--- a/src/HOL/Complex.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Complex.thy Wed Aug 10 21:24:26 2011 +0200
@@ -340,16 +340,10 @@
subsection {* Completeness of the Complexes *}
interpretation Re: bounded_linear "Re"
-apply (unfold_locales, simp, simp)
-apply (rule_tac x=1 in exI)
-apply (simp add: complex_norm_def)
-done
+ by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
interpretation Im: bounded_linear "Im"
-apply (unfold_locales, simp, simp)
-apply (rule_tac x=1 in exI)
-apply (simp add: complex_norm_def)
-done
+ by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
lemma tendsto_Complex [tendsto_intros]:
assumes "(f ---> a) net" and "(g ---> b) net"
@@ -518,11 +512,8 @@
by (simp add: norm_mult power2_eq_square)
interpretation cnj: bounded_linear "cnj"
-apply (unfold_locales)
-apply (rule complex_cnj_add)
-apply (rule complex_cnj_scaleR)
-apply (rule_tac x=1 in exI, simp)
-done
+ using complex_cnj_add complex_cnj_scaleR
+ by (rule bounded_linear_intro [where K=1], simp)
subsection{*The Functions @{term sgn} and @{term arg}*}
--- a/src/HOL/Library/FrechetDeriv.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Library/FrechetDeriv.thy Wed Aug 10 21:24:26 2011 +0200
@@ -28,29 +28,17 @@
lemma FDERIV_bounded_linear: "FDERIV f x :> D \<Longrightarrow> bounded_linear D"
by (simp add: fderiv_def)
-lemma bounded_linear_zero:
- "bounded_linear (\<lambda>x::'a::real_normed_vector. 0::'b::real_normed_vector)"
-proof
- show "(0::'b) = 0 + 0" by simp
- fix r show "(0::'b) = scaleR r 0" by simp
- have "\<forall>x::'a. norm (0::'b) \<le> norm x * 0" by simp
- thus "\<exists>K. \<forall>x::'a. norm (0::'b) \<le> norm x * K" ..
-qed
+lemma bounded_linear_zero: "bounded_linear (\<lambda>x. 0)"
+ by (rule bounded_linear_intro [where K=0], simp_all)
lemma FDERIV_const: "FDERIV (\<lambda>x. k) x :> (\<lambda>h. 0)"
-by (simp add: fderiv_def bounded_linear_zero)
+ by (simp add: fderiv_def bounded_linear_zero)
-lemma bounded_linear_ident:
- "bounded_linear (\<lambda>x::'a::real_normed_vector. x)"
-proof
- fix x y :: 'a show "x + y = x + y" by simp
- fix r and x :: 'a show "scaleR r x = scaleR r x" by simp
- have "\<forall>x::'a. norm x \<le> norm x * 1" by simp
- thus "\<exists>K. \<forall>x::'a. norm x \<le> norm x * K" ..
-qed
+lemma bounded_linear_ident: "bounded_linear (\<lambda>x. x)"
+ by (rule bounded_linear_intro [where K=1], simp_all)
lemma FDERIV_ident: "FDERIV (\<lambda>x. x) x :> (\<lambda>h. h)"
-by (simp add: fderiv_def bounded_linear_ident)
+ by (simp add: fderiv_def bounded_linear_ident)
subsection {* Addition *}
--- a/src/HOL/Library/Inner_Product.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Library/Inner_Product.thy Wed Aug 10 21:24:26 2011 +0200
@@ -123,8 +123,7 @@
unfolding power2_sum power2_norm_eq_inner
by (simp add: inner_add inner_commute)
show "0 \<le> norm x + norm y"
- unfolding norm_eq_sqrt_inner
- by (simp add: add_nonneg_nonneg)
+ unfolding norm_eq_sqrt_inner by simp
qed
have "sqrt (a\<twosuperior> * inner x x) = \<bar>a\<bar> * sqrt (inner x x)"
by (simp add: real_sqrt_mult_distrib)
@@ -217,7 +216,7 @@
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_complex_def by (simp add: right_distrib)
show "0 \<le> inner x x"
- unfolding inner_complex_def by (simp add: add_nonneg_nonneg)
+ unfolding inner_complex_def by simp
show "inner x x = 0 \<longleftrightarrow> x = 0"
unfolding inner_complex_def
by (simp add: add_nonneg_eq_0_iff complex_Re_Im_cancel_iff)
--- a/src/HOL/Library/Product_Vector.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Library/Product_Vector.thy Wed Aug 10 21:24:26 2011 +0200
@@ -435,27 +435,21 @@
subsection {* Pair operations are linear *}
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
+ using fst_add fst_scaleR
+ by (rule bounded_linear_intro [where K=1], simp add: norm_prod_def)
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
+ using snd_add snd_scaleR
+ by (rule bounded_linear_intro [where K=1], simp add: norm_prod_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: real_sum_squared_expand x y)
apply (simp add: mult_nonneg_nonneg x y)
-apply (simp add: add_nonneg_nonneg x y)
+apply (simp add: x y)
done
lemma bounded_linear_Pair:
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Wed Aug 10 21:24:26 2011 +0200
@@ -355,9 +355,11 @@
lemma \<pi>_inj_on: "inj_on (\<pi>::nat\<Rightarrow>'n::finite) {..<CARD('n)}"
using bij_betw_pi[where 'n='n] by (simp add: bij_betw_def)
-instantiation cart :: (real_basis,finite) real_basis
+instantiation cart :: (euclidean_space, finite) euclidean_space
begin
+definition "dimension (t :: ('a ^ 'b) itself) = CARD('b) * DIM('a)"
+
definition "(basis i::'a^'b) =
(if i < (CARD('b) * DIM('a))
then (\<chi> j::'b. if j = \<pi>(i div DIM('a)) then basis (i mod DIM('a)) else 0)
@@ -417,133 +419,84 @@
finally show ?thesis by simp
qed
-instance
-proof
- let ?b = "basis :: nat \<Rightarrow> 'a^'b"
- let ?b' = "basis :: nat \<Rightarrow> 'a"
+lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
+ by (rule dimension_cart_def)
- have setsum_basis:
- "\<And>f. (\<Sum>x\<in>range basis. f (x::'a)) = f 0 + (\<Sum>i<DIM('a). f (basis i))"
- unfolding range_basis apply (subst setsum.insert)
- by (auto simp: basis_eq_0_iff setsum.insert setsum_reindex[OF basis_inj])
+lemma all_less_DIM_cart:
+ fixes m n :: nat
+ shows "(\<forall>i<DIM('a^'b). P i) \<longleftrightarrow> (\<forall>x::'b. \<forall>i<DIM('a). P (i + \<pi>' x * DIM('a)))"
+unfolding DIM_cart
+apply safe
+apply (drule spec, erule mp, erule linear_less_than_times [OF pi'_range])
+apply (erule split_CARD_DIM, simp)
+done
- have inj: "inj_on ?b {..<CARD('b)*DIM('a)}"
- by (auto intro!: inj_onI elim!: split_CARD_DIM split: split_if_asm
- simp add: Cart_eq basis_eq_pi' all_conj_distrib basis_neq_0
- inj_on_iff[OF basis_inj])
- moreover
- hence indep: "independent (?b ` {..<CARD('b) * DIM('a)})"
- proof (rule independent_eq_inj_on[THEN iffD2], safe elim!: split_CARD_DIM del: notI)
- fix j and i :: 'b and u :: "'a^'b \<Rightarrow> real" assume "j < DIM('a)"
- let ?x = "j + \<pi>' i * DIM('a)"
- show "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) \<noteq> ?b ?x"
- unfolding Cart_eq not_all
- proof
- have "(\<lambda>j. j + \<pi>' i*DIM('a))`({..<DIM('a)}-{j}) =
- {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)} - {?x}"(is "?S = ?I - _")
- proof safe
- fix y assume "y \<in> ?I"
- moreover def k \<equiv> "y - \<pi>' i*DIM('a)"
- ultimately have "k < DIM('a)" and "y = k + \<pi>' i * DIM('a)" by auto
- moreover assume "y \<notin> ?S"
- ultimately show "y = j + \<pi>' i * DIM('a)" by auto
- qed auto
+lemma eq_pi_iff:
+ fixes x :: "'c::finite"
+ shows "i < CARD('c::finite) \<Longrightarrow> x = \<pi> i \<longleftrightarrow> \<pi>' x = i"
+ by auto
+
+lemma all_less_mult:
+ fixes m n :: nat
+ shows "(\<forall>i<(m * n). P i) \<longleftrightarrow> (\<forall>i<m. \<forall>j<n. P (j + i * n))"
+apply safe
+apply (drule spec, erule mp, erule (1) linear_less_than_times)
+apply (erule split_times_into_modulo, simp)
+done
+
+lemma inner_if:
+ "inner (if a then x else y) z = (if a then inner x z else inner y z)"
+ "inner x (if a then y else z) = (if a then inner x y else inner x z)"
+ by simp_all
- have "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) $ i =
- (\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k $ i)" by simp
- also have "\<dots> = (\<Sum>k\<in>?S. u(?b k) *\<^sub>R ?b k $ i)"
- unfolding `?S = ?I - {?x}`
- proof (safe intro!: setsum_mono_zero_cong_right)
- fix y assume "y \<in> {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)}"
- moreover have "Suc (\<pi>' i) * DIM('a) \<le> CARD('b) * DIM('a)"
- unfolding mult_le_cancel2 using pi'_range[of i] by simp
- ultimately show "y < CARD('b) * DIM('a)" by simp
- next
- fix y assume "y < CARD('b) * DIM('a)"
- with split_CARD_DIM guess l k . note y = this
- moreover assume "u (?b y) *\<^sub>R ?b y $ i \<noteq> 0"
- ultimately show "y \<in> {\<pi>' i*DIM('a)..<Suc (\<pi>' i) * DIM('a)}"
- by (auto simp: basis_eq_pi' split: split_if_asm)
- qed simp
- also have "\<dots> = (\<Sum>k\<in>{..<DIM('a)} - {j}. u (?b (k + \<pi>' i*DIM('a))) *\<^sub>R (?b' k))"
- by (subst setsum_reindex) (auto simp: basis_eq_pi' intro!: inj_onI)
- also have "\<dots> \<noteq> ?b ?x $ i"
- proof -
- note independent_eq_inj_on[THEN iffD1, OF basis_inj independent_basis, rule_format]
- note this[of j "\<lambda>v. u (\<chi> ka::'b. if ka = i then v else (0\<Colon>'a))"]
- thus ?thesis by (simp add: `j < DIM('a)` basis_eq pi'_range)
- qed
- finally show "(\<Sum>k\<in>{..<CARD('b) * DIM('a)} - {?x}. u(?b k) *\<^sub>R ?b k) $ i \<noteq> ?b ?x $ i" .
- qed
- qed
- ultimately
- show "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
- by (auto intro!: exI[of _ "CARD('b) * DIM('a)"] simp: basis_cart_def)
-
- from indep have exclude_0: "0 \<notin> ?b ` {..<CARD('b) * DIM('a)}"
- using dependent_0[of "?b ` {..<CARD('b) * DIM('a)}"] by blast
-
- show "span (range ?b) = UNIV"
- proof -
- { fix x :: "'a^'b"
- let "?if i y" = "(\<chi> k::'b. if k = i then ?b' y else (0\<Colon>'a))"
- have The_if: "\<And>i j. j < DIM('a) \<Longrightarrow> (THE k. (?if i j) $ k \<noteq> 0) = i"
- by (rule the_equality) (simp_all split: split_if_asm add: basis_neq_0)
- { fix x :: 'a
- have "x \<in> span (range basis)"
- using span_basis by (auto simp: range_basis)
- hence "\<exists>u. (\<Sum>x<DIM('a). u (?b' x) *\<^sub>R ?b' x) = x"
- by (subst (asm) span_finite) (auto simp: setsum_basis) }
- hence "\<forall>i. \<exists>u. (\<Sum>x<DIM('a). u (?b' x) *\<^sub>R ?b' x) = i" by auto
- then obtain u where u: "\<forall>i. (\<Sum>x<DIM('a). u i (?b' x) *\<^sub>R ?b' x) = i"
- by (auto dest: choice)
- have "\<exists>u. \<forall>i. (\<Sum>j<DIM('a). u (?if i j) *\<^sub>R ?b' j) = x $ i"
- apply (rule exI[of _ "\<lambda>v. let i = (THE i. v$i \<noteq> 0) in u (x$i) (v$i)"])
- using The_if u by simp }
- moreover
- have "\<And>i::'b. {..<CARD('b)} \<inter> {x. i = \<pi> x} = {\<pi>' i}"
- using pi'_range[where 'n='b] by auto
- moreover
- have "range ?b = {0} \<union> ?b ` {..<CARD('b) * DIM('a)}"
- by (auto simp: image_def basis_cart_def)
- ultimately
- show ?thesis
- by (auto simp add: Cart_eq setsum_reindex[OF inj] range_basis
- setsum_mult_product basis_eq if_distrib setsum_cases span_finite
- setsum_reindex[OF basis_inj])
- qed
+instance proof
+ show "0 < DIM('a ^ 'b)"
+ unfolding dimension_cart_def
+ by (intro mult_pos_pos zero_less_card_finite DIM_positive)
+next
+ fix i :: nat
+ assume "DIM('a ^ 'b) \<le> i" thus "basis i = (0::'a^'b)"
+ unfolding dimension_cart_def basis_cart_def
+ by simp
+next
+ show "\<forall>i<DIM('a ^ 'b). \<forall>j<DIM('a ^ 'b).
+ (basis i :: 'a ^ 'b) \<bullet> basis j = (if i = j then 1 else 0)"
+ apply (simp add: inner_vector_def)
+ apply safe
+ apply (erule split_CARD_DIM, simp add: basis_eq_pi')
+ apply (simp add: inner_if setsum_delta cong: if_cong)
+ apply (simp add: basis_orthonormal)
+ apply (elim split_CARD_DIM, simp add: basis_eq_pi')
+ apply (simp add: inner_if setsum_delta cong: if_cong)
+ apply (clarsimp simp add: basis_orthonormal)
+ done
+next
+ fix x :: "'a ^ 'b"
+ show "(\<forall>i<DIM('a ^ 'b). inner (basis i) x = 0) \<longleftrightarrow> x = 0"
+ unfolding all_less_DIM_cart
+ unfolding inner_vector_def
+ apply (simp add: basis_eq_pi')
+ apply (simp add: inner_if setsum_delta cong: if_cong)
+ apply (simp add: euclidean_all_zero)
+ apply (simp add: Cart_eq)
+ done
qed
end
-lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a::real_basis)"
-proof (safe intro!: dimension_eq elim!: split_times_into_modulo del: notI)
- fix i j assume *: "i < CARD('b)" "j < DIM('a)"
- hence A: "(i * DIM('a) + j) div DIM('a) = i"
- by (subst div_add1_eq) simp
- from * have B: "(i * DIM('a) + j) mod DIM('a) = j"
- unfolding mod_mult_self3 by simp
- show "basis (j + i * DIM('a)) \<noteq> (0::'a^'b)" unfolding basis_cart_def
- using * basis_finite[where 'a='a]
- linear_less_than_times[of i "CARD('b)" j "DIM('a)"]
- by (auto simp: A B field_simps Cart_eq basis_eq_0_iff)
-qed (auto simp: basis_cart_def)
-
lemma if_distr: "(if P then f else g) $ i = (if P then f $ i else g $ i)" by simp
lemma split_dimensions'[consumes 1]:
- assumes "k < DIM('a::real_basis^'b)"
- obtains i j where "i < CARD('b::finite)" and "j < DIM('a::real_basis)" and "k = j + i * DIM('a::real_basis)"
+ assumes "k < DIM('a::euclidean_space^'b)"
+ obtains i j where "i < CARD('b::finite)" and "j < DIM('a::euclidean_space)" and "k = j + i * DIM('a::euclidean_space)"
using split_times_into_modulo[OF assms[simplified]] .
lemma cart_euclidean_bound[intro]:
- assumes j:"j < DIM('a::{real_basis})"
- shows "j + \<pi>' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::real_basis)"
+ assumes j:"j < DIM('a::euclidean_space)"
+ shows "j + \<pi>' (i::'b::finite) * DIM('a) < CARD('b) * DIM('a::euclidean_space)"
using linear_less_than_times[OF pi'_range j, of i] .
-instance cart :: (real_basis_with_inner,finite) real_basis_with_inner ..
-
-lemma (in real_basis) forall_CARD_DIM:
+lemma (in euclidean_space) forall_CARD_DIM:
"(\<forall>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<forall>(i::'b::finite) j. j<DIM('a) \<longrightarrow> P (j + \<pi>' i * DIM('a)))"
(is "?l \<longleftrightarrow> ?r")
proof (safe elim!: split_times_into_modulo)
@@ -557,7 +510,7 @@
show "P (j + i * DIM('a))" by simp
qed
-lemma (in real_basis) exists_CARD_DIM:
+lemma (in euclidean_space) exists_CARD_DIM:
"(\<exists>i<CARD('b) * DIM('a). P i) \<longleftrightarrow> (\<exists>i::'b::finite. \<exists>j<DIM('a). P (j + \<pi>' i * DIM('a)))"
using forall_CARD_DIM[where 'b='b, of "\<lambda>x. \<not> P x"] by blast
@@ -572,7 +525,7 @@
lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD
lemma cart_euclidean_nth[simp]:
- fixes x :: "('a::real_basis_with_inner, 'b::finite) cart"
+ fixes x :: "('a::euclidean_space, 'b::finite) cart"
assumes j:"j < DIM('a)"
shows "x $$ (j + \<pi>' i * DIM('a)) = x $ i $$ j"
unfolding euclidean_component_def inner_vector_def basis_eq_pi'[OF j] if_distrib cond_application_beta
@@ -606,22 +559,6 @@
thus "x = y \<and> i = j" using * by simp
qed simp
-instance cart :: (euclidean_space,finite) euclidean_space
-proof (default, safe elim!: split_dimensions')
- let ?b = "basis :: nat \<Rightarrow> 'a^'b"
- have if_distrib_op: "\<And>f P Q a b c d.
