d" "l - d' > a" by metis
from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e2" by metis
from th0[rule_format, of "l - d'"] d' have "f (l - d') \<in> e2" by auto
moreover
have "f (l - d') \<in> e1" using ale1[rule_format, of "l -d'"] d' by auto
ultimately have False using e12 alb d' by auto}
moreover
{assume le1: "f l \<in> e1"
from le1 fa fb e12 alb have lb: "l \<noteq> b" by metis
hence blp: "b - l > 0" using alb by arith
from e1[rule_format, OF le1] obtain e where
e: "e > 0" "\<forall>y. dist y (f l) < e \<longrightarrow> y \<in> e1" by metis
from dst[OF alb e(1)] obtain d where
d: "d > 0" "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> dist (f y) (f l) < e" by metis
have "\<exists>d'. d' < d \<and> d' >0" using d(1) by dlo
then obtain d' where d': "d' > 0" "d' < d" by metis
from d e have th0: "\<forall>y. \<bar>y - l\<bar> < d \<longrightarrow> f y \<in> e1" by auto
hence "\<forall>y. l \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" using d' by auto
with ale1 have "\<forall>y. a \<le> y \<and> y \<le> l + d' \<longrightarrow> f y \<in> e1" by auto
with l d' have False
by (auto simp add: isLub_def isUb_def setle_def setge_def leastP_def) }
ultimately show ?thesis using alb by metis
qed
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 *}
lemma square_bound_lemma: "(x::real) < (1 + x) * (1 + x)"
proof-
have "(x + 1/2)^2 + 3/4 > 0" using zero_le_power2[of "x+1/2"] by arith
thus ?thesis by (simp add: field_simps power2_eq_square)
qed
lemma square_continuous: "0 < (e::real) ==> \<exists>d. 0 < d \<and> (\<forall>y. abs(y - x) < d \<longrightarrow> abs(y * y - x * x) < e)"
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)
apply (rule_tac x="s" in exI)
apply auto
apply (erule_tac x=y in allE)
apply auto
done
lemma real_le_lsqrt: "0 <= x \<Longrightarrow> 0 <= y \<Longrightarrow> x <= y^2 ==> sqrt x <= y"
using real_sqrt_le_iff[of x "y^2"] by simp
lemma real_le_rsqrt: "x^2 \<le> y \<Longrightarrow> x \<le> sqrt y"
using real_sqrt_le_mono[of "x^2" y] by simp
lemma real_less_rsqrt: "x^2 < y \<Longrightarrow> x < sqrt y"
using real_sqrt_less_mono[of "x^2" y] by simp
lemma sqrt_even_pow2: assumes n: "even n"
shows "sqrt(2 ^ n) = 2 ^ (n div 2)"
proof-
from n obtain m where m: "n = 2*m" unfolding even_mult_two_ex ..
from m have "sqrt(2 ^ n) = sqrt ((2 ^ m) ^ 2)"
by (simp only: power_mult[symmetric] mult_commute)
then show ?thesis using m by simp
qed
lemma real_div_sqrt: "0 <= x ==> x / sqrt(x) = sqrt(x)"
apply (cases "x = 0", simp_all)
using sqrt_divide_self_eq[of x]
apply (simp add: inverse_eq_divide field_simps)
done
text{* Hence derive more interesting properties of the norm. *}
lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
by (simp add: norm_vector_def setL2_right_distrib abs_mult)
lemma norm_eq_0_dot: "(norm x = 0) \<longleftrightarrow> (inner x x = (0::real))"
by (simp add: norm_vector_def setL2_def power2_eq_square)
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
by vector
lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_ssub_ldistrib)
lemma vector_mul_rcancel[simp]: "a *s x = b *s x \<longleftrightarrow> (a::real) = b \<or> x = 0"
by (metis eq_iff_diff_eq_0 vector_mul_eq_0 vector_sub_rdistrib)
lemma vector_mul_lcancel_imp: "a \<noteq> (0::real) ==> a *s x = a *s y ==> (x = y)"
by (metis vector_mul_lcancel)
lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
by (metis vector_mul_rcancel)
lemma norm_cauchy_schwarz:
shows "inner x y <= norm x * norm y"
using Cauchy_Schwarz_ineq2[of x y] by auto
lemma norm_cauchy_schwarz_abs:
shows "\<bar>inner x y\<bar> \<le> norm x * norm y"
by (rule Cauchy_Schwarz_ineq2)
lemma norm_triangle_sub:
fixes x y :: "'a::real_normed_vector"
shows "norm x \<le> norm y + norm (x - y)"
using norm_triangle_ineq[of "y" "x - y"] by (simp add: field_simps)
lemma component_le_norm: "\<bar>x$i\<bar> <= norm x"
apply (simp add: norm_vector_def)
apply (rule member_le_setL2, simp_all)
done
lemma norm_bound_component_le: "norm x <= e ==> \<bar>x$i\<bar> <= e"
by (metis component_le_norm order_trans)
lemma norm_bound_component_lt: "norm x < e ==> \<bar>x$i\<bar> < e"
by (metis component_le_norm basic_trans_rules(21))
lemma norm_le_l1: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
by (simp add: norm_vector_def setL2_le_setsum)
lemma real_abs_norm: "\<bar>norm x\<bar> = norm x"
by (rule abs_norm_cancel)
lemma real_abs_sub_norm: "\<bar>norm x - norm y\<bar> <= norm(x - y)"
by (rule norm_triangle_ineq3)
lemma norm_le: "norm(x) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
by (simp add: norm_eq_sqrt_inner)
lemma norm_lt: "norm(x) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
by (simp add: norm_eq_sqrt_inner)
lemma norm_eq: "norm(x) = norm (y) \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
apply(subst order_eq_iff) unfolding norm_le by auto
lemma norm_eq_1: "norm(x) = 1 \<longleftrightarrow> x \<bullet> x = 1"
unfolding norm_eq_sqrt_inner by auto
text{* Squaring equations and inequalities involving norms. *}
lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
by (simp add: norm_eq_sqrt_inner)
lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
by (auto simp add: norm_eq_sqrt_inner)
lemma real_abs_le_square_iff: "\<bar>x\<bar> \<le> \<bar>y\<bar> \<longleftrightarrow> (x::real)^2 \<le> y^2"
proof
assume "\<bar>x\<bar> \<le> \<bar>y\<bar>"
then have "\<bar>x\<bar>\<twosuperior> \<le> \<bar>y\<bar>\<twosuperior>" by (rule power_mono, simp)
then show "x\<twosuperior> \<le> y\<twosuperior>" by simp
next
assume "x\<twosuperior> \<le> y\<twosuperior>"
then have "sqrt (x\<twosuperior>) \<le> sqrt (y\<twosuperior>)" by (rule real_sqrt_le_mono)
then show "\<bar>x\<bar> \<le> \<bar>y\<bar>" by simp
qed
lemma norm_le_square: "norm(x) <= a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x <= a^2"
apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
done
lemma norm_ge_square: "norm(x) >= a \<longleftrightarrow> a <= 0 \<or> x \<bullet> x >= a ^ 2"
apply (simp add: dot_square_norm real_abs_le_square_iff[symmetric])
using norm_ge_zero[of x]
apply arith
done
lemma norm_lt_square: "norm(x) < a \<longleftrightarrow> 0 < a \<and> x \<bullet> x < a^2"
by (metis not_le norm_ge_square)
lemma norm_gt_square: "norm(x) > a \<longleftrightarrow> a < 0 \<or> x \<bullet> x > a^2"
by (metis norm_le_square not_less)
text{* Dot product in terms of the norm rather than conversely. *}
lemmas inner_simps = inner.add_left inner.add_right inner.diff_right inner.diff_left
inner.scaleR_left inner.scaleR_right
lemma dot_norm: "x \<bullet> y = (norm(x + y) ^2 - norm x ^ 2 - norm y ^ 2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute by auto
lemma dot_norm_neg: "x \<bullet> y = ((norm x ^ 2 + norm y ^ 2) - norm(x - y) ^ 2) / 2"
unfolding power2_norm_eq_inner inner_simps inner_commute by(auto simp add:algebra_simps)
text{* Equality of vectors in terms of @{term "op \<bullet>"} products. *}
lemma vector_eq: "x = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y \<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs then show ?rhs by simp
next
assume ?rhs
then have "x \<bullet> x - x \<bullet> y = 0 \<and> x \<bullet> y - y \<bullet> y = 0" by simp
hence "x \<bullet> (x - y) = 0 \<and> y \<bullet> (x - y) = 0" by (simp add: inner_simps inner_commute)
then have "(x - y) \<bullet> (x - y) = 0" by (simp add: field_simps inner_simps inner_commute)
then show "x = y" by (simp)
qed
subsection{* General linear decision procedure for normed spaces. *}
lemma norm_cmul_rule_thm:
fixes x :: "'a::real_normed_vector"
shows "b >= norm(x) ==> \<bar>c\<bar> * b >= norm(scaleR c x)"
unfolding norm_scaleR
apply (erule mult_mono1)
apply simp
done
(* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
lemma norm_add_rule_thm:
fixes x1 x2 :: "'a::real_normed_vector"
shows "norm x1 \<le> b1 \<Longrightarrow> norm x2 \<le> b2 \<Longrightarrow> norm (x1 + x2) \<le> b1 + b2"
by (rule order_trans [OF norm_triangle_ineq add_mono])
lemma ge_iff_diff_ge_0: "(a::'a::linordered_ring) \<ge> b == a - b \<ge> 0"
by (simp add: field_simps)
lemma pth_1:
fixes x :: "'a::real_normed_vector"
shows "x == scaleR 1 x" by simp
lemma pth_2:
fixes x :: "'a::real_normed_vector"
shows "x - y == x + -y" by (atomize (full)) simp
lemma pth_3:
fixes x :: "'a::real_normed_vector"
shows "- x == scaleR (-1) x" by simp
lemma pth_4:
fixes x :: "'a::real_normed_vector"
shows "scaleR 0 x == 0" and "scaleR c 0 = (0::'a)" by simp_all
lemma pth_5:
fixes x :: "'a::real_normed_vector"
shows "scaleR c (scaleR d x) == scaleR (c * d) x" by simp
lemma pth_6:
fixes x :: "'a::real_normed_vector"
shows "scaleR c (x + y) == scaleR c x + scaleR c y"
by (simp add: scaleR_right_distrib)
lemma pth_7:
fixes x :: "'a::real_normed_vector"
shows "0 + x == x" and "x + 0 == x" by simp_all
lemma pth_8:
fixes x :: "'a::real_normed_vector"
shows "scaleR c x + scaleR d x == scaleR (c + d) x"
by (simp add: scaleR_left_distrib)
lemma pth_9:
fixes x :: "'a::real_normed_vector" shows
"(scaleR c x + z) + scaleR d x == scaleR (c + d) x + z"
"scaleR c x + (scaleR d x + z) == scaleR (c + d) x + z"
"(scaleR c x + w) + (scaleR d x + z) == scaleR (c + d) x + (w + z)"
by (simp_all add: algebra_simps)
lemma pth_a:
fixes x :: "'a::real_normed_vector"
shows "scaleR 0 x + y == y" by simp
lemma pth_b:
fixes x :: "'a::real_normed_vector" shows
"scaleR c x + scaleR d y == scaleR c x + scaleR d y"
"(scaleR c x + z) + scaleR d y == scaleR c x + (z + scaleR d y)"
"scaleR c x + (scaleR d y + z) == scaleR c x + (scaleR d y + z)"
"(scaleR c x + w) + (scaleR d y + z) == scaleR c x + (w + (scaleR d y + z))"
by (simp_all add: algebra_simps)
lemma pth_c:
fixes x :: "'a::real_normed_vector" shows
"scaleR c x + scaleR d y == scaleR d y + scaleR c x"
"(scaleR c x + z) + scaleR d y == scaleR d y + (scaleR c x + z)"
"scaleR c x + (scaleR d y + z) == scaleR d y + (scaleR c x + z)"
"(scaleR c x + w) + (scaleR d y + z) == scaleR d y + ((scaleR c x + w) + z)"
by (simp_all add: algebra_simps)
lemma pth_d:
fixes x :: "'a::real_normed_vector"
shows "x + 0 == x" by simp
lemma norm_imp_pos_and_ge:
fixes x :: "'a::real_normed_vector"
shows "norm x == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
by atomize auto
lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
lemma norm_pths:
fixes x :: "'a::real_normed_vector" shows
"x = y \<longleftrightarrow> norm (x - y) \<le> 0"
"x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
using norm_ge_zero[of "x - y"] by auto
use "normarith.ML"
method_setup norm = {* Scan.succeed (SIMPLE_METHOD' o NormArith.norm_arith_tac)
*} "Proves simple linear statements about vector norms"
text{* Hence more metric properties. *}
lemma dist_triangle_alt:
fixes x y z :: "'a::metric_space"
shows "dist y z <= dist x y + dist x z"
by (rule dist_triangle3)
lemma dist_pos_lt:
fixes x y :: "'a::metric_space"
shows "x \<noteq> y ==> 0 < dist x y"
by (simp add: zero_less_dist_iff)
lemma dist_nz:
fixes x y :: "'a::metric_space"
shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
by (simp add: zero_less_dist_iff)
lemma dist_triangle_le:
fixes x y z :: "'a::metric_space"
shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
by (rule order_trans [OF dist_triangle2])
lemma dist_triangle_lt:
fixes x y z :: "'a::metric_space"
shows "dist x z + dist y z < e ==> dist x y < e"
by (rule le_less_trans [OF dist_triangle2])
lemma dist_triangle_half_l:
fixes x1 x2 y :: "'a::metric_space"
shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
by (rule dist_triangle_lt [where z=y], simp)
lemma dist_triangle_half_r:
fixes x1 x2 y :: "'a::metric_space"
shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
by (rule dist_triangle_half_l, simp_all add: dist_commute)
lemma norm_triangle_half_r:
shows "norm (y - x1) < e / 2 \<Longrightarrow> norm (y - x2) < e / 2 \<Longrightarrow> norm (x1 - x2) < e"
using dist_triangle_half_r unfolding dist_norm[THEN sym] by auto
lemma norm_triangle_half_l: assumes "norm (x - y) < e / 2" "norm (x' - (y)) < e / 2"
shows "norm (x - x') < e"
using dist_triangle_half_l[OF assms[unfolded dist_norm[THEN sym]]]
unfolding dist_norm[THEN sym] .
lemma norm_triangle_le: "norm(x) + norm y <= e ==> norm(x + y) <= e"
by (metis order_trans norm_triangle_ineq)
lemma norm_triangle_lt: "norm(x) + norm(y) < e ==> norm(x + y) < e"
by (metis basic_trans_rules(21) norm_triangle_ineq)
lemma dist_triangle_add:
fixes x y x' y' :: "'a::real_normed_vector"
shows "dist (x + y) (x' + y') <= dist x x' + dist y y'"
by norm
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
unfolding dist_norm vector_ssub_ldistrib[symmetric] norm_mul ..
lemma dist_triangle_add_half:
fixes x x' y y' :: "'a::real_normed_vector"
shows "dist x x' < e / 2 \<Longrightarrow> dist y y' < e / 2 \<Longrightarrow> dist(x + y) (x' + y') < e"
by norm
lemma setsum_component [simp]:
fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
by (cases "finite S", induct S set: finite, simp_all)
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
by (simp add: Cart_eq)
lemma setsum_clauses:
shows "setsum f {} = 0"
and "finite S \<Longrightarrow> setsum f (insert x S) =
(if x \<in> S then setsum f S else f x + setsum f S)"
by (auto simp add: insert_absorb)
lemma setsum_cmul:
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
by (simp add: Cart_eq setsum_right_distrib)
lemma setsum_norm:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
proof(induct rule: finite_induct[OF fS])
case 1 thus ?case by simp
next
case (2 x S)
from "2.hyps" have "norm (setsum f (insert x S)) \<le> norm (f x) + norm (setsum f S)" by (simp add: norm_triangle_ineq)
also have "\<dots> \<le> norm (f x) + setsum (\<lambda>x. norm(f x)) S"
using "2.hyps" by simp
finally show ?case using "2.hyps" by simp
qed
lemma setsum_norm_le:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
shows "norm (setsum f S) \<le> setsum g S"
proof-
from fg have "setsum (\<lambda>x. norm(f x)) S <= setsum g S"
by - (rule setsum_mono, simp)
then show ?thesis using setsum_norm[OF fS, of f] fg
by arith
qed
lemma setsum_norm_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes fS: "finite S"
and K: "\<forall>x \<in> S. norm (f x) \<le> K"
shows "norm (setsum f S) \<le> of_nat (card S) * K"
using setsum_norm_le[OF fS K] setsum_constant[symmetric]
by simp
lemma setsum_vmul:
fixes f :: "'a \<Rightarrow> 'b::semiring_0"
assumes fS: "finite S"
shows "setsum f S *s v = setsum (\<lambda>x. f x *s v) S"
proof(induct rule: finite_induct[OF fS])
case 1 then show ?case by simp
next
case (2 x F)
from "2.hyps" have "setsum f (insert x F) *s v = (f x + setsum f F) *s v"
by simp
also have "\<dots> = f x *s v + setsum f F *s v"
by (simp add: vector_sadd_rdistrib)
also have "\<dots> = setsum (\<lambda>x. f x *s v) (insert x F)" using "2.hyps" by simp
finally show ?case .
