(* Title: HOL/Multivariate_Analysis/Euclidean_Space.thy
Author: Johannes Hölzl, TU München
Author: Brian Huffman, Portland State University
*)
header {* Finite-Dimensional Inner Product Spaces *}
theory Euclidean_Space
imports
Complex_Main
"~~/src/HOL/Library/Inner_Product"
"~~/src/HOL/Library/Product_Vector"
begin
subsection {* Type class of Euclidean spaces *}
class euclidean_space = real_inner +
fixes dimension :: "'a itself \<Rightarrow> nat"
fixes basis :: "nat \<Rightarrow> 'a"
assumes DIM_positive [intro]:
"0 < dimension TYPE('a)"
assumes basis_zero [simp]:
"dimension TYPE('a) \<le> i \<Longrightarrow> basis i = 0"
assumes basis_orthonormal:
"\<forall>i<dimension TYPE('a). \<forall>j<dimension TYPE('a).
inner (basis i) (basis j) = (if i = j then 1 else 0)"
assumes euclidean_all_zero:
"(\<forall>i<dimension TYPE('a). inner (basis i) x = 0) \<longleftrightarrow> (x = 0)"
syntax "_type_dimension" :: "type => nat" ("(1DIM/(1'(_')))")
translations "DIM('t)" == "CONST dimension (TYPE('t))"
lemma (in euclidean_space) dot_basis:
"inner (basis i) (basis j) = (if i = j \<and> i < DIM('a) then 1 else 0)"
proof (cases "(i < DIM('a) \<and> j < DIM('a))")
case False
hence "inner (basis i) (basis j) = 0" by auto
thus ?thesis using False by auto
next
case True thus ?thesis using basis_orthonormal by auto
qed
lemma (in euclidean_space) basis_eq_0_iff [simp]:
"basis i = 0 \<longleftrightarrow> DIM('a) \<le> i"
proof -
have "inner (basis i) (basis i) = 0 \<longleftrightarrow> DIM('a) \<le> i"
by (simp add: dot_basis)
thus ?thesis by simp
qed
lemma (in euclidean_space) norm_basis [simp]:
"norm (basis i) = (if i < DIM('a) then 1 else 0)"
unfolding norm_eq_sqrt_inner dot_basis by simp
lemma (in euclidean_space) basis_neq_0 [intro]:
assumes "i<DIM('a)" shows "(basis i) \<noteq> 0"
using assms by simp
subsubsection {* Projecting components *}
definition (in euclidean_space) euclidean_component (infixl "$$" 90)
where "x $$ i = inner (basis i) x"
lemma bounded_linear_euclidean_component:
"bounded_linear (\<lambda>x. euclidean_component x i)"
unfolding euclidean_component_def
by (rule inner.bounded_linear_right)
interpretation euclidean_component:
bounded_linear "\<lambda>x. euclidean_component x i"
by (rule bounded_linear_euclidean_component)
lemma euclidean_eqI:
fixes x y :: "'a::euclidean_space"
assumes "\<And>i. i < DIM('a) \<Longrightarrow> x $$ i = y $$ i" shows "x = y"
proof -
from assms have "\<forall>i<DIM('a). (x - y) $$ i = 0"
by (simp add: euclidean_component.diff)
then show "x = y"
unfolding euclidean_component_def euclidean_all_zero by simp
qed
lemma euclidean_eq:
fixes x y :: "'a::euclidean_space"
shows "x = y \<longleftrightarrow> (\<forall>i<DIM('a). x $$ i = y $$ i)"
by (auto intro: euclidean_eqI)
lemma (in euclidean_space) basis_component [simp]:
"basis i $$ j = (if i = j \<and> i < DIM('a) then 1 else 0)"
unfolding euclidean_component_def dot_basis by auto
lemma (in euclidean_space) basis_at_neq_0 [intro]:
"i < DIM('a) \<Longrightarrow> basis i $$ i \<noteq> 0"
by simp
lemma (in euclidean_space) euclidean_component_ge [simp]:
assumes "i \<ge> DIM('a)" shows "x $$ i = 0"
unfolding euclidean_component_def basis_zero[OF assms] by simp
lemma euclidean_scaleR:
shows "(a *\<^sub>R x) $$ i = a * (x$$i)"
unfolding euclidean_component_def by auto
lemmas euclidean_simps =
euclidean_component.add
euclidean_component.diff
euclidean_scaleR
euclidean_component.minus
euclidean_component.setsum
basis_component
lemma euclidean_representation:
fixes x :: "'a::euclidean_space"
shows "x = (\<Sum>i<DIM('a). (x$$i) *\<^sub>R basis i)"
apply (rule euclidean_eqI)
apply (simp add: euclidean_component.setsum euclidean_component.scaleR)
apply (simp add: if_distrib setsum_delta cong: if_cong)
done
subsubsection {* Binder notation for vectors *}
definition (in euclidean_space) Chi (binder "\<chi>\<chi> " 10) where
"(\<chi>\<chi> i. f i) = (\<Sum>i<DIM('a). f i *\<^sub>R basis i)"