- f (if P then a else b) (if Q then c else d) =
- (if P then if Q then f a c else f a d else if Q then f b c else f b d)"
- by simp
-
- fix i j k l
- assume "i < CARD('b)" "k < CARD('b)" "j < DIM('a)" "l < DIM('a)"
- thus "?b (j + i * DIM('a)) \<bullet> ?b (l + k * DIM('a)) =
- (if j + i * DIM('a) = l + k * DIM('a) then 1 else 0)"
- using inj_on_iff[OF \<pi>_inj_on[where 'n='b], of k i]
- by (auto simp add: basis_eq inner_vector_def if_distrib_op[of inner] setsum_cases basis_orthonormal mult_split_eq)
-qed
-
instance cart :: (ordered_euclidean_space,finite) ordered_euclidean_space
proof
fix x y::"'a^'b"
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy Wed Aug 10 21:24:26 2011 +0200
@@ -6,7 +6,10 @@
header {* Convex sets, functions and related things. *}
theory Convex_Euclidean_Space
-imports Topology_Euclidean_Space Convex "~~/src/HOL/Library/Set_Algebras"
+imports
+ Topology_Euclidean_Space
+ "~~/src/HOL/Library/Convex"
+ "~~/src/HOL/Library/Set_Algebras"
begin
@@ -3054,7 +3057,7 @@
apply(rule,rule,rule,rule,rule,rule,rule,rule,rule) apply(erule_tac exE)+
apply(rule_tac x="\<lambda>n. u *\<^sub>R xb n + v *\<^sub>R xc n" in exI) apply(rule,rule)
apply(rule assms[unfolded convex_def, rule_format]) prefer 6
- apply(rule Lim_add) apply(rule_tac [1-2] Lim_cmul) by auto
+ by (auto intro: tendsto_intros)
lemma convex_interior:
fixes s :: "'a::real_normed_vector set"
--- a/src/HOL/Multivariate_Analysis/Derivative.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy Wed Aug 10 21:24:26 2011 +0200
@@ -73,7 +73,7 @@
apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE)
unfolding dist_norm netlimit_at_vector[of x] by (auto simp add: diff_diff_eq add.commute) qed qed
-subsection {* These are the only cases we'll care about, probably. *}
+text {* These are the only cases we'll care about, probably. *}
lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
@@ -83,7 +83,7 @@
bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within unfolding within_UNIV by auto
-subsection {* More explicit epsilon-delta forms. *}
+text {* More explicit epsilon-delta forms. *}
lemma has_derivative_within':
"(f has_derivative f')(at x within s) \<longleftrightarrow> bounded_linear f' \<and>
@@ -133,17 +133,18 @@
"(f has_derivative f') net \<Longrightarrow> linear f'"
by (rule derivative_linear [THEN bounded_linear_imp_linear])
-subsection {* Combining theorems. *}
+subsubsection {* Combining theorems. *}
lemma (in bounded_linear) has_derivative: "(f has_derivative f) net"
unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
- unfolding diff by(simp add: Lim_const)
+ unfolding diff by (simp add: tendsto_const)
lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp
lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
- unfolding has_derivative_def apply(rule,rule bounded_linear_zero) by(simp add: Lim_const)
+ unfolding has_derivative_def
+ by (rule, rule bounded_linear_zero, simp add: tendsto_const)
lemma (in bounded_linear) cmul: shows "bounded_linear (\<lambda>x. (c::real) *\<^sub>R f x)"
proof -
@@ -156,7 +157,8 @@
lemma has_derivative_cmul: assumes "(f has_derivative f') net" shows "((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net"
unfolding has_derivative_def apply(rule,rule bounded_linear.cmul)
- using assms[unfolded has_derivative_def] using Lim_cmul[OF assms[unfolded has_derivative_def,THEN conjunct2]]
+ using assms[unfolded has_derivative_def]
+ using scaleR.tendsto[OF tendsto_const assms[unfolded has_derivative_def,THEN conjunct2]]
unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto
lemma has_derivative_cmul_eq: assumes "c \<noteq> 0"
@@ -171,34 +173,35 @@
lemma has_derivative_neg_eq: "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net \<longleftrightarrow> (f has_derivative f') net"
apply(rule, drule_tac[!] has_derivative_neg) by auto
-lemma has_derivative_add: assumes "(f has_derivative f') net" "(g has_derivative g') net"
- shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net" proof-
+lemma has_derivative_add:
+ assumes "(f has_derivative f') net" and "(g has_derivative g') net"
+ shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net"
+proof-
note as = assms[unfolded has_derivative_def]
show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
- using Lim_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
- by (auto simp add:algebra_simps scaleR_right_diff_distrib scaleR_right_distrib) qed
+ using tendsto_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
+ by (auto simp add:algebra_simps scaleR_right_diff_distrib scaleR_right_distrib)
+qed
lemma has_derivative_add_const:"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
apply(drule has_derivative_add) apply(rule has_derivative_const) by auto
lemma has_derivative_sub:
- "(f has_derivative f') net \<Longrightarrow> (g has_derivative g') net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_derivative (\<lambda>h. f'(h) - g'(h))) net"
- apply(drule has_derivative_add) apply(drule has_derivative_neg,assumption) by(simp add:algebra_simps)
+ assumes "(f has_derivative f') net" and "(g has_derivative g') net"
+ shows "((\<lambda>x. f(x) - g(x)) has_derivative (\<lambda>h. f'(h) - g'(h))) net"
+ unfolding diff_minus by (intro has_derivative_add has_derivative_neg assms)
-lemma has_derivative_setsum: assumes "finite s" "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
+lemma has_derivative_setsum:
+ assumes "finite s" and "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
- apply(induct_tac s rule:finite_subset_induct[where A=s]) apply(rule assms(1))
-proof- fix x F assume as:"finite F" "x \<notin> F" "x\<in>s" "((\<lambda>x. \<Sum>a\<in>F. f a x) has_derivative (\<lambda>h. \<Sum>a\<in>F. f' a h)) net"
- thus "((\<lambda>xa. \<Sum>a\<in>insert x F. f a xa) has_derivative (\<lambda>h. \<Sum>a\<in>insert x F. f' a h)) net"
- unfolding setsum_insert[OF as(1,2)] apply-apply(rule has_derivative_add) apply(rule assms(2)[rule_format]) by auto
-qed(auto intro!: has_derivative_const)
+ using assms by (induct, simp_all add: has_derivative_const has_derivative_add)
lemma has_derivative_setsum_numseg:
"\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> ((f i) has_derivative (f' i)) net \<Longrightarrow>
((\<lambda>x. setsum (\<lambda>i. f i x) {m..n::nat}) has_derivative (\<lambda>h. setsum (\<lambda>i. f' i h) {m..n})) net"
- apply(rule has_derivative_setsum) by auto
+ by (rule has_derivative_setsum) simp_all
-subsection {* somewhat different results for derivative of scalar multiplier. *}
+text {* Somewhat different results for derivative of scalar multiplier. *}
(** move **)
lemma linear_vmul_component:
@@ -211,7 +214,8 @@
unfolding euclidean_component_def
by (rule inner.bounded_linear_right)
-lemma has_derivative_vmul_component: fixes c::"'a::real_normed_vector \<Rightarrow> 'b::euclidean_space" and v::"'c::real_normed_vector"
+lemma has_derivative_vmul_component:
+ fixes c::"'a::real_normed_vector \<Rightarrow> 'b::euclidean_space" and v::"'c::real_normed_vector"
assumes "(c has_derivative c') net"
shows "((\<lambda>x. c(x)$$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$$k *\<^sub>R v)) net" proof-
have *:"\<And>y. (c y $$ k *\<^sub>R v - (c (netlimit net) $$ k *\<^sub>R v + c' (y - netlimit net) $$ k *\<^sub>R v)) =
@@ -222,7 +226,8 @@
apply (rule bounded_linear_compose [OF scaleR.bounded_linear_left])
apply (rule bounded_linear_compose [OF bounded_linear_euclidean_component])
apply (rule derivative_linear [OF assms])
- apply(subst scaleR_zero_left[THEN sym, of v]) unfolding scaleR_scaleR apply(rule Lim_vmul)
+ apply(subst scaleR_zero_left[THEN sym, of v]) unfolding scaleR_scaleR
+ apply (intro tendsto_intros)
using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net")
apply(rule,assumption,rule disjI2,rule,rule) proof-
have *:"\<And>x. x - 0 = (x::'a)" by auto
@@ -261,11 +266,13 @@
apply(drule Lim_inner[where a=v]) unfolding o_def
by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed
-lemmas has_derivative_intros = has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id has_derivative_const
- has_derivative_neg has_derivative_vmul_component has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul
- bounded_linear.has_derivative has_derivative_lift_dot
+lemmas has_derivative_intros =
+ has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id
+ has_derivative_const has_derivative_neg has_derivative_vmul_component
+ has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul
+ bounded_linear.has_derivative has_derivative_lift_dot
-subsection {* limit transformation for derivatives. *}
+subsubsection {* Limit transformation for derivatives *}
lemma has_derivative_transform_within:
assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x within s)"
@@ -287,7 +294,7 @@
apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto
-subsection {* differentiability. *}
+subsection {* Differentiability *}
no_notation Deriv.differentiable (infixl "differentiable" 60)
@@ -303,22 +310,28 @@
lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
unfolding differentiable_def using has_derivative_at_within by blast
-lemma differentiable_within_open: assumes "a \<in> s" "open s" shows
- "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
+lemma differentiable_within_open: (* TODO: delete *)
+ assumes "a \<in> s" and "open s"
+ shows "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
using assms by (simp only: at_within_interior interior_open)
-lemma differentiable_on_eq_differentiable_at: "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
- unfolding differentiable_on_def by(auto simp add: differentiable_within_open)
+lemma differentiable_on_eq_differentiable_at:
+ "open s \<Longrightarrow> (f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x))"
+ unfolding differentiable_on_def
+ by (auto simp add: at_within_interior interior_open)
lemma differentiable_transform_within:
- assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable (at x within s)"
+ assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
+ assumes "f differentiable (at x within s)"
shows "g differentiable (at x within s)"
- using assms(4) unfolding differentiable_def by(auto intro!: has_derivative_transform_within[OF assms(1-3)])
+ using assms(4) unfolding differentiable_def
+ by (auto intro!: has_derivative_transform_within[OF assms(1-3)])
lemma differentiable_transform_at:
assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
shows "g differentiable at x"
- using assms(3) unfolding differentiable_def using has_derivative_transform_at[OF assms(1-2)] by auto
+ using assms(3) unfolding differentiable_def
+ using has_derivative_transform_at[OF assms(1-2)] by auto
subsection {* Frechet derivative and Jacobian matrix. *}
@@ -330,34 +343,50 @@
lemma linear_frechet_derivative:
shows "f differentiable net \<Longrightarrow> linear(frechet_derivative f net)"
- unfolding frechet_derivative_works has_derivative_def by (auto intro: bounded_linear_imp_linear)
+ unfolding frechet_derivative_works has_derivative_def
+ by (auto intro: bounded_linear_imp_linear)
-subsection {* Differentiability implies continuity. *}
+subsection {* Differentiability implies continuity *}
-lemma Lim_mul_norm_within: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
+lemma Lim_mul_norm_within:
+ fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
- unfolding Lim_within apply(rule,rule) apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
- apply(rule_tac x="min d 1" in exI) apply rule defer apply(rule,erule_tac x=x in ballE) unfolding dist_norm diff_0_right
+ unfolding Lim_within apply(rule,rule)
+ apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
+ apply(rule_tac x="min d 1" in exI) apply rule defer
+ apply(rule,erule_tac x=x in ballE) unfolding dist_norm diff_0_right
by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])
-lemma differentiable_imp_continuous_within: assumes "f differentiable (at x within s)"
- shows "continuous (at x within s) f" proof-
- from assms guess f' unfolding differentiable_def has_derivative_within .. note f'=this
+lemma differentiable_imp_continuous_within:
+ assumes "f differentiable (at x within s)"
+ shows "continuous (at x within s) f"
+proof-
+ from assms guess f' unfolding differentiable_def has_derivative_within ..
+ note f'=this
then interpret bounded_linear f' by auto
have *:"\<And>xa. x\<noteq>xa \<Longrightarrow> (f' \<circ> (\<lambda>y. y - x)) xa + norm (xa - x) *\<^sub>R ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) - ((f' \<circ> (\<lambda>y. y - x)) x + 0) = f xa - f x"
using zero by auto
have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
apply(rule continuous_within_compose) apply(rule continuous_intros)+
by(rule linear_continuous_within[OF f'[THEN conjunct1]])
- show ?thesis unfolding continuous_within using f'[THEN conjunct2, THEN Lim_mul_norm_within]
- apply-apply(drule Lim_add) apply(rule **[unfolded continuous_within]) unfolding Lim_within and dist_norm
- apply(rule,rule) apply(erule_tac x=e in allE) apply(erule impE|assumption)+ apply(erule exE,rule_tac x=d in exI)
- by(auto simp add:zero * elim!:allE) qed
+ show ?thesis unfolding continuous_within
+ using f'[THEN conjunct2, THEN Lim_mul_norm_within]
+ apply- apply(drule tendsto_add)
+ apply(rule **[unfolded continuous_within])
+ unfolding Lim_within and dist_norm
+ apply (rule, rule)
+ apply (erule_tac x=e in allE)
+ apply (erule impE | assumption)+
+ apply (erule exE, rule_tac x=d in exI)
+ by (auto simp add: zero * elim!: allE)
+qed
-lemma differentiable_imp_continuous_at: "f differentiable at x \<Longrightarrow> continuous (at x) f"
+lemma differentiable_imp_continuous_at:
+ "f differentiable at x \<Longrightarrow> continuous (at x) f"
by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])
-lemma differentiable_imp_continuous_on: "f differentiable_on s \<Longrightarrow> continuous_on s f"
+lemma differentiable_imp_continuous_on:
+ "f differentiable_on s \<Longrightarrow> continuous_on s f"
unfolding differentiable_on_def continuous_on_eq_continuous_within
using differentiable_imp_continuous_within by blast
@@ -369,39 +398,56 @@
"f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
unfolding differentiable_def using has_derivative_within_subset by blast
-lemma differentiable_on_subset: "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
+lemma differentiable_on_subset:
+ "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
unfolding differentiable_on_def using differentiable_within_subset by blast
lemma differentiable_on_empty: "f differentiable_on {}"
unfolding differentiable_on_def by auto
-subsection {* Several results are easier using a "multiplied-out" variant. *)
-(* (I got this idea from Dieudonne's proof of the chain rule). *}
+text {* Several results are easier using a "multiplied-out" variant.
+(I got this idea from Dieudonne's proof of the chain rule). *}
lemma has_derivative_within_alt:
"(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
(\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm(f(y) - f(x) - f'(y - x)) \<le> e * norm(y - x))" (is "?lhs \<longleftrightarrow> ?rhs")
-proof assume ?lhs thus ?rhs unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
- unfolding Lim_within apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
- apply(erule exE,rule_tac x=d in exI) apply(erule conjE,rule,assumption,rule,rule) proof-
+proof
+ assume ?lhs thus ?rhs
+ unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
+ unfolding Lim_within
+ apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
+ apply(erule exE,rule_tac x=d in exI)
+ apply(erule conjE,rule,assumption,rule,rule)
+ proof-
fix x y e d assume as:"0 < e" "0 < d" "norm (y - x) < d" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
then interpret bounded_linear f' by auto
show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
- case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero) next
+ case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero)
+ next
case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
unfolding dist_norm diff_0_right using as(3)
using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded dist_norm]
by (auto simp add: linear_0 linear_sub)
- thus ?thesis by(auto simp add:algebra_simps) qed qed next
- assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within apply-apply(erule conjE,rule,assumption)
- apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer apply(erule exE,rule_tac x=d in exI)
- apply(erule conjE,rule,assumption,rule,rule) unfolding dist_norm diff_0_right norm_scaleR
- apply(erule_tac x=xa in ballE,erule impE) proof-
+ thus ?thesis by(auto simp add:algebra_simps)
+ qed
+ qed
+next
+ assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within
+ apply-apply(erule conjE,rule,assumption)
+ apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer
+ apply(erule exE,rule_tac x=d in exI)
+ apply(erule conjE,rule,assumption,rule,rule)
+ unfolding dist_norm diff_0_right norm_scaleR
+ apply(erule_tac x=xa in ballE,erule impE)
+ proof-
fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
"norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
- apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps) qed auto qed
+ apply(rule_tac le_less_trans[of _ "e/2"])
+ by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps)
+ qed auto
+qed
lemma has_derivative_at_alt:
"(f has_derivative f') (at x) \<longleftrightarrow> bounded_linear f' \<and>
@@ -411,11 +457,14 @@
subsection {* The chain rule. *}
lemma diff_chain_within:
- assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at (f x) within (f ` s))"
+ assumes "(f has_derivative f') (at x within s)"
+ assumes "(g has_derivative g') (at (f x) within (f ` s))"
shows "((g o f) has_derivative (g' o f'))(at x within s)"
- unfolding has_derivative_within_alt apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
+ unfolding has_derivative_within_alt
+ apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
- apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) proof(rule,rule)
+ apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption)
+proof(rule,rule)
note assms = assms[unfolded has_derivative_within_alt]
fix e::real assume "0<e"
guess B1 using bounded_linear.pos_bounded[OF assms(1)[THEN conjunct1, rule_format]] .. note B1 = this
@@ -436,50 +485,74 @@
hence 1:"norm (f y - f x - f' (y - x)) \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x))" using d1 d2 d by auto
have "norm (f y - f x) \<le> norm (f y - f x - f' (y - x)) + norm (f' (y - x))"
- using norm_triangle_sub[of "f y - f x" "f' (y - x)"] by(auto simp add:algebra_simps)
- also have "\<dots> \<le> norm (f y - f x - f' (y - x)) + B1 * norm (y - x)" apply(rule add_left_mono) using B1 by(auto simp add:algebra_simps)
- also have "\<dots> \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)" apply(rule add_right_mono) using d1 d2 d as by auto
+ using norm_triangle_sub[of "f y - f x" "f' (y - x)"]
+ by(auto simp add:algebra_simps)
+ also have "\<dots> \<le> norm (f y - f x - f' (y - x)) + B1 * norm (y - x)"
+ apply(rule add_left_mono) using B1 by(auto simp add:algebra_simps)
+ also have "\<dots> \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)"
+ apply(rule add_right_mono) using d1 d2 d as by auto
also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto
- hence "norm (f y - f x) \<le> d * (1 + B1)" apply- apply(rule order_trans,assumption,rule mult_right_mono) using as B1 by auto
+ hence "norm (f y - f x) \<le> d * (1 + B1)" apply-
+ apply(rule order_trans,assumption,rule mult_right_mono)
+ using as B1 by auto
also have "\<dots> < de" using d B1 by(auto simp add:field_simps)
finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
- apply-apply(rule de[THEN conjunct2,rule_format]) using `y\<in>s` using d as by auto
+ apply-apply(rule de[THEN conjunct2,rule_format])
+ using `y\<in>s` using d as by auto
also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto
- also have "\<dots> \<le> e / 2 * norm (y - x)" apply(rule mult_left_mono) using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq)
+ also have "\<dots> \<le> e / 2 * norm (y - x)" apply(rule mult_left_mono)
+ using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq)
finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
interpret g': bounded_linear g' using assms(2) by auto
interpret f': bounded_linear f' using assms(1) by auto
have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
by(auto simp add:algebra_simps f'.diff g'.diff g'.add)
- also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:algebra_simps)
- also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))" apply(rule mult_left_mono) using as d1 d2 d B2 by auto
+ also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2
+ by (auto simp add: algebra_simps)
+ also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))"
+ apply (rule mult_left_mono) using as d1 d2 d B2 by auto
also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
- have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \<le> e * norm (y - x)" using 5 4 by auto
- thus "norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub) by assumption qed qed
+ have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \<le> e * norm (y - x)"
+ using 5 4 by auto
+ thus "norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)"
+ unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub)
+ by assumption
+ qed
+qed
lemma diff_chain_at:
"(f has_derivative f') (at x) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow> ((g o f) has_derivative (g' o f')) (at x)"
- using diff_chain_within[of f f' x UNIV g g'] using has_derivative_within_subset[of g g' "f x" UNIV "range f"] unfolding within_UNIV by auto
+ using diff_chain_within[of f f' x UNIV g g']
+ using has_derivative_within_subset[of g g' "f x" UNIV "range f"]
+ unfolding within_UNIV by auto
subsection {* Composition rules stated just for differentiability. *}
-lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector filter)"
+lemma differentiable_const [intro]:
+ "(\<lambda>z. c) differentiable (net::'a::real_normed_vector filter)"
unfolding differentiable_def using has_derivative_const by auto
-lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector filter)"
+lemma differentiable_id [intro]:
+ "(\<lambda>z. z) differentiable (net::'a::real_normed_vector filter)"
unfolding differentiable_def using has_derivative_id by auto
-lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector filter)"
- unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto
+lemma differentiable_cmul [intro]:
+ "f differentiable net \<Longrightarrow>
+ (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector filter)"
+ unfolding differentiable_def
+ apply(erule exE, drule has_derivative_cmul) by auto
-lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector filter)"
- unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto
+lemma differentiable_neg [intro]:
+ "f differentiable net \<Longrightarrow>
+ (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector filter)"
+ unfolding differentiable_def
+ apply(erule exE, drule has_derivative_neg) by auto
lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
\<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector filter)"
@@ -488,14 +561,18 @@
lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
\<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector filter)"
- unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
- apply(rule has_derivative_sub) by auto
+ unfolding differentiable_def apply(erule exE)+
+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
+ apply(rule has_derivative_sub) by auto
lemma differentiable_setsum:
assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
- shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net" proof-
+ shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net"
+proof-
guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
- thus ?thesis unfolding differentiable_def apply- apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto qed
+ thus ?thesis unfolding differentiable_def apply-
+ apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto
+qed
lemma differentiable_setsum_numseg:
shows "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> (f i) differentiable net \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) {m::nat..n}) differentiable net"
@@ -517,63 +594,102 @@
limit point from any direction. But OK for nontrivial intervals etc.