qed
(* FIXME : Problem thm setsum_vmul[of _ "f:: 'a \<Rightarrow> real ^'n"] ---
Get rid of *s and use real_vector instead! Also prove that ^ creates a real_vector !! *)
(* FIXME: Here too need stupid finiteness assumption on T!!! *)
lemma setsum_group:
assumes fS: "finite S" and fT: "finite T" and fST: "f ` S \<subseteq> T"
shows "setsum (\<lambda>y. setsum g {x. x\<in> S \<and> f x = y}) T = setsum g S"
apply (subst setsum_image_gen[OF fS, of g f])
apply (rule setsum_mono_zero_right[OF fT fST])
by (auto intro: setsum_0')
lemma vsum_norm_allsubsets_bound:
fixes f:: "'a \<Rightarrow> real ^'n"
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e"
proof-
let ?d = "real CARD('n)"
let ?nf = "\<lambda>x. norm (f x)"
let ?U = "UNIV :: 'n set"
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"
by (rule setsum_commute)
have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
have "setsum ?nf P \<le> setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P"
apply (rule setsum_mono)
by (rule norm_le_l1)
also have "\<dots> \<le> 2 * ?d * e"
unfolding th0 th1
proof(rule setsum_bounded)
fix i assume i: "i \<in> ?U"
let ?Pp = "{x. x\<in> P \<and> f x $ i \<ge> 0}"
let ?Pn = "{x. x \<in> P \<and> f x $ i < 0}"
have thp: "P = ?Pp \<union> ?Pn" by auto
have thp0: "?Pp \<inter> ?Pn ={}" by auto
have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i] fPs[OF PpP]
by (auto intro: abs_le_D1)
have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i] fPs[OF PnP]
by (auto simp add: setsum_negf intro: abs_le_D1)
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"
apply (subst thp)
apply (rule setsum_Un_zero)
using fP thp0 by auto
also have "\<dots> \<le> 2*e" using Pne Ppe by arith
finally show "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P \<le> 2*e" .
qed
finally show ?thesis .
qed
lemma dot_lsum: "finite S \<Longrightarrow> setsum f S \<bullet> y = setsum (\<lambda>x. f x \<bullet> y) S "
apply(induct rule: finite_induct) by(auto simp add: inner_simps)
lemma dot_rsum: "finite S \<Longrightarrow> y \<bullet> setsum f S = setsum (\<lambda>x. y \<bullet> f x) S "
apply(induct rule: finite_induct) by(auto simp add: inner_simps)
subsection{* Basis vectors in coordinate directions. *}
definition "basis k = (\<chi> i. if i = k then 1 else 0)"
lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)"
unfolding basis_def by simp
lemma delta_mult_idempotent:
"(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)
lemma norm_basis:
shows "norm (basis k :: real ^'n) = 1"
apply (simp add: basis_def norm_eq_sqrt_inner) unfolding inner_vector_def
apply (vector delta_mult_idempotent)
using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] by auto
lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1"
by (rule norm_basis)
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
apply (rule exI[where x="c *s basis arbitrary"])
by (simp only: norm_mul norm_basis)
lemma vector_choose_dist: assumes e: "0 <= e"
shows "\<exists>(y::real^'n). dist x y = e"
proof-
from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e"
by blast
then have "dist x (x - c) = e" by (simp add: dist_norm)
then show ?thesis by blast
qed
lemma basis_inj: "inj (basis :: 'n \<Rightarrow> real ^'n)"
by (simp add: inj_on_def Cart_eq)
lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
by auto
lemma basis_expansion:
"setsum (\<lambda>i. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
lemma basis_expansion_unique:
"setsum (\<lambda>i. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)"
by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong)
lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
by auto
lemma dot_basis:
shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i) = (x$i)"
unfolding inner_vector_def by (auto simp add: basis_def cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
lemma inner_basis:
fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n"
shows "inner (basis i) x = inner 1 (x $ i)"
and "inner x (basis i) = inner (x $ i) 1"
unfolding inner_vector_def basis_def
by (auto simp add: cond_application_beta cond_value_iff setsum_delta cong del: if_weak_cong)
lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
by (auto simp add: Cart_eq)
lemma basis_nonzero:
shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
by (simp add: basis_eq_0)
lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = z"
proof
assume "\<forall>x. x \<bullet> y = x \<bullet> z"
hence "\<forall>x. x \<bullet> (y - z) = 0" by (simp add: inner_simps)
hence "(y - z) \<bullet> (y - z) = 0" ..
thus "y = z" by simp
qed simp
lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = y"
proof
assume "\<forall>z. x \<bullet> z = y \<bullet> z"
hence "\<forall>z. (x - y) \<bullet> z = 0" by (simp add: inner_simps)
hence "(x - y) \<bullet> (x - y) = 0" ..
thus "x = y" by simp
qed simp
subsection{* Orthogonality. *}
definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
lemma orthogonal_basis:
shows "orthogonal (basis i) x \<longleftrightarrow> x$i = (0::real)"
by (auto simp add: orthogonal_def inner_vector_def basis_def cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
lemma orthogonal_basis_basis:
shows "orthogonal (basis i :: real^'n) (basis j) \<longleftrightarrow> i \<noteq> j"
unfolding orthogonal_basis[of i] basis_component[of j] by simp
(* FIXME : Maybe some of these require less than comm_ring, but not all*)
lemma orthogonal_clauses:
"orthogonal a (0::real ^'n)"
"orthogonal a x ==> orthogonal a (c *\<^sub>R x)"
"orthogonal a x ==> orthogonal a (-x)"
"orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x + y)"
"orthogonal a x \<Longrightarrow> orthogonal a y ==> orthogonal a (x - y)"
"orthogonal 0 a"
"orthogonal x a ==> orthogonal (c *\<^sub>R x) a"
"orthogonal x a ==> orthogonal (-x) a"
"orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x + y) a"
"orthogonal x a \<Longrightarrow> orthogonal y a ==> orthogonal (x - y) a"
unfolding orthogonal_def inner_simps by auto
lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
by (simp add: orthogonal_def inner_commute)
subsection{* Linear functions. *}
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)"
lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *s x) = c *s f x"
shows "linear f" using assms unfolding linear_def by auto
lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
by (vector linear_def Cart_eq field_simps)
lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
lemma linear_compose_add: "linear (f :: 'a ^'n \<Rightarrow> 'a::semiring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
by (vector linear_def Cart_eq field_simps)
lemma linear_compose_sub: "linear (f :: 'a ^'n \<Rightarrow> 'a::ring_1 ^'m) \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
by (vector linear_def Cart_eq field_simps)
lemma linear_compose: "linear f \<Longrightarrow> linear g ==> linear (g o f)"
by (simp add: linear_def)
lemma linear_id: "linear id" by (simp add: linear_def id_def)
lemma linear_zero: "linear (\<lambda>x. 0::'a::semiring_1 ^ 'n)" by (simp add: linear_def)
lemma linear_compose_setsum:
assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a :: 'a::semiring_1 ^ 'n \<Rightarrow> 'a ^'m)"
shows "linear(\<lambda>x. setsum (\<lambda>a. f a x :: 'a::semiring_1 ^'m) S)"
using lS
apply (induct rule: finite_induct[OF fS])
by (auto simp add: linear_zero intro: linear_compose_add)
lemma linear_vmul_component:
fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
assumes lf: "linear f"
shows "linear (\<lambda>x. f x $ k *s v)"
using lf
apply (auto simp add: linear_def )
by (vector field_simps)+
lemma linear_0: "linear f ==> f 0 = (0::'a::semiring_1 ^'n)"
unfolding linear_def
apply clarsimp
apply (erule allE[where x="0::'a"])
apply simp
done
lemma linear_cmul: "linear f ==> f(c*s x) = c *s f x" by (simp add: linear_def)
lemma linear_neg: "linear (f :: 'a::ring_1 ^'n \<Rightarrow> _) ==> f (-x) = - f x"
unfolding vector_sneg_minus1
using linear_cmul[of f] by auto
lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
lemma linear_sub: "linear (f::'a::ring_1 ^'n \<Rightarrow> _) ==> f(x - y) = f x - f y"
by (simp add: diff_def linear_add linear_neg)
lemma linear_setsum:
fixes f:: "'a::semiring_1^'n \<Rightarrow> _"
assumes lf: "linear f" and fS: "finite S"
shows "f (setsum g S) = setsum (f o g) S"
proof (induct rule: finite_induct[OF fS])
case 1 thus ?case by (simp add: linear_0[OF lf])
next
case (2 x F)
have "f (setsum g (insert x F)) = f (g x + setsum g F)" using "2.hyps"
by simp
also have "\<dots> = f (g x) + f (setsum g F)" using linear_add[OF lf] by simp
also have "\<dots> = setsum (f o g) (insert x F)" using "2.hyps" by simp
finally show ?case .
qed
lemma linear_setsum_mul:
fixes f:: "'a ^'n \<Rightarrow> 'a::semiring_1^'m"
assumes lf: "linear f" and fS: "finite S"
shows "f (setsum (\<lambda>i. c i *s v i) S) = setsum (\<lambda>i. c i *s f (v i)) S"
using linear_setsum[OF lf fS, of "\<lambda>i. c i *s v i" , unfolded o_def]
linear_cmul[OF lf] by simp
lemma linear_injective_0:
assumes lf: "linear (f:: 'a::ring_1 ^ 'n \<Rightarrow> _)"
shows "inj f \<longleftrightarrow> (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
proof-
have "inj f \<longleftrightarrow> (\<forall> x y. f x = f y \<longrightarrow> x = y)" by (simp add: inj_on_def)
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f x - f y = 0 \<longrightarrow> x - y = 0)" by simp
also have "\<dots> \<longleftrightarrow> (\<forall> x y. f (x - y) = 0 \<longrightarrow> x - y = 0)"
by (simp add: linear_sub[OF lf])
also have "\<dots> \<longleftrightarrow> (\<forall> x. f x = 0 \<longrightarrow> x = 0)" by auto
finally show ?thesis .
qed
lemma linear_bounded:
fixes f:: "real ^'m \<Rightarrow> real ^'n"
assumes lf: "linear f"
shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
proof-
let ?S = "UNIV:: 'm set"
let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
have fS: "finite ?S" by simp
{fix x:: "real ^ 'm"
let ?g = "(\<lambda>i. (x$i) *s (basis i) :: real ^ 'm)"
have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
by (simp only: basis_expansion)
also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf]
by auto
finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
{fix i assume i: "i \<in> ?S"
from component_le_norm[of x i]
have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
unfolding norm_mul
apply (simp only: mult_commute)
apply (rule mult_mono)
by (auto simp add: field_simps) }
then have th: "\<forall>i\<in> ?S. norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x" by metis
from setsum_norm_le[OF fS, of "\<lambda>i. (x$i) *s (f (basis i))", OF th]
have "norm (f x) \<le> ?B * norm x" unfolding th0 setsum_left_distrib by metis}
then show ?thesis by blast
qed
lemma linear_bounded_pos:
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
proof-
from linear_bounded[OF lf] obtain B where
B: "\<forall>x. norm (f x) \<le> B * norm x" by blast
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
{assume C: "B < 0"
have "norm (1::real ^ 'n) > 0" by simp
with C have "B * norm (1:: real ^ 'n) < 0"
by (simp add: mult_less_0_iff)
with B[rule_format, of 1] norm_ge_zero[of "f 1"] have False by simp
}
then have Bp: "B \<ge> 0" by ferrack
{fix x::"real ^ 'n"
have "norm (f x) \<le> ?K * norm x"
using B[rule_format, of x] norm_ge_zero[of x] norm_ge_zero[of "f x"] Bp
apply (auto simp add: field_simps split add: abs_split)
apply (erule order_trans, simp)
done
}
then show ?thesis using Kp by blast
qed
lemma smult_conv_scaleR: "c *s x = scaleR c x"
unfolding vector_scalar_mult_def vector_scaleR_def by simp
lemma linear_conv_bounded_linear:
fixes f :: "real ^ _ \<Rightarrow> real ^ _"
shows "linear f \<longleftrightarrow> bounded_linear f"
proof
assume "linear f"
show "bounded_linear f"
proof
fix x y show "f (x + y) = f x + f y"
using `linear f` unfolding linear_def by simp
next
fix r x show "f (scaleR r x) = scaleR r (f x)"
using `linear f` unfolding linear_def
by (simp add: smult_conv_scaleR)
next
have "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
using `linear f` by (rule linear_bounded)
thus "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
by (simp add: mult_commute)
qed
next
assume "bounded_linear f"
then interpret f: bounded_linear f .
show "linear f"
unfolding linear_def smult_conv_scaleR
by (simp add: f.add f.scaleR)
qed
lemma bounded_linearI': fixes f::"real^'n \<Rightarrow> real^'m"
assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *s x) = c *s f x"
shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym]
by(rule linearI[OF assms])
subsection{* Bilinear functions. *}
definition "bilinear f \<longleftrightarrow> (\<forall>x. linear(\<lambda>y. f x y)) \<and> (\<forall>y. linear(\<lambda>x. f x y))"
lemma bilinear_ladd: "bilinear h ==> h (x + y) z = (h x z) + (h y z)"
by (simp add: bilinear_def linear_def)
lemma bilinear_radd: "bilinear h ==> h x (y + z) = (h x y) + (h x z)"
by (simp add: bilinear_def linear_def)
lemma bilinear_lmul: "bilinear h ==> h (c *s x) y = c *s (h x y)"
by (simp add: bilinear_def linear_def)
lemma bilinear_rmul: "bilinear h ==> h x (c *s y) = c *s (h x y)"
by (simp add: bilinear_def linear_def)
lemma bilinear_lneg: "bilinear h ==> h (- (x:: 'a::ring_1 ^ 'n)) y = -(h x y)"
by (simp only: vector_sneg_minus1 bilinear_lmul)
lemma bilinear_rneg: "bilinear h ==> h x (- (y:: 'a::ring_1 ^ 'n)) = - h x y"
by (simp only: vector_sneg_minus1 bilinear_rmul)
lemma (in ab_group_add) eq_add_iff: "x = x + y \<longleftrightarrow> y = 0"
using add_imp_eq[of x y 0] by auto
lemma bilinear_lzero:
fixes h :: "'a::ring^'n \<Rightarrow> _" assumes bh: "bilinear h" shows "h 0 x = 0"
using bilinear_ladd[OF bh, of 0 0 x]
by (simp add: eq_add_iff field_simps)
lemma bilinear_rzero:
fixes h :: "'a::ring^_ \<Rightarrow> _" assumes bh: "bilinear h" shows "h x 0 = 0"
using bilinear_radd[OF bh, of x 0 0 ]
by (simp add: eq_add_iff field_simps)
lemma bilinear_lsub: "bilinear h ==> h (x - (y:: 'a::ring_1 ^ _)) z = h x z - h y z"
by (simp add: diff_def bilinear_ladd bilinear_lneg)
lemma bilinear_rsub: "bilinear h ==> h z (x - (y:: 'a::ring_1 ^ _)) = h z x - h z y"
by (simp add: diff_def bilinear_radd bilinear_rneg)
lemma bilinear_setsum:
fixes h:: "'a ^_ \<Rightarrow> 'a::semiring_1^_\<Rightarrow> 'a ^ _"
assumes bh: "bilinear h" and fS: "finite S" and fT: "finite T"
shows "h (setsum f S) (setsum g T) = setsum (\<lambda>(i,j). h (f i) (g j)) (S \<times> T) "
proof-
have "h (setsum f S) (setsum g T) = setsum (\<lambda>x. h (f x) (setsum g T)) S"
apply (rule linear_setsum[unfolded o_def])
using bh fS by (auto simp add: bilinear_def)
also have "\<dots> = setsum (\<lambda>x. setsum (\<lambda>y. h (f x) (g y)) T) S"
apply (rule setsum_cong, simp)
apply (rule linear_setsum[unfolded o_def])
using bh fT by (auto simp add: bilinear_def)
finally show ?thesis unfolding setsum_cartesian_product .
qed
lemma bilinear_bounded:
fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^'k"
assumes bh: "bilinear h"
shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
proof-
let ?M = "UNIV :: 'm set"
let ?N = "UNIV :: 'n set"
let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
have fM: "finite ?M" and fN: "finite ?N" by simp_all
{fix x:: "real ^ 'm" and y :: "real^'n"
have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$i) *s basis i) ?M) (setsum (\<lambda>i. (y$i) *s basis i) ?N))" unfolding basis_expansion ..
also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$i) *s basis i) ((y$j) *s basis j)) (?M \<times> ?N))" unfolding bilinear_setsum[OF bh fM fN] ..
finally have th: "norm (h x y) = \<dots>" .
have "norm (h x y) \<le> ?B * norm x * norm y"
apply (simp add: setsum_left_distrib th)
apply (rule setsum_norm_le)
using fN fM
apply simp
apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh] field_simps)
apply (rule mult_mono)
apply (auto simp add: zero_le_mult_iff component_le_norm)
apply (rule mult_mono)
apply (auto simp add: zero_le_mult_iff component_le_norm)
done}
then show ?thesis by metis
qed
lemma bilinear_bounded_pos:
fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^'k"
assumes bh: "bilinear h"
shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
proof-
from bilinear_bounded[OF bh] obtain B where
B: "\<forall>x y. norm (h x y) \<le> B * norm x * norm y" by blast
let ?K = "\<bar>B\<bar> + 1"
have Kp: "?K > 0" by arith
have KB: "B < ?K" by arith
{fix x::"real ^'m" and y :: "real ^'n"
from KB Kp
have "B * norm x * norm y \<le> ?K * norm x * norm y"
apply -
apply (rule mult_right_mono, rule mult_right_mono)
by auto
then have "norm (h x y) \<le> ?K * norm x * norm y"
using B[rule_format, of x y] by simp}
with Kp show ?thesis by blast
qed
lemma bilinear_conv_bounded_bilinear:
fixes h :: "real ^ _ \<Rightarrow> real ^ _ \<Rightarrow> real ^ _"
shows "bilinear h \<longleftrightarrow> bounded_bilinear h"
proof
assume "bilinear h"
show "bounded_bilinear h"
proof
fix x y z show "h (x + y) z = h x z + h y z"
using `bilinear h` unfolding bilinear_def linear_def by simp
next
fix x y z show "h x (y + z) = h x y + h x z"
using `bilinear h` unfolding bilinear_def linear_def by simp
next
fix r x y show "h (scaleR r x) y = scaleR r (h x y)"
using `bilinear h` unfolding bilinear_def linear_def
by (simp add: smult_conv_scaleR)
next
fix r x y show "h x (scaleR r y) = scaleR r (h x y)"
using `bilinear h` unfolding bilinear_def linear_def
by (simp add: smult_conv_scaleR)
next
have "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
using `bilinear h` by (rule bilinear_bounded)
thus "\<exists>K. \<forall>x y. norm (h x y) \<le> norm x * norm y * K"
by (simp add: mult_ac)
qed
next
assume "bounded_bilinear h"
then interpret h: bounded_bilinear h .