lemma euclidean_lambda_beta [simp]:
"((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ j = (if j < DIM('a) then f j else 0)"
by (auto simp: euclidean_component.setsum euclidean_component.scaleR
Chi_def if_distrib setsum_cases intro!: setsum_cong)
lemma euclidean_lambda_beta':
"j < DIM('a) \<Longrightarrow> ((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ j = f j"
by simp
lemma euclidean_lambda_beta'':"(\<forall>j < DIM('a::euclidean_space). P j (((\<chi>\<chi> i. f i)::'a) $$ j)) \<longleftrightarrow>
(\<forall>j < DIM('a::euclidean_space). P j (f j))" by auto
lemma euclidean_beta_reduce[simp]:
"(\<chi>\<chi> i. x $$ i) = (x::'a::euclidean_space)"
by (simp add: euclidean_eq)
lemma euclidean_lambda_beta_0[simp]:
"((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ 0 = f 0"
by (simp add: DIM_positive)
lemma euclidean_inner:
"inner x (y::'a) = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) * (y $$ i))"
by (subst (1 2) euclidean_representation,
simp add: inner_left.setsum inner_right.setsum
dot_basis if_distrib setsum_cases mult_commute)
lemma component_le_norm: "\<bar>x$$i\<bar> \<le> norm (x::'a::euclidean_space)"
unfolding euclidean_component_def
by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
subsection {* Class instances *}
subsubsection {* Type @{typ real} *}
instantiation real :: euclidean_space
begin
definition
"dimension (t::real itself) = 1"
definition [simp]:
"basis i = (if i = 0 then 1 else (0::real))"
lemma DIM_real [simp]: "DIM(real) = 1"
by (rule dimension_real_def)
instance
by default simp+
end
subsubsection {* Type @{typ complex} *}
instantiation complex :: euclidean_space
begin
definition
"dimension (t::complex itself) = 2"
definition
"basis i = (if i = 0 then 1 else if i = 1 then ii else 0)"
lemma all_less_Suc: "(\<forall>i<Suc n. P i) \<longleftrightarrow> (\<forall>i<n. P i) \<and> P n"
by (auto simp add: less_Suc_eq)
instance proof
show "0 < DIM(complex)"
unfolding dimension_complex_def by simp
next
fix i :: nat
assume "DIM(complex) \<le> i" thus "basis i = (0::complex)"
unfolding dimension_complex_def basis_complex_def by simp
next
show "\<forall>i<DIM(complex). \<forall>j<DIM(complex).
inner (basis i::complex) (basis j) = (if i = j then 1 else 0)"
unfolding dimension_complex_def basis_complex_def inner_complex_def
by (simp add: numeral_2_eq_2 all_less_Suc)
next
fix x :: complex
show "(\<forall>i<DIM(complex). inner (basis i) x = 0) \<longleftrightarrow> x = 0"
unfolding dimension_complex_def basis_complex_def inner_complex_def
by (simp add: numeral_2_eq_2 all_less_Suc complex_eq_iff)
qed
end
lemma DIM_complex[simp]: "DIM(complex) = 2"
by (rule dimension_complex_def)
subsubsection {* Type @{typ "'a \<times> 'b"} *}
instantiation prod :: (euclidean_space, euclidean_space) euclidean_space
begin
definition
"dimension (t::('a \<times> 'b) itself) = DIM('a) + DIM('b)"
definition
"basis i = (if i < DIM('a) then (basis i, 0) else (0, basis (i - DIM('a))))"
lemma all_less_sum:
fixes m n :: nat
shows "(\<forall>i<(m + n). P i) \<longleftrightarrow> (\<forall>i<m. P i) \<and> (\<forall>i<n. P (m + i))"
by (induct n, simp, simp add: all_less_Suc)
instance proof
show "0 < DIM('a \<times> 'b)"
unfolding dimension_prod_def by (intro add_pos_pos DIM_positive)
next
fix i :: nat
assume "DIM('a \<times> 'b) \<le> i" thus "basis i = (0::'a \<times> 'b)"
unfolding dimension_prod_def basis_prod_def zero_prod_def
by simp
next
show "\<forall>i<DIM('a \<times> 'b). \<forall>j<DIM('a \<times> 'b).
inner (basis i::'a \<times> 'b) (basis j) = (if i = j then 1 else 0)"
unfolding dimension_prod_def basis_prod_def inner_prod_def
unfolding all_less_sum prod_eq_iff
by (simp add: basis_orthonormal)
next
fix x :: "'a \<times> 'b"
show "(\<forall>i<DIM('a \<times> 'b). inner (basis i) x = 0) \<longleftrightarrow> x = 0"
unfolding dimension_prod_def basis_prod_def inner_prod_def
unfolding all_less_sum prod_eq_iff
by (simp add: euclidean_all_zero)
qed
end
end