*}
-lemma frechet_derivative_unique_within: fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)"
- "(\<forall>i<DIM('a). \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R basis i) \<in> s)" shows "f' = f''" proof-
+lemma frechet_derivative_unique_within:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "(f has_derivative f') (at x within s)"
+ assumes "(f has_derivative f'') (at x within s)"
+ assumes "(\<forall>i<DIM('a). \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R basis i) \<in> s)"
+ shows "f' = f''"
+proof-
note as = assms(1,2)[unfolded has_derivative_def]
- then interpret f': bounded_linear f' by auto from as interpret f'': bounded_linear f'' by auto
- have "x islimpt s" unfolding islimpt_approachable proof(rule,rule)
- fix e::real assume "0<e" guess d using assms(3)[rule_format,OF DIM_positive `e>0`] ..
- thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x="x + d *\<^sub>R basis 0" in bexI)
- unfolding dist_norm by auto qed
- hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within) unfolding trivial_limit_within by simp
- show ?thesis apply(rule linear_eq_stdbasis) unfolding linear_conv_bounded_linear
- apply(rule as(1,2)[THEN conjunct1])+ proof(rule,rule,rule ccontr)
+ then interpret f': bounded_linear f' by auto
+ from as interpret f'': bounded_linear f'' by auto
+ have "x islimpt s" unfolding islimpt_approachable
+ proof(rule,rule)
+ fix e::real assume "0<e" guess d
+ using assms(3)[rule_format,OF DIM_positive `e>0`] ..
+ thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e"
+ apply(rule_tac x="x + d *\<^sub>R basis 0" in bexI)
+ unfolding dist_norm by auto
+ qed
+ hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within)
+ unfolding trivial_limit_within by simp
+ show ?thesis apply(rule linear_eq_stdbasis)
+ unfolding linear_conv_bounded_linear
+ apply(rule as(1,2)[THEN conjunct1])+
+ proof(rule,rule,rule ccontr)
fix i assume i:"i<DIM('a)" def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
- assume "f' (basis i) \<noteq> f'' (basis i)" hence "e>0" unfolding e_def by auto
- guess d using Lim_sub[OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
+ assume "f' (basis i) \<noteq> f'' (basis i)"
+ hence "e>0" unfolding e_def by auto
+ guess d using tendsto_diff [OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
guess c using assms(3)[rule_format,OF i d[THEN conjunct1]] .. note c=this
have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R basis i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R basis i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R basis i)) + f'' (c *\<^sub>R basis i)))"
unfolding scaleR_right_distrib by auto
also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"
- unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto
- also have "\<dots> = e" unfolding e_def using c[THEN conjunct1] using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] by (auto simp add: add.commute ab_diff_minus)
- finally show False using c using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"]
- unfolding dist_norm unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
- scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib using i by auto qed qed
+ unfolding f'.scaleR f''.scaleR
+ unfolding scaleR_right_distrib scaleR_minus_right by auto
+ also have "\<dots> = e" unfolding e_def using c[THEN conjunct1]
+ using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"]
+ by (auto simp add: add.commute ab_diff_minus)
+ finally show False using c
+ using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis i"]
+ unfolding dist_norm
+ unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
+ scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
+ using i by auto
+ qed
+qed
lemma frechet_derivative_unique_at:
shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
unfolding FDERIV_conv_has_derivative [symmetric]
by (rule FDERIV_unique)
-lemma continuous_isCont: "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def
+lemma continuous_isCont: "isCont f x = continuous (at x) f"
+ unfolding isCont_def LIM_def
unfolding continuous_at Lim_at unfolding dist_nz by auto
-lemma frechet_derivative_unique_within_closed_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" (is "x\<in>?I") and
- "(f has_derivative f' ) (at x within {a..b})" and
- "(f has_derivative f'') (at x within {a..b})"
- shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(3,4))+ proof(rule,rule,rule,rule)
+lemma frechet_derivative_unique_within_closed_interval:
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" (is "x\<in>?I")
+ assumes "(f has_derivative f' ) (at x within {a..b})"
+ assumes "(f has_derivative f'') (at x within {a..b})"
+ shows "f' = f''"
+ apply(rule frechet_derivative_unique_within)
+ apply(rule assms(3,4))+
+proof(rule,rule,rule,rule)
fix e::real and i assume "e>0" and i:"i<DIM('a)"
- thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a..b}" proof(cases "x$$i=a$$i")
- case True thus ?thesis apply(rule_tac x="(min (b$$i - a$$i) e) / 2" in exI)
+ thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a..b}"
+ proof(cases "x$$i=a$$i")
+ case True thus ?thesis
+ apply(rule_tac x="(min (b$$i - a$$i) e) / 2" in exI)
using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
- unfolding mem_interval euclidean_simps basis_component using i by(auto simp add:field_simps)
+ unfolding mem_interval euclidean_simps basis_component
+ using i by (auto simp add: field_simps)
next note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
case False moreover have "a $$ i < x $$ i" using False * by auto
- moreover { have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> a$$i *2 + x$$i - a$$i" by auto
- also have "\<dots> = a$$i + x$$i" by auto also have "\<dots> \<le> 2 * x$$i" using * by auto
- finally have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> x $$ i * 2" by auto }
+ moreover {
+ have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> a$$i *2 + x$$i - a$$i"
+ by auto
+ also have "\<dots> = a$$i + x$$i" by auto
+ also have "\<dots> \<le> 2 * x$$i" using * by auto
+ finally have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> x $$ i * 2" by auto
+ }
moreover have "min (x $$ i - a $$ i) e \<ge> 0" using * and `e>0` by auto
hence "x $$ i * 2 \<le> b $$ i * 2 + min (x $$ i - a $$ i) e" using * by auto
- ultimately show ?thesis apply(rule_tac x="- (min (x$$i - a$$i) e) / 2" in exI)
+ ultimately show ?thesis
+ apply(rule_tac x="- (min (x$$i - a$$i) e) / 2" in exI)
using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
- unfolding mem_interval euclidean_simps basis_component using i by(auto simp add:field_simps) qed qed
+ unfolding mem_interval euclidean_simps basis_component
+ using i by (auto simp add: field_simps)
+ qed
+qed
-lemma frechet_derivative_unique_within_open_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "x \<in> {a<..<b}" "(f has_derivative f' ) (at x within {a<..<b})"
- "(f has_derivative f'') (at x within {a<..<b})"
+lemma frechet_derivative_unique_within_open_interval:
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "x \<in> {a<..<b}"
+ assumes "(f has_derivative f' ) (at x within {a<..<b})"
+ assumes "(f has_derivative f'') (at x within {a<..<b})"
shows "f' = f''"
proof -
from assms(1) have *: "at x within {a<..<b} = at x"
@@ -587,8 +703,10 @@
apply(rule frechet_derivative_unique_at[of f],assumption)
unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto
-lemma frechet_derivative_within_closed_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" "(f has_derivative f') (at x within {a.. b})"
+lemma frechet_derivative_within_closed_interval:
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "\<forall>i<DIM('a). a$$i < b$$i" and "x \<in> {a..b}"
+ assumes "(f has_derivative f') (at x within {a.. b})"
shows "frechet_derivative f (at x within {a.. b}) = f'"
apply(rule frechet_derivative_unique_within_closed_interval[where f=f])
apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
@@ -660,11 +778,13 @@
have ***: "\<And>y y1 y2 d dx::real.
(y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
show False apply(rule ***[OF **, where dx="d * ?D k $$ j" and d="\<bar>?D k $$ j\<bar> / 2 * \<bar>d\<bar>"])
- using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
- unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding algebra_simps by (auto intro: mult_pos_pos)
+ using *[of "-d"] and *[of d] and d[THEN conjunct1] and j
+ unfolding mult_minus_left
+ unfolding abs_mult diff_minus_eq_add scaleR.minus_left
+ unfolding algebra_simps by (auto intro: mult_pos_pos)
qed
-subsection {* In particular if we have a mapping into @{typ "real"}. *}
+text {* In particular if we have a mapping into @{typ "real"}. *}
lemma differential_zero_maxmin:
fixes f::"'a\<Colon>euclidean_space \<Rightarrow> real"
@@ -673,7 +793,8 @@
and mono: "(\<forall>y\<in>s. f y \<le> f x) \<or> (\<forall>y\<in>s. f x \<le> f y)"
shows "f' = (\<lambda>v. 0)"
proof -
- obtain e where e:"e>0" "ball x e \<subseteq> s" using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
+ obtain e where e:"e>0" "ball x e \<subseteq> s"
+ using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
with differential_zero_maxmin_component[where 'b=real, of 0 e x f, simplified]
have "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j)) = (0::'a)"
unfolding differentiable_def using mono deriv by auto
@@ -685,273 +806,431 @@
qed
lemma rolle: fixes f::"real\<Rightarrow>real"
- assumes "a < b" "f a = f b" "continuous_on {a..b} f"
- "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
- shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)" proof-
- have "\<exists>x\<in>{a<..<b}. ((\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x))" proof-
- have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto hence *:"{a .. b}\<noteq>{}" by auto
+ assumes "a < b" and "f a = f b" and "continuous_on {a..b} f"
+ assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
+ shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)"
+proof-
+ have "\<exists>x\<in>{a<..<b}. ((\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x))"
+ proof-
+ have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto
+ hence *:"{a .. b}\<noteq>{}" by auto
guess d using continuous_attains_sup[OF compact_interval * assms(3)] .. note d=this
guess c using continuous_attains_inf[OF compact_interval * assms(3)] .. note c=this
- show ?thesis proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
- case True thus ?thesis apply(erule_tac disjE) apply(rule_tac x=d in bexI)
- apply(rule_tac[3] x=c in bexI) using d c by auto next def e \<equiv> "(a + b) /2"
+ show ?thesis
+ proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
+ case True thus ?thesis
+ apply(erule_tac disjE) apply(rule_tac x=d in bexI)
+ apply(rule_tac[3] x=c in bexI)
+ using d c by auto
+ next
+ def e \<equiv> "(a + b) /2"
case False hence "f d = f c" using d c assms(2) by auto
- hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d" using c d apply- apply(erule_tac x=x in ballE)+ by auto
- thus ?thesis apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto qed qed
+ hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d"
+ using c d apply- apply(erule_tac x=x in ballE)+ by auto
+ thus ?thesis
+ apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto
+ qed
+ qed
then guess x .. note x=this
- hence "f' x = (\<lambda>v. 0)" apply(rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
+ hence "f' x = (\<lambda>v. 0)"
+ apply(rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
defer apply(rule open_interval)
apply(rule assms(4)[unfolded has_derivative_at[THEN sym],THEN bspec[where x=x]],assumption)
unfolding o_def apply(erule disjE,rule disjI2) by auto
thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
- apply(drule_tac x=v in fun_cong) using x(1) by auto qed
+ apply(drule_tac x=v in fun_cong) using x(1) by auto
+qed
subsection {* One-dimensional mean value theorem. *}
lemma mvt: fixes f::"real \<Rightarrow> real"
- assumes "a < b" "continuous_on {a .. b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
- shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))" proof-
+ assumes "a < b" and "continuous_on {a .. b} f"
+ assumes "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
+ shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))"
+proof-
have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
- apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"]) defer
- apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ proof
+ apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"])
+ defer
+ apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+
+ proof
fix x assume x:"x \<in> {a<..<b}"
show "((\<lambda>x. f x - (f b - f a) / (b - a) * x) has_derivative (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)"
by(rule has_derivative_intros assms(3)[rule_format,OF x]
- has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+
+ has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+
qed(insert assms(1), auto simp add:field_simps)
- then guess x .. thus ?thesis apply(rule_tac x=x in bexI) apply(drule fun_cong[of _ _ "b - a"]) by auto qed
+ then guess x ..
+ thus ?thesis apply(rule_tac x=x in bexI)
+ apply(drule fun_cong[of _ _ "b - a"]) by auto
+qed
-lemma mvt_simple: fixes f::"real \<Rightarrow> real"
- assumes "a<b" "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
+lemma mvt_simple:
+ fixes f::"real \<Rightarrow> real"
+ assumes "a<b" and "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
- apply(rule mvt) apply(rule assms(1), rule differentiable_imp_continuous_on)
- unfolding differentiable_on_def differentiable_def defer proof
+ apply(rule mvt)
+ apply(rule assms(1), rule differentiable_imp_continuous_on)
+ unfolding differentiable_on_def differentiable_def defer
+proof
fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)"
unfolding has_derivative_within_open[OF x open_interval,THEN sym]
- apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using x by auto qed(insert assms(2), auto)
+ apply(rule has_derivative_within_subset)
+ apply(rule assms(2)[rule_format])
+ using x by auto
+qed(insert assms(2), auto)
-lemma mvt_very_simple: fixes f::"real \<Rightarrow> real"
- assumes "a \<le> b" "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
- shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)" proof(cases "a = b")
+lemma mvt_very_simple:
+ fixes f::"real \<Rightarrow> real"
+ assumes "a \<le> b" and "\<forall>x\<in>{a..b}. (f has_derivative f'(x)) (at x within {a..b})"
+ shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)"
+proof (cases "a = b")
interpret bounded_linear "f' b" using assms(2) assms(1) by auto
case True thus ?thesis apply(rule_tac x=a in bexI)
using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
unfolding True using zero by auto next
- case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto qed
+ case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto
+qed
-subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
+text {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}
-lemma mvt_general: fixes f::"real\<Rightarrow>'a::euclidean_space"
- assumes "a<b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
- shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))" proof-
+lemma mvt_general:
+ fixes f::"real\<Rightarrow>'a::euclidean_space"
+ assumes "a<b" and "continuous_on {a..b} f"
+ assumes "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
+ shows "\<exists>x\<in>{a<..<b}. norm(f b - f a) \<le> norm(f'(x) (b - a))"
+proof-
have "\<exists>x\<in>{a<..<b}. (op \<bullet> (f b - f a) \<circ> f) b - (op \<bullet> (f b - f a) \<circ> f) a = (f b - f a) \<bullet> f' x (b - a)"
- apply(rule mvt) apply(rule assms(1)) apply(rule continuous_on_inner continuous_on_intros assms(2))+
- unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto
+ apply(rule mvt) apply(rule assms(1))
+ apply(rule continuous_on_inner continuous_on_intros assms(2))+
+ unfolding o_def apply(rule,rule has_derivative_lift_dot)
+ using assms(3) by auto
then guess x .. note x=this
show ?thesis proof(cases "f a = f b")
case False
- have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2" by(simp add: power2_eq_square)
+ have "norm (f b - f a) * norm (f b - f a) = norm (f b - f a)^2"
+ by (simp add: power2_eq_square)
also have "\<dots> = (f b - f a) \<bullet> (f b - f a)" unfolding power2_norm_eq_inner ..