show "bilinear h"
unfolding bilinear_def linear_conv_bounded_linear
using h.bounded_linear_left h.bounded_linear_right
by simp
qed
subsection{* Adjoints. *}
definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
lemma adjoint_works_lemma:
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
proof-
let ?N = "UNIV :: 'n set"
let ?M = "UNIV :: 'm set"
have fN: "finite ?N" by simp
have fM: "finite ?M" by simp
{fix y:: "real ^ 'm"
let ?w = "(\<chi> i. (f (basis i) \<bullet> y)) :: real ^ 'n"
{fix x
have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *s basis i) ?N) \<bullet> y"
by (simp only: basis_expansion)
also have "\<dots> = (setsum (\<lambda>i. (x$i) *s f (basis i)) ?N) \<bullet> y"
unfolding linear_setsum[OF lf fN]
by (simp add: linear_cmul[OF lf])
finally have "f x \<bullet> y = x \<bullet> ?w"
apply (simp only: )
apply (simp add: inner_vector_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
done}
}
then show ?thesis unfolding adjoint_def
some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
by metis
qed
lemma adjoint_works:
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "x \<bullet> adjoint f y = f x \<bullet> y"
using adjoint_works_lemma[OF lf] by metis
lemma adjoint_linear:
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "linear (adjoint f)"
unfolding linear_def vector_eq_ldot[where 'a="real^'n", symmetric] apply safe
unfolding inner_simps smult_conv_scaleR adjoint_works[OF lf] by auto
lemma adjoint_clauses:
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "x \<bullet> adjoint f y = f x \<bullet> y"
and "adjoint f y \<bullet> x = y \<bullet> f x"
by (simp_all add: adjoint_works[OF lf] inner_commute)
lemma adjoint_adjoint:
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "adjoint (adjoint f) = f"
apply (rule ext)
by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
lemma adjoint_unique:
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
shows "f' = adjoint f"
apply (rule ext)
using u
by (simp add: vector_eq_rdot[where 'a="real^'n", symmetric] adjoint_clauses[OF lf])
subsection {* Matrix operations *}
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m" (infixl "**" 70)
where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm" (infixl "*v" 70)
where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n " (infixl "v*" 70)
where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
definition transpose where
"(transpose::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
by (vector matrix_matrix_mult_def setsum_addf[symmetric] field_simps)
lemma matrix_mul_lid:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "mat 1 ** A = A"
apply (simp add: matrix_matrix_mult_def mat_def)
apply vector
by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite] mult_1_left mult_zero_left if_True UNIV_I)
lemma matrix_mul_rid:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "A ** mat 1 = A"
apply (simp add: matrix_matrix_mult_def mat_def)
apply vector
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)
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
apply (subst setsum_commute)
apply simp
done
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
apply (subst setsum_commute)
apply simp
done
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
apply (vector matrix_vector_mult_def mat_def)
by (simp add: cond_value_iff cond_application_beta
setsum_delta' cong del: if_weak_cong)
lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
by (simp add: matrix_matrix_mult_def transpose_def Cart_eq mult_commute)
lemma matrix_eq:
fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
apply auto
apply (subst Cart_eq)
apply clarify
apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq cong del: if_weak_cong)
apply (erule_tac x="basis ia" in allE)
apply (erule_tac x="i" in allE)
by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
lemma matrix_vector_mul_component:
shows "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
by (simp add: matrix_vector_mult_def inner_vector_def)
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
apply (simp add: inner_vector_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
apply (subst setsum_commute)
by simp
lemma transpose_mat: "transpose (mat n) = mat n"
by (vector transpose_def mat_def)
lemma transpose_transpose: "transpose(transpose A) = A"
by (vector transpose_def)
lemma row_transpose:
fixes A:: "'a::semiring_1^_^_"
shows "row i (transpose A) = column i A"
by (simp add: row_def column_def transpose_def Cart_eq)
lemma column_transpose:
fixes A:: "'a::semiring_1^_^_"
shows "column i (transpose A) = row i A"
by (simp add: row_def column_def transpose_def Cart_eq)
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
by (auto simp add: rows_def columns_def row_transpose intro: set_ext)
lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" by (metis transpose_transpose rows_transpose)
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
by (simp add: matrix_vector_mult_def inner_vector_def)
lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
lemma vector_componentwise:
"(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))"
apply (subst basis_expansion[symmetric])
by (vector Cart_eq setsum_component)
lemma linear_componentwise:
fixes f:: "'a::ring_1 ^'m \<Rightarrow> 'a ^ _"
assumes lf: "linear f"
shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
proof-
let ?M = "(UNIV :: 'm set)"
let ?N = "(UNIV :: 'n set)"
have fM: "finite ?M" by simp
have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
unfolding vector_smult_component[symmetric]
unfolding setsum_component[of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
..
then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
qed
text{* Inverse matrices (not necessarily square) *}
definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
(SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
text{* Correspondence between matrices and linear operators. *}
definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
where "matrix f = (\<chi> i j. (f(basis j))$i)"
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ _))"
by (simp add: linear_def matrix_vector_mult_def Cart_eq field_simps setsum_right_distrib setsum_addf)
lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n)"
apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
apply clarify
apply (rule linear_componentwise[OF lf, symmetric])
done
lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n))" by (simp add: ext matrix_works)
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A"
by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
lemma matrix_compose:
assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> 'a^'m)"
and lg: "linear (g::'a::comm_ring_1^'m \<Rightarrow> 'a^_)"
shows "matrix (g o f) = matrix g ** matrix f"
using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def transpose_def Cart_eq mult_commute)
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
apply (rule adjoint_unique[symmetric])
apply (rule matrix_vector_mul_linear)
apply (simp add: transpose_def inner_vector_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
apply (subst setsum_commute)
apply (auto simp add: mult_ac)
done
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
shows "matrix(adjoint f) = transpose(matrix f)"
apply (subst matrix_vector_mul[OF lf])
unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
subsection{* Interlude: Some properties of real sets *}
lemma seq_mono_lemma: assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n" and "\<forall>n \<ge> m. e n <= e m"
shows "\<forall>n \<ge> m. d n < e m"
using prems apply auto
apply (erule_tac x="n" in allE)
apply (erule_tac x="n" in allE)
apply auto
done
lemma real_convex_bound_lt:
assumes xa: "(x::real) < a" and ya: "y < a" and u: "0 <= u" and v: "0 <= v"
and uv: "u + v = 1"
shows "u * x + v * y < a"
proof-
have uv': "u = 0 \<longrightarrow> v \<noteq> 0" using u v uv by arith
have "a = a * (u + v)" unfolding uv by simp
hence th: "u * a + v * a = a" by (simp add: field_simps)
from xa u have "u \<noteq> 0 \<Longrightarrow> u*x < u*a" by (simp add: mult_strict_left_mono)
from ya v have "v \<noteq> 0 \<Longrightarrow> v * y < v * a" by (simp add: mult_strict_left_mono)
from xa ya u v have "u * x + v * y < u * a + v * a"
apply (cases "u = 0", simp_all add: uv')
apply(rule mult_strict_left_mono)
using uv' apply simp_all
apply (rule add_less_le_mono)
apply(rule mult_strict_left_mono)
apply simp_all
apply (rule mult_left_mono)
apply simp_all
done
thus ?thesis unfolding th .
qed
lemma real_convex_bound_le:
assumes xa: "(x::real) \<le> a" and ya: "y \<le> a" and u: "0 <= u" and v: "0 <= v"
and uv: "u + v = 1"
shows "u * x + v * y \<le> a"
proof-
from xa ya u v have "u * x + v * y \<le> u * a + v * a" by (simp add: add_mono mult_left_mono)
also have "\<dots> \<le> (u + v) * a" by (simp add: field_simps)
finally show ?thesis unfolding uv by simp
qed
lemma infinite_enumerate: assumes fS: "infinite S"
shows "\<exists>r. subseq r \<and> (\<forall>n. r n \<in> S)"
unfolding subseq_def
using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
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)"
apply auto
apply (rule_tac x="d/2" in exI)
apply auto
done
lemma triangle_lemma:
assumes x: "0 <= (x::real)" and y:"0 <= y" and z: "0 <= z" and xy: "x^2 <= y^2 + z^2"
shows "x <= y + z"
proof-
have "y^2 + z^2 \<le> y^2 + 2*y*z + z^2" using z y by (simp add: mult_nonneg_nonneg)
with xy have th: "x ^2 \<le> (y+z)^2" by (simp add: power2_eq_square field_simps)
from y z have yz: "y + z \<ge> 0" by arith
from power2_le_imp_le[OF th yz] show ?thesis .
qed
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
(\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
let ?S = "(UNIV :: 'n set)"
{assume H: "?rhs"
then have ?lhs by auto}
moreover
{assume H: "?lhs"
then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
{fix i
from f have "P i (f i)" by metis
then have "P i (?x$i)" by auto
}
hence "\<forall>i. P i (?x$i)" by metis
hence ?rhs by metis }
ultimately show ?thesis by metis
qed
lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
lemma vec_add: "vec(x + y) = vec x + vec y" by (vector vec_def)
lemma vec_sub: "vec(x - y) = vec x - vec y" by (vector vec_def)
lemma vec_cmul: "vec(c* x) = c *s vec x " by (vector vec_def)
lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
lemma vec_setsum: assumes fS: "finite S"
shows "vec(setsum f S) = setsum (vec o f) S"
apply (induct rule: finite_induct[OF fS])
apply (simp)
apply (auto simp add: vec_add)
done
text{* Pasting vectors. *}
lemma linear_fstcart[intro]: "linear fstcart"
by (auto simp add: linear_def Cart_eq)
lemma linear_sndcart[intro]: "linear sndcart"
by (auto simp add: linear_def Cart_eq)
lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"
by (simp add: Cart_eq)
lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b::finite + 'c::finite)) + fstcart y"
by (simp add: Cart_eq)
lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b::finite + 'c::finite))"
by (simp add: Cart_eq)
lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^(_ + _))"
by (simp add: Cart_eq)
lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^(_ + _)) - fstcart y"
by (simp add: Cart_eq)
lemma fstcart_setsum:
fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
assumes fS: "finite S"
shows "fstcart (setsum f S) = setsum (\<lambda>i. fstcart (f i)) S"
by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"
by (simp add: Cart_eq)
lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^(_ + _)) + sndcart y"
by (simp add: Cart_eq)
lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^(_ + _))"
by (simp add: Cart_eq)
lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^(_ + _))"
by (simp add: Cart_eq)
lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^(_ + _)) - sndcart y"
by (simp add: Cart_eq)
lemma sndcart_setsum:
fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
assumes fS: "finite S"
shows "sndcart (setsum f S) = setsum (\<lambda>i. sndcart (f i)) S"
by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
by (simp add: pastecart_eq)
lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
by (simp add: pastecart_eq)
lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
by (simp add: pastecart_eq)
lemma pastecart_neg[simp]: "pastecart (- (x::'a::ring_1^_)) (- y) = - pastecart x y"
unfolding vector_sneg_minus1 pastecart_cmul ..
lemma pastecart_sub: "pastecart (x1::'a::ring_1^_) y1 - pastecart x2 y2 = pastecart (x1 - x2) (y1 - y2)"
by (simp add: diff_def pastecart_neg[symmetric] del: pastecart_neg)
lemma pastecart_setsum:
fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
assumes fS: "finite S"
shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
by (simp add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS])
lemma setsum_Plus:
"\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
(\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
unfolding Plus_def
by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
lemma setsum_UNIV_sum:
fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
apply (subst UNIV_Plus_UNIV [symmetric])
apply (rule setsum_Plus [OF finite finite])
done
lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))"
proof-
have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
by simp
have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
by (simp add: inner_vector_def setsum_UNIV_sum pastecart_def setsum_nonneg)
then show ?thesis
unfolding th0
unfolding norm_eq_sqrt_inner real_sqrt_le_iff id_def
by (simp add: inner_vector_def)
qed
lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
unfolding dist_norm by (metis fstcart_sub[symmetric] norm_fstcart)
lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"
proof-
have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
by simp
have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
by (simp add: inner_vector_def setsum_UNIV_sum pastecart_def setsum_nonneg)
then show ?thesis
unfolding th0
unfolding norm_eq_sqrt_inner real_sqrt_le_iff id_def
by (simp add: inner_vector_def)
qed
lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
unfolding dist_norm by (metis sndcart_sub[symmetric] norm_sndcart)
lemma dot_pastecart: "(pastecart (x1::real^'n) (x2::real^'m)) \<bullet> (pastecart y1 y2) = x1 \<bullet> y1 + x2 \<bullet> y2"
by (simp add: inner_vector_def setsum_UNIV_sum pastecart_def)
text {* TODO: move to NthRoot *}
lemma sqrt_add_le_add_sqrt:
assumes x: "0 \<le> x" and y: "0 \<le> y"
shows "sqrt (x + y) \<le> sqrt x + sqrt y"
apply (rule power2_le_imp_le)
apply (simp add: real_sum_squared_expand add_nonneg_nonneg x y)
apply (simp add: mult_nonneg_nonneg x y)
apply (simp add: add_nonneg_nonneg x y)
done
lemma norm_pastecart: "norm (pastecart x y) <= norm x + norm y"
unfolding norm_vector_def setL2_def setsum_UNIV_sum
by (simp add: sqrt_add_le_add_sqrt setsum_nonneg)
subsection {* A generic notion of "hull" (convex, affine, conic hull and closure). *}
definition hull :: "'a set set \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "hull" 75) where
"S hull s = Inter {t. t \<in> S \<and> s \<subseteq> t}"
lemma hull_same: "s \<in> S \<Longrightarrow> S hull s = s"
unfolding hull_def by auto
lemma hull_in: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) \<in> S"
unfolding hull_def subset_iff by auto
lemma hull_eq: "(\<And>T. T \<subseteq> S ==> Inter T \<in> S) ==> (S hull s) = s \<longleftrightarrow> s \<in> S"
using hull_same[of s S] hull_in[of S s] by metis
lemma hull_hull: "S hull (S hull s) = S hull s"
unfolding hull_def by blast
lemma hull_subset[intro]: "s \<subseteq> (S hull s)"
unfolding hull_def by blast
lemma hull_mono: " s \<subseteq> t ==> (S hull s) \<subseteq> (S hull t)"
unfolding hull_def by blast
lemma hull_antimono: "S \<subseteq> T ==> (T hull s) \<subseteq> (S hull s)"
unfolding hull_def by blast
lemma hull_minimal: "s \<subseteq> t \<Longrightarrow> t \<in> S ==> (S hull s) \<subseteq> t"
unfolding hull_def by blast
lemma subset_hull: "t \<in> S ==> S hull s \<subseteq> t \<longleftrightarrow> s \<subseteq> t"
unfolding hull_def by blast
