--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Thu Aug 11 09:11:15 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Thu Aug 11 13:05:56 2011 -0700
@@ -15,22 +15,75 @@
subsection {* Type class of Euclidean spaces *}
class euclidean_space = real_inner +
+ fixes Basis :: "'a set"
+ assumes nonempty_Basis [simp]: "Basis \<noteq> {}"
+ assumes finite_Basis [simp]: "finite Basis"
+ assumes inner_Basis:
+ "\<lbrakk>u \<in> Basis; v \<in> Basis\<rbrakk> \<Longrightarrow> inner u v = (if u = v then 1 else 0)"
+ assumes euclidean_all_zero_iff:
+ "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> (x = 0)"
+
+ -- "FIXME: make this a separate definition"
fixes dimension :: "'a itself \<Rightarrow> nat"
+ assumes dimension_def: "dimension TYPE('a) = card Basis"
+
+ -- "FIXME: eventually basis function can be removed"
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)"
+ assumes image_basis: "basis ` {..<dimension TYPE('a)} = Basis"
+ assumes basis_finite: "basis ` {dimension TYPE('a)..} = {0}"
syntax "_type_dimension" :: "type => nat" ("(1DIM/(1'(_')))")
translations "DIM('t)" == "CONST dimension (TYPE('t))"
+lemma (in euclidean_space) norm_Basis: "u \<in> Basis \<Longrightarrow> norm u = 1"
+ unfolding norm_eq_sqrt_inner by (simp add: inner_Basis)
+
+lemma (in euclidean_space) sgn_Basis: "u \<in> Basis \<Longrightarrow> sgn u = u"
+ unfolding sgn_div_norm by (simp add: norm_Basis scaleR_one)
+
+lemma (in euclidean_space) Basis_zero [simp]: "0 \<notin> Basis"
+proof
+ assume "0 \<in> Basis" thus "False"
+ using inner_Basis [of 0 0] by simp
+qed
+
+lemma (in euclidean_space) nonzero_Basis: "u \<in> Basis \<Longrightarrow> u \<noteq> 0"
+ by clarsimp
+
+text {* Lemmas related to @{text basis} function. *}
+
+lemma (in euclidean_space) euclidean_all_zero:
+ "(\<forall>i<DIM('a). inner (basis i) x = 0) \<longleftrightarrow> (x = 0)"
+ using euclidean_all_zero_iff [of x, folded image_basis]
+ unfolding ball_simps by (simp add: Ball_def inner_commute)
+
+lemma (in euclidean_space) basis_zero [simp]:
+ "DIM('a) \<le> i \<Longrightarrow> basis i = 0"
+ using basis_finite by auto
+
+lemma (in euclidean_space) DIM_positive [intro]: "0 < DIM('a)"
+ unfolding dimension_def by (simp add: card_gt_0_iff)
+
+lemma (in euclidean_space) basis_inj [simp, intro]: "inj_on basis {..<DIM('a)}"
+ by (simp add: inj_on_iff_eq_card image_basis dimension_def [symmetric])
+
+lemma (in euclidean_space) basis_in_Basis [simp]:
+ "basis i \<in> Basis \<longleftrightarrow> i < DIM('a)"
+ by (cases "i < DIM('a)", simp add: image_basis [symmetric], simp)
+
+lemma (in euclidean_space) Basis_elim:
+ assumes "u \<in> Basis" obtains i where "i < DIM('a)" and "u = basis i"
+ using assms unfolding image_basis [symmetric] by fast
+
+lemma (in euclidean_space) basis_orthonormal:
+ "\<forall>i<DIM('a). \<forall>j<DIM('a).
+ inner (basis i) (basis j) = (if i = j then 1 else 0)"
+ apply clarify
+ apply (simp add: inner_Basis)
+ apply (simp add: basis_inj [unfolded inj_on_def])
+ done
+
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))")
@@ -161,6 +214,9 @@
begin
definition
+ "Basis = {1::real}"
+
+definition
"dimension (t::real itself) = 1"
definition [simp]:
@@ -170,42 +226,44 @@
by (rule dimension_real_def)
instance
- by default simp+
+ by default (auto simp add: Basis_real_def)
end
subsubsection {* Type @{typ complex} *}
+ (* TODO: move these to Complex.thy/Inner_Product.thy *)
+
+lemma norm_ii [simp]: "norm ii = 1"
+ unfolding i_def by simp
+
+lemma complex_inner_1 [simp]: "inner 1 x = Re x"
+ unfolding inner_complex_def by simp
+
+lemma complex_inner_1_right [simp]: "inner x 1 = Re x"
+ unfolding inner_complex_def by simp
+
+lemma complex_inner_ii_left [simp]: "inner ii x = Im x"
+ unfolding inner_complex_def by simp
+
+lemma complex_inner_ii_right [simp]: "inner x ii = Im x"
+ unfolding inner_complex_def by simp
+
instantiation complex :: euclidean_space
begin
+definition Basis_complex_def:
+ "Basis = {1, ii}"
+
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
+instance
+ by default (auto simp add: Basis_complex_def dimension_complex_def
+ basis_complex_def intro: complex_eqI split: split_if_asm)
end
@@ -214,40 +272,50 @@
subsubsection {* Type @{typ "'a \<times> 'b"} *}
+lemma disjoint_iff: "A \<inter> B = {} \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
+ by auto (* TODO: move elsewhere *)
+
instantiation prod :: (euclidean_space, euclidean_space) euclidean_space
begin
definition
+ "Basis = (\<lambda>u. (u, 0)) ` Basis \<union> (\<lambda>v. (0, v)) ` Basis"
+
+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)
+ show "(Basis :: ('a \<times> 'b) set) \<noteq> {}"
+ unfolding Basis_prod_def by simp
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
+ show "finite (Basis :: ('a \<times> 'b) set)"
+ unfolding Basis_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)
+ fix u v :: "'a \<times> 'b"
+ assume "u \<in> Basis" and "v \<in> Basis"
+ thus "inner u v = (if u = v then 1 else 0)"
+ unfolding Basis_prod_def inner_prod_def
+ by (auto simp add: inner_Basis split: split_if_asm)
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)
+ show "(\<forall>u\<in>Basis. inner x u = 0) \<longleftrightarrow> x = 0"
+ unfolding Basis_prod_def ball_Un ball_simps
+ by (simp add: inner_prod_def prod_eq_iff euclidean_all_zero_iff)
+next
+ show "DIM('a \<times> 'b) = card (Basis :: ('a \<times> 'b) set)"
+ unfolding dimension_prod_def Basis_prod_def
+ by (simp add: card_Un_disjoint disjoint_iff
+ card_image inj_on_def dimension_def)
+next
+ show "basis ` {..<DIM('a \<times> 'b)} = (Basis :: ('a \<times> 'b) set)"
+ by (auto simp add: Basis_prod_def dimension_prod_def basis_prod_def
+ image_def elim!: Basis_elim)
+next
+ show "basis ` {DIM('a \<times> 'b)..} = {0::('a \<times> 'b)}"
+ by (auto simp add: dimension_prod_def basis_prod_def prod_eq_iff image_def)
qed
end