- also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)" using x unfolding inner_simps by (auto simp add: inner_diff_left)
- also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))" by(rule norm_cauchy_schwarz)
- finally show ?thesis using False x(1) by(auto simp add: real_mult_left_cancel) next
- case True thus ?thesis using assms(1) apply(rule_tac x="(a + b) /2" in bexI) by auto qed qed
+ also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)"
+ using x unfolding inner_simps by (auto simp add: inner_diff_left)
+ also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))"
+ by (rule norm_cauchy_schwarz)
+ finally show ?thesis using False x(1)
+ by (auto simp add: real_mult_left_cancel)
+ next
+ case True thus ?thesis using assms(1)
+ apply (rule_tac x="(a + b) /2" in bexI) by auto
+ qed
+qed
-subsection {* Still more general bound theorem. *}
+text {* Still more general bound theorem. *}
-lemma differentiable_bound: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "convex s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
- shows "norm(f x - f y) \<le> B * norm(x - y)" proof-
+lemma differentiable_bound:
+ fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)"
+ assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
+ shows "norm(f x - f y) \<le> B * norm(x - y)"
+proof-
let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
- using assms(1)[unfolded convex_alt,rule_format,OF x y] unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib by(auto simp add:algebra_simps)
- hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply- apply(rule continuous_on_intros continuous_on_vmul)+
- unfolding continuous_on_eq_continuous_within apply(rule,rule differentiable_imp_continuous_within)
+ using assms(1)[unfolded convex_alt,rule_format,OF x y]
+ unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib
+ by (auto simp add: algebra_simps)
+ hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply-
+ apply(rule continuous_on_intros continuous_on_vmul)+
+ unfolding continuous_on_eq_continuous_within
+ apply(rule,rule differentiable_imp_continuous_within)
unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
- apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) by auto
- have 2:"\<forall>u\<in>{0<..<1}. ((f \<circ> ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u)" proof rule case goal1
+ apply(rule has_derivative_within_subset)
+ apply(rule assms(2)[rule_format]) by auto
+ have 2:"\<forall>u\<in>{0<..<1}. ((f \<circ> ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u)"
+ proof rule
+ case goal1
let ?u = "x + u *\<^sub>R (y - x)"
have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})"
apply(rule diff_chain_within) apply(rule has_derivative_intros)+
- apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using goal1 * by auto
- thus ?case unfolding has_derivative_within_open[OF goal1 open_interval] by auto qed
+ apply(rule has_derivative_within_subset)
+ apply(rule assms(2)[rule_format]) using goal1 * by auto
+ thus ?case
+ unfolding has_derivative_within_open[OF goal1 open_interval] by auto
+ qed
guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
- have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y" proof- case goal1
+ have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y"
+ proof-
+ case goal1
have "norm (f' x y) \<le> onorm (f' x) * norm y"
using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
- also have "\<dots> \<le> B * norm y" apply(rule mult_right_mono)
- using assms(3)[rule_format,OF goal1] by(auto simp add:field_simps)
- finally show ?case by simp qed
+ also have "\<dots> \<le> B * norm y"
+ apply(rule mult_right_mono)
+ using assms(3)[rule_format,OF goal1]
+ by(auto simp add:field_simps)
+ finally show ?case by simp
+ qed
have "norm (f x - f y) = norm ((f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 1 - (f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 0)"
by(auto simp add:norm_minus_commute)
also have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
- finally show ?thesis by(auto simp add:norm_minus_commute) qed
+ finally show ?thesis by(auto simp add:norm_minus_commute)
+qed
-lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
- assumes "convex s" "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)" "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
+lemma differentiable_bound_real:
+ fixes f::"real \<Rightarrow> real"
+ assumes "convex s" and "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)"
+ assumes "\<forall>x\<in>s. onorm(f' x) \<le> B" and x:"x\<in>s" and y:"y\<in>s"
shows "norm(f x - f y) \<le> B * norm(x - y)"
using differentiable_bound[of s f f' B x y]
unfolding Ball_def image_iff o_def using assms by auto
-subsection {* In particular. *}
+text {* In particular. *}
-lemma has_derivative_zero_constant: fixes f::"real\<Rightarrow>real"
+lemma has_derivative_zero_constant:
+ fixes f::"real\<Rightarrow>real"
assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
- shows "\<exists>c. \<forall>x\<in>s. f x = c" proof(cases "s={}")
+ shows "\<exists>c. \<forall>x\<in>s. f x = c"
+proof(cases "s={}")
case False then obtain x where "x\<in>s" by auto
have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
- thus ?case using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
- unfolding onorm_const by auto qed
- thus ?thesis apply(rule_tac x="f x" in exI) by auto qed auto
+ thus ?case
+ using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
+ unfolding onorm_const by auto qed
+ thus ?thesis apply(rule_tac x="f x" in exI) by auto
+qed auto
lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
- assumes "convex s" "a \<in> s" "f a = c" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
- shows "f x = c" using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
+ assumes "convex s" and "a \<in> s" and "f a = c"
+ assumes "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" and "x\<in>s"
+ shows "f x = c"
+ using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto
subsection {* Differentiability of inverse function (most basic form). *}
-lemma has_derivative_inverse_basic: fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
- assumes "(f has_derivative f') (at (g y))" "bounded_linear g'" "g' \<circ> f' = id" "continuous (at y) g"
- "open t" "y \<in> t" "\<forall>z\<in>t. f(g z) = z"
- shows "(g has_derivative g') (at y)" proof-
- interpret f': bounded_linear f' using assms unfolding has_derivative_def by auto
+lemma has_derivative_inverse_basic:
+ fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
+ assumes "(f has_derivative f') (at (g y))"
+ assumes "bounded_linear g'" and "g' \<circ> f' = id" and "continuous (at y) g"
+ assumes "open t" and "y \<in> t" and "\<forall>z\<in>t. f(g z) = z"
+ shows "(g has_derivative g') (at y)"
+proof-
+ interpret f': bounded_linear f'
+ using assms unfolding has_derivative_def by auto
interpret g': bounded_linear g' using assms by auto
guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
(* have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
- have lem1:"\<forall>e>0. \<exists>d>0. \<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y - g'(z - y)) \<le> e * norm(g z - g y)" proof(rule,rule) case goal1
+ have lem1:"\<forall>e>0. \<exists>d>0. \<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y - g'(z - y)) \<le> e * norm(g z - g y)"
+ proof(rule,rule)
+ case goal1
have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
- thus ?case apply(rule_tac x=d in exI) apply rule defer proof(rule,rule)
- fix z assume as:"norm (z - y) < d" hence "z\<in>t" using d2 d unfolding dist_norm by auto
+ thus ?case apply(rule_tac x=d in exI) apply rule defer
+ proof(rule,rule)
+ fix z assume as:"norm (z - y) < d" hence "z\<in>t"
+ using d2 d unfolding dist_norm by auto
have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
- unfolding g'.diff f'.diff unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
- unfolding assms(7)[rule_format,OF `z\<in>t`] apply(subst norm_minus_cancel[THEN sym]) by auto
- also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C" by(rule C[THEN conjunct2,rule_format])
- also have "\<dots> \<le> (e / C) * norm (g z - g y) * C" apply(rule mult_right_mono)
- apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]]) apply(cases "z=y") defer
- apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format]) using as d C d0 by auto
- also have "\<dots> \<le> e * norm (g z - g y)" using C by(auto simp add:field_simps)
- finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)" by simp qed auto qed
- have *:"(0::real) < 1 / 2" by auto guess d using lem1[rule_format,OF *] .. note d=this def B\<equiv>"C*2"
+ unfolding g'.diff f'.diff
+ unfolding assms(3)[unfolded o_def id_def, THEN fun_cong]
+ unfolding assms(7)[rule_format,OF `z\<in>t`]
+ apply(subst norm_minus_cancel[THEN sym]) by auto
+ also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C"
+ by (rule C [THEN conjunct2, rule_format])
+ also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
+ apply(rule mult_right_mono)
+ apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]])
+ apply(cases "z=y") defer
+ apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format])
+ using as d C d0 by auto
+ also have "\<dots> \<le> e * norm (g z - g y)"
+ using C by (auto simp add: field_simps)
+ finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
+ by simp
+ qed auto
+ qed
+ have *:"(0::real) < 1 / 2" by auto
+ guess d using lem1[rule_format,OF *] .. note d=this
+ def B\<equiv>"C*2"
have "B>0" unfolding B_def using C by auto
- have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)" proof(rule,rule) case goal1
- have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))" by(rule norm_triangle_sub)
- also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)" apply(rule add_left_mono) using d and goal1 by auto
- also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)" apply(rule add_right_mono) using C by auto
- finally show ?case unfolding B_def by(auto simp add:field_simps) qed
- show ?thesis unfolding has_derivative_at_alt proof(rule,rule assms,rule,rule) case goal1
+ have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)"
+ proof(rule,rule) case goal1
+ have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
+ by(rule norm_triangle_sub)
+ also have "\<dots> \<le> norm(g' (z - y)) + 1 / 2 * norm (g z - g y)"
+ apply(rule add_left_mono) using d and goal1 by auto
+ also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
+ apply(rule add_right_mono) using C by auto
+ finally show ?case unfolding B_def by(auto simp add:field_simps)
+ qed
+ show ?thesis unfolding has_derivative_at_alt
+ proof(rule,rule assms,rule,rule) case goal1
hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
guess d' using lem1[rule_format,OF *] .. note d'=this
guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
- show ?case apply(rule_tac x=k in exI,rule) defer proof(rule,rule) fix z assume as:"norm(z - y) < k"
- hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)" using d' k by auto
- also have "\<dots> \<le> e * norm(z - y)" unfolding times_divide_eq_left pos_divide_le_eq[OF `B>0`]
- using lem2[THEN spec[where x=z]] using k as using `e>0` by(auto simp add:field_simps)
- finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)" by simp qed(insert k, auto) qed qed
+ show ?case
+ apply(rule_tac x=k in exI,rule) defer
+ proof(rule,rule)
+ fix z assume as:"norm(z - y) < k"
+ hence "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
+ using d' k by auto
+ also have "\<dots> \<le> e * norm(z - y)"
+ unfolding times_divide_eq_left pos_divide_le_eq[OF `B>0`]
+ using lem2[THEN spec[where x=z]] using k as using `e>0`
+ by (auto simp add: field_simps)
+ finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
+ by simp qed(insert k, auto)
+ qed
+qed
-subsection {* Simply rewrite that based on the domain point x. *}
+text {* Simply rewrite that based on the domain point x. *}
-lemma has_derivative_inverse_basic_x: fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
+lemma has_derivative_inverse_basic_x:
+ fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
"continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
shows "(g has_derivative g') (at (f(x)))"
apply(rule has_derivative_inverse_basic) using assms by auto
-subsection {* This is the version in Dieudonne', assuming continuity of f and g. *}
+text {* This is the version in Dieudonne', assuming continuity of f and g. *}
-lemma has_derivative_inverse_dieudonne: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+lemma has_derivative_inverse_dieudonne:
+ fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
(**) "x\<in>s" "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
shows "(g has_derivative g') (at (f x))"
apply(rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
- using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)] continuous_on_eq_continuous_at[OF assms(2)] by auto
+ using assms(3-6) unfolding continuous_on_eq_continuous_at[OF assms(1)]
+ continuous_on_eq_continuous_at[OF assms(2)] by auto
-subsection {* Here's the simplest way of not assuming much about g. *}
+text {* Here's the simplest way of not assuming much about g. *}
-lemma has_derivative_inverse: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+lemma has_derivative_inverse:
+ fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
"\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
- shows "(g has_derivative g') (at (f x))" proof-
+ shows "(g has_derivative g') (at (f x))"
+proof-
{ fix y assume "y\<in>interior (f ` s)"
- then obtain x where "x\<in>s" and *:"y = f x" unfolding image_iff using interior_subset by auto
- have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] .. } note * = this
- show ?thesis apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
- apply(rule continuous_on_interior[OF _ assms(3)]) apply(rule continuous_on_inverse[OF assms(4,1)])
- apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ by(rule, rule *, assumption) qed
+ then obtain x where "x\<in>s" and *:"y = f x"
+ unfolding image_iff using interior_subset by auto
+ have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] ..
+ } note * = this
+ show ?thesis
+ apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
+ apply(rule continuous_on_interior[OF _ assms(3)])
+ apply(rule continuous_on_inverse[OF assms(4,1)])
+ apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+
+ by(rule, rule *, assumption)
+qed
subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}
-lemma brouwer_surjective: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
+lemma brouwer_surjective:
+ fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
assumes "compact t" "convex t" "t \<noteq> {}" "continuous_on t f"
"\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
- shows "\<exists>y\<in>t. f y = x" proof-
- have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y" by(auto simp add:algebra_simps)
- show ?thesis unfolding * apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
- apply(rule continuous_on_intros assms)+ using assms(4-6) by auto qed
+ shows "\<exists>y\<in>t. f y = x"
+proof-
+ have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y"
+ by(auto simp add:algebra_simps)
+ show ?thesis
+ unfolding *
+ apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
+ apply(rule continuous_on_intros assms)+ using assms(4-6) by auto
+qed
-lemma brouwer_surjective_cball: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
+lemma brouwer_surjective_cball:
+ fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
assumes "0 < e" "continuous_on (cball a e) f"
"\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
- shows "\<exists>y\<in>cball a e. f y = x" apply(rule brouwer_surjective) apply(rule compact_cball convex_cball)+
- unfolding cball_eq_empty using assms by auto
+ shows "\<exists>y\<in>cball a e. f y = x"
+ apply(rule brouwer_surjective)
+ apply(rule compact_cball convex_cball)+
+ unfolding cball_eq_empty using assms by auto
text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}
-lemma sussmann_open_mapping: fixes f::"'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
+lemma sussmann_open_mapping:
+ fixes f::"'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
assumes "open s" "continuous_on s f" "x \<in> s"
"(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
"t \<subseteq> s" "x \<in> interior t"
- shows "f x \<in> interior (f ` t)" proof-
- interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
+ shows "f x \<in> interior (f ` t)"
+proof-
+ interpret f':bounded_linear f'
+ using assms unfolding has_derivative_def by auto
interpret g':bounded_linear g' using assms by auto
- guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this hence *:"1/(2*B)>0" by(auto intro!: divide_pos_pos)
+ guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this
+ hence *:"1/(2*B)>0" by (auto intro!: divide_pos_pos)
guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
- have *:"0<e0/B" "0<e1/B" apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
+ have *:"0<e0/B" "0<e1/B"
+ apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
guess e using real_lbound_gt_zero[OF *] .. note e=this
have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
- prefer 3 apply(rule,rule) proof-
- show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))" unfolding g'.diff
+ prefer 3 apply(rule,rule)
+ proof-
+ show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
+ unfolding g'.diff
apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
- apply(rule continuous_on_subset[OF assms(2)]) apply(rule,unfold image_iff,erule bexE) proof-
+ apply(rule continuous_on_subset[OF assms(2)])
+ apply(rule,unfold image_iff,erule bexE)
+ proof-
fix y z assume as:"y \<in>cball (f x) e" "z = x + (g' y - g' (f x))"
- have "dist x z = norm (g' (f x) - g' y)" unfolding as(2) and dist_norm by auto
- also have "\<dots> \<le> norm (f x - y) * B" unfolding g'.diff[THEN sym] using B by auto
- also have "\<dots> \<le> e * B" using as(1)[unfolded mem_cball dist_norm] using B by auto
+ have "dist x z = norm (g' (f x) - g' y)"
+ unfolding as(2) and dist_norm by auto
+ also have "\<dots> \<le> norm (f x - y) * B"
+ unfolding g'.diff[THEN sym] using B by auto
+ also have "\<dots> \<le> e * B"
+ using as(1)[unfolded mem_cball dist_norm] using B by auto
also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
finally have "z\<in>cball x e1" unfolding mem_cball by force
- thus "z \<in> s" using e1 assms(7) by auto qed next
+ thus "z \<in> s" using e1 assms(7) by auto
+ qed
+ next
fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
- also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using as(2)[unfolded mem_cball dist_norm] and B unfolding norm_minus_commute by auto
+ also have "\<dots> \<le> e * B" apply(rule mult_right_mono)
+ using as(2)[unfolded mem_cball dist_norm] and B
+ unfolding norm_minus_commute by auto
also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
- have **:"f x + f' (x + g' (z - f x) - x) = z" using assms(6)[unfolded o_def id_def,THEN cong] by auto
+ have **:"f x + f' (x + g' (z - f x) - x) = z"
+ using assms(6)[unfolded o_def id_def,THEN cong] by auto
have "norm (f x - (y + (z - f (x + g' (z - f x))))) \<le> norm (f (x + g' (z - f x)) - z) + norm (f x - y)"
- using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"] by(auto simp add:algebra_simps)
- also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0[THEN conjunct2,rule_format,OF *] unfolding algebra_simps ** by auto
- also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2" using as(1)[unfolded mem_cball dist_norm] by auto
- also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2" using * and B by(auto simp add:field_simps)
+ using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"]
+ by (auto simp add: algebra_simps)
+ also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)"
+ using e0[THEN conjunct2,rule_format,OF *]
+ unfolding algebra_simps ** by auto
+ also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2"
+ using as(1)[unfolded mem_cball dist_norm] by auto
+ also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2"
+ using * and B by (auto simp add: field_simps)
also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
- also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono) using as(2)[unfolded mem_cball dist_norm] unfolding norm_minus_commute by auto
- finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e" unfolding mem_cball dist_norm by auto
+ also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono)
+ using as(2)[unfolded mem_cball dist_norm]
+ unfolding norm_minus_commute by auto
+ finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e"
+ unfolding mem_cball dist_norm by auto
qed(insert e, auto) note lem = this
show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
- apply(rule,rule divide_pos_pos) prefer 3 proof
- fix y assume "y \<in> ball (f x) (e/2)" hence *:"y\<in>cball (f x) (e/2)" by auto
+ apply(rule,rule divide_pos_pos) prefer 3
+ proof
+ fix y assume "y \<in> ball (f x) (e/2)"
+ hence *:"y\<in>cball (f x) (e/2)" by auto
guess z using lem[rule_format,OF *] .. note z=this
- hence "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by(auto simp add:field_simps)
- also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using z(1) unfolding mem_cball dist_norm norm_minus_commute using B by auto
+ hence "norm (g' (z - f x)) \<le> norm (z - f x) * B"
+ using B by (auto simp add: field_simps)
+ also have "\<dots> \<le> e * B"
+ apply (rule mult_right_mono) using z(1)
+ unfolding mem_cball dist_norm norm_minus_commute using B by auto
also have "\<dots> \<le> e1" using e B unfolding less_divide_eq by auto
- finally have "x + g'(z - f x) \<in> t" apply- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
+ finally have "x + g'(z - f x) \<in> t" apply-
+ apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format])
unfolding mem_cball dist_norm by auto
- thus "y \<in> f ` t" using z by auto qed(insert e, auto) qed
+ thus "y \<in> f ` t" using z by auto
+ qed(insert e, auto)
+qed
text {* Hence the following eccentric variant of the inverse function theorem. *)
(* This has no continuity assumptions, but we do need the inverse function. *)
@@ -960,7 +1239,8 @@
(* move before left_inverse_linear in Euclidean_Space*)
- lemma right_inverse_linear: fixes f::"'a::euclidean_space => 'a"
+ lemma right_inverse_linear:
+ fixes f::"'a::euclidean_space => 'a"
assumes lf: "linear f" and gf: "f o g = id"
shows "linear g"
proof-
@@ -973,289 +1253,495 @@
with h(1) show ?thesis by blast
qed
-lemma has_derivative_inverse_strong: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
- assumes "open s" "x \<in> s" "continuous_on s f"
- "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" "f' o g' = id"
- shows "(g has_derivative g') (at (f x))" proof-
- have linf:"bounded_linear f'" using assms(5) unfolding has_derivative_def by auto
- hence ling:"bounded_linear g'" unfolding linear_conv_bounded_linear[THEN sym]
- apply- apply(rule right_inverse_linear) using assms(6) by auto
- moreover have "g' \<circ> f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[THEN sym]
+lemma has_derivative_inverse_strong:
+ fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
+ assumes "open s" and "x \<in> s" and "continuous_on s f"
+ assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" and "f' o g' = id"
+ shows "(g has_derivative g') (at (f x))"
+proof-
+ have linf:"bounded_linear f'"
+ using assms(5) unfolding has_derivative_def by auto
+ hence ling:"bounded_linear g'"
+ unfolding linear_conv_bounded_linear[THEN sym]
+ apply- apply(rule right_inverse_linear) using assms(6) by auto
+ moreover have "g' \<circ> f' = id" using assms(6) linf ling
+ unfolding linear_conv_bounded_linear[THEN sym]
using linear_inverse_left by auto
- moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)" apply(rule,rule,rule,rule sussmann_open_mapping )
+ moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)"
+ apply(rule,rule,rule,rule sussmann_open_mapping )
apply(rule assms ling)+ by auto
- have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof(rule,rule)
+ have "continuous (at (f x)) g" unfolding continuous_at Lim_at
+ proof(rule,rule)
fix e::real assume "e>0"
- hence "f x \<in> interior (f ` (ball x e \<inter> s))" using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
+ hence "f x \<in> interior (f ` (ball x e \<inter> s))"
+ using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
then guess d unfolding mem_interior .. note d=this
show "\<exists>d>0. \<forall>y. 0 < dist y (f x) \<and> dist y (f x) < d \<longrightarrow> dist (g y) (g (f x)) < e"
- apply(rule_tac x=d in exI) apply(rule,rule d[THEN conjunct1]) proof(rule,rule) case goal1
- hence "g y \<in> g ` f ` (ball x e \<inter> s)" using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
+ apply(rule_tac x=d in exI)
+ apply(rule,rule d[THEN conjunct1])
+ proof(rule,rule) case goal1
+ hence "g y \<in> g ` f ` (ball x e \<inter> s)"
+ using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
by(auto simp add:dist_commute)
hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
- thus "dist (g y) (g (f x)) < e" using assms(4)[rule_format,OF `x\<in>s`] by(auto simp add:dist_commute) qed qed
- moreover have "f x \<in> interior (f ` s)" apply(rule sussmann_open_mapping)
- apply(rule assms ling)+ using interior_open[OF assms(1)] and `x\<in>s` by auto
- moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y" proof- case goal1
- hence "y\<in>f ` s" using interior_subset by auto then guess z unfolding image_iff ..