lemma hull_unique: "s \<subseteq> t \<Longrightarrow> t \<in> S \<Longrightarrow> (\<And>t'. s \<subseteq> t' \<Longrightarrow> t' \<in> S ==> t \<subseteq> t')
==> (S hull s = t)"
unfolding hull_def by auto
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"
using hull_minimal[of S "{x. P x}" Q]
by (auto simp add: subset_eq Collect_def mem_def)
lemma hull_inc: "x \<in> S \<Longrightarrow> x \<in> P hull S" by (metis hull_subset subset_eq)
lemma hull_union_subset: "(S hull s) \<union> (S hull t) \<subseteq> (S hull (s \<union> t))"
unfolding Un_subset_iff by (metis hull_mono Un_upper1 Un_upper2)
lemma hull_union: assumes T: "\<And>T. T \<subseteq> S ==> Inter T \<in> S"
shows "S hull (s \<union> t) = S hull (S hull s \<union> S hull t)"
apply rule
apply (rule hull_mono)
unfolding Un_subset_iff
apply (metis hull_subset Un_upper1 Un_upper2 subset_trans)
apply (rule hull_minimal)
apply (metis hull_union_subset)
apply (metis hull_in T)
done
lemma hull_redundant_eq: "a \<in> (S hull s) \<longleftrightarrow> (S hull (insert a s) = S hull s)"
unfolding hull_def by blast
lemma hull_redundant: "a \<in> (S hull s) ==> (S hull (insert a s) = S hull s)"
by (metis hull_redundant_eq)
text{* Archimedian properties and useful consequences. *}
lemma real_arch_simple: "\<exists>n. x <= real (n::nat)"
using reals_Archimedean2[of x] apply auto by (rule_tac x="Suc n" in exI, auto)
lemmas real_arch_lt = reals_Archimedean2
lemmas real_arch = reals_Archimedean3
lemma real_arch_inv: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
using reals_Archimedean
apply (auto simp add: field_simps)
apply (subgoal_tac "inverse (real n) > 0")
apply arith
apply simp
done
lemma real_pow_lbound: "0 <= x ==> 1 + real n * x <= (1 + x) ^ n"
proof(induct n)
case 0 thus ?case by simp
next
case (Suc n)
hence h: "1 + real n * x \<le> (1 + x) ^ n" by simp
from h have p: "1 \<le> (1 + x) ^ n" using Suc.prems by simp
from h have "1 + real n * x + x \<le> (1 + x) ^ n + x" by simp
also have "\<dots> \<le> (1 + x) ^ Suc n" apply (subst diff_le_0_iff_le[symmetric])
apply (simp add: field_simps)
using mult_left_mono[OF p Suc.prems] by simp
finally show ?case by (simp add: real_of_nat_Suc field_simps)
qed
lemma real_arch_pow: assumes x: "1 < (x::real)" shows "\<exists>n. y < x^n"
proof-
from x have x0: "x - 1 > 0" by arith
from real_arch[OF x0, rule_format, of y]
obtain n::nat where n:"y < real n * (x - 1)" by metis
from x0 have x00: "x- 1 \<ge> 0" by arith
from real_pow_lbound[OF x00, of n] n
have "y < x^n" by auto
then show ?thesis by metis
qed
lemma real_arch_pow2: "\<exists>n. (x::real) < 2^ n"
using real_arch_pow[of 2 x] by simp
lemma real_arch_pow_inv: assumes y: "(y::real) > 0" and x1: "x < 1"
shows "\<exists>n. x^n < y"
proof-
{assume x0: "x > 0"
from x0 x1 have ix: "1 < 1/x" by (simp add: field_simps)
from real_arch_pow[OF ix, of "1/y"]
obtain n where n: "1/y < (1/x)^n" by blast
then
have ?thesis using y x0 by (auto simp add: field_simps power_divide) }
moreover
{assume "\<not> x > 0" with y x1 have ?thesis apply auto by (rule exI[where x=1], auto)}
ultimately show ?thesis by metis
qed
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)"
by (metis real_arch_inv)
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"
apply (rule forall_pos_mono)
apply auto
apply (atomize)
apply (erule_tac x="n - 1" in allE)
apply auto
done
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"
shows "x = 0"
proof-
{assume "x \<noteq> 0" with x0 have xp: "x > 0" by arith
from real_arch[OF xp, rule_format, of c] obtain n::nat where n: "c < real n * x" by blast
with xc[rule_format, of n] have "n = 0" by arith
with n c have False by simp}
then show ?thesis by blast
qed
(* ------------------------------------------------------------------------- *)
(* Geometric progression. *)
(* ------------------------------------------------------------------------- *)
lemma sum_gp_basic: "((1::'a::{field}) - x) * setsum (\<lambda>i. x^i) {0 .. n} = (1 - x^(Suc n))"
(is "?lhs = ?rhs")
proof-
{assume x1: "x = 1" hence ?thesis by simp}
moreover
{assume x1: "x\<noteq>1"
hence x1': "x - 1 \<noteq> 0" "1 - x \<noteq> 0" "x - 1 = - (1 - x)" "- (1 - x) \<noteq> 0" by auto
from geometric_sum[OF x1, of "Suc n", unfolded x1']
have "(- (1 - x)) * setsum (\<lambda>i. x^i) {0 .. n} = - (1 - x^(Suc n))"
unfolding atLeastLessThanSuc_atLeastAtMost
using x1' apply (auto simp only: field_simps)
apply (simp add: field_simps)
done
then have ?thesis by (simp add: field_simps) }
ultimately show ?thesis by metis
qed
lemma sum_gp_multiplied: assumes mn: "m <= n"
shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
(is "?lhs = ?rhs")
proof-
let ?S = "{0..(n - m)}"
from mn have mn': "n - m \<ge> 0" by arith
let ?f = "op + m"
have i: "inj_on ?f ?S" unfolding inj_on_def by auto
have f: "?f ` ?S = {m..n}"
using mn apply (auto simp add: image_iff Bex_def) by arith
have th: "op ^ x o op + m = (\<lambda>i. x^m * x^i)"
by (rule ext, simp add: power_add power_mult)
from setsum_reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})" by simp
then show ?thesis unfolding sum_gp_basic using mn
by (simp add: field_simps power_add[symmetric])
qed
lemma sum_gp: "setsum (op ^ (x::'a::{field})) {m .. n} =
(if n < m then 0 else if x = 1 then of_nat ((n + 1) - m)
else (x^ m - x^ (Suc n)) / (1 - x))"
proof-
{assume nm: "n < m" hence ?thesis by simp}
moreover
{assume "\<not> n < m" hence nm: "m \<le> n" by arith
{assume x: "x = 1" hence ?thesis by simp}
moreover
{assume x: "x \<noteq> 1" hence nz: "1 - x \<noteq> 0" by simp
from sum_gp_multiplied[OF nm, of x] nz have ?thesis by (simp add: field_simps)}
ultimately have ?thesis by metis
}
ultimately show ?thesis by metis
qed
lemma sum_gp_offset: "setsum (op ^ (x::'a::{field})) {m .. m+n} =
(if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
unfolding sum_gp[of x m "m + n"] power_Suc
by (simp add: field_simps power_add)
subsection{* A bit of linear algebra. *}
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 )"
definition "span S = (subspace hull S)"
definition "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
abbreviation "independent s == ~(dependent s)"
(* Closure properties of subspaces. *)
lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
lemma subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
lemma subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
by (metis subspace_def)
lemma subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *s x \<in> S"
by (metis subspace_def)
lemma subspace_neg: "subspace S \<Longrightarrow> (x::'a::ring_1^_) \<in> S \<Longrightarrow> - x \<in> S"
by (metis vector_sneg_minus1 subspace_mul)
lemma subspace_sub: "subspace S \<Longrightarrow> (x::'a::ring_1^_) \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
by (metis diff_def subspace_add subspace_neg)
lemma subspace_setsum:
assumes sA: "subspace A" and fB: "finite B"
and f: "\<forall>x\<in> B. f x \<in> A"
shows "setsum f B \<in> A"
using fB f sA
apply(induct rule: finite_induct[OF fB])
by (simp add: subspace_def sA, auto simp add: sA subspace_add)
lemma subspace_linear_image:
assumes lf: "linear (f::'a::semiring_1^_ \<Rightarrow> _)" and sS: "subspace S"
shows "subspace(f ` S)"
using lf sS linear_0[OF lf]
unfolding linear_def subspace_def
apply (auto simp add: image_iff)
apply (rule_tac x="x + y" in bexI, auto)
apply (rule_tac x="c*s x" in bexI, auto)
done
lemma subspace_linear_preimage: "linear (f::'a::semiring_1^_ \<Rightarrow> _) ==> subspace S ==> subspace {x. f x \<in> S}"
by (auto simp add: subspace_def linear_def linear_0[of f])
lemma subspace_trivial: "subspace {0::'a::semiring_1 ^_}"
by (simp add: subspace_def)
lemma subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
by (simp add: subspace_def)
lemma span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
by (metis span_def hull_mono)
lemma subspace_span: "subspace(span S)"
unfolding span_def
apply (rule hull_in[unfolded mem_def])
apply (simp only: subspace_def Inter_iff Int_iff subset_eq)
apply auto
apply (erule_tac x="X" in ballE)
apply (simp add: mem_def)
apply blast
apply (erule_tac x="X" in ballE)
apply (erule_tac x="X" in ballE)
apply (erule_tac x="X" in ballE)
apply (clarsimp simp add: mem_def)
apply simp
apply simp
apply simp
apply (erule_tac x="X" in ballE)
apply (erule_tac x="X" in ballE)
apply (simp add: mem_def)
apply simp
apply simp
done
lemma span_clauses:
"a \<in> S ==> a \<in> span S"
"0 \<in> span S"
"x\<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
"x \<in> span S \<Longrightarrow> c *s x \<in> span S"
by (metis span_def hull_subset subset_eq)
(metis subspace_span subspace_def)+
lemma span_induct: assumes SP: "\<And>x. x \<in> S ==> P x"
and P: "subspace P" and x: "x \<in> span S" shows "P x"
proof-
from SP have SP': "S \<subseteq> P" by (simp add: mem_def subset_eq)
from P have P': "P \<in> subspace" by (simp add: mem_def)
from x hull_minimal[OF SP' P', unfolded span_def[symmetric]]
show "P x" by (metis mem_def subset_eq)
qed
lemma span_empty: "span {} = {(0::'a::semiring_0 ^ _)}"
apply (simp add: span_def)
apply (rule hull_unique)
apply (auto simp add: mem_def subspace_def)
unfolding mem_def[of "0::'a^_", symmetric]
apply simp
done
lemma independent_empty: "independent {}"
by (simp add: dependent_def)
lemma independent_mono: "independent A \<Longrightarrow> B \<subseteq> A ==> independent B"
apply (clarsimp simp add: dependent_def span_mono)
apply (subgoal_tac "span (B - {a}) \<le> span (A - {a})")
apply force
apply (rule span_mono)
apply auto
done
lemma span_subspace: "A \<subseteq> B \<Longrightarrow> B \<le> span A \<Longrightarrow> subspace B \<Longrightarrow> span A = B"
by (metis order_antisym span_def hull_minimal mem_def)
lemma span_induct': assumes SP: "\<forall>x \<in> S. P x"
and P: "subspace P" shows "\<forall>x \<in> span S. P x"
using span_induct SP P by blast
inductive span_induct_alt_help for S:: "'a::semiring_1^_ \<Rightarrow> bool"
where
span_induct_alt_help_0: "span_induct_alt_help S 0"
| 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)"
lemma span_induct_alt':
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"
proof-
{fix x:: "'a^'n" assume x: "span_induct_alt_help S x"
have "h x"
apply (rule span_induct_alt_help.induct[OF x])
apply (rule h0)
apply (rule hS, assumption, assumption)
done}
note th0 = this
{fix x assume x: "x \<in> span S"
have "span_induct_alt_help S x"
proof(rule span_induct[where x=x and S=S])
show "x \<in> span S" using x .
next
fix x assume xS : "x \<in> S"
from span_induct_alt_help_S[OF xS span_induct_alt_help_0, of 1]
show "span_induct_alt_help S x" by simp
next
have "span_induct_alt_help S 0" by (rule span_induct_alt_help_0)
moreover
{fix x y assume h: "span_induct_alt_help S x" "span_induct_alt_help S y"
from h
have "span_induct_alt_help S (x + y)"
apply (induct rule: span_induct_alt_help.induct)
apply simp
unfolding add_assoc
apply (rule span_induct_alt_help_S)
apply assumption
apply simp
done}
moreover
{fix c x assume xt: "span_induct_alt_help S x"
then have "span_induct_alt_help S (c*s x)"
apply (induct rule: span_induct_alt_help.induct)
apply (simp add: span_induct_alt_help_0)
apply (simp add: vector_smult_assoc vector_add_ldistrib)
apply (rule span_induct_alt_help_S)
apply assumption
apply simp
done
}
ultimately show "subspace (span_induct_alt_help S)"
unfolding subspace_def mem_def Ball_def by blast
qed}
with th0 show ?thesis by blast
qed
lemma span_induct_alt:
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"
shows "h x"
using span_induct_alt'[of h S] h0 hS x by blast
(* Individual closure properties. *)
lemma span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
lemma span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
lemma span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
by (metis subspace_add subspace_span)
lemma span_mul: "x \<in> span S ==> (c *s x) \<in> span S"
by (metis subspace_span subspace_mul)
lemma span_neg: "x \<in> span S ==> - (x::'a::ring_1^_) \<in> span S"
by (metis subspace_neg subspace_span)
lemma span_sub: "(x::'a::ring_1^_) \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
by (metis subspace_span subspace_sub)
lemma span_setsum: "finite A \<Longrightarrow> \<forall>x \<in> A. f x \<in> span S ==> setsum f A \<in> span S"
by (rule subspace_setsum, rule subspace_span)
lemma span_add_eq: "(x::'a::ring_1^_) \<in> span S \<Longrightarrow> x + y \<in> span S \<longleftrightarrow> y \<in> span S"
apply (auto simp only: span_add span_sub)
apply (subgoal_tac "(x + y) - x \<in> span S", simp)
by (simp only: span_add span_sub)
(* Mapping under linear image. *)
lemma span_linear_image: assumes lf: "linear (f::'a::semiring_1 ^ _ => _)"
shows "span (f ` S) = f ` (span S)"
proof-
{fix x
assume x: "x \<in> span (f ` S)"
have "x \<in> f ` span S"
apply (rule span_induct[where x=x and S = "f ` S"])
apply (clarsimp simp add: image_iff)
apply (frule span_superset)
apply blast
apply (simp only: mem_def)
apply (rule subspace_linear_image[OF lf])
apply (rule subspace_span)
apply (rule x)
done}
moreover
{fix x assume x: "x \<in> span S"
have th0:"(\<lambda>a. f a \<in> span (f ` S)) = {x. f x \<in> span (f ` S)}" apply (rule set_ext)
unfolding mem_def Collect_def ..
have "f x \<in> span (f ` S)"
apply (rule span_induct[where S=S])
apply (rule span_superset)
apply simp
apply (subst th0)
apply (rule subspace_linear_preimage[OF lf subspace_span, of "f ` S"])
apply (rule x)
done}
ultimately show ?thesis by blast
qed
(* The key breakdown property. *)
lemma span_breakdown:
assumes bS: "(b::'a::ring_1 ^ _) \<in> S" and aS: "a \<in> span S"
shows "\<exists>k. a - k*s b \<in> span (S - {b})" (is "?P a")
proof-
{fix x assume xS: "x \<in> S"
{assume ab: "x = b"
then have "?P x"
apply simp
apply (rule exI[where x="1"], simp)
by (rule span_0)}
moreover
{assume ab: "x \<noteq> b"
then have "?P x" using xS
apply -
apply (rule exI[where x=0])
apply (rule span_superset)
by simp}
ultimately have "?P x" by blast}
moreover have "subspace ?P"
unfolding subspace_def
apply auto
apply (simp add: mem_def)
apply (rule exI[where x=0])
using span_0[of "S - {b}"]
apply (simp add: mem_def)
apply (clarsimp simp add: mem_def)
apply (rule_tac x="k + ka" in exI)
apply (subgoal_tac "x + y - (k + ka) *s b = (x - k*s b) + (y - ka *s b)")
apply (simp only: )
apply (rule span_add[unfolded mem_def])
apply assumption+
apply (vector field_simps)
apply (clarsimp simp add: mem_def)
apply (rule_tac x= "c*k" in exI)
apply (subgoal_tac "c *s x - (c * k) *s b = c*s (x - k*s b)")
apply (simp only: )
apply (rule span_mul[unfolded mem_def])
apply assumption
by (vector field_simps)
ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
qed
lemma span_breakdown_eq:
"(x::'a::ring_1^_) \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *s a) \<in> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
{assume x: "x \<in> span (insert a S)"
from x span_breakdown[of "a" "insert a S" "x"]
have ?rhs apply clarsimp
apply (rule_tac x= "k" in exI)
apply (rule set_rev_mp[of _ "span (S - {a})" _])
apply assumption
apply (rule span_mono)
apply blast
done}
moreover
{ fix k assume k: "x - k *s a \<in> span S"
have eq: "x = (x - k *s a) + k *s a" by vector
have "(x - k *s a) + k *s a \<in> span (insert a S)"
apply (rule span_add)
apply (rule set_rev_mp[of _ "span S" _])
apply (rule k)
apply (rule span_mono)
apply blast
apply (rule span_mul)
apply (rule span_superset)
apply blast
done
then have ?lhs using eq by metis}
ultimately show ?thesis by blast
qed
(* Hence some "reversal" results.*)
lemma in_span_insert:
assumes a: "(a::'a::field^_) \<in> span (insert b S)" and na: "a \<notin> span S"
shows "b \<in> span (insert a S)"
proof-
from span_breakdown[of b "insert b S" a, OF insertI1 a]
obtain k where k: "a - k*s b \<in> span (S - {b})" by auto
{assume k0: "k = 0"
with k have "a \<in> span S"
apply (simp)
apply (rule set_rev_mp)
apply assumption
apply (rule span_mono)
apply blast
done
with na have ?thesis by blast}
moreover
{assume k0: "k \<noteq> 0"
have eq: "b = (1/k) *s a - ((1/k) *s a - b)" by vector
from k0 have eq': "(1/k) *s (a - k*s b) = (1/k) *s a - b"
by (vector field_simps)
from k have "(1/k) *s (a - k*s b) \<in> span (S - {b})"
by (rule span_mul)
hence th: "(1/k) *s a - b \<in> span (S - {b})"
unfolding eq' .
