--- a/src/HOL/IsaMakefile Fri Feb 20 07:41:41 2009 -0800
+++ b/src/HOL/IsaMakefile Fri Feb 20 08:02:11 2009 -0800
@@ -340,6 +340,7 @@
Library/Poly_Deriv.thy \
Library/Polynomial.thy \
Library/Product_plus.thy \
+ Library/Product_Vector.thy \
Library/Enum.thy Library/Float.thy $(SRC)/Tools/float.ML $(SRC)/HOL/Tools/float_arith.ML \
Library/reify_data.ML Library/reflection.ML
@cd Library; $(ISABELLE_TOOL) usedir $(OUT)/HOL Library
--- a/src/HOL/Library/Library.thy Fri Feb 20 07:41:41 2009 -0800
+++ b/src/HOL/Library/Library.thy Fri Feb 20 08:02:11 2009 -0800
@@ -41,7 +41,7 @@
Poly_Deriv
Polynomial
Primes
- Product_plus
+ Product_Vector
Quickcheck
Quicksort
Quotient
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Product_Vector.thy Fri Feb 20 08:02:11 2009 -0800
@@ -0,0 +1,273 @@
+(* Title: HOL/Library/Product_Vector.thy
+ Author: Brian Huffman
+*)
+
+header {* Cartesian Products as Vector Spaces *}
+
+theory Product_Vector
+imports Inner_Product Product_plus
+begin
+
+subsection {* Product is a real vector space *}
+
+instantiation "*" :: (real_vector, real_vector) real_vector
+begin
+
+definition scaleR_prod_def:
+ "scaleR r A = (scaleR r (fst A), scaleR r (snd A))"
+
+lemma fst_scaleR [simp]: "fst (scaleR r A) = scaleR r (fst A)"
+ unfolding scaleR_prod_def by simp
+
+lemma snd_scaleR [simp]: "snd (scaleR r A) = scaleR r (snd A)"
+ unfolding scaleR_prod_def by simp
+
+lemma scaleR_Pair [simp]: "scaleR r (a, b) = (scaleR r a, scaleR r b)"
+ unfolding scaleR_prod_def by simp
+
+instance proof
+ fix a b :: real and x y :: "'a \<times> 'b"
+ show "scaleR a (x + y) = scaleR a x + scaleR a y"
+ by (simp add: expand_prod_eq scaleR_right_distrib)
+ show "scaleR (a + b) x = scaleR a x + scaleR b x"
+ by (simp add: expand_prod_eq scaleR_left_distrib)
+ show "scaleR a (scaleR b x) = scaleR (a * b) x"
+ by (simp add: expand_prod_eq)
+ show "scaleR 1 x = x"
+ by (simp add: expand_prod_eq)
+qed
+
+end
+
+subsection {* Product is a normed vector space *}
+
+instantiation
+ "*" :: (real_normed_vector, real_normed_vector) real_normed_vector
+begin
+
+definition norm_prod_def:
+ "norm x = sqrt ((norm (fst x))\<twosuperior> + (norm (snd x))\<twosuperior>)"
+
+definition sgn_prod_def:
+ "sgn (x::'a \<times> 'b) = scaleR (inverse (norm x)) x"
+
+lemma norm_Pair: "norm (a, b) = sqrt ((norm a)\<twosuperior> + (norm b)\<twosuperior>)"
+ unfolding norm_prod_def by simp
+
+instance proof
+ fix r :: real and x y :: "'a \<times> 'b"
+ show "0 \<le> norm x"
+ unfolding norm_prod_def by simp
+ show "norm x = 0 \<longleftrightarrow> x = 0"
+ unfolding norm_prod_def
+ by (simp add: expand_prod_eq)
+ show "norm (x + y) \<le> norm x + norm y"
+ unfolding norm_prod_def
+ apply (rule order_trans [OF _ real_sqrt_sum_squares_triangle_ineq])
+ apply (simp add: add_mono power_mono norm_triangle_ineq)
+ done
+ show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
+ unfolding norm_prod_def
+ apply (simp add: norm_scaleR power_mult_distrib)
+ apply (simp add: right_distrib [symmetric])
+ apply (simp add: real_sqrt_mult_distrib)
+ done
+ show "sgn x = scaleR (inverse (norm x)) x"
+ by (rule sgn_prod_def)
+qed
+
+end
+
+subsection {* Product is an inner product space *}
+
+instantiation "*" :: (real_inner, real_inner) real_inner