- thus ?case using assms(4) by auto qed
- ultimately show ?thesis apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)]) using assms by auto qed
+ thus "dist (g y) (g (f x)) < e"
+ using assms(4)[rule_format,OF `x\<in>s`]
+ by (auto simp add: dist_commute)
+ qed
+ qed
+ moreover have "f x \<in> interior (f ` s)"
+ apply(rule sussmann_open_mapping)
+ apply(rule assms ling)+
+ using interior_open[OF assms(1)] and `x\<in>s` by auto
+ moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y"
+ proof- case goal1
+ hence "y\<in>f ` s" using interior_subset by auto
+ then guess z unfolding image_iff ..
+ thus ?case using assms(4) by auto
+ qed
+ ultimately show ?thesis
+ apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)])
+ using assms by auto
+qed
-subsection {* A rewrite based on the other domain. *}
+text {* A rewrite based on the other domain. *}
-lemma has_derivative_inverse_strong_x: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
- assumes "open s" "g y \<in> s" "continuous_on s f"
- "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))" "f' o g' = id" "f(g y) = y"
+lemma has_derivative_inverse_strong_x:
+ fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
+ assumes "open s" and "g y \<in> s" and "continuous_on s f"
+ assumes "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))"
+ assumes "f' o g' = id" and "f(g y) = y"
shows "(g has_derivative g') (at y)"
using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp
-subsection {* On a region. *}
+text {* On a region. *}
-lemma has_derivative_inverse_on: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
- assumes "open s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)" "\<forall>x\<in>s. g(f x) = x" "f'(x) o g'(x) = id" "x\<in>s"
+lemma has_derivative_inverse_on:
+ fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
+ assumes "open s" and "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
+ assumes "\<forall>x\<in>s. g(f x) = x" and "f'(x) o g'(x) = id" and "x\<in>s"
shows "(g has_derivative g'(x)) (at (f x))"
- apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f]) apply(rule assms)+
+ apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f])
+ apply(rule assms)+
unfolding continuous_on_eq_continuous_at[OF assms(1)]
- apply(rule,rule differentiable_imp_continuous_at) unfolding differentiable_def using assms by auto
+ apply(rule,rule differentiable_imp_continuous_at)
+ unfolding differentiable_def using assms by auto
-subsection {* Invertible derivative continous at a point implies local injectivity. *)
-(* It's only for this we need continuity of the derivative, except of course *)
-(* if we want the fact that the inverse derivative is also continuous. So if *)
-(* we know for some other reason that the inverse function exists, it's OK. *}
+text {* Invertible derivative continous at a point implies local
+injectivity. It's only for this we need continuity of the derivative,
+except of course if we want the fact that the inverse derivative is
+also continuous. So if we know for some other reason that the inverse
+function exists, it's OK. *}
-lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
- using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g] by(auto simp add:algebra_simps)
+lemma bounded_linear_sub:
+ "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
+ using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g]
+ by (auto simp add: algebra_simps)
-lemma has_derivative_locally_injective: fixes f::"'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
+lemma has_derivative_locally_injective:
+ fixes f::"'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
"\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
"\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
- obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)" proof-
+ obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)"
+proof-
interpret bounded_linear g' using assms by auto
note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
have "g' (f' a (\<chi>\<chi> i.1)) = (\<chi>\<chi> i.1)" "(\<chi>\<chi> i.1) \<noteq> (0::'n)" defer
apply(subst euclidean_eq) using f'g' by auto
- hence *:"0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp
+ hence *:"0 < onorm g'"
+ unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp
def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
guess d1 using assms(6)[rule_format,OF *] .. note d1=this
from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
- guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] .. note d2=this
- guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d = this
- show ?thesis proof show "a\<in>ball a d" using d by auto
- show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x" proof(intro strip)
+ guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] ..
+ note d2=this
+ guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] ..
+ note d = this
+ show ?thesis
+ proof
+ show "a\<in>ball a d" using d by auto
+ show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x"
+ proof (intro strip)
fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
- def ph \<equiv> "\<lambda>w. w - g'(f w - f x)" have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
- unfolding ph_def o_def unfolding diff using f'g' by(auto simp add:algebra_simps)
+ def ph \<equiv> "\<lambda>w. w - g'(f w - f x)"
+ have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
+ unfolding ph_def o_def unfolding diff using f'g'
+ by (auto simp add: algebra_simps)
have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
- apply(rule_tac[!] ballI) proof- fix u assume u:"u \<in> ball a d" hence "u\<in>s" using d d2 by auto
- have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)" unfolding o_def and diff using f'g' by auto
+ apply(rule_tac[!] ballI)
+ proof-
+ fix u assume u:"u \<in> ball a d"
+ hence "u\<in>s" using d d2 by auto
+ have *:"(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
+ unfolding o_def and diff using f'g' by auto
show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
- unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
- apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
- apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
+ unfolding ph' * apply(rule diff_chain_within) defer
+ apply(rule bounded_linear.has_derivative[OF assms(3)])
+ apply(rule has_derivative_intros) defer
+ apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
+ apply(rule has_derivative_at_within)
+ using assms(5) and `u\<in>s` `a\<in>s`
by(auto intro!: has_derivative_intros derivative_linear)
- have **:"bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)" apply(rule_tac[!] bounded_linear_sub)
- apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
- have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)" unfolding * apply(rule onorm_compose)
- unfolding linear_conv_bounded_linear by(rule assms(3) **)+
- also have "\<dots> \<le> onorm g' * k" apply(rule mult_left_mono)
- using d1[THEN conjunct2,rule_format,of u] using onorm_neg[OF **(1)[unfolded linear_linear]]
- using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]] by(auto simp add:algebra_simps)
+ have **:"bounded_linear (\<lambda>x. f' u x - f' a x)"
+ "bounded_linear (\<lambda>x. f' a x - f' u x)"
+ apply(rule_tac[!] bounded_linear_sub)
+ apply(rule_tac[!] derivative_linear)
+ using assms(5) `u\<in>s` `a\<in>s` by auto
+ have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)"
+ unfolding * apply(rule onorm_compose)
+ unfolding linear_conv_bounded_linear by(rule assms(3) **)+
+ also have "\<dots> \<le> onorm g' * k"
+ apply(rule mult_left_mono)
+ using d1[THEN conjunct2,rule_format,of u]
+ using onorm_neg[OF **(1)[unfolded linear_linear]]
+ using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]]
+ by (auto simp add: algebra_simps)
also have "\<dots> \<le> 1/2" unfolding k_def by auto
- finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption qed
- moreover have "norm (ph y - ph x) = norm (y - x)" apply(rule arg_cong[where f=norm])
+ finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption
+ qed
+ moreover have "norm (ph y - ph x) = norm (y - x)"
+ apply(rule arg_cong[where f=norm])
unfolding ph_def using diff unfolding as by auto
- ultimately show "x = y" unfolding norm_minus_commute by auto qed qed auto qed
+ ultimately show "x = y" unfolding norm_minus_commute by auto
+ qed
+ qed auto
+qed
subsection {* Uniformly convergent sequence of derivatives. *}
-lemma has_derivative_sequence_lipschitz_lemma: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
- assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
- shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)" proof(default)+
+lemma has_derivative_sequence_lipschitz_lemma:
+ fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ assumes "convex s"
+ assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ assumes "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
+ shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
+proof (default)+
fix m n x y assume as:"N\<le>m" "N\<le>n" "x\<in>s" "y\<in>s"
show "norm((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm(x - y)"
- apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)]) apply(rule_tac[!] ballI) proof-
- fix x assume "x\<in>s" show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within s)"
+ apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)])
+ apply(rule_tac[!] ballI)
+ proof-
+ fix x assume "x\<in>s"
+ show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within s)"
by(rule has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
- { fix h have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
- using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"] unfolding norm_minus_commute by(auto simp add:algebra_simps)
- also have "\<dots> \<le> e * norm h+ e * norm h" using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h] assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
+ { fix h
+ have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
+ using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"]
+ unfolding norm_minus_commute by (auto simp add: algebra_simps)
+ also have "\<dots> \<le> e * norm h+ e * norm h"
+ using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h]
+ using assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
by(auto simp add:field_simps)
finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
- thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e" apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
- unfolding linear_conv_bounded_linear using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear] by auto qed qed
+ thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
+ apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
+ unfolding linear_conv_bounded_linear
+ using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear]
+ by auto
+ qed
+qed
-lemma has_derivative_sequence_lipschitz: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
- assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)" "0 < e"
- shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)" proof(rule,rule)
+lemma has_derivative_sequence_lipschitz:
+ fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ assumes "convex s"
+ assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ assumes "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
+ assumes "0 < e"
+ shows "\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)"
+proof(rule,rule)
case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
guess N using assms(3)[rule_format,OF *(2)] ..
- thus ?case apply(rule_tac x=N in exI) apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *]) using assms by auto qed
+ thus ?case
+ apply(rule_tac x=N in exI)
+ apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *])
+ using assms by auto
+qed
-lemma has_derivative_sequence: fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
- assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
- "x0 \<in> s" "((\<lambda>n. f n x0) ---> l) sequentially"
- shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)" proof-
+lemma has_derivative_sequence:
+ fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ assumes "convex s"
+ assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ assumes "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
+ assumes "x0 \<in> s" and "((\<lambda>n. f n x0) ---> l) sequentially"
+ shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and>
+ (g has_derivative g'(x)) (at x within s)"
+proof-
have lem1:"\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f m x - f n x) - (f m y - f n y)) \<le> e * norm(x - y)"
- apply(rule has_derivative_sequence_lipschitz[where e="42::nat"]) apply(rule assms)+ by auto
- have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially" apply(rule bchoice) unfolding convergent_eq_cauchy proof
- fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)" proof(cases "x=x0")
- case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto next
- case False show ?thesis unfolding Cauchy_def proof(rule,rule)
- fix e::real assume "e>0" hence *:"e/2>0" "e/2/norm(x-x0)>0" using False by(auto intro!:divide_pos_pos)
+ apply(rule has_derivative_sequence_lipschitz[where e="42::nat"])
+ apply(rule assms)+ by auto
+ have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially"
+ apply(rule bchoice) unfolding convergent_eq_cauchy
+ proof
+ fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)"
+ proof(cases "x=x0")
+ case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto
+ next
+ case False show ?thesis unfolding Cauchy_def
+ proof(rule,rule)
+ fix e::real assume "e>0"
+ hence *:"e/2>0" "e/2/norm(x-x0)>0"
+ using False by (auto intro!: divide_pos_pos)
guess M using convergent_imp_cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
guess N using lem1[rule_format,OF *(2)] .. note N = this
- show " \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e" apply(rule_tac x="max M N" in exI) proof(default+)
+ show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
+ apply(rule_tac x="max M N" in exI)
+ proof(default+)
fix m n assume as:"max M N \<le>m" "max M N\<le>n"
have "dist (f m x) (f n x) \<le> norm (f m x0 - f n x0) + norm (f m x - f n x - (f m x0 - f n x0))"
unfolding dist_norm by(rule norm_triangle_sub)
- also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2" using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False by auto
- also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_right_mono) using as and M[rule_format] unfolding dist_norm by auto
- finally show "dist (f m x) (f n x) < e" by auto qed qed qed qed
+ also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2"
+ using N[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False
+ by auto
+ also have "\<dots> < e / 2 + e / 2"
+ apply(rule add_strict_right_mono)
+ using as and M[rule_format] unfolding dist_norm by auto
+ finally show "dist (f m x) (f n x) < e" by auto
+ qed
+ qed
+ qed
+ qed
then guess g .. note g = this
- have lem2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f n x - f n y) - (g x - g y)) \<le> e * norm(x - y)" proof(rule,rule)
- fix e::real assume *:"e>0" guess N using lem1[rule_format,OF *] .. note N=this
- show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply(rule_tac x=N in exI) proof(default+)
+ have lem2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm((f n x - f n y) - (g x - g y)) \<le> e * norm(x - y)"
+ proof(rule,rule)
+ fix e::real assume *:"e>0"
+ guess N using lem1[rule_format,OF *] .. note N=this
+ show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>y\<in>s. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
+ apply(rule_tac x=N in exI)
+ proof(default+)
fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
- have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
- unfolding eventually_sequentially apply(rule_tac x=N in exI) proof(rule,rule)
- fix m assume "N\<le>m" thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
- using N[rule_format, of n m x y] and as by(auto simp add:algebra_simps) qed
- thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply-
+ have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially"
+ unfolding eventually_sequentially
+ apply(rule_tac x=N in exI)
+ proof(rule,rule)
+ fix m assume "N\<le>m"
+ thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
+ using N[rule_format, of n m x y] and as
+ by (auto simp add: algebra_simps)
+ qed
+ thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
+ apply-
apply(rule Lim_norm_ubound[OF trivial_limit_sequentially, where f="\<lambda>m. (f n x - f n y) - (f m x - f m y)"])
- apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption qed qed
+ apply(rule tendsto_intros g[rule_format] as)+ by assumption
+ qed
+ qed
show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
- apply(rule,rule,rule g[rule_format],assumption) proof fix x assume "x\<in>s"
- have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially" unfolding Lim_sequentially proof(rule,rule,rule)
- fix u and e::real assume "e>0" show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e" proof(cases "u=0")
+ apply(rule,rule,rule g[rule_format],assumption)
+ proof fix x assume "x\<in>s"
+ have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially"
+ unfolding Lim_sequentially
+ proof(rule,rule,rule)
+ fix u and e::real assume "e>0"
+ show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e"
+ proof(cases "u=0")
case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
show ?thesis apply(rule_tac x=N in exI) unfolding True
- using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto next
- case False hence *:"e / 2 / norm u > 0" using `e>0` by(auto intro!: divide_pos_pos)
+ using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto
+ next
+ case False hence *:"e / 2 / norm u > 0"
+ using `e>0` by (auto intro!: divide_pos_pos)
guess N using assms(3)[rule_format,OF *] .. note N=this
- show ?thesis apply(rule_tac x=N in exI) proof(rule,rule) case goal1
- show ?case unfolding dist_norm using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
- by (auto simp add:field_simps) qed qed qed
- show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule)
+ show ?thesis apply(rule_tac x=N in exI)
+ proof(rule,rule) case goal1
+ show ?case unfolding dist_norm
+ using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
+ by (auto simp add:field_simps)
+ qed
+ qed
+ qed
+ show "bounded_linear (g' x)"
+ unfolding linear_linear linear_def
+ apply(rule,rule,rule) defer
+ proof(rule,rule)
fix x' y z::"'m" and c::real
note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
- show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'" apply(rule tendsto_unique[OF trivial_limit_sequentially])
+ show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'"
+ apply(rule tendsto_unique[OF trivial_limit_sequentially])
apply(rule lem3[rule_format])
unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
- apply(rule Lim_cmul) by(rule lem3[rule_format])
- show "g' x (y + z) = g' x y + g' x z" apply(rule tendsto_unique[OF trivial_limit_sequentially])
- apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
- apply(rule Lim_add) by(rule lem3[rule_format])+ qed
- show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)" proof(rule,rule) case goal1
- have *:"e/3>0" using goal1 by auto guess N1 using assms(3)[rule_format,OF *] .. note N1=this
+ apply (intro tendsto_intros) by(rule lem3[rule_format])
+ show "g' x (y + z) = g' x y + g' x z"
+ apply(rule tendsto_unique[OF trivial_limit_sequentially])
+ apply(rule lem3[rule_format])
+ unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
+ apply(rule tendsto_add) by(rule lem3[rule_format])+
+ qed
+ show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
+ proof(rule,rule) case goal1
+ have *:"e/3>0" using goal1 by auto
+ guess N1 using assms(3)[rule_format,OF *] .. note N1=this
guess N2 using lem2[rule_format,OF *] .. note N2=this
guess d1 using assms(2)[unfolded has_derivative_within_alt, rule_format,OF `x\<in>s`, of "max N1 N2",THEN conjunct2,rule_format,OF *] .. note d1=this
- show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1]) proof(rule,rule)
- fix y assume as:"y \<in> s" "norm (y - x) < d1" let ?N ="max N1 N2"
- have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)" apply(subst norm_minus_cancel[THEN sym])
- using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto moreover
- have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)" using d1 and as by auto ultimately
+ show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1])
+ proof(rule,rule)
+ fix y assume as:"y \<in> s" "norm (y - x) < d1"
+ let ?N ="max N1 N2"
+ have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)"
+ apply(subst norm_minus_cancel[THEN sym])
+ using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto
+ moreover
+ have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)"
+ using d1 and as by auto
+ ultimately
have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
- using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
- by (auto simp add:algebra_simps) moreover
- have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)" using N1 `x\<in>s` by auto
+ using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
+ by (auto simp add:algebra_simps)
+ moreover
+ have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)"
+ using N1 `x\<in>s` by auto
ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
- using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"] by(auto simp add:algebra_simps)
- qed qed qed qed
+ using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"]
+ by(auto simp add:algebra_simps)
+ qed
+ qed
+ qed
+qed
-subsection {* Can choose to line up antiderivatives if we want. *}
+text {* Can choose to line up antiderivatives if we want. *}
-lemma has_antiderivative_sequence: fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
- assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm h"
- shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof(cases "s={}")
- case False then obtain a where "a\<in>s" by auto have *:"\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x" by auto
- show ?thesis apply(rule *) apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
- apply(rule,rule) apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)
- apply(rule `a\<in>s`) by(auto intro!: Lim_const) qed auto
+lemma has_antiderivative_sequence:
+ fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ assumes "convex s"
+ assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ assumes "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm h"
+ shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
+proof(cases "s={}")
+ case False then obtain a where "a\<in>s" by auto
+ have *:"\<And>P Q. \<exists>g. \<forall>x\<in>s. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>s. Q g x" by auto
+ show ?thesis
+ apply(rule *)
+ apply(rule has_derivative_sequence[OF assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
+ apply(rule,rule)
+ apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)
+ apply(rule `a\<in>s`) by(auto intro!: tendsto_const)
+qed auto
-lemma has_antiderivative_limit: fixes g'::"'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
- assumes "convex s" "\<forall>e>0. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> e * norm(h))"
- shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)" proof-
+lemma has_antiderivative_limit:
+ fixes g'::"'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
+ assumes "convex s"
+ assumes "\<forall>e>0. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> e * norm(h))"
+ shows "\<exists>g. \<forall>x\<in>s. (g has_derivative g'(x)) (at x within s)"
+proof-
have *:"\<forall>n. \<exists>f f'. \<forall>x\<in>s. (f has_derivative (f' x)) (at x within s) \<and> (\<forall>h. norm(f' x h - g' x h) \<le> inverse (real (Suc n)) * norm(h))"
- apply(rule) using assms(2) apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
- guess f using *[THEN choice] .. note * = this guess f' using *[THEN choice] .. note f=this
- show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer proof(rule,rule)
+ apply(rule) using assms(2)
+ apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
+ guess f using *[THEN choice] .. note * = this
+ guess f' using *[THEN choice] .. note f=this
+ show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer
+ proof(rule,rule)
fix e::real assume "0<e" guess N using reals_Archimedean[OF `e>0`] .. note N=this
- show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h" apply(rule_tac x=N in exI) proof(default+) case goal1
+ show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
+ apply(rule_tac x=N in exI)
+ proof(default+)
+ case goal1
have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
using goal1(1) by(auto simp add:field_simps)
- show ?case using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
- apply(rule order_trans) using N * apply(cases "h=0") by auto qed qed(insert f,auto) qed
+ show ?case
+ using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]]
+ apply(rule order_trans) using N * apply(cases "h=0") by auto
+ qed
+ qed(insert f,auto)
+qed
subsection {* Differentiation of a series. *}
definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"
-lemma has_derivative_series: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
- assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
- "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(setsum (\<lambda>i. f' i x h) (k \<inter> {0..n}) - g' x h) \<le> e * norm(h)"
- "x\<in>s" "((\<lambda>n. f n x) sums_seq l) k"
+lemma has_derivative_series:
+ fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
+ assumes "convex s"
+ assumes "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
+ assumes "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(setsum (\<lambda>i. f' i x h) (k \<inter> {0..n}) - g' x h) \<le> e * norm(h)"
+ assumes "x\<in>s" and "((\<lambda>n. f n x) sums_seq l) k"
shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) sums_seq (g x)) k \<and> (g has_derivative g'(x)) (at x within s)"
- unfolding sums_seq_def apply(rule has_derivative_sequence[OF assms(1) _ assms(3)]) apply(rule,rule)
- apply(rule has_derivative_setsum) defer apply(rule,rule assms(2)[rule_format],assumption)
+ unfolding sums_seq_def
+ apply(rule has_derivative_sequence[OF assms(1) _ assms(3)])
+ apply(rule,rule)
+ apply(rule has_derivative_setsum) defer
+ apply(rule,rule assms(2)[rule_format],assumption)
using assms(4-5) unfolding sums_seq_def by auto
subsection {* Derivative with composed bilinear function. *}
lemma has_derivative_bilinear_within:
- assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at x within s)" "bounded_bilinear h"
- shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)" proof-
- have "(g ---> g x) (at x within s)" apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
- using assms(2) unfolding differentiable_def by auto moreover
- interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
- interpret g':bounded_linear g' using assms unfolding has_derivative_def by auto
- interpret h:bounded_bilinear h using assms by auto
- have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)" unfolding f'.zero[THEN sym]
- apply(rule Lim_linear[of "\<lambda>y. y - x" 0 "at x within s" f']) using Lim_sub[OF Lim_within_id Lim_const, of x x s]
+ assumes "(f has_derivative f') (at x within s)"
+ assumes "(g has_derivative g') (at x within s)"
+ assumes "bounded_bilinear h"
+ shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)"
+proof-
+ have "(g ---> g x) (at x within s)"
+ apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
+ using assms(2) unfolding differentiable_def by auto
+ moreover
+ interpret f':bounded_linear f'
+ using assms unfolding has_derivative_def by auto
+ interpret g':bounded_linear g'
+ using assms unfolding has_derivative_def by auto
+ interpret h:bounded_bilinear h
+ using assms by auto
+ have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)"
+ unfolding f'.zero[THEN sym]
+ using bounded_linear.tendsto [of f' "\<lambda>y. y - x" 0 "at x within s"]
+ using tendsto_diff [OF Lim_within_id tendsto_const, of x x s]
unfolding id_def using assms(1) unfolding has_derivative_def by auto
hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
- using Lim_add[OF Lim_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"] by auto ultimately
+ using tendsto_add[OF tendsto_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"]
+ by auto
+ ultimately
have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
+ h ((1/ (norm (x' - x))) *\<^sub>R (f x' - (f x + f' (x' - x)))) (g x')) ---> h (f x) 0 + h 0 (g x)) (at x within s)"
- apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) using assms(1-2) unfolding has_derivative_within by auto
+ apply-apply(rule tendsto_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)])
+ using assms(1-2) unfolding has_derivative_within by auto
guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
guess C using f'.pos_bounded .. note C=this
guess D using g'.pos_bounded .. note D=this
have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
- have **:"((\<lambda>y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)" unfolding Lim_within proof(rule,rule) case goal1
+ have **:"((\<lambda>y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)"
+ unfolding Lim_within
+ proof(rule,rule) case goal1
hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
- thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) proof(rule,rule,erule conjE)
+ thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule)
+ proof(rule,rule,erule conjE)
fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
- also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono)
- apply(rule mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
- also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps)
- also have "\<dots> < e * norm (y - x)" apply(rule mult_strict_right_mono)
- using as(3)[unfolded dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps)
+ also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B"
+ apply(rule mult_right_mono)
+ apply(rule mult_mono) using B C D
+ by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
+ also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)"
+ by (auto simp add: field_simps)
+ also have "\<dots> < e * norm (y - x)"
+ apply(rule mult_strict_right_mono)
+ using as(3)[unfolded dist_norm] and as(2)
+ unfolding pos_less_divide_eq[OF bcd] by (auto simp add: field_simps)
finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
- unfolding dist_norm apply-apply(cases "y = x") by(auto simp add:field_simps) qed qed
+ unfolding dist_norm apply-apply(cases "y = x")
+ by(auto simp add: field_simps)
+ qed
+ qed
have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))"
apply (rule bounded_linear_add)
apply (rule bounded_linear_compose [OF h.bounded_linear_right `bounded_linear g'`])
apply (rule bounded_linear_compose [OF h.bounded_linear_left `bounded_linear f'`])
done
- thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within
+ thus ?thesis using tendsto_add[OF * **] unfolding has_derivative_within
unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
- scaleR_right_diff_distrib h.zero_right h.zero_left by(auto simp add:field_simps) qed
+ scaleR_right_diff_distrib h.zero_right h.zero_left
+ by(auto simp add:field_simps)
+qed
lemma has_derivative_bilinear_at:
- assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "bounded_bilinear h"
+ assumes "(f has_derivative f') (at x)"
+ assumes "(g has_derivative g') (at x)"
+ assumes "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
- using has_derivative_bilinear_within[of f f' x UNIV g g' h] unfolding within_UNIV using assms by auto
+ using has_derivative_bilinear_within[of f f' x UNIV g g' h]
+ unfolding within_UNIV using assms by auto
subsection {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
@@ -1265,14 +1751,20 @@
definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"
-lemma vector_derivative_works: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
+lemma vector_derivative_works:
+ fixes f::"real \<Rightarrow> 'a::real_normed_vector"
shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
-proof assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
+proof
+ assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
then interpret bounded_linear f' by auto
thus ?r unfolding vector_derivative_def has_vector_derivative_def
apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
using f' unfolding scaleR[THEN sym] by auto
-next assume ?r thus ?l unfolding vector_derivative_def has_vector_derivative_def differentiable_def by auto qed
+next
+ assume ?r thus ?l
+ unfolding vector_derivative_def has_vector_derivative_def differentiable_def
+ by auto
+qed
lemma vector_derivative_unique_at:
assumes "(f has_vector_derivative f') (at x)"
@@ -1285,16 +1777,26 @@
thus ?thesis unfolding fun_eq_iff by auto
qed
-lemma vector_derivative_unique_within_closed_interval: fixes f::"real \<Rightarrow> 'n::ordered_euclidean_space"
- assumes "a < b" "x \<in> {a..b}"
- "(f has_vector_derivative f') (at x within {a..b})"
- "(f has_vector_derivative f'') (at x within {a..b})" shows "f' = f''" proof-
+lemma vector_derivative_unique_within_closed_interval:
+ fixes f::"real \<Rightarrow> 'n::ordered_euclidean_space"
+ assumes "a < b" and "x \<in> {a..b}"
+ assumes "(f has_vector_derivative f') (at x within {a..b})"
+ assumes "(f has_vector_derivative f'') (at x within {a..b})"
+ shows "f' = f''"
+proof-
have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
apply(rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
- using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2) by auto
- show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
- hence "(\<lambda>x. x *\<^sub>R f') 1 = (\<lambda>x. x *\<^sub>R f'') 1" using * by (auto simp: fun_eq_iff)
- ultimately show False unfolding o_def by auto qed qed
+ using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2)
+ by auto
+ show ?thesis
+ proof(rule ccontr)
+ assume "f' \<noteq> f''"
+ moreover
+ hence "(\<lambda>x. x *\<^sub>R f') 1 = (\<lambda>x. x *\<^sub>R f'') 1"
+ using * by (auto simp: fun_eq_iff)
+ ultimately show False unfolding o_def by auto
+ qed
+qed
lemma vector_derivative_at:
shows "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
@@ -1302,8 +1804,10 @@
unfolding vector_derivative_works[THEN sym] differentiable_def
unfolding has_vector_derivative_def by auto
-lemma vector_derivative_within_closed_interval: fixes f::"real \<Rightarrow> 'a::ordered_euclidean_space"
- assumes "a < b" "x \<in> {a..b}" "(f has_vector_derivative f') (at x within {a..b})"
+lemma vector_derivative_within_closed_interval:
+ fixes f::"real \<Rightarrow> 'a::ordered_euclidean_space"
+ assumes "a < b" and "x \<in> {a..b}"
+ assumes "(f has_vector_derivative f') (at x within {a..b})"
shows "vector_derivative f (at x within {a..b}) = f'"
apply(rule vector_derivative_unique_within_closed_interval)
using vector_derivative_works[unfolded differentiable_def]
@@ -1320,71 +1824,95 @@
lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
unfolding has_vector_derivative_def using has_derivative_id by auto
-lemma has_vector_derivative_cmul: "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
- unfolding has_vector_derivative_def apply(drule has_derivative_cmul) by(auto simp add:algebra_simps)
+lemma has_vector_derivative_cmul:
+ "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
+ unfolding has_vector_derivative_def apply(drule has_derivative_cmul)
+ by (auto simp add: algebra_simps)
-lemma has_vector_derivative_cmul_eq: assumes "c \<noteq> 0"
+lemma has_vector_derivative_cmul_eq:
+ assumes "c \<noteq> 0"
shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
apply(rule has_vector_derivative_cmul) using assms by auto
lemma has_vector_derivative_neg:
- "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
+ "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto
lemma has_vector_derivative_add:
- assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
+ assumes "(f has_vector_derivative f') net"
+ assumes "(g has_vector_derivative g') net"
shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto
lemma has_vector_derivative_sub:
- assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
+ assumes "(f has_vector_derivative f') net"
+ assumes "(g has_vector_derivative g') net"
shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto
lemma has_vector_derivative_bilinear_within:
- assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at x within s)" "bounded_bilinear h"
- shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)" proof-
+ assumes "(f has_vector_derivative f') (at x within s)"
+ assumes "(g has_vector_derivative g') (at x within s)"
+ assumes "bounded_bilinear h"
+ shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)"
+proof-
interpret bounded_bilinear h using assms by auto
show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
unfolding o_def has_vector_derivative_def
- using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed
+ using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib
+ by auto
+qed
lemma has_vector_derivative_bilinear_at:
- assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at x)" "bounded_bilinear h"
+ assumes "(f has_vector_derivative f') (at x)"
+ assumes "(g has_vector_derivative g') (at x)"
+ assumes "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
apply(rule has_vector_derivative_bilinear_within[where s=UNIV, unfolded within_UNIV]) using assms by auto
-lemma has_vector_derivative_at_within: "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
- unfolding has_vector_derivative_def apply(rule has_derivative_at_within) by auto
+lemma has_vector_derivative_at_within:
+ "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within s)"
+ unfolding has_vector_derivative_def
+ by (rule has_derivative_at_within) auto
lemma has_vector_derivative_transform_within:
- assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x within s)"
+ assumes "0 < d" and "x \<in> s" and "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'"
+ assumes "(f has_vector_derivative f') (at x within s)"
shows "(g has_vector_derivative f') (at x within s)"
- using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within)
+ using assms unfolding has_vector_derivative_def
+ by (rule has_derivative_transform_within)
lemma has_vector_derivative_transform_at:
- assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x)"
+ assumes "0 < d" and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"
+ assumes "(f has_vector_derivative f') (at x)"
shows "(g has_vector_derivative f') (at x)"
- using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_at)
+ using assms unfolding has_vector_derivative_def
+ by (rule has_derivative_transform_at)
lemma has_vector_derivative_transform_within_open:
- assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_vector_derivative f') (at x)"
+ assumes "open s" and "x \<in> s" and "\<forall>y\<in>s. f y = g y"
+ assumes "(f has_vector_derivative f') (at x)"
shows "(g has_vector_derivative f') (at x)"
- using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_within_open)
+ using assms unfolding has_vector_derivative_def
+ by (rule has_derivative_transform_within_open)
lemma vector_diff_chain_at:
- assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at (f x))"
+ assumes "(f has_vector_derivative f') (at x)"
+ assumes "(g has_vector_derivative g') (at (f x))"
shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
- using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
+ using assms(2) unfolding has_vector_derivative_def apply-
+ apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
unfolding o_def scaleR.scaleR_left by auto
lemma vector_diff_chain_within:
- assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at (f x) within f ` s)"
+ assumes "(f has_vector_derivative f') (at x within s)"
+ assumes "(g has_vector_derivative g') (at (f x) within f ` s)"
shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
- using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
+ using assms(2) unfolding has_vector_derivative_def apply-
+ apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
unfolding o_def scaleR.scaleR_left by auto
end
--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy Wed Aug 10 21:24:26 2011 +0200
@@ -1,1761 +1,107 @@
(* Title: HOL/Multivariate_Analysis/Euclidean_Space.thy
- Author: Amine Chaieb, University of Cambridge
+ Author: Johannes Hölzl, TU München
+ Author: Brian Huffman, Portland State University
*)
-header {* (Real) Vectors in Euclidean space, and elementary linear algebra.*}
+header {* Finite-Dimensional Inner Product Spaces *}
theory Euclidean_Space
imports
Complex_Main
- "~~/src/HOL/Library/Infinite_Set"
"~~/src/HOL/Library/Inner_Product"
- L2_Norm
- "~~/src/HOL/Library/Convex"
-uses
- "~~/src/HOL/Library/positivstellensatz.ML" (* FIXME duplicate use!? *)
- ("normarith.ML")
-begin
-
-lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
- by auto
-
-notation inner (infix "\<bullet>" 70)
-
-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 "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
- then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
- {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
- let ?d' = "min (d/2) ((l - a)/2)"
- have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
- by (simp add: min_max.less_infI2)
- then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
- 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 "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
- then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
- 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. *}
-
-(* FIXME: same as norm_scaleR
-lemma norm_mul[simp]: "norm(a *\<^sub>R 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: setL2_def power2_eq_square)
-
-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 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_left_mono)
- 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)
-*} "prove simple linear statements about vector norms"
-
-
-text{* Hence more metric properties. *}
-
-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_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_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_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_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 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)
-
-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. *}
-
-context real_inner
+ "~~/src/HOL/Library/Product_Vector"
begin
-definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
-
-lemma orthogonal_clauses:
- "orthogonal a 0"
- "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
- "orthogonal a x \<Longrightarrow> orthogonal a (-x)"
- "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
- "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
- "orthogonal 0 a"
- "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
- "orthogonal x a \<Longrightarrow> orthogonal (-x) a"
- "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
- "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
- unfolding orthogonal_def inner_simps inner_add_left inner_add_right inner_diff_left inner_diff_right (*FIXME*) by auto
-
-end
-
-lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
- by (simp add: orthogonal_def inner_commute)
-
-subsection{* Linear functions. *}
-
-definition
- linear :: "('a::real_vector \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" where
- "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
-
-lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
- shows "linear f" using assms unfolding linear_def by auto
-
-lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
- by (simp add: linear_def algebra_simps)
-
-lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
- by (simp add: linear_def)
-
-lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
- by (simp add: linear_def algebra_simps)
-
-lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
- by (simp add: linear_def algebra_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)" by (simp add: linear_def)
-
-lemma linear_compose_setsum:
- assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
- shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
- using lS
- apply (induct rule: finite_induct[OF fS])
- by (auto simp add: linear_zero intro: linear_compose_add)
-
-lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
- unfolding linear_def
- apply clarsimp
- apply (erule allE[where x="0::'a"])
- apply simp
- done
-
-lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x" by (simp add: linear_def)
-
-lemma linear_neg: "linear f ==> f (-x) = - f x"
- using linear_cmul [where c="-1"] by simp
-
-lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
-
-lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
- by (simp add: diff_minus linear_add linear_neg)
-
-lemma linear_setsum:
- 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:
- assumes lf: "linear f" and fS: "finite S"
- shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
- using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def]
- linear_cmul[OF lf] by simp
-
-lemma linear_injective_0:
- assumes lf: "linear f"
- 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
-
-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 *\<^sub>R x) y = c *\<^sub>R (h x y)"
- by (simp add: bilinear_def linear_def)
-
-lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
- by (simp add: bilinear_def linear_def)
-
-lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
- by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
-
-lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
- by (simp only: scaleR_minus1_left [symmetric] 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:
- 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:
- 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) z = h x z - h y z"
- by (simp add: diff_minus bilinear_ladd bilinear_lneg)
-
-lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
- by (simp add: diff_minus bilinear_radd bilinear_rneg)
-
-lemma bilinear_setsum:
- 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
-
-subsection{* Adjoints. *}
-
-definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
-
-lemma adjoint_unique:
- assumes "\<forall>x y. inner (f x) y = inner x (g y)"
- shows "adjoint f = g"
-unfolding adjoint_def
-proof (rule some_equality)
- show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
-next
- fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
- hence "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
- hence "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
- hence "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
- hence "\<forall>y. h y = g y" by simp
- thus "h = g" by (simp add: ext)
-qed
-
-lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
-
-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 assms apply auto
- apply (erule_tac x="n" in allE)
- apply (erule_tac x="n" in allE)
- apply auto
- done
-
-
-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
-
-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
-
-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
-
-subsection {* 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 (in real_vector)
- subspace :: "'a set \<Rightarrow> bool" where
- "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 *\<^sub>R x \<in>S )"
-
-definition (in real_vector) "span S = (subspace hull S)"
-definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
-abbreviation (in real_vector) "independent s == ~(dependent s)"
-
-text {* Closure properties of subspaces. *}
-
-lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
-
-lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
- by (metis subspace_def)
-
-lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
- by (metis subspace_def)
-
-lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
- by (metis scaleR_minus1_left subspace_mul)
-
-lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
- by (metis diff_minus subspace_add subspace_neg)
-
-lemma (in real_vector) 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" 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 *\<^sub>R x" in bexI, auto)
- done
-
-lemma subspace_linear_preimage: "linear f ==> 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}"
- by (simp add: subspace_def)
-
-lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
- by (simp add: subspace_def)
-
-lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
- by (metis span_def hull_mono)
-
-lemma (in real_vector) 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 (in real_vector) 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 *\<^sub>R x \<in> span S"
- by (metis span_def hull_subset subset_eq)
- (metis subspace_span subspace_def)+
-
-lemma (in real_vector) 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[simp]: "span {} = {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 (in real_vector) independent_empty[intro]: "independent {}"
- by (simp add: dependent_def)
-
-lemma dependent_single[simp]:
- "dependent {x} \<longleftrightarrow> x = 0"
- unfolding dependent_def by auto
-
-lemma (in real_vector) 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 (in real_vector) 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 (in real_vector) 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 (in real_vector) span_induct_alt_help for S:: "'a \<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 *\<^sub>R x + z)"
-
-lemma span_induct_alt':
- assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" shows "\<forall>x \<in> span S. h x"
-proof-
- {fix x:: "'a" 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"
+subsection {* Type class of Euclidean spaces *}
- 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 *\<^sub>R x)"
- apply (induct rule: span_induct_alt_help.induct)
- apply (simp add: span_induct_alt_help_0)
- apply (simp add: scaleR_right_distrib)
- 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" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
- shows "h x"
-using span_induct_alt'[of h S] h0 hS x by blast
-
-text {* Individual closure properties. *}
-
-lemma span_span: "span (span A) = span A"
- unfolding span_def hull_hull ..