from k
have ?thesis
apply (subst eq)
apply (rule span_sub)
apply (rule span_mul)
apply (rule span_superset)
apply blast
apply (rule set_rev_mp)
apply (rule th)
apply (rule span_mono)
using na by blast}
ultimately show ?thesis by blast
qed
lemma in_span_delete:
assumes a: "(a::'a::field^_) \<in> span S"
and na: "a \<notin> span (S-{b})"
shows "b \<in> span (insert a (S - {b}))"
apply (rule in_span_insert)
apply (rule set_rev_mp)
apply (rule a)
apply (rule span_mono)
apply blast
apply (rule na)
done
(* Transitivity property. *)
lemma span_trans:
assumes x: "(x::'a::ring_1^_) \<in> span S" and y: "y \<in> span (insert x S)"
shows "y \<in> span S"
proof-
from span_breakdown[of x "insert x S" y, OF insertI1 y]
obtain k where k: "y -k*s x \<in> span (S - {x})" by auto
have eq: "y = (y - k *s x) + k *s x" by vector
show ?thesis
apply (subst eq)
apply (rule span_add)
apply (rule set_rev_mp)
apply (rule k)
apply (rule span_mono)
apply blast
apply (rule span_mul)
by (rule x)
qed
(* ------------------------------------------------------------------------- *)
(* An explicit expansion is sometimes needed. *)
(* ------------------------------------------------------------------------- *)
lemma span_explicit:
"span P = {y::'a::semiring_1^_. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *s v) S = y}"
(is "_ = ?E" is "_ = {y. ?h y}" is "_ = {y. \<exists>S u. ?Q S u y}")
proof-
{fix x assume x: "x \<in> ?E"
then obtain S u where fS: "finite S" and SP: "S\<subseteq>P" and u: "setsum (\<lambda>v. u v *s v) S = x"
by blast
have "x \<in> span P"
unfolding u[symmetric]
apply (rule span_setsum[OF fS])
using span_mono[OF SP]
by (auto intro: span_superset span_mul)}
moreover
have "\<forall>x \<in> span P. x \<in> ?E"
unfolding mem_def Collect_def
proof(rule span_induct_alt')
show "?h 0"
apply (rule exI[where x="{}"]) by simp
next
fix c x y
assume x: "x \<in> P" and hy: "?h y"
from hy obtain S u where fS: "finite S" and SP: "S\<subseteq>P"
and u: "setsum (\<lambda>v. u v *s v) S = y" by blast
let ?S = "insert x S"
let ?u = "\<lambda>y. if y = x then (if x \<in> S then u y + c else c)
else u y"
from fS SP x have th0: "finite (insert x S)" "insert x S \<subseteq> P" by blast+
{assume xS: "x \<in> S"
have S1: "S = (S - {x}) \<union> {x}"
and Sss:"finite (S - {x})" "finite {x}" "(S -{x}) \<inter> {x} = {}" using xS fS by auto
have "setsum (\<lambda>v. ?u v *s v) ?S =(\<Sum>v\<in>S - {x}. u v *s v) + (u x + c) *s x"
using xS
by (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]]
setsum_clauses(2)[OF fS] cong del: if_weak_cong)
also have "\<dots> = (\<Sum>v\<in>S. u v *s v) + c *s x"
apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
by (vector field_simps)
also have "\<dots> = c*s x + y"
by (simp add: add_commute u)
finally have "setsum (\<lambda>v. ?u v *s v) ?S = c*s x + y" .
then have "?Q ?S ?u (c*s x + y)" using th0 by blast}
moreover
{assume xS: "x \<notin> S"
have th00: "(\<Sum>v\<in>S. (if v = x then c else u v) *s v) = y"
unfolding u[symmetric]
apply (rule setsum_cong2)
using xS by auto
have "?Q ?S ?u (c*s x + y)" using fS xS th0
by (simp add: th00 setsum_clauses add_commute cong del: if_weak_cong)}
ultimately have "?Q ?S ?u (c*s x + y)"
by (cases "x \<in> S", simp, simp)
then show "?h (c*s x + y)"
apply -
apply (rule exI[where x="?S"])
apply (rule exI[where x="?u"]) by metis
qed
ultimately show ?thesis by blast
qed
lemma dependent_explicit:
"dependent P \<longleftrightarrow> (\<exists>S u. finite S \<and> S \<subseteq> P \<and> (\<exists>(v::'a::{idom,field}^_) \<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *s v) S = 0))" (is "?lhs = ?rhs")
proof-
{assume dP: "dependent P"
then obtain a S u where aP: "a \<in> P" and fS: "finite S"
and SP: "S \<subseteq> P - {a}" and ua: "setsum (\<lambda>v. u v *s v) S = a"
unfolding dependent_def span_explicit by blast
let ?S = "insert a S"
let ?u = "\<lambda>y. if y = a then - 1 else u y"
let ?v = a
from aP SP have aS: "a \<notin> S" by blast
from fS SP aP have th0: "finite ?S" "?S \<subseteq> P" "?v \<in> ?S" "?u ?v \<noteq> 0" by auto
have s0: "setsum (\<lambda>v. ?u v *s v) ?S = 0"
using fS aS
apply (simp add: vector_smult_lneg setsum_clauses field_simps)
apply (subst (2) ua[symmetric])
apply (rule setsum_cong2)
by auto
with th0 have ?rhs
apply -
apply (rule exI[where x= "?S"])
apply (rule exI[where x= "?u"])
by clarsimp}
moreover
{fix S u v assume fS: "finite S"
and SP: "S \<subseteq> P" and vS: "v \<in> S" and uv: "u v \<noteq> 0"
and u: "setsum (\<lambda>v. u v *s v) S = 0"
let ?a = v
let ?S = "S - {v}"
let ?u = "\<lambda>i. (- u i) / u v"
have th0: "?a \<in> P" "finite ?S" "?S \<subseteq> P" using fS SP vS by auto
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"
using fS vS uv
by (simp add: setsum_diff1 vector_smult_lneg divide_inverse
vector_smult_assoc field_simps)
also have "\<dots> = ?a"
unfolding setsum_cmul u
using uv by (simp add: vector_smult_lneg)
finally have "setsum (\<lambda>v. ?u v *s v) ?S = ?a" .
with th0 have ?lhs
unfolding dependent_def span_explicit
apply -
apply (rule bexI[where x= "?a"])
apply simp_all
apply (rule exI[where x= "?S"])
by auto}
ultimately show ?thesis by blast
qed
lemma span_finite:
assumes fS: "finite S"
shows "span S = {(y::'a::semiring_1^_). \<exists>u. setsum (\<lambda>v. u v *s v) S = y}"
(is "_ = ?rhs")
proof-
{fix y assume y: "y \<in> span S"
from y obtain S' u where fS': "finite S'" and SS': "S' \<subseteq> S" and
u: "setsum (\<lambda>v. u v *s v) S' = y" unfolding span_explicit by blast
let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
from setsum_restrict_set[OF fS, of "\<lambda>v. u v *s v" S', symmetric] SS'
have "setsum (\<lambda>v. ?u v *s v) S = setsum (\<lambda>v. u v *s v) S'"
unfolding cond_value_iff cond_application_beta
by (simp add: cond_value_iff inf_absorb2 cong del: if_weak_cong)
hence "setsum (\<lambda>v. ?u v *s v) S = y" by (metis u)
hence "y \<in> ?rhs" by auto}
moreover
{fix y u assume u: "setsum (\<lambda>v. u v *s v) S = y"
then have "y \<in> span S" using fS unfolding span_explicit by auto}
ultimately show ?thesis by blast
qed
(* Standard bases are a spanning set, and obviously finite. *)
lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n | i. i \<in> (UNIV :: 'n set)} = UNIV"
apply (rule set_ext)
apply auto
apply (subst basis_expansion[symmetric])
apply (rule span_setsum)
apply simp
apply auto
apply (rule span_mul)
apply (rule span_superset)
apply (auto simp add: Collect_def mem_def)
done
lemma finite_stdbasis: "finite {basis i ::real^'n |i. i\<in> (UNIV:: 'n set)}" (is "finite ?S")
proof-
have eq: "?S = basis ` UNIV" by blast
show ?thesis unfolding eq by auto
qed
lemma card_stdbasis: "card {basis i ::real^'n |i. i\<in> (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _")
proof-
have eq: "?S = basis ` UNIV" by blast
show ?thesis unfolding eq using card_image[OF basis_inj] by simp
qed
lemma independent_stdbasis_lemma:
assumes x: "(x::'a::semiring_1 ^ _) \<in> span (basis ` S)"
and iS: "i \<notin> S"
shows "(x$i) = 0"
proof-
let ?U = "UNIV :: 'n set"
let ?B = "basis ` S"
let ?P = "\<lambda>(x::'a^_). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
{fix x::"'a^_" assume xS: "x\<in> ?B"
from xS have "?P x" by auto}
moreover
have "subspace ?P"
by (auto simp add: subspace_def Collect_def mem_def)
ultimately show ?thesis
using x span_induct[of ?B ?P x] iS by blast
qed
lemma independent_stdbasis: "independent {basis i ::real^'n |i. i\<in> (UNIV :: 'n set)}"
proof-
let ?I = "UNIV :: 'n set"
let ?b = "basis :: _ \<Rightarrow> real ^'n"
let ?B = "?b ` ?I"
have eq: "{?b i|i. i \<in> ?I} = ?B"
by auto
{assume d: "dependent ?B"
then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
unfolding dependent_def by auto
have eq1: "?B - {?b k} = ?B - ?b ` {k}" by simp
have eq2: "?B - {?b k} = ?b ` (?I - {k})"
unfolding eq1
apply (rule inj_on_image_set_diff[symmetric])
apply (rule basis_inj) using k(1) by auto
from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
from independent_stdbasis_lemma[OF th0, of k, simplified]
have False by simp}
then show ?thesis unfolding eq dependent_def ..
qed
(* This is useful for building a basis step-by-step. *)
lemma independent_insert:
"independent(insert (a::'a::field ^_) S) \<longleftrightarrow>
(if a \<in> S then independent S
else independent S \<and> a \<notin> span S)" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
{assume aS: "a \<in> S"
hence ?thesis using insert_absorb[OF aS] by simp}
moreover
{assume aS: "a \<notin> S"
{assume i: ?lhs
then have ?rhs using aS
apply simp
apply (rule conjI)
apply (rule independent_mono)
apply assumption
apply blast
by (simp add: dependent_def)}
moreover
{assume i: ?rhs
have ?lhs using i aS
apply simp
apply (auto simp add: dependent_def)
apply (case_tac "aa = a", auto)
apply (subgoal_tac "insert a S - {aa} = insert a (S - {aa})")
apply simp
apply (subgoal_tac "a \<in> span (insert aa (S - {aa}))")
apply (subgoal_tac "insert aa (S - {aa}) = S")
apply simp
apply blast
apply (rule in_span_insert)
apply assumption
apply blast
apply blast
done}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed
(* The degenerate case of the Exchange Lemma. *)
lemma mem_delete: "x \<in> (A - {a}) \<longleftrightarrow> x \<noteq> a \<and> x \<in> A"
by blast
lemma span_span: "span (span A) = span A"
unfolding span_def hull_hull ..
lemma span_inc: "S \<subseteq> span S"
by (metis subset_eq span_superset)
lemma spanning_subset_independent:
assumes BA: "B \<subseteq> A" and iA: "independent (A::('a::field ^_) set)"
and AsB: "A \<subseteq> span B"
shows "A = B"
proof
from BA show "B \<subseteq> A" .
next
from span_mono[OF BA] span_mono[OF AsB]
have sAB: "span A = span B" unfolding span_span by blast
{fix x assume x: "x \<in> A"
from iA have th0: "x \<notin> span (A - {x})"
unfolding dependent_def using x by blast
from x have xsA: "x \<in> span A" by (blast intro: span_superset)
have "A - {x} \<subseteq> A" by blast
hence th1:"span (A - {x}) \<subseteq> span A" by (metis span_mono)
{assume xB: "x \<notin> B"
from xB BA have "B \<subseteq> A -{x}" by blast
hence "span B \<subseteq> span (A - {x})" by (metis span_mono)
with th1 th0 sAB have "x \<notin> span A" by blast
with x have False by (metis span_superset)}
then have "x \<in> B" by blast}
then show "A \<subseteq> B" by blast
qed
(* The general case of the Exchange Lemma, the key to what follows. *)
lemma exchange_lemma:
assumes f:"finite (t:: ('a::field^'n) set)" and i: "independent s"
and sp:"s \<subseteq> span t"
shows "\<exists>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
using f i sp
proof(induct "card (t - s)" arbitrary: s t rule: less_induct)
case less
note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
let ?P = "\<lambda>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
let ?ths = "\<exists>t'. ?P t'"
{assume st: "s \<subseteq> t"
from st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
by (auto intro: span_superset)}
moreover
{assume st: "t \<subseteq> s"
from spanning_subset_independent[OF st s sp]
st ft span_mono[OF st] have ?ths apply - apply (rule exI[where x=t])
by (auto intro: span_superset)}
moreover
{assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
from b have "t - {b} - s \<subset> t - s" by blast
then have cardlt: "card (t - {b} - s) < card (t - s)" using ft
by (auto intro: psubset_card_mono)
from b ft have ct0: "card t \<noteq> 0" by auto
{assume stb: "s \<subseteq> span(t -{b})"
from ft have ftb: "finite (t -{b})" by auto
from less(1)[OF cardlt ftb s stb]
obtain u where u: "card u = card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" and fu: "finite u" by blast
let ?w = "insert b u"
have th0: "s \<subseteq> insert b u" using u by blast
from u(3) b have "u \<subseteq> s \<union> t" by blast
then have th1: "insert b u \<subseteq> s \<union> t" using u b by blast
have bu: "b \<notin> u" using b u by blast
from u(1) ft b have "card u = (card t - 1)" by auto
then
have th2: "card (insert b u) = card t"
using card_insert_disjoint[OF fu bu] ct0 by auto
from u(4) have "s \<subseteq> span u" .
also have "\<dots> \<subseteq> span (insert b u)" apply (rule span_mono) by blast
finally have th3: "s \<subseteq> span (insert b u)" .
from th0 th1 th2 th3 fu have th: "?P ?w" by blast
from th have ?ths by blast}
moreover
{assume stb: "\<not> s \<subseteq> span(t -{b})"
from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
have ab: "a \<noteq> b" using a b by blast
have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
using cardlt ft a b by auto
have ft': "finite (insert a (t - {b}))" using ft by auto
{fix x assume xs: "x \<in> s"
have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
from b(1) have "b \<in> span t" by (simp add: span_superset)
have bs: "b \<in> span (insert a (t - {b}))" apply(rule in_span_delete)
using a sp unfolding subset_eq by auto
from xs sp have "x \<in> span t" by blast
with span_mono[OF t]
have x: "x \<in> span (insert b (insert a (t - {b})))" ..
from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))" .}
then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast
from less(1)[OF mlt ft' s sp'] obtain u where
u: "card u = card (insert a (t -{b}))" "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
"s \<subseteq> span u" by blast
from u a b ft at ct0 have "?P u" by auto
then have ?ths by blast }
ultimately have ?ths by blast
}
ultimately
show ?ths by blast
qed
(* This implies corresponding size bounds. *)
lemma independent_span_bound:
assumes f: "finite t" and i: "independent (s::('a::field^_) set)" and sp:"s \<subseteq> span t"
shows "finite s \<and> card s \<le> card t"
by (metis exchange_lemma[OF f i sp] finite_subset card_mono)
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
proof-
have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
show ?thesis unfolding eq
apply (rule finite_imageI)
apply (rule finite)
done
qed
lemma independent_bound:
fixes S:: "(real^'n) set"
shows "independent S \<Longrightarrow> finite S \<and> card S <= CARD('n)"
apply (subst card_stdbasis[symmetric])
apply (rule independent_span_bound)
apply (rule finite_Atleast_Atmost_nat)
apply assumption
unfolding span_stdbasis
apply (rule subset_UNIV)
done
lemma dependent_biggerset: "(finite (S::(real ^'n) set) ==> card S > CARD('n)) ==> dependent S"
by (metis independent_bound not_less)
(* Hence we can create a maximal independent subset. *)
lemma maximal_independent_subset_extend:
assumes sv: "(S::(real^'n) set) \<subseteq> V" and iS: "independent S"
shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
using sv iS
proof(induct "CARD('n) - card S" arbitrary: S rule: less_induct)
case less
note sv = `S \<subseteq> V` and i = `independent S`
let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
let ?ths = "\<exists>x. ?P x"
let ?d = "CARD('n)"
{assume "V \<subseteq> span S"
then have ?ths using sv i by blast }
moreover
{assume VS: "\<not> V \<subseteq> span S"
from VS obtain a where a: "a \<in> V" "a \<notin> span S" by blast
from a have aS: "a \<notin> S" by (auto simp add: span_superset)
have th0: "insert a S \<subseteq> V" using a sv by blast
from independent_insert[of a S] i a
have th1: "independent (insert a S)" by auto
have mlt: "?d - card (insert a S) < ?d - card S"
using aS a independent_bound[OF th1]
by auto
from less(1)[OF mlt th0 th1]
obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
by blast
from B have "?P B" by auto
then have ?ths by blast}
ultimately show ?ths by blast
qed
lemma maximal_independent_subset:
"\<exists>(B:: (real ^'n) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty)
(* Notion of dimension. *)
definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n))"
lemma basis_exists: "\<exists>B. (B :: (real ^'n) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = dim V)"
unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (card B = n)"]
using maximal_independent_subset[of V] independent_bound
by auto
(* Consequences of independence or spanning for cardinality. *)
lemma independent_card_le_dim:
assumes "(B::(real ^'n) set) \<subseteq> V" and "independent B" shows "card B \<le> dim V"
proof -
from basis_exists[of V] `B \<subseteq> V`
obtain B' where "independent B'" and "B \<subseteq> span B'" and "card B' = dim V" by blast
with independent_span_bound[OF _ `independent B` `B \<subseteq> span B'`] independent_bound[of B']
show ?thesis by auto
qed
lemma span_card_ge_dim: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
by (metis basis_exists[of V] independent_span_bound subset_trans)
lemma basis_card_eq_dim:
"B \<subseteq> (V:: (real ^'n) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_bound)
lemma dim_unique: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> card B = n \<Longrightarrow> dim V = n"
by (metis basis_card_eq_dim)
(* More lemmas about dimension. *)
lemma dim_univ: "dim (UNIV :: (real^'n) set) = CARD('n)"
apply (rule dim_unique[of "{basis i |i. i\<in> (UNIV :: 'n set)}"])
by (auto simp only: span_stdbasis card_stdbasis finite_stdbasis independent_stdbasis)
lemma dim_subset:
"(S:: (real ^'n) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
using basis_exists[of T] basis_exists[of S]
by (metis independent_card_le_dim subset_trans)
lemma dim_subset_univ: "dim (S:: (real^'n) set) \<le> CARD('n)"
by (metis dim_subset subset_UNIV dim_univ)
(* Converses to those. *)
lemma card_ge_dim_independent:
assumes BV:"(B::(real ^'n) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
shows "V \<subseteq> span B"
proof-
{fix a assume aV: "a \<in> V"
{assume aB: "a \<notin> span B"
then have iaB: "independent (insert a B)" using iB aV BV by (simp add: independent_insert)
from aV BV have th0: "insert a B \<subseteq> V" by blast
from aB have "a \<notin>B" by (auto simp add: span_superset)
with independent_card_le_dim[OF th0 iaB] dVB independent_bound[OF iB] have False by auto }
then have "a \<in> span B" by blast}
then show ?thesis by blast
qed
lemma card_le_dim_spanning:
assumes BV: "(B:: (real ^'n) set) \<subseteq> V" and VB: "V \<subseteq> span B"
and fB: "finite B" and dVB: "dim V \<ge> card B"
shows "independent B"
proof-
{fix a assume a: "a \<in> B" "a \<in> span (B -{a})"
from a fB have c0: "card B \<noteq> 0" by auto
from a fB have cb: "card (B -{a}) = card B - 1" by auto
from BV a have th0: "B -{a} \<subseteq> V" by blast
{fix x assume x: "x \<in> V"
from a have eq: "insert a (B -{a}) = B" by blast
from x VB have x': "x \<in> span B" by blast
from span_trans[OF a(2), unfolded eq, OF x']
have "x \<in> span (B -{a})" . }
then have th1: "V \<subseteq> span (B -{a})" by blast
have th2: "finite (B -{a})" using fB by auto
from span_card_ge_dim[OF th0 th1 th2]
have c: "dim V \<le> card (B -{a})" .