+begin
+
+definition inner_prod_def:
+ "inner x y = inner (fst x) (fst y) + inner (snd x) (snd y)"
+
+lemma inner_Pair [simp]: "inner (a, b) (c, d) = inner a c + inner b d"
+ unfolding inner_prod_def by simp
+
+instance proof
+ fix r :: real
+ fix x y z :: "'a::real_inner * 'b::real_inner"
+ show "inner x y = inner y x"
+ unfolding inner_prod_def
+ by (simp add: inner_commute)
+ show "inner (x + y) z = inner x z + inner y z"
+ unfolding inner_prod_def
+ by (simp add: inner_left_distrib)
+ show "inner (scaleR r x) y = r * inner x y"
+ unfolding inner_prod_def
+ by (simp add: inner_scaleR_left right_distrib)
+ show "0 \<le> inner x x"
+ unfolding inner_prod_def
+ by (intro add_nonneg_nonneg inner_ge_zero)
+ show "inner x x = 0 \<longleftrightarrow> x = 0"
+ unfolding inner_prod_def expand_prod_eq
+ by (simp add: add_nonneg_eq_0_iff)
+ show "norm x = sqrt (inner x x)"
+ unfolding norm_prod_def inner_prod_def
+ by (simp add: power2_norm_eq_inner)
+qed
+
+end
+
+subsection {* Pair operations are linear and continuous *}
+
+interpretation fst!: bounded_linear fst
+ apply (unfold_locales)
+ apply (rule fst_add)
+ apply (rule fst_scaleR)
+ apply (rule_tac x="1" in exI, simp add: norm_Pair)
+ done
+
+interpretation snd!: bounded_linear snd
+ apply (unfold_locales)
+ apply (rule snd_add)
+ apply (rule snd_scaleR)
+ apply (rule_tac x="1" in exI, simp add: norm_Pair)
+ done
+
+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 bounded_linear_Pair:
+ assumes f: "bounded_linear f"
+ assumes g: "bounded_linear g"
+ shows "bounded_linear (\<lambda>x. (f x, g x))"
+proof
+ interpret f: bounded_linear f by fact
+ interpret g: bounded_linear g by fact
+ fix x y and r :: real
+ show "(f (x + y), g (x + y)) = (f x, g x) + (f y, g y)"
+ by (simp add: f.add g.add)
+ show "(f (r *\<^sub>R x), g (r *\<^sub>R x)) = r *\<^sub>R (f x, g x)"
+ by (simp add: f.scaleR g.scaleR)
+ obtain Kf where "0 < Kf" and norm_f: "\<And>x. norm (f x) \<le> norm x * Kf"
+ using f.pos_bounded by fast
+ obtain Kg where "0 < Kg" and norm_g: "\<And>x. norm (g x) \<le> norm x * Kg"
+ using g.pos_bounded by fast
+ have "\<forall>x. norm (f x, g x) \<le> norm x * (Kf + Kg)"
+ apply (rule allI)
+ apply (simp add: norm_Pair)
+ apply (rule order_trans [OF sqrt_add_le_add_sqrt], simp, simp)
+ apply (simp add: right_distrib)
+ apply (rule add_mono [OF norm_f norm_g])
+ done
+ then show "\<exists>K. \<forall>x. norm (f x, g x) \<le> norm x * K" ..
+qed
+
+text {*
+ TODO: The next three proofs are nearly identical to each other.
+ Is there a good way to factor out the common parts?
+*}
+
+lemma LIMSEQ_Pair:
+ assumes "X ----> a" and "Y ----> b"
+ shows "(\<lambda>n. (X n, Y n)) ----> (a, b)"
+proof (rule LIMSEQ_I)
+ fix r :: real assume "0 < r"
+ then have "0 < r / sqrt 2" (is "0 < ?s")
+ by (simp add: divide_pos_pos)
+ obtain M where M: "\<forall>n\<ge>M. norm (X n - a) < ?s"
+ using LIMSEQ_D [OF `X ----> a` `0 < ?s`] ..
+ obtain N where N: "\<forall>n\<ge>N. norm (Y n - b) < ?s"
+ using LIMSEQ_D [OF `Y ----> b` `0 < ?s`] ..
+ have "\<forall>n\<ge>max M N. norm ((X n, Y n) - (a, b)) < r"
+ using M N by (simp add: real_sqrt_sum_squares_less norm_Pair)
+ then show "\<exists>n0. \<forall>n\<ge>n0. norm ((X n, Y n) - (a, b)) < r" ..