-
-lemma (in real_vector) span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
-
-lemma (in real_vector) span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
-
-lemma span_inc: "S \<subseteq> span S"
- by (metis subset_eq span_superset)
-
-lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
- unfolding dependent_def apply(rule_tac x=0 in bexI)
- using assms span_0 by auto
-
-lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
- by (metis subspace_add subspace_span)
-
-lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
- by (metis subspace_span subspace_mul)
-
-lemma span_neg: "x \<in> span S ==> - x \<in> span S"
- by (metis subspace_neg subspace_span)
-
-lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
- by (metis subspace_span subspace_sub)
-
-lemma (in real_vector) 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 \<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)
-
-text {* Mapping under linear image. *}
-
-lemma span_linear_image: assumes lf: "linear f"
- 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_eqI)
- 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
-
-text {* The key breakdown property. *}
-
-lemma span_breakdown:
- assumes bS: "b \<in> S" and aS: "a \<in> span S"
- shows "\<exists>k. a - k *\<^sub>R 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) *\<^sub>R b = (x - k*\<^sub>R b) + (y - ka *\<^sub>R b)")
- apply (simp only: )
- apply (rule span_add[unfolded mem_def])
- apply assumption+
- apply (simp add: algebra_simps)
- apply (clarsimp simp add: mem_def)
- apply (rule_tac x= "c*k" in exI)
- apply (subgoal_tac "c *\<^sub>R x - (c * k) *\<^sub>R b = c*\<^sub>R (x - k*\<^sub>R b)")
- apply (simp only: )
- apply (rule span_mul[unfolded mem_def])
- apply assumption
- by (simp add: algebra_simps)
- ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
-qed
-
-lemma span_breakdown_eq:
- "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R 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 *\<^sub>R a \<in> span S"
- have eq: "x = (x - k *\<^sub>R a) + k *\<^sub>R a" by simp
- have "(x - k *\<^sub>R a) + k *\<^sub>R 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
-
-text {* Hence some "reversal" results. *}
-
-lemma in_span_insert:
- assumes a: "a \<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*\<^sub>R 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) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
- from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
- by (simp add: algebra_simps)
- from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})"
- by (rule span_mul)
- hence th: "(1/k) *\<^sub>R 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 \<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
-
-text {* Transitivity property. *}
-
-lemma span_trans:
- assumes x: "x \<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*\<^sub>R x \<in> span (S - {x})" by auto
- have eq: "y = (y - k *\<^sub>R x) + k *\<^sub>R x" by simp
- 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
-
-lemma span_insert_0[simp]: "span (insert 0 S) = span S"
- using span_mono[of S "insert 0 S"] by (auto intro: span_trans span_0)
-
-text {* An explicit expansion is sometimes needed. *}
-
-lemma span_explicit:
- "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R 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 *\<^sub>R 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 *\<^sub>R 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 *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R 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 *\<^sub>R v) + c *\<^sub>R x"
- apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
- by (simp add: algebra_simps)
- also have "\<dots> = c*\<^sub>R x + y"
- by (simp add: add_commute u)
- finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
- then have "?Q ?S ?u (c*\<^sub>R 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) *\<^sub>R v) = y"
- unfolding u[symmetric]
- apply (rule setsum_cong2)
- using xS by auto
- have "?Q ?S ?u (c*\<^sub>R 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*\<^sub>R x + y)"
- by (cases "x \<in> S", simp, simp)
- then show "?h (c*\<^sub>R 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\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R 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 *\<^sub>R 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 *\<^sub>R v) ?S = 0"
- using fS aS
- apply (simp add: 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 *\<^sub>R 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 *\<^sub>R v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
- using fS vS uv
- by (simp add: setsum_diff1 divide_inverse field_simps)
- also have "\<dots> = ?a"
- unfolding scaleR_right.setsum [symmetric] u
- using uv by simp
- finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
- with th0 have ?lhs
- unfolding dependent_def span_explicit
- apply -
- apply (rule bexI[where x= "?a"])
- apply (simp_all del: scaleR_minus_left)
- apply (rule exI[where x= "?S"])
- by (auto simp del: scaleR_minus_left)}
- ultimately show ?thesis by blast
-qed
-
-
-lemma span_finite:
- assumes fS: "finite S"
- shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R 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 *\<^sub>R v) S' = y" unfolding span_explicit by blast
- let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
- have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
- using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
- hence "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
- hence "y \<in> ?rhs" by auto}
- moreover
- {fix y u assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
- then have "y \<in> span S" using fS unfolding span_explicit by auto}
- ultimately show ?thesis by blast
-qed
-
-lemma Int_Un_cancel: "(A \<union> B) \<inter> A = A" "(A \<union> B) \<inter> B = B" by auto
-
-lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
-proof safe
- fix x assume "x \<in> span (A \<union> B)"
- then obtain S u where S: "finite S" "S \<subseteq> A \<union> B" and x: "x = (\<Sum>v\<in>S. u v *\<^sub>R v)"
- unfolding span_explicit by auto
-
- let ?Sa = "\<Sum>v\<in>S\<inter>A. u v *\<^sub>R v"
- let ?Sb = "(\<Sum>v\<in>S\<inter>(B - A). u v *\<^sub>R v)"
- show "x \<in> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
- proof
- show "x = (case (?Sa, ?Sb) of (a, b) \<Rightarrow> a + b)"
- unfolding x using S
- by (simp, subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
-
- from S have "?Sa \<in> span A" unfolding span_explicit
- by (auto intro!: exI[of _ "S \<inter> A"])
- moreover from S have "?Sb \<in> span B" unfolding span_explicit
- by (auto intro!: exI[of _ "S \<inter> (B - A)"])
- ultimately show "(?Sa, ?Sb) \<in> span A \<times> span B" by simp
- qed
-next
- fix a b assume "a \<in> span A" and "b \<in> span B"
- then obtain Sa ua Sb ub where span:
- "finite Sa" "Sa \<subseteq> A" "a = (\<Sum>v\<in>Sa. ua v *\<^sub>R v)"
- "finite Sb" "Sb \<subseteq> B" "b = (\<Sum>v\<in>Sb. ub v *\<^sub>R v)"
- unfolding span_explicit by auto
- let "?u v" = "(if v \<in> Sa then ua v else 0) + (if v \<in> Sb then ub v else 0)"
- from span have "finite (Sa \<union> Sb)" "Sa \<union> Sb \<subseteq> A \<union> B"
- and "a + b = (\<Sum>v\<in>(Sa\<union>Sb). ?u v *\<^sub>R v)"
- unfolding setsum_addf scaleR_left_distrib
- by (auto simp add: if_distrib cond_application_beta setsum_cases Int_Un_cancel)
- thus "a + b \<in> span (A \<union> B)"
- unfolding span_explicit by (auto intro!: exI[of _ ?u])
-qed
-
-text {* This is useful for building a basis step-by-step. *}
-
-lemma independent_insert:
- "independent(insert a 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
-
-text {* 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 spanning_subset_independent:
- assumes BA: "B \<subseteq> A" and iA: "independent A"
- 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
-
-text {* The general case of the Exchange Lemma, the key to what follows. *}
-
-lemma exchange_lemma:
- assumes f:"finite t" 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
-
-text {* This implies corresponding size bounds. *}
-
-lemma independent_span_bound:
- assumes f: "finite t" and i: "independent s" 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
-
-subsection{* Euclidean Spaces as Typeclass*}
-
-class real_basis = real_vector +
+class euclidean_space = real_inner +
+ fixes dimension :: "'a itself \<Rightarrow> nat"
fixes basis :: "nat \<Rightarrow> 'a"
- assumes span_basis: "span (range basis) = UNIV"
- assumes dimension_exists: "\<exists>d>0.
- basis ` {d..} = {0} \<and>
- independent (basis ` {..<d}) \<and>
- inj_on basis {..<d}"
-
-definition (in real_basis) dimension :: "'a itself \<Rightarrow> nat" where
- "dimension x =
- (THE d. basis ` {d..} = {0} \<and> independent (basis ` {..<d}) \<and> inj_on basis {..<d})"
+ 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 real_basis) dimensionI:
- assumes "\<And>d. \<lbrakk> 0 < d; basis ` {d..} = {0}; independent (basis ` {..<d});
- inj_on basis {..<d} \<rbrakk> \<Longrightarrow> P d"
- shows "P DIM('a)"
-proof -
- obtain d where "0 < d" and d: "basis ` {d..} = {0} \<and>
- independent (basis ` {..<d}) \<and> inj_on basis {..<d}" (is "?P d")
- using dimension_exists by auto
- show ?thesis unfolding dimension_def
- proof (rule theI2)
- fix d' assume "?P d'"
- with d have "basis d' = 0" "basis d = 0" and
- "d < d' \<Longrightarrow> basis d \<noteq> 0"
- "d' < d \<Longrightarrow> basis d' \<noteq> 0"
- using dependent_0 by auto
- thus "d' = d" by (cases rule: linorder_cases) auto
- moreover have "P d" using assms[of d] `0 < d` d by auto
- ultimately show "P d'" by simp
- next show "?P d" using `?P d` .
- qed
-qed
-
-lemma (in real_basis) dimension_eq:
- assumes "\<And>i. i < d \<Longrightarrow> basis i \<noteq> 0"
- assumes "\<And>i. d \<le> i \<Longrightarrow> basis i = 0"
- shows "DIM('a) = d"
-proof (rule dimensionI)
- let ?b = "basis :: nat \<Rightarrow> 'a"
- fix d' assume *: "?b ` {d'..} = {0}" "independent (?b ` {..<d'})"
- show "d' = d"
- proof (cases rule: linorder_cases)
- assume "d' < d" hence "basis d' \<noteq> 0" using assms by auto
- with * show ?thesis by auto
- next
- assume "d < d'" hence "basis d \<noteq> 0" using * dependent_0 by auto
- with assms(2) `d < d'` show ?thesis by auto
- qed
-qed
-
-lemma (in real_basis)
- shows basis_finite: "basis ` {DIM('a)..} = {0}"
- and independent_basis: "independent (basis ` {..<DIM('a)})"
- and DIM_positive[intro]: "(DIM('a) :: nat) > 0"
- and basis_inj[simp, intro]: "inj_on basis {..<DIM('a)}"
- by (auto intro!: dimensionI)
-
-lemma (in real_basis) basis_eq_0_iff: "basis j = 0 \<longleftrightarrow> DIM('a) \<le> j"
-proof
- show "DIM('a) \<le> j \<Longrightarrow> basis j = 0" using basis_finite by auto
-next
- have "j < DIM('a) \<Longrightarrow> basis j \<noteq> 0"
- using independent_basis by (auto intro!: dependent_0)
- thus "basis j = 0 \<Longrightarrow> DIM('a) \<le> j" unfolding not_le[symmetric] by blast
-qed
-
-lemma (in real_basis) range_basis:
- "range basis = insert 0 (basis ` {..<DIM('a)})"
-proof -
- have *: "UNIV = {..<DIM('a)} \<union> {DIM('a)..}" by auto
- show ?thesis unfolding * image_Un basis_finite by auto
-qed
-
-lemma (in real_basis) range_basis_finite[intro]:
- "finite (range basis)"
- unfolding range_basis by auto
-
-lemma (in real_basis) basis_neq_0[intro]:
- assumes "i<DIM('a)" shows "(basis i) \<noteq> 0"
-proof(rule ccontr) assume "\<not> basis i \<noteq> (0::'a)"
- hence "(0::'a) \<in> basis ` {..<DIM('a)}" using assms by auto
- from dependent_0[OF this] show False using independent_basis by auto
-qed
-
-lemma (in real_basis) basis_zero[simp]:
- assumes"i \<ge> DIM('a)" shows "basis i = 0"
-proof-
- have "(basis i::'a) \<in> basis ` {DIM('a)..}" using assms by auto
- thus ?thesis unfolding basis_finite by fastsimp
-qed
-
-lemma basis_representation:
- "\<exists>u. x = (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R (v\<Colon>'a\<Colon>real_basis))"
-proof -
- have "x\<in>UNIV" by auto from this[unfolded span_basis[THEN sym]]
- have "\<exists>u. (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R v) = x"
- unfolding range_basis span_insert_0 apply(subst (asm) span_finite) by auto
- thus ?thesis by fastsimp
-qed
-
-lemma span_basis'[simp]:"span ((basis::nat=>'a) ` {..<DIM('a::real_basis)}) = UNIV"
- apply(subst span_basis[symmetric]) unfolding range_basis by auto
-
-lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::real_basis)}) = DIM('a)"
- apply(subst card_image) using basis_inj by auto
-
-lemma in_span_basis: "(x::'a::real_basis) \<in> span (basis ` {..<DIM('a)})"
- unfolding span_basis' ..
-
-lemma independent_eq_inj_on:
- fixes D :: nat and f :: "nat \<Rightarrow> 'c::real_vector" assumes *: "inj_on f {..<D}"
- shows "independent (f ` {..<D}) \<longleftrightarrow> (\<forall>a u. a < D \<longrightarrow> (\<Sum>i\<in>{..<D}-{a}. u (f i) *\<^sub>R f i) \<noteq> f a)"
-proof -
- from * have eq: "\<And>i. i < D \<Longrightarrow> f ` {..<D} - {f i} = f`({..<D} - {i})"
- and inj: "\<And>i. inj_on f ({..<D} - {i})"
- by (auto simp: inj_on_def)
- have *: "\<And>i. finite (f ` {..<D} - {i})" by simp
- show ?thesis unfolding dependent_def span_finite[OF *]
- by (auto simp: eq setsum_reindex[OF inj])
-qed
-
-class real_basis_with_inner = real_inner + real_basis
-begin
-
-definition euclidean_component (infixl "$$" 90) where
- "x $$ i = inner (basis i) x"
-
-definition Chi (binder "\<chi>\<chi> " 10) where
- "(\<chi>\<chi> i. f i) = (\<Sum>i<DIM('a). f i *\<^sub>R basis i)"
-
-lemma basis_at_neq_0[intro]:
- "i < DIM('a) \<Longrightarrow> basis i $$ i \<noteq> 0"
- unfolding euclidean_component_def by (auto intro!: basis_neq_0)
-
-lemma euclidean_component_ge[simp]:
- assumes "i \<ge> DIM('a)" shows "x $$ i = 0"
- unfolding euclidean_component_def basis_zero[OF assms] by auto
-
-lemma euclidean_scaleR:
- shows "(a *\<^sub>R x) $$ i = a * (x$$i)"
- unfolding euclidean_component_def by auto
-
-end
-
-interpretation euclidean_component: additive "\<lambda>x. euclidean_component x i"
-proof qed (simp add: euclidean_component_def inner_right.add)
-
-subsection{* Euclidean Spaces as Typeclass *}
-
-class euclidean_space = real_basis_with_inner +
- assumes basis_orthonormal: "\<forall>i<DIM('a). \<forall>j<DIM('a).