from c c0 dVB cb have False by simp}
then show ?thesis unfolding dependent_def by blast
qed
lemma card_eq_dim: "(B:: (real ^'n) set) \<subseteq> V \<Longrightarrow> card B = dim V \<Longrightarrow> finite B \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
by (metis order_eq_iff card_le_dim_spanning
card_ge_dim_independent)
(* ------------------------------------------------------------------------- *)
(* More general size bound lemmas. *)
(* ------------------------------------------------------------------------- *)
lemma independent_bound_general:
"independent (S:: (real^'n) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
by (metis independent_card_le_dim independent_bound subset_refl)
lemma dependent_biggerset_general: "(finite (S:: (real^'n) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
using independent_bound_general[of S] by (metis linorder_not_le)
lemma dim_span: "dim (span (S:: (real ^'n) set)) = dim S"
proof-
have th0: "dim S \<le> dim (span S)"
by (auto simp add: subset_eq intro: dim_subset span_superset)
from basis_exists[of S]
obtain B where B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
have bSS: "B \<subseteq> span S" using B(1) by (metis subset_eq span_inc)
have sssB: "span S \<subseteq> span B" using span_mono[OF B(3)] by (simp add: span_span)
from span_card_ge_dim[OF bSS sssB fB(1)] th0 show ?thesis
using fB(2) by arith
qed
lemma subset_le_dim: "(S:: (real ^'n) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
by (metis dim_span dim_subset)
lemma span_eq_dim: "span (S:: (real ^'n) set) = span T ==> dim S = dim T"
by (metis dim_span)
lemma spans_image:
assumes lf: "linear (f::'a::semiring_1^_ \<Rightarrow> _)" and VB: "V \<subseteq> span B"
shows "f ` V \<subseteq> span (f ` B)"
unfolding span_linear_image[OF lf]
by (metis VB image_mono)
lemma dim_image_le:
fixes f :: "real^'n \<Rightarrow> real^'m"
assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n) set)"
proof-
from basis_exists[of S] obtain B where
B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" by blast
from B have fB: "finite B" "card B = dim S" using independent_bound by blast+
have "dim (f ` S) \<le> card (f ` B)"
apply (rule span_card_ge_dim)
using lf B fB by (auto simp add: span_linear_image spans_image subset_image_iff)
also have "\<dots> \<le> dim S" using card_image_le[OF fB(1)] fB by simp
finally show ?thesis .
qed
(* Relation between bases and injectivity/surjectivity of map. *)
lemma spanning_surjective_image:
assumes us: "UNIV \<subseteq> span (S:: ('a::semiring_1 ^_) set)"
and lf: "linear f" and sf: "surj f"
shows "UNIV \<subseteq> span (f ` S)"
proof-
have "UNIV \<subseteq> f ` UNIV" using sf by (auto simp add: surj_def)
also have " \<dots> \<subseteq> span (f ` S)" using spans_image[OF lf us] .
finally show ?thesis .
qed
lemma independent_injective_image:
assumes iS: "independent (S::('a::semiring_1^_) set)" and lf: "linear f" and fi: "inj f"
shows "independent (f ` S)"
proof-
{fix a assume a: "a \<in> S" "f a \<in> span (f ` S - {f a})"
have eq: "f ` S - {f a} = f ` (S - {a})" using fi
by (auto simp add: inj_on_def)
from a have "f a \<in> f ` span (S -{a})"
unfolding eq span_linear_image[OF lf, of "S - {a}"] by blast
hence "a \<in> span (S -{a})" using fi by (auto simp add: inj_on_def)
with a(1) iS have False by (simp add: dependent_def) }
then show ?thesis unfolding dependent_def by blast
qed
(* ------------------------------------------------------------------------- *)
(* Picking an orthogonal replacement for a spanning set. *)
(* ------------------------------------------------------------------------- *)
(* FIXME : Move to some general theory ?*)
definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
lemma vector_sub_project_orthogonal: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
unfolding inner_simps smult_conv_scaleR by auto
lemma basis_orthogonal:
fixes B :: "(real ^'n) set"
assumes fB: "finite B"
shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
(is " \<exists>C. ?P B C")
proof(induct rule: finite_induct[OF fB])
case 1 thus ?case apply (rule exI[where x="{}"]) by (auto simp add: pairwise_def)
next
case (2 a B)
note fB = `finite B` and aB = `a \<notin> B`
from `\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C`
obtain C where C: "finite C" "card C \<le> card B"
"span C = span B" "pairwise orthogonal C" by blast
let ?a = "a - setsum (\<lambda>x. (x \<bullet> a / (x \<bullet> x)) *s x) C"
let ?C = "insert ?a C"
from C(1) have fC: "finite ?C" by simp
from fB aB C(1,2) have cC: "card ?C \<le> card (insert a B)" by (simp add: card_insert_if)
{fix x k
have th0: "\<And>(a::'b::comm_ring) b c. a - (b - c) = c + (a - b)" by (simp add: field_simps)
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"
apply (simp only: vector_ssub_ldistrib th0)
apply (rule span_add_eq)
apply (rule span_mul)
apply (rule span_setsum[OF C(1)])
apply clarify
apply (rule span_mul)
by (rule span_superset)}
then have SC: "span ?C = span (insert a B)"
unfolding expand_set_eq span_breakdown_eq C(3)[symmetric] by auto
thm pairwise_def
{fix x y assume xC: "x \<in> ?C" and yC: "y \<in> ?C" and xy: "x \<noteq> y"
{assume xa: "x = ?a" and ya: "y = ?a"
have "orthogonal x y" using xa ya xy by blast}
moreover
{assume xa: "x = ?a" and ya: "y \<noteq> ?a" "y \<in> C"
from ya have Cy: "C = insert y (C - {y})" by blast
have fth: "finite (C - {y})" using C by simp
have "orthogonal x y"
using xa ya
unfolding orthogonal_def xa inner_simps diff_eq_0_iff_eq
apply simp
apply (subst Cy)
using C(1) fth
apply (simp only: setsum_clauses) unfolding smult_conv_scaleR
apply (auto simp add: inner_simps inner_commute[of y a] dot_lsum[OF fth])
apply (rule setsum_0')
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
by auto}
moreover
{assume xa: "x \<noteq> ?a" "x \<in> C" and ya: "y = ?a"
from xa have Cx: "C = insert x (C - {x})" by blast
have fth: "finite (C - {x})" using C by simp
have "orthogonal x y"
using xa ya
unfolding orthogonal_def ya inner_simps diff_eq_0_iff_eq
apply simp
apply (subst Cx)
using C(1) fth
apply (simp only: setsum_clauses) unfolding smult_conv_scaleR
apply (subst inner_commute[of x])
apply (auto simp add: inner_simps inner_commute[of x a] dot_rsum[OF fth])
apply (rule setsum_0')
apply clarsimp
apply (rule C(4)[unfolded pairwise_def orthogonal_def, rule_format])
by auto}
moreover
{assume xa: "x \<in> C" and ya: "y \<in> C"
have "orthogonal x y" using xa ya xy C(4) unfolding pairwise_def by blast}
ultimately have "orthogonal x y" using xC yC by blast}
then have CPO: "pairwise orthogonal ?C" unfolding pairwise_def by blast
from fC cC SC CPO have "?P (insert a B) ?C" by blast
then show ?case by blast
qed
lemma orthogonal_basis_exists:
fixes V :: "(real ^'n) set"
shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (card B = dim V) \<and> pairwise orthogonal B"
proof-
from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "card B = dim V" by blast
from B have fB: "finite B" "card B = dim V" using independent_bound by auto
from basis_orthogonal[OF fB(1)] obtain C where
C: "finite C" "card C \<le> card B" "span C = span B" "pairwise orthogonal C" by blast
from C B
have CSV: "C \<subseteq> span V" by (metis span_inc span_mono subset_trans)
from span_mono[OF B(3)] C have SVC: "span V \<subseteq> span C" by (simp add: span_span)
from card_le_dim_spanning[OF CSV SVC C(1)] C(2,3) fB
have iC: "independent C" by (simp add: dim_span)
from C fB have "card C \<le> dim V" by simp
moreover have "dim V \<le> card C" using span_card_ge_dim[OF CSV SVC C(1)]
by (simp add: dim_span)
ultimately have CdV: "card C = dim V" using C(1) by simp
from C B CSV CdV iC show ?thesis by auto
qed
lemma span_eq: "span S = span T \<longleftrightarrow> S \<subseteq> span T \<and> T \<subseteq> span S"
using span_inc[unfolded subset_eq] using span_mono[of T "span S"] span_mono[of S "span T"]
by(auto simp add: span_span)
(* ------------------------------------------------------------------------- *)
(* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *)
(* ------------------------------------------------------------------------- *)
lemma span_not_univ_orthogonal:
assumes sU: "span S \<noteq> UNIV"
shows "\<exists>(a:: real ^'n). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
proof-
from sU obtain a where a: "a \<notin> span S" by blast
from orthogonal_basis_exists obtain B where
B: "independent B" "B \<subseteq> span S" "S \<subseteq> span B" "card B = dim S" "pairwise orthogonal B"
by blast
from B have fB: "finite B" "card B = dim S" using independent_bound by auto
from span_mono[OF B(2)] span_mono[OF B(3)]
have sSB: "span S = span B" by (simp add: span_span)
let ?a = "a - setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *s b) B"
have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *s b) B \<in> span S"
unfolding sSB
apply (rule span_setsum[OF fB(1)])
apply clarsimp
apply (rule span_mul)
by (rule span_superset)
with a have a0:"?a \<noteq> 0" by auto
have "\<forall>x\<in>span B. ?a \<bullet> x = 0"
proof(rule span_induct')
show "subspace (\<lambda>x. ?a \<bullet> x = 0)" by (auto simp add: subspace_def mem_def inner_simps smult_conv_scaleR)
next
{fix x assume x: "x \<in> B"
from x have B': "B = insert x (B - {x})" by blast
have fth: "finite (B - {x})" using fB by simp
have "?a \<bullet> x = 0"
apply (subst B') using fB fth
unfolding setsum_clauses(2)[OF fth]
apply simp unfolding inner_simps smult_conv_scaleR
apply (clarsimp simp add: inner_simps smult_conv_scaleR dot_lsum)
apply (rule setsum_0', rule ballI)
unfolding inner_commute
by (auto simp add: x field_simps intro: B(5)[unfolded pairwise_def orthogonal_def, rule_format])}
then show "\<forall>x \<in> B. ?a \<bullet> x = 0" by blast
qed
with a0 show ?thesis unfolding sSB by (auto intro: exI[where x="?a"])
qed
lemma span_not_univ_subset_hyperplane:
assumes SU: "span S \<noteq> (UNIV ::(real^'n) set)"
shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
using span_not_univ_orthogonal[OF SU] by auto
lemma lowdim_subset_hyperplane:
assumes d: "dim S < CARD('n::finite)"
shows "\<exists>(a::real ^'n). a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
proof-
{assume "span S = UNIV"
hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp
hence "dim S = CARD('n)" by (simp add: dim_span dim_univ)
with d have False by arith}
hence th: "span S \<noteq> UNIV" by blast
from span_not_univ_subset_hyperplane[OF th] show ?thesis .
qed
(* We can extend a linear basis-basis injection to the whole set. *)
lemma linear_indep_image_lemma:
assumes lf: "linear f" and fB: "finite B"
and ifB: "independent (f ` B)"
and fi: "inj_on f B" and xsB: "x \<in> span B"
and fx: "f (x::'a::field^_) = 0"
shows "x = 0"
using fB ifB fi xsB fx
proof(induct arbitrary: x rule: finite_induct[OF fB])
case 1 thus ?case by (auto simp add: span_empty)
next
case (2 a b x)
have fb: "finite b" using "2.prems" by simp
have th0: "f ` b \<subseteq> f ` (insert a b)"
apply (rule image_mono) by blast
from independent_mono[ OF "2.prems"(2) th0]
have ifb: "independent (f ` b)" .
have fib: "inj_on f b"
apply (rule subset_inj_on [OF "2.prems"(3)])
by blast
from span_breakdown[of a "insert a b", simplified, OF "2.prems"(4)]
obtain k where k: "x - k*s a \<in> span (b -{a})" by blast
have "f (x - k*s a) \<in> span (f ` b)"
unfolding span_linear_image[OF lf]
apply (rule imageI)
using k span_mono[of "b-{a}" b] by blast
hence "f x - k*s f a \<in> span (f ` b)"
by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
hence th: "-k *s f a \<in> span (f ` b)"
using "2.prems"(5) by (simp add: vector_smult_lneg)
{assume k0: "k = 0"
from k0 k have "x \<in> span (b -{a})" by simp
then have "x \<in> span b" using span_mono[of "b-{a}" b]
by blast}
moreover
{assume k0: "k \<noteq> 0"
from span_mul[OF th, of "- 1/ k"] k0
have th1: "f a \<in> span (f ` b)"
by (auto simp add: vector_smult_assoc)
from inj_on_image_set_diff[OF "2.prems"(3), of "insert a b " "{a}", symmetric]
have tha: "f ` insert a b - f ` {a} = f ` (insert a b - {a})" by blast
from "2.prems"(2)[unfolded dependent_def bex_simps(10), rule_format, of "f a"]
have "f a \<notin> span (f ` b)" using tha
using "2.hyps"(2)
"2.prems"(3) by auto
with th1 have False by blast
then have "x \<in> span b" by blast}
ultimately have xsb: "x \<in> span b" by blast
from "2.hyps"(3)[OF fb ifb fib xsb "2.prems"(5)]
show "x = 0" .
qed
(* We can extend a linear mapping from basis. *)
lemma linear_independent_extend_lemma:
assumes fi: "finite B" and ib: "independent B"
shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g ((x::'a::field^'n) + y) = g x + g y)
\<and> (\<forall>x\<in> span B. \<forall>c. g (c*s x) = c *s g x)
\<and> (\<forall>x\<in> B. g x = f x)"
using ib fi
proof(induct rule: finite_induct[OF fi])
case 1 thus ?case by (auto simp add: span_empty)
next
case (2 a b)
from "2.prems" "2.hyps" have ibf: "independent b" "finite b"
by (simp_all add: independent_insert)
from "2.hyps"(3)[OF ibf] obtain g where
g: "\<forall>x\<in>span b. \<forall>y\<in>span b. g (x + y) = g x + g y"
"\<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
let ?h = "\<lambda>z. SOME k. (z - k *s a) \<in> span b"
{fix z assume z: "z \<in> span (insert a b)"
have th0: "z - ?h z *s a \<in> span b"
apply (rule someI_ex)
unfolding span_breakdown_eq[symmetric]
using z .