+qed
+
+lemma Cauchy_Pair:
+ assumes "Cauchy X" and "Cauchy Y"
+ shows "Cauchy (\<lambda>n. (X n, Y n))"
+proof (rule CauchyI)
+ fix r :: real assume "0 < r"
+ then have "0 < r / sqrt 2" (is "0 < ?s")
+ by (simp add: divide_pos_pos)
+ obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < ?s"
+ using CauchyD [OF `Cauchy X` `0 < ?s`] ..
+ obtain N where N: "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (Y m - Y n) < ?s"
+ using CauchyD [OF `Cauchy Y` `0 < ?s`] ..
+ have "\<forall>m\<ge>max M N. \<forall>n\<ge>max M N. norm ((X m, Y m) - (X n, Y n)) < r"
+ using M N by (simp add: real_sqrt_sum_squares_less norm_Pair)
+ then show "\<exists>n0. \<forall>m\<ge>n0. \<forall>n\<ge>n0. norm ((X m, Y m) - (X n, Y n)) < r" ..
+qed
+
+lemma LIM_Pair:
+ assumes "f -- x --> a" and "g -- x --> b"
+ shows "(\<lambda>x. (f x, g x)) -- x --> (a, b)"
+proof (rule LIM_I)
+ fix r :: real assume "0 < r"
+ then have "0 < r / sqrt 2" (is "0 < ?e")
+ by (simp add: divide_pos_pos)
+ obtain s where s: "0 < s"
+ "\<forall>z. z \<noteq> x \<and> norm (z - x) < s \<longrightarrow> norm (f z - a) < ?e"
+ using LIM_D [OF `f -- x --> a` `0 < ?e`] by fast
+ obtain t where t: "0 < t"
+ "\<forall>z. z \<noteq> x \<and> norm (z - x) < t \<longrightarrow> norm (g z - b) < ?e"
+ using LIM_D [OF `g -- x --> b` `0 < ?e`] by fast
+ have "0 < min s t \<and>
+ (\<forall>z. z \<noteq> x \<and> norm (z - x) < min s t \<longrightarrow> norm ((f z, g z) - (a, b)) < r)"
+ using s t by (simp add: real_sqrt_sum_squares_less norm_Pair)
+ then show
+ "\<exists>s>0. \<forall>z. z \<noteq> x \<and> norm (z - x) < s \<longrightarrow> norm ((f z, g z) - (a, b)) < r" ..
+qed
+
+lemma isCont_Pair [simp]:
+ "\<lbrakk>isCont f x; isCont g x\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. (f x, g x)) x"
+ unfolding isCont_def by (rule LIM_Pair)
+
+
+subsection {* Product is a complete vector space *}
+
+instance "*" :: (banach, banach) banach
+proof
+ fix X :: "nat \<Rightarrow> 'a \<times> 'b" assume "Cauchy X"
+ have 1: "(\<lambda>n. fst (X n)) ----> lim (\<lambda>n. fst (X n))"
+ using fst.Cauchy [OF `Cauchy X`]
+ by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
+ have 2: "(\<lambda>n. snd (X n)) ----> lim (\<lambda>n. snd (X n))"
+ using snd.Cauchy [OF `Cauchy X`]
+ by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
+ have "X ----> (lim (\<lambda>n. fst (X n)), lim (\<lambda>n. snd (X n)))"
+ using LIMSEQ_Pair [OF 1 2] by simp
+ then show "convergent X"
+ by (rule convergentI)
+qed
+
+subsection {* Frechet derivatives involving pairs *}
+
+lemma FDERIV_Pair:
+ assumes f: "FDERIV f x :> f'" and g: "FDERIV g x :> g'"
+ shows "FDERIV (\<lambda>x. (f x, g x)) x :> (\<lambda>h. (f' h, g' h))"
+apply (rule FDERIV_I)
+apply (rule bounded_linear_Pair)
+apply (rule FDERIV_bounded_linear [OF f])
+apply (rule FDERIV_bounded_linear [OF g])
+apply (simp add: norm_Pair)
+apply (rule real_LIM_sandwich_zero)
+apply (rule LIM_add_zero)
+apply (rule FDERIV_D [OF f])
+apply (rule FDERIV_D [OF g])
+apply (rename_tac h)
+apply (simp add: divide_nonneg_pos)
+apply (rename_tac h)
+apply (subst add_divide_distrib [symmetric])
+apply (rule divide_right_mono [OF _ norm_ge_zero])
+apply (rule order_trans [OF sqrt_add_le_add_sqrt])
+apply simp
+apply simp
+apply simp
+done
+
+end