- inner (basis i) (basis j) = (if i = j then 1 else 0)"
-
lemma (in euclidean_space) dot_basis:
- "inner (basis i) (basis j) = (if i = j \<and> i<DIM('a) then 1 else 0)"
+ "inner (basis i) (basis j) = (if i = j \<and> i < DIM('a) then 1 else 0)"
proof (cases "(i < DIM('a) \<and> j < DIM('a))")
case False
- hence "basis i = 0 \<or> basis j = 0"
- using basis_finite by fastsimp
- hence "inner (basis i) (basis j) = 0" by (rule disjE) simp_all
+ hence "inner (basis i) (basis j) = 0" by auto
thus ?thesis using False by auto
next
case True thus ?thesis using basis_orthonormal by auto
qed
-lemma (in euclidean_space) basis_component[simp]:
+lemma (in euclidean_space) basis_eq_0_iff [simp]:
+ "basis i = 0 \<longleftrightarrow> DIM('a) \<le> i"
+proof -
+ have "inner (basis i) (basis i) = 0 \<longleftrightarrow> DIM('a) \<le> i"
+ by (simp add: dot_basis)
+ thus ?thesis by simp
+qed
+
+lemma (in euclidean_space) norm_basis [simp]:
+ "norm (basis i) = (if i < DIM('a) then 1 else 0)"
+ unfolding norm_eq_sqrt_inner dot_basis by simp
+
+lemma (in euclidean_space) basis_neq_0 [intro]:
+ assumes "i<DIM('a)" shows "(basis i) \<noteq> 0"
+ using assms by simp
+
+subsubsection {* Projecting components *}
+
+definition (in euclidean_space) euclidean_component (infixl "$$" 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$$i)"
+ unfolding euclidean_component_def by auto
+
lemmas euclidean_simps =
euclidean_component.add
euclidean_component.diff
@@ -1765,34 +111,22 @@
basis_component
lemma euclidean_representation:
- "(x\<Colon>'a\<Colon>euclidean_space) = (\<Sum>i\<in>{..<DIM('a)}. (x$$i) *\<^sub>R basis i)"
-proof-
- from basis_representation[of x] guess u ..
- hence *:"x = (\<Sum>i\<in>{..<DIM('a)}. u (basis i) *\<^sub>R (basis i))"
- by (simp add: setsum_reindex)
- show ?thesis apply(subst *)
- proof(safe intro!: setsum_cong2)
- fix i assume i: "i < DIM('a)"
- hence "x$$i = (\<Sum>x<DIM('a). (if i = x then u (basis x) else 0))"
- by (auto simp: euclidean_simps * intro!: setsum_cong)
- also have "... = u (basis i)" using i by (auto simp: setsum_cases)
- finally show "u (basis i) *\<^sub>R basis i = x $$ i *\<^sub>R basis i" by simp
- qed
-qed
+ fixes x :: "'a::euclidean_space"
+ shows "x = (\<Sum>i<DIM('a). (x$$i) *\<^sub>R basis i)"
+ apply (rule euclidean_eqI)
+ apply (simp add: euclidean_component.setsum euclidean_component.scaleR)
+ apply (simp add: if_distrib setsum_delta cong: if_cong)
+ done
-lemma euclidean_eq:
- fixes x y :: "'a\<Colon>euclidean_space"
- shows "x = y \<longleftrightarrow> (\<forall>i<DIM('a). x$$i = y$$i)" (is "?l = ?r")
-proof safe
- assume "\<forall>i<DIM('a). x $$ i = y $$ i"
- thus "x = y"
- by (subst (1 2) euclidean_representation) auto
-qed
+subsubsection {* Binder notation for vectors *}
+
+definition (in euclidean_space) Chi (binder "\<chi>\<chi> " 10) where
+ "(\<chi>\<chi> i. f i) = (\<Sum>i<DIM('a). f i *\<^sub>R basis i)"
-lemma euclidean_lambda_beta[simp]:
+lemma euclidean_lambda_beta [simp]:
"((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ j = (if j < DIM('a) then f j else 0)"
- by (auto simp: euclidean_simps Chi_def if_distrib setsum_cases
- intro!: setsum_cong)
+ 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) $$ j = f j"
@@ -1803,1716 +137,119 @@
lemma euclidean_beta_reduce[simp]:
"(\<chi>\<chi> i. x $$ i) = (x::'a::euclidean_space)"
- by (subst euclidean_eq) (simp add: euclidean_lambda_beta)
+ by (simp add: euclidean_eq)
lemma euclidean_lambda_beta_0[simp]:
"((\<chi>\<chi> i. f i)::'a::euclidean_space) $$ 0 = f 0"
by (simp add: DIM_positive)
lemma euclidean_inner:
- "x \<bullet> (y::'a) = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) \<bullet> (y $$ i))"
-proof -
- have "x \<bullet> y = (\<Sum>i<DIM('a). x $$ i *\<^sub>R basis i) \<bullet>
- (\<Sum>i<DIM('a). y $$ i *\<^sub>R (basis i :: 'a))"
- by (subst (1 2) euclidean_representation) simp
- also have "\<dots> = (\<Sum>i<DIM('a::euclidean_space). (x $$ i) \<bullet> (y $$ i))"
- unfolding inner_left.setsum inner_right.setsum
- by (auto simp add: dot_basis if_distrib setsum_cases intro!: setsum_cong)
- finally show ?thesis .
-qed
-
-lemma norm_basis[simp]:"norm (basis i::'a::euclidean_space) = (if i<DIM('a) then 1 else 0)"
- unfolding norm_eq_sqrt_inner dot_basis by auto
+ "inner x (y::'a) = (\<Sum>i<DIM('a::euclidean_space). (x $$ 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
- apply(rule order_trans[OF real_inner_class.Cauchy_Schwarz_ineq2]) by auto
-
-lemma norm_bound_component_le: "norm (x::'a::euclidean_space) \<le> e \<Longrightarrow> \<bar>x$$i\<bar> <= e"
- by (metis component_le_norm order_trans)
-
-lemma norm_bound_component_lt: "norm (x::'a::euclidean_space) < e \<Longrightarrow> \<bar>x$$i\<bar> < e"
- by (metis component_le_norm basic_trans_rules(21))
-
-lemma norm_le_l1: "norm (x::'a::euclidean_space) \<le> (\<Sum>i<DIM('a). \<bar>x $$ i\<bar>)"
- apply (subst euclidean_representation[of x])
- apply (rule order_trans[OF setsum_norm])
- by (auto intro!: setsum_mono)
+ by (rule order_trans [OF Cauchy_Schwarz_ineq2]) simp
-lemma setsum_norm_allsubsets_bound:
- fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
- assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
- shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real DIM('n) * e"
-proof-
- let ?d = "real DIM('n)"
- let ?nf = "\<lambda>x. norm (f x)"
- let ?U = "{..<DIM('n)}"
- have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $$ 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 $$ i\<bar>) ?Pp \<le> e"
- using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i] fPs[OF PpP]
- unfolding euclidean_component.setsum by(auto intro: abs_le_D1)
- have Pne: "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn \<le> e"
- using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i] fPs[OF PnP]
- unfolding euclidean_component.setsum euclidean_component.minus
- 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
+subsection {* Class instances *}
-lemma choice_iff': "(\<forall>x<d. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x<d. P x (f x))" by metis
-
-lemma lambda_skolem': "(\<forall>i<DIM('a::euclidean_space). \<exists>x. P i x) \<longleftrightarrow>
- (\<exists>x::'a. \<forall>i<DIM('a). P i (x$$i))" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
- let ?S = "{..<DIM('a)}"
- {assume H: "?rhs"
- then have ?lhs by auto}
- moreover
- {assume H: "?lhs"
- then obtain f where f:"\<forall>i<DIM('a). P i (f i)" unfolding choice_iff' by metis
- let ?x = "(\<chi>\<chi> i. (f i)) :: 'a"
- {fix i assume i:"i<DIM('a)"
- with f have "P i (f i)" by metis
- then have "P i (?x$$i)" using i by auto
- }
- hence "\<forall>i<DIM('a). P i (?x$$i)" by metis
- hence ?rhs by metis }
- ultimately show ?thesis by metis
-qed
-
-subsection {* An ordering on euclidean spaces that will allow us to talk about intervals *}
-
-class ordered_euclidean_space = ord + euclidean_space +
- assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i \<le> y $$ i)"
- and eucl_less: "x < y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i < y $$ i)"
-
-lemma eucl_less_not_refl[simp, intro!]: "\<not> x < (x::'a::ordered_euclidean_space)"
- unfolding eucl_less[where 'a='a] by auto
+subsubsection {* Type @{typ real} *}
-lemma euclidean_trans[trans]:
- fixes x y z :: "'a::ordered_euclidean_space"
- shows "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
- and "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
- and "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
- by (force simp: eucl_less[where 'a='a] eucl_le[where 'a='a])+
-
-subsection {* Linearity and Bilinearity continued *}
-
-lemma linear_bounded:
- fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes lf: "linear f"
- shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
-proof-
- let ?S = "{..<DIM('a)}"
- let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
- have fS: "finite ?S" by simp
- {fix x:: "'a"
- let ?g = "(\<lambda> i. (x$$i) *\<^sub>R (basis i) :: 'a)"
- have "norm (f x) = norm (f (setsum (\<lambda>i. (x$$i) *\<^sub>R (basis i)) ?S))"
- apply(subst euclidean_representation[of x]) ..
- also have "\<dots> = norm (setsum (\<lambda> i. (x$$i) *\<^sub>R 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$$i) *\<^sub>R f (basis i))?S)" .
- {fix i assume i: "i \<in> ?S"
- from component_le_norm[of x i]
- have "norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x"
- unfolding norm_scaleR
- apply (simp only: mult_commute)
- apply (rule mult_mono)
- by (auto simp add: field_simps) }
- then have th: "\<forall>i\<in> ?S. norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x" by metis
- from setsum_norm_le[OF fS, of "\<lambda>i. (x$$i) *\<^sub>R (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
+instantiation real :: euclidean_space
+begin
-lemma linear_bounded_pos:
- fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
- 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 "((\<chi>\<chi> i. 1)::'a) \<noteq> 0" unfolding euclidean_eq[where 'a='a]
- by(auto intro!:exI[where x=0] simp add:euclidean_component.zero)
- hence "norm ((\<chi>\<chi> i. 1)::'a) > 0" by auto
- with C have "B * norm ((\<chi>\<chi> i. 1)::'a) < 0"
- by (simp add: mult_less_0_iff)
- with B[rule_format, of "(\<chi>\<chi> i. 1)::'a"] norm_ge_zero[of "f ((\<chi>\<chi> i. 1)::'a)"] have False by simp
- }
- then have Bp: "B \<ge> 0" by (metis not_leE)
- {fix x::"'a"
- 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 linear_conv_bounded_linear:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
- 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
- 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"
- by (simp add: f.add f.scaleR linear_def)
-qed
+definition
+ "dimension (t::real itself) = 1"
-lemma bounded_linearI': fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
- shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym]
- by(rule linearI[OF assms])
-
+definition [simp]:
+ "basis i = (if i = 0 then 1 else (0::real))"
-lemma bilinear_bounded:
- fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
- assumes bh: "bilinear h"
- shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
-proof-
- let ?M = "{..<DIM('m)}"
- let ?N = "{..<DIM('n)}"
- 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:: "'m" and y :: "'n"
- have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$$i) *\<^sub>R basis i) ?M) (setsum (\<lambda>i. (y$$i) *\<^sub>R basis i) ?N))"
- apply(subst euclidean_representation[where 'a='m])
- apply(subst euclidean_representation[where 'a='n]) ..
- also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$$i) *\<^sub>R basis i) ((y$$j) *\<^sub>R 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 simp del: scaleR_scaleR)
- 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:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
- 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::'a and y::'b
- 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 DIM_real [simp]: "DIM(real) = 1"
+ by (rule dimension_real_def)
-lemma bilinear_conv_bounded_bilinear:
- fixes h :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
- 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
- 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
- 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 {* We continue. *}
-
-lemma independent_bound:
- fixes S:: "('a::euclidean_space) set"
- shows "independent S \<Longrightarrow> finite S \<and> card S <= DIM('a::euclidean_space)"
- using independent_span_bound[of "(basis::nat=>'a) ` {..<DIM('a)}" S] by auto
-
-lemma dependent_biggerset: "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
- by (metis independent_bound not_less)
-
-text {* Hence we can create a maximal independent subset. *}
+instance
+ by default simp+
-lemma maximal_independent_subset_extend:
- assumes sv: "(S::('a::euclidean_space) 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 "DIM('a) - 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 = "DIM('a)"
- {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
+end
-lemma maximal_independent_subset:
- "\<exists>(B:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
- by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"] empty_subsetI independent_empty)
-
-
-text {* 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 :: ('a::euclidean_space) 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
-
-text {* Consequences of independence or spanning for cardinality. *}
-
-lemma independent_card_le_dim:
- assumes "(B::('a::euclidean_space) 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::('a::euclidean_space) 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:: ('a::euclidean_space) 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::('a::euclidean_space) 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)
-
-text {* More lemmas about dimension. *}
+subsubsection {* Type @{typ complex} *}
-lemma dim_UNIV: "dim (UNIV :: ('a::euclidean_space) set) = DIM('a)"
- apply (rule dim_unique[of "(basis::nat=>'a) ` {..<DIM('a)}"])
- using independent_basis by auto
-
-lemma dim_subset:
- "(S:: ('a::euclidean_space) 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:: ('a::euclidean_space) set) \<le> DIM('a)"
- by (metis dim_subset subset_UNIV dim_UNIV)
-
-text {* Converses to those. *}
-
-lemma card_ge_dim_independent:
- assumes BV:"(B::('a::euclidean_space) 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
+instantiation complex :: euclidean_space
+begin
-lemma card_le_dim_spanning:
- assumes BV: "(B:: ('a::euclidean_space) 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:: ('a::euclidean_space) 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)
-
-text {* More general size bound lemmas. *}
-
-lemma independent_bound_general:
- "independent (S:: ('a::euclidean_space) 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:: ('a::euclidean_space) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
- using independent_bound_general[of S] by (metis linorder_not_le)
+definition
+ "dimension (t::complex itself) = 2"
-lemma dim_span: "dim (span (S:: ('a::euclidean_space) 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:: ('a::euclidean_space) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
- by (metis dim_span dim_subset)
-
-lemma span_eq_dim: "span (S:: ('a::euclidean_space) set) = span T ==> dim S = dim T"
- by (metis dim_span)
-
-lemma spans_image:
- assumes lf: "linear f" 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 :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S)"
-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
+definition
+ "basis i = (if i = 0 then 1 else if i = 1 then ii else 0)"
-text {* Relation between bases and injectivity/surjectivity of map. *}
-
-lemma spanning_surjective_image:
- assumes us: "UNIV \<subseteq> span S"
- 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 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)
-lemma independent_injective_image:
- assumes iS: "independent S" 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
-
-text {* 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::'a::euclidean_space) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
- unfolding inner_simps by auto
-
-lemma basis_orthogonal:
- fixes B :: "('a::euclidean_space) 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)
+instance proof
+ show "0 < DIM(complex)"
+ unfolding dimension_complex_def by simp
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)) *\<^sub>R 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::'a) b c. a - (b - c) = c + (a - b)" by (simp add: field_simps)
- have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
- apply (simp only: scaleR_right_diff_distrib 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 set_eq_iff 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)
- 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)
- 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 :: "('a::euclidean_space) 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)
-
-text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
-
-lemma span_not_univ_orthogonal: fixes S::"('a::euclidean_space) set"
- assumes sU: "span S \<noteq> UNIV"
- shows "\<exists>(a::'a). 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)) *\<^sub>R b) B"
- have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R 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)
+ fix i :: nat
+ assume "DIM(complex) \<le> i" thus "basis i = (0::complex)"
+ unfolding dimension_complex_def basis_complex_def by simp
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
- apply (clarsimp simp add: inner_simps 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 ::('a::euclidean_space) 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: fixes S::"('a::euclidean_space) set"
- assumes d: "dim S < DIM('a)"
- shows "\<exists>(a::'a). a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
-proof-
- {assume "span S = UNIV"
- hence "dim (span S) = dim (UNIV :: ('a) set)" by simp
- hence "dim S = DIM('a)" 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
-
-text {* 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 = 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)
+ 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
- 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*\<^sub>R a \<in> span (b -{a})" by blast
- have "f (x - k*\<^sub>R 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*\<^sub>R f a \<in> span (f ` b)"
- by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
- hence th: "-k *\<^sub>R f a \<in> span (f ` b)"
- using "2.prems"(5) by simp
- {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
- 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(8), 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" .
+ 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
-text {* We can extend a linear mapping from basis. *}
-
-lemma linear_independent_extend_lemma:
- fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
- assumes fi: "finite B" and ib: "independent B"
- shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y)
- \<and> (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R 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 *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast
- let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
- {fix z assume z: "z \<in> span (insert a b)"
- have th0: "z - ?h z *\<^sub>R a \<in> span b"
- apply (rule someI_ex)
- unfolding span_breakdown_eq[symmetric]
- using z .
- {fix k assume k: "z - k *\<^sub>R a \<in> span b"
- have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
- by (simp add: field_simps scaleR_left_distrib [symmetric])
- from span_sub[OF th0 k]
- have khz: "(k - ?h z) *\<^sub>R 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
- 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 *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)" by blast}
- note h = this
- let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R 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) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>R a)"
- by (simp add: algebra_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: scaleR_left_distrib)}
- moreover
- {fix x:: "'a" and c:: real assume x: "x \<in> span (insert a b)"
- have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
- by (simp add: algebra_simps)
- have hc: "?h (c *\<^sub>R 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 *\<^sub>R x) = c*\<^sub>R ?g x"
- unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
- by (simp add: algebra_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:