{fix k assume k: "z - k *s a \<in> span b"
have eq: "z - ?h z *s a - (z - k*s a) = (k - ?h z) *s a"
by (simp add: field_simps vector_sadd_rdistrib[symmetric])
from span_sub[OF th0 k]
have khz: "(k - ?h z) *s a \<in> span b" by (simp add: eq)
{assume "k \<noteq> ?h z" hence k0: "k - ?h z \<noteq> 0" by simp
from k0 span_mul[OF khz, of "1 /(k - ?h z)"]
have "a \<in> span b" by (simp add: vector_smult_assoc)
with "2.prems"(1) "2.hyps"(2) have False
by (auto simp add: dependent_def)}
then have "k = ?h z" by blast}
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}
note h = this
let ?g = "\<lambda>z. ?h z *s f a + g (z - ?h z *s a)"
{fix x y assume x: "x \<in> span (insert a b)" and y: "y \<in> span (insert a b)"
have tha: "\<And>(x::'a^'n) y a k l. (x + y) - (k + l) *s a = (x - k *s a) + (y - l *s a)"
by (vector field_simps)
have addh: "?h (x + y) = ?h x + ?h y"
apply (rule conjunct2[OF h, rule_format, symmetric])
apply (rule span_add[OF x y])
unfolding tha
by (metis span_add x y conjunct1[OF h, rule_format])
have "?g (x + y) = ?g x + ?g y"
unfolding addh tha
g(1)[rule_format,OF conjunct1[OF h, OF x] conjunct1[OF h, OF y]]
by (simp add: vector_sadd_rdistrib)}
moreover
{fix x:: "'a^'n" and c:: 'a assume x: "x \<in> span (insert a b)"
have tha: "\<And>(x::'a^'n) c k a. c *s x - (c * k) *s a = c *s (x - k *s a)"
by (vector field_simps)
have hc: "?h (c *s x) = c * ?h x"
apply (rule conjunct2[OF h, rule_format, symmetric])
apply (metis span_mul x)
by (metis tha span_mul x conjunct1[OF h])
have "?g (c *s x) = c*s ?g x"
unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
by (vector field_simps)}
moreover
{fix x assume x: "x \<in> (insert a b)"
{assume xa: "x = a"
have ha1: "1 = ?h a"
apply (rule conjunct2[OF h, rule_format])
apply (metis span_superset insertI1)
using conjunct1[OF h, OF span_superset, OF insertI1]
by (auto simp add: span_0)
from xa ha1[symmetric] have "?g x = f x"
apply simp
using g(2)[rule_format, OF span_0, of 0]
by simp}
moreover
{assume xb: "x \<in> b"
have h0: "0 = ?h x"
apply (rule conjunct2[OF h, rule_format])
apply (metis span_superset x)
apply simp
apply (metis span_superset xb)
done
have "?g x = f x"
by (simp add: h0[symmetric] g(3)[rule_format, OF xb])}
ultimately have "?g x = f x" using x by blast }
ultimately show ?case apply - apply (rule exI[where x="?g"]) by blast
qed
lemma linear_independent_extend:
assumes iB: "independent (B:: (real ^'n) set)"
shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
proof-
from maximal_independent_subset_extend[of B UNIV] iB
obtain C where C: "B \<subseteq> C" "independent C" "\<And>x. x \<in> span C" by auto
from C(2) independent_bound[of C] linear_independent_extend_lemma[of C f]
obtain g where g: "(\<forall>x\<in> span C. \<forall>y\<in> span C. g (x + y) = g x + g y)
\<and> (\<forall>x\<in> span C. \<forall>c. g (c*s x) = c *s g x)
\<and> (\<forall>x\<in> C. g x = f x)" by blast
from g show ?thesis unfolding linear_def using C
apply clarsimp by blast
qed
(* Can construct an isomorphism between spaces of same dimension. *)
lemma card_le_inj: assumes fA: "finite A" and fB: "finite B"
and c: "card A \<le> card B" shows "(\<exists>f. f ` A \<subseteq> B \<and> inj_on f A)"
using fB c
proof(induct arbitrary: B rule: finite_induct[OF fA])
case 1 thus ?case by simp
next
case (2 x s t)
thus ?case
proof(induct rule: finite_induct[OF "2.prems"(1)])
case 1 then show ?case by simp
next
case (2 y t)
from "2.prems"(1,2,5) "2.hyps"(1,2) have cst:"card s \<le> card t" by simp
from "2.prems"(3) [OF "2.hyps"(1) cst] obtain f where
f: "f ` s \<subseteq> t \<and> inj_on f s" by blast
from f "2.prems"(2) "2.hyps"(2) show ?case
apply -
apply (rule exI[where x = "\<lambda>z. if z = x then y else f z"])
by (auto simp add: inj_on_def)
qed
qed
lemma card_subset_eq: assumes fB: "finite B" and AB: "A \<subseteq> B" and
c: "card A = card B"
shows "A = B"
proof-
from fB AB have fA: "finite A" by (auto intro: finite_subset)
from fA fB have fBA: "finite (B - A)" by auto
have e: "A \<inter> (B - A) = {}" by blast
have eq: "A \<union> (B - A) = B" using AB by blast
from card_Un_disjoint[OF fA fBA e, unfolded eq c]
have "card (B - A) = 0" by arith
hence "B - A = {}" unfolding card_eq_0_iff using fA fB by simp
with AB show "A = B" by blast
qed
lemma subspace_isomorphism:
assumes s: "subspace (S:: (real ^'n) set)"
and t: "subspace (T :: (real ^'m) set)"
and d: "dim S = dim T"
shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
proof-
from basis_exists[of S] independent_bound obtain B where
B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S" and fB: "finite B" by blast
from basis_exists[of T] independent_bound obtain C where
C: "C \<subseteq> T" "independent C" "T \<subseteq> span C" "card C = dim T" and fC: "finite C" by blast
from B(4) C(4) card_le_inj[of B C] d obtain f where
f: "f ` B \<subseteq> C" "inj_on f B" using `finite B` `finite C` by auto
from linear_independent_extend[OF B(2)] obtain g where
g: "linear g" "\<forall>x\<in> B. g x = f x" by blast
from inj_on_iff_eq_card[OF fB, of f] f(2)
have "card (f ` B) = card B" by simp
with B(4) C(4) have ceq: "card (f ` B) = card C" using d
by simp
have "g ` B = f ` B" using g(2)
by (auto simp add: image_iff)
also have "\<dots> = C" using card_subset_eq[OF fC f(1) ceq] .
finally have gBC: "g ` B = C" .
have gi: "inj_on g B" using f(2) g(2)
by (auto simp add: inj_on_def)
note g0 = linear_indep_image_lemma[OF g(1) fB, unfolded gBC, OF C(2) gi]
{fix x y assume x: "x \<in> S" and y: "y \<in> S" and gxy:"g x = g y"
from B(3) x y have x': "x \<in> span B" and y': "y \<in> span B" by blast+
from gxy have th0: "g (x - y) = 0" by (simp add: linear_sub[OF g(1)])
have th1: "x - y \<in> span B" using x' y' by (metis span_sub)
have "x=y" using g0[OF th1 th0] by simp }
then have giS: "inj_on g S"
unfolding inj_on_def by blast
from span_subspace[OF B(1,3) s]
have "g ` S = span (g ` B)" by (simp add: span_linear_image[OF g(1)])
also have "\<dots> = span C" unfolding gBC ..
also have "\<dots> = T" using span_subspace[OF C(1,3) t] .
finally have gS: "g ` S = T" .
from g(1) gS giS show ?thesis by blast
qed
(* linear functions are equal on a subspace if they are on a spanning set. *)
lemma subspace_kernel:
assumes lf: "linear (f::'a::semiring_1 ^_ \<Rightarrow> _)"
shows "subspace {x. f x = 0}"
apply (simp add: subspace_def)
by (simp add: linear_add[OF lf] linear_cmul[OF lf] linear_0[OF lf])
lemma linear_eq_0_span:
assumes lf: "linear f" and f0: "\<forall>x\<in>B. f x = 0"
shows "\<forall>x \<in> span B. f x = (0::'a::semiring_1 ^_)"
proof
fix x assume x: "x \<in> span B"
let ?P = "\<lambda>x. f x = 0"
from subspace_kernel[OF lf] have "subspace ?P" unfolding Collect_def .
with x f0 span_induct[of B "?P" x] show "f x = 0" by blast
qed
lemma linear_eq_0:
assumes lf: "linear f" and SB: "S \<subseteq> span B" and f0: "\<forall>x\<in>B. f x = 0"
shows "\<forall>x \<in> S. f x = (0::'a::semiring_1^_)"
by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
lemma linear_eq:
assumes lf: "linear (f::'a::ring_1^_ \<Rightarrow> _)" and lg: "linear g" and S: "S \<subseteq> span B"
and fg: "\<forall> x\<in> B. f x = g x"
shows "\<forall>x\<in> S. f x = g x"
proof-
let ?h = "\<lambda>x. f x - g x"
from fg have fg': "\<forall>x\<in> B. ?h x = 0" by simp
from linear_eq_0[OF linear_compose_sub[OF lf lg] S fg']
show ?thesis by simp
qed
lemma linear_eq_stdbasis:
assumes lf: "linear (f::'a::ring_1^'m \<Rightarrow> 'a^'n)" and lg: "linear g"
and fg: "\<forall>i. f (basis i) = g(basis i)"
shows "f = g"
proof-
let ?U = "UNIV :: 'm set"
let ?I = "{basis i:: 'a^'m|i. i \<in> ?U}"
{fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
from equalityD2[OF span_stdbasis]
have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
from linear_eq[OF lf lg IU] fg x
have "f x = g x" unfolding Collect_def Ball_def mem_def by metis}
then show ?thesis by (auto intro: ext)
qed
(* Similar results for bilinear functions. *)
lemma bilinear_eq:
assumes bf: "bilinear (f:: 'a::ring^_ \<Rightarrow> 'a^_ \<Rightarrow> 'a^_)"
and bg: "bilinear g"
and SB: "S \<subseteq> span B" and TC: "T \<subseteq> span C"
and fg: "\<forall>x\<in> B. \<forall>y\<in> C. f x y = g x y"
shows "\<forall>x\<in>S. \<forall>y\<in>T. f x y = g x y "
proof-
let ?P = "\<lambda>x. \<forall>y\<in> span C. f x y = g x y"
from bf bg have sp: "subspace ?P"
unfolding bilinear_def linear_def subspace_def bf bg
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])
have "\<forall>x \<in> span B. \<forall>y\<in> span C. f x y = g x y"
apply -
apply (rule ballI)
apply (rule span_induct[of B ?P])
defer
apply (rule sp)
apply assumption
apply (clarsimp simp add: Ball_def)
apply (rule_tac P="\<lambda>y. f xa y = g xa y" and S=C in span_induct)
using fg
apply (auto simp add: subspace_def)
using bf bg unfolding bilinear_def linear_def
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])
then show ?thesis using SB TC by (auto intro: ext)
qed
lemma bilinear_eq_stdbasis:
assumes bf: "bilinear (f:: 'a::ring_1^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^_)"
and bg: "bilinear g"
and fg: "\<forall>i j. f (basis i) (basis j) = g (basis i) (basis j)"
shows "f = g"
proof-
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
from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
qed
(* Detailed theorems about left and right invertibility in general case. *)
lemma left_invertible_transpose:
"(\<exists>(B). B ** transpose (A) = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). A ** B = mat 1)"
by (metis matrix_transpose_mul transpose_mat transpose_transpose)
lemma right_invertible_transpose:
"(\<exists>(B). transpose (A) ** B = mat (1::'a::comm_semiring_1)) \<longleftrightarrow> (\<exists>(B). B ** A = mat 1)"
by (metis matrix_transpose_mul transpose_mat transpose_transpose)
lemma linear_injective_left_inverse:
assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and fi: "inj f"
shows "\<exists>g. linear g \<and> g o f = id"
proof-
from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
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
from h(2)
have th: "\<forall>i. (h \<circ> f) (basis i) = id (basis i)"
using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
by auto
from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
have "h o f = id" .
then show ?thesis using h(1) by blast
qed
lemma linear_surjective_right_inverse:
assumes lf: "linear (f:: real ^'m \<Rightarrow> real ^'n)" and sf: "surj f"
shows "\<exists>g. linear g \<and> f o g = id"
proof-
from linear_independent_extend[OF independent_stdbasis]
obtain h:: "real ^'n \<Rightarrow> real ^'m" where
h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> (UNIV :: 'n set)}. h x = inv f x" by blast
from h(2)
have th: "\<forall>i. (f o h) (basis i) = id (basis i)"
using sf
apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
apply (erule_tac x="basis i" in allE)
by auto
from linear_eq_stdbasis[OF linear_compose[OF h(1) lf] linear_id th]
have "f o h = id" .
then show ?thesis using h(1) by blast
qed
lemma matrix_left_invertible_injective:
"(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
proof-
{fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hence "x = y"
unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
moreover
{assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
hence i: "inj (op *v A)" unfolding inj_on_def by auto
from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
obtain g where g: "linear g" "g o op *v A = id" by blast
have "matrix g ** A = mat 1"
unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
using g(2) by (simp add: o_def id_def stupid_ext)
then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
ultimately show ?thesis by blast
qed
lemma matrix_left_invertible_ker:
"(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
unfolding matrix_left_invertible_injective
using linear_injective_0[OF matrix_vector_mul_linear, of A]
by (simp add: inj_on_def)
lemma matrix_right_invertible_surjective:
"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
proof-
{fix B :: "real ^'m^'n" assume AB: "A ** B = mat 1"
{fix x :: "real ^ 'm"
have "A *v (B *v x) = x"
by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
hence "surj (op *v A)" unfolding surj_def by metis }
moreover
{assume sf: "surj (op *v A)"
from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
by blast
have "A ** (matrix g) = mat 1"
unfolding matrix_eq matrix_vector_mul_lid
matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
using g(2) unfolding o_def stupid_ext[symmetric] id_def
.
hence "\<exists>B. A ** (B::real^'m^'n) = mat 1" by blast
}
ultimately show ?thesis unfolding surj_def by blast
qed
lemma matrix_left_invertible_independent_columns:
fixes A :: "real^'n^'m"
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))"
(is "?lhs \<longleftrightarrow> ?rhs")
proof-
let ?U = "UNIV :: 'n set"
{assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
{fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
and i: "i \<in> ?U"
let ?x = "\<chi> i. c i"
have th0:"A *v ?x = 0"
using c
unfolding matrix_mult_vsum Cart_eq
by auto
from k[rule_format, OF th0] i
have "c i = 0" by (vector Cart_eq)}
hence ?rhs by blast}
moreover
{assume H: ?rhs
{fix x assume x: "A *v x = 0"
let ?c = "\<lambda>i. ((x$i ):: real)"
from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
have "x = 0" by vector}}
ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
qed
lemma matrix_right_invertible_independent_rows:
fixes A :: "real^'n^'m"
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))"
unfolding left_invertible_transpose[symmetric]
matrix_left_invertible_independent_columns
by (simp add: column_transpose)
lemma matrix_right_invertible_span_columns:
"(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
proof-
let ?U = "UNIV :: 'm set"
have fU: "finite ?U" by simp
have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
apply (subst eq_commute) ..
have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
{assume h: ?lhs
{fix x:: "real ^'n"
from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
have "x \<in> span (columns A)"
unfolding y[symmetric]
apply (rule span_setsum[OF fU])
apply clarify
apply (rule span_mul)
apply (rule span_superset)
unfolding columns_def
by blast}
then have ?rhs unfolding rhseq by blast}
moreover
{assume h:?rhs
let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
{fix y have "?P y"
proof(rule span_induct_alt[of ?P "columns A"])
show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
by (rule exI[where x=0], simp)
next
fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
unfolding columns_def by blast
from y2 obtain x:: "real ^'m" where
x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
show "?P (c*s y1 + y2)"
proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] cond_value_iff right_distrib cond_application_beta cong del: if_weak_cong)
fix j
have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
else (x$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)
by (simp add: field_simps)
have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
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)$j))) ?U"
apply (rule setsum_cong[OF refl])
using th by blast
also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
by (simp add: setsum_addf)
also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
unfolding setsum_delta[OF fU]
using i(1) by simp
finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
qed
next
show "y \<in> span (columns A)" unfolding h by blast
qed}
then have ?lhs unfolding lhseq ..}
ultimately show ?thesis by blast
qed
lemma matrix_left_invertible_span_rows:
"(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
unfolding right_invertible_transpose[symmetric]
unfolding columns_transpose[symmetric]
unfolding matrix_right_invertible_span_columns
..
(* An injective map real^'n->real^'n is also surjective. *)
lemma linear_injective_imp_surjective:
assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and fi: "inj f"
shows "surj f"
proof-
let ?U = "UNIV :: (real ^'n) set"
from basis_exists[of ?U] obtain B
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" "card B = dim ?U"
by blast
from B(4) have d: "dim ?U = card B" by simp
have th: "?U \<subseteq> span (f ` B)"
apply (rule card_ge_dim_independent)
apply blast
apply (rule independent_injective_image[OF B(2) lf fi])
apply (rule order_eq_refl)
apply (rule sym)
unfolding d
apply (rule card_image)
apply (rule subset_inj_on[OF fi])
by blast
from th show ?thesis
unfolding span_linear_image[OF lf] surj_def
using B(3) by blast
qed
(* And vice versa. *)
lemma surjective_iff_injective_gen:
assumes fS: "finite S" and fT: "finite T" and c: "card S = card T"
and ST: "f ` S \<subseteq> T"
shows "(\<forall>y \<in> T. \<exists>x \<in> S. f x = y) \<longleftrightarrow> inj_on f S" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
{assume h: "?lhs"
{fix x y assume x: "x \<in> S" and y: "y \<in> S" and f: "f x = f y"
from x fS have S0: "card S \<noteq> 0" by auto
{assume xy: "x \<noteq> y"
have th: "card S \<le> card (f ` (S - {y}))"
unfolding c
apply (rule card_mono)
apply (rule finite_imageI)
using fS apply simp
using h xy x y f unfolding subset_eq image_iff
apply auto
apply (case_tac "xa = f x")
apply (rule bexI[where x=x])
apply auto
done
also have " \<dots> \<le> card (S -{y})"
apply (rule card_image_le)
using fS by simp
also have "\<dots> \<le> card S - 1" using y fS by simp
finally have False using S0 by arith }
then have "x = y" by blast}
then have ?rhs unfolding inj_on_def by blast}
moreover
{assume h: ?rhs
have "f ` S = T"
apply (rule card_subset_eq[OF fT ST])
unfolding card_image[OF h] using c .
then have ?lhs by blast}
ultimately show ?thesis by blast
qed
lemma linear_surjective_imp_injective:
assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f"
shows "inj f"
proof-
let ?U = "UNIV :: (real ^'n) set"
from basis_exists[of ?U] obtain B
where B: "B \<subseteq> ?U" "independent B" "?U \<subseteq> span B" and d: "card B = dim ?U"
by blast
{fix x assume x: "x \<in> span B" and fx: "f x = 0"
from B(2) have fB: "finite B" using independent_bound by auto
have fBi: "independent (f ` B)"
apply (rule card_le_dim_spanning[of "f ` B" ?U])
apply blast
using sf B(3)
unfolding span_linear_image[OF lf] surj_def subset_eq image_iff
apply blast
using fB apply (blast intro: finite_imageI)
unfolding d[symmetric]
apply (rule card_image_le)
apply (rule fB)
done
have th0: "dim ?U \<le> card (f ` B)"
apply (rule span_card_ge_dim)
apply blast
unfolding span_linear_image[OF lf]
apply (rule subset_trans[where B = "f ` UNIV"])
using sf unfolding surj_def apply blast
apply (rule image_mono)
apply (rule B(3))
apply (metis finite_imageI fB)
done
moreover have "card (f ` B) \<le> card B"
by (rule card_image_le, rule fB)
ultimately have th1: "card B = card (f ` B)" unfolding d by arith
have fiB: "inj_on f B"
unfolding surjective_iff_injective_gen[OF fB finite_imageI[OF fB] th1 subset_refl, symmetric] by blast
from linear_indep_image_lemma[OF lf fB fBi fiB x] fx
have "x = 0" by blast}
note th = this
from th show ?thesis unfolding linear_injective_0[OF lf]
using B(3) by blast
qed
(* Hence either is enough for isomorphism. *)
lemma left_right_inverse_eq:
assumes fg: "f o g = id" and gh: "g o h = id"
shows "f = h"
proof-
have "f = f o (g o h)" unfolding gh by simp
also have "\<dots> = (f o g) o h" by (simp add: o_assoc)
finally show "f = h" unfolding fg by simp
qed
lemma isomorphism_expand:
"f o g = id \<and> g o f = id \<longleftrightarrow> (\<forall>x. f(g x) = x) \<and> (\<forall>x. g(f x) = x)"
by (simp add: expand_fun_eq o_def id_def)
lemma linear_injective_isomorphism:
assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'n)" and fi: "inj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
unfolding isomorphism_expand[symmetric]
using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
by (metis left_right_inverse_eq)
lemma linear_surjective_isomorphism:
assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and sf: "surj f"
shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
unfolding isomorphism_expand[symmetric]
using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
by (metis left_right_inverse_eq)
(* Left and right inverses are the same for R^N->R^N. *)
lemma linear_inverse_left:
assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and lf': "linear f'"
shows "f o f' = id \<longleftrightarrow> f' o f = id"
proof-
{fix f f':: "real ^'n \<Rightarrow> real ^'n"
assume lf: "linear f" "linear f'" and f: "f o f' = id"
from f have sf: "surj f"
apply (auto simp add: o_def stupid_ext[symmetric] id_def surj_def)
by metis
from linear_surjective_isomorphism[OF lf(1) sf] lf f
have "f' o f = id" unfolding stupid_ext[symmetric] o_def id_def
by metis}
then show ?thesis using lf lf' by metis
qed
(* Moreover, a one-sided inverse is automatically linear. *)
lemma left_inverse_linear:
assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and gf: "g o f = id"
shows "linear g"
proof-
from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
by metis
from linear_injective_isomorphism[OF lf fi]
obtain h:: "real ^'n \<Rightarrow> real ^'n" where
h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
have "h = g" apply (rule ext) using gf h(2,3)
apply (simp add: o_def id_def stupid_ext[symmetric])
by metis
with h(1) show ?thesis by blast
qed
lemma right_inverse_linear:
assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and gf: "f o g = id"
shows "linear g"
proof-
from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
by metis
from linear_surjective_isomorphism[OF lf fi]
obtain h:: "real ^'n \<Rightarrow> real ^'n" where
h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
have "h = g" apply (rule ext) using gf h(2,3)
apply (simp add: o_def id_def stupid_ext[symmetric])
by metis
with h(1) show ?thesis by blast
qed
(* The same result in terms of square matrices. *)
lemma matrix_left_right_inverse:
fixes A A' :: "real ^'n^'n"
shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
proof-
{fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
have sA: "surj (op *v A)"
unfolding surj_def
apply clarify
apply (rule_tac x="(A' *v y)" in exI)
by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
obtain f' :: "real ^'n \<Rightarrow> real ^'n"
where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
have th: "matrix f' ** A = mat 1"
by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hence "matrix f' ** A = A' ** A" by simp
hence "A' ** A = mat 1" by (simp add: th)}
then show ?thesis by blast
qed
(* Considering an n-element vector as an n-by-1 or 1-by-n matrix. *)
definition "rowvector v = (\<chi> i j. (v$j))"
definition "columnvector v = (\<chi> i j. (v$i))"
lemma transpose_columnvector:
"transpose(columnvector v) = rowvector v"
by (simp add: transpose_def rowvector_def columnvector_def Cart_eq)
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
by (simp add: transpose_def columnvector_def rowvector_def Cart_eq)
lemma dot_rowvector_columnvector:
"columnvector (A *v v) = A ** columnvector v"
by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
lemma dot_matrix_product: "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vector_def)
lemma dot_matrix_vector_mul:
fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
shows "(A *v x) \<bullet> (B *v y) =
(((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
unfolding dot_matrix_product transpose_columnvector[symmetric]
dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
(* Infinity norm. *)
definition "infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
by auto
lemma infnorm_set_image:
"{abs(x$i) |i. i\<in> (UNIV :: _ set)} =
(\<lambda>i. abs(x$i)) ` (UNIV)" by blast
lemma infnorm_set_lemma:
shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"
and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
unfolding infnorm_set_image
by (auto intro: finite_imageI)
lemma infnorm_pos_le: "0 \<le> infnorm (x::real^'n)"
unfolding infnorm_def
unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
unfolding infnorm_set_image
by auto
lemma infnorm_triangle: "infnorm ((x::real^'n) + y) \<le> infnorm x + infnorm y"
proof-
have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
have th2: "\<And>x (y::real). abs(x + y) - abs(x) <= abs(y)" by arith
show ?thesis
unfolding infnorm_def
unfolding Sup_finite_le_iff[ OF infnorm_set_lemma]
apply (subst diff_le_eq[symmetric])
unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
unfolding infnorm_set_image bex_simps
apply (subst th)
unfolding th1
unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps bex_simps
apply simp
apply (metis th2)
done
qed
lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n) = 0"
proof-
have "infnorm x <= 0 \<longleftrightarrow> x = 0"
unfolding infnorm_def
unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps
by vector
then show ?thesis using infnorm_pos_le[of x] by simp
qed
lemma infnorm_0: "infnorm 0 = 0"
by (simp add: infnorm_eq_0)
lemma infnorm_neg: "infnorm (- x) = infnorm x"
unfolding infnorm_def
apply (rule cong[of "Sup" "Sup"])
apply blast
apply (rule set_ext)
apply auto
done
lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
proof-
have "y - x = - (x - y)" by simp
then show ?thesis by (metis infnorm_neg)
qed
lemma real_abs_sub_infnorm: "\<bar> infnorm x - infnorm y\<bar> \<le> infnorm (x - y)"
proof-
have th: "\<And>(nx::real) n ny. nx <= n + ny \<Longrightarrow> ny <= n + nx ==> \<bar>nx - ny\<bar> <= n"
by arith
from infnorm_triangle[of "x - y" " y"] infnorm_triangle[of "x - y" "-x"]
have ths: "infnorm x \<le> infnorm (x - y) + infnorm y"
"infnorm y \<le> infnorm (x - y) + infnorm x"
by (simp_all add: field_simps infnorm_neg diff_def[symmetric])
from th[OF ths] show ?thesis .
qed
lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
using infnorm_pos_le[of x] by arith
lemma component_le_infnorm:
shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
proof-
let ?U = "UNIV :: 'n set"
let ?S = "{\<bar>x$i\<bar> |i. i\<in> ?U}"
have fS: "finite ?S" unfolding image_Collect[symmetric]
apply (rule finite_imageI) unfolding Collect_def mem_def by simp
have S0: "?S \<noteq> {}" by blast
have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
from Sup_finite_in[OF fS S0]
show ?thesis unfolding infnorm_def infnorm_set_image
by (metis Sup_finite_ge_iff finite finite_imageI UNIV_not_empty image_is_empty
rangeI real_le_refl)
qed
lemma infnorm_mul_lemma: "infnorm(a *s x) <= \<bar>a\<bar> * infnorm x"
apply (subst infnorm_def)
unfolding Sup_finite_le_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps
apply (simp add: abs_mult)
apply (rule allI)
apply (cut_tac component_le_infnorm[of x])
apply (rule mult_mono)
apply auto
done
lemma infnorm_mul: "infnorm(a *s x) = abs a * infnorm x"
proof-
{assume a0: "a = 0" hence ?thesis by (simp add: infnorm_0) }
moreover
{assume a0: "a \<noteq> 0"
from a0 have th: "(1/a) *s (a *s x) = x"
by (simp add: vector_smult_assoc)
from a0 have ap: "\<bar>a\<bar> > 0" by arith
from infnorm_mul_lemma[of "1/a" "a *s x"]
have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*s x)"
unfolding th by simp
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)
then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*s x)"
using ap by (simp add: field_simps)
with infnorm_mul_lemma[of a x] have ?thesis by arith }
ultimately show ?thesis by blast
qed
lemma infnorm_pos_lt: "infnorm x > 0 \<longleftrightarrow> x \<noteq> 0"
using infnorm_pos_le[of x] infnorm_eq_0[of x] by arith
(* Prove that it differs only up to a bound from Euclidean norm. *)
lemma infnorm_le_norm: "infnorm x \<le> norm x"
unfolding infnorm_def Sup_finite_le_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image ball_simps
by (metis component_le_norm)
lemma card_enum: "card {1 .. n} = n" by auto
lemma norm_le_infnorm: "norm(x) <= sqrt(real CARD('n)) * infnorm(x::real ^'n)"
proof-
let ?d = "CARD('n)"
have "real ?d \<ge> 0" by simp
hence d2: "(sqrt (real ?d))^2 = real ?d"
by (auto intro: real_sqrt_pow2)
have th: "sqrt (real ?d) * infnorm x \<ge> 0"
by (simp add: zero_le_mult_iff infnorm_pos_le)
have th1: "x \<bullet> x \<le> (sqrt (real ?d) * infnorm x)^2"
unfolding power_mult_distrib d2
unfolding real_of_nat_def inner_vector_def
apply (subst power2_abs[symmetric])
apply (rule setsum_bounded)
apply(auto simp add: power2_eq_square[symmetric])
apply (subst power2_abs[symmetric])
apply (rule power_mono)
unfolding infnorm_def Sup_finite_ge_iff[OF infnorm_set_lemma]
unfolding infnorm_set_image bex_simps apply(rule_tac x=i in exI) by auto
from real_le_lsqrt[OF inner_ge_zero th th1]
show ?thesis unfolding norm_eq_sqrt_inner id_def .
qed
(* Equality in Cauchy-Schwarz and triangle inequalities. *)
lemma norm_cauchy_schwarz_eq: "(x::real ^'n) \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *s y = norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
{assume h: "x = 0"
hence ?thesis by simp}
moreover
{assume h: "y = 0"
hence ?thesis by simp}
moreover
{assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
from inner_eq_zero_iff[of "norm y *s x - norm x *s y"]
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)"
using x y
unfolding inner_simps smult_conv_scaleR
unfolding power2_norm_eq_inner[symmetric] power2_eq_square diff_eq_0_iff_eq apply (simp add: inner_commute)
apply (simp add: field_simps) by metis
also have "\<dots> \<longleftrightarrow> (2 * norm x * norm y * (norm x * norm y - x \<bullet> y) = 0)" using x y
by (simp add: field_simps inner_commute)
also have "\<dots> \<longleftrightarrow> ?lhs" using x y
apply simp
by metis
finally have ?thesis by blast}
ultimately show ?thesis by blast
qed
lemma norm_cauchy_schwarz_abs_eq:
fixes x y :: "real ^ 'n"
shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
norm x *s y = norm y *s x \<or> norm(x) *s y = - norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
have "?rhs \<longleftrightarrow> norm x *s y = norm y *s x \<or> norm (- x) *s y = norm y *s (- x)"
apply simp by vector
also have "\<dots> \<longleftrightarrow>(x \<bullet> y = norm x * norm y \<or>
(-x) \<bullet> y = norm x * norm y)"
unfolding norm_cauchy_schwarz_eq[symmetric]
unfolding norm_minus_cancel
norm_mul by blast
also have "\<dots> \<longleftrightarrow> ?lhs"
unfolding th[OF mult_nonneg_nonneg, OF norm_ge_zero[of x] norm_ge_zero[of y]] inner_simps by auto
finally show ?thesis ..
qed
lemma norm_triangle_eq:
fixes x y :: "real ^ 'n"
shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
proof-
{assume x: "x =0 \<or> y =0"
hence ?thesis by (cases "x=0", simp_all)}
moreover
{assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
hence "norm x \<noteq> 0" "norm y \<noteq> 0"
by simp_all
hence n: "norm x > 0" "norm y > 0"
using norm_ge_zero[of x] norm_ge_zero[of y]
by arith+
have th: "\<And>(a::real) b c. a + b + c \<noteq> 0 ==> (a = b + c \<longleftrightarrow> a^2 = (b + c)^2)" by algebra
have "norm(x + y) = norm x + norm y \<longleftrightarrow> norm(x + y)^ 2 = (norm x + norm y) ^2"
apply (rule th) using n norm_ge_zero[of "x + y"]
by arith
also have "\<dots> \<longleftrightarrow> norm x *s y = norm y *s x"
unfolding norm_cauchy_schwarz_eq[symmetric]
unfolding power2_norm_eq_inner inner_simps
by (simp add: power2_norm_eq_inner[symmetric] power2_eq_square inner_commute field_simps)
finally have ?thesis .}
ultimately show ?thesis by blast
qed
(* Collinearity.*)
definition "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *s u)"
lemma collinear_empty: "collinear {}" by (simp add: collinear_def)
lemma collinear_sing: "collinear {(x::'a::ring_1^_)}"
apply (simp add: collinear_def)
apply (rule exI[where x=0])
by simp
lemma collinear_2: "collinear {(x::'a::ring_1^_),y}"
apply (simp add: collinear_def)
apply (rule exI[where x="x - y"])
apply auto
apply (rule exI[where x=0], simp)
apply (rule exI[where x=1], simp)
apply (rule exI[where x="- 1"], simp add: vector_sneg_minus1[symmetric])
apply (rule exI[where x=0], simp)
done
lemma collinear_lemma: "collinear {(0::real^_),x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *s x)" (is "?lhs \<longleftrightarrow> ?rhs")
proof-
{assume "x=0 \<or> y = 0" hence ?thesis
by (cases "x = 0", simp_all add: collinear_2 insert_commute)}
moreover
{assume x: "x \<noteq> 0" and y: "y \<noteq> 0"
{assume h: "?lhs"
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
from u[rule_format, of x 0] u[rule_format, of y 0]
obtain cx and cy where
cx: "x = cx*s u" and cy: "y = cy*s u"
by auto
from cx x have cx0: "cx \<noteq> 0" by auto
from cy y have cy0: "cy \<noteq> 0" by auto
let ?d = "cy / cx"
from cx cy cx0 have "y = ?d *s x"
by (simp add: vector_smult_assoc)
hence ?rhs using x y by blast}
moreover
{assume h: "?rhs"
then obtain c where c: "y = c*s x" using x y by blast
have ?lhs unfolding collinear_def c
apply (rule exI[where x=x])
apply auto
apply (rule exI[where x="- 1"], simp only: vector_smult_lneg vector_smult_lid)
apply (rule exI[where x= "-c"], simp only: vector_smult_lneg)
apply (rule exI[where x=1], simp)
apply (rule exI[where x="1 - c"], simp add: vector_smult_lneg vector_sub_rdistrib)
apply (rule exI[where x="c - 1"], simp add: vector_smult_lneg vector_sub_rdistrib)
done}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed
lemma norm_cauchy_schwarz_equal:
fixes x y :: "real ^ 'n"
shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
unfolding norm_cauchy_schwarz_abs_eq
apply (cases "x=0", simp_all add: collinear_2)
apply (cases "y=0", simp_all add: collinear_2 insert_commute)
unfolding collinear_lemma
apply simp
apply (subgoal_tac "norm x \<noteq> 0")
apply (subgoal_tac "norm y \<noteq> 0")
apply (rule iffI)
apply (cases "norm x *s y = norm y *s x")
apply (rule exI[where x="(1/norm x) * norm y"])
apply (drule sym)
unfolding vector_smult_assoc[symmetric]
apply (simp add: vector_smult_assoc field_simps)
apply (rule exI[where x="(1/norm x) * - norm y"])
apply clarify
apply (drule sym)
unfolding vector_smult_assoc[symmetric]
apply (simp add: vector_smult_assoc field_simps)
apply (erule exE)
apply (erule ssubst)
unfolding vector_smult_assoc
unfolding norm_mul
apply (subgoal_tac "norm x * c = \<bar>c\<bar> * norm x \<or> norm x * c = - \<bar>c\<bar> * norm x")
apply (case_tac "c <= 0", simp add: field_simps)
apply (simp add: field_simps)
apply (case_tac "c <= 0", simp add: field_simps)
apply (simp add: field_simps)
apply simp
apply simp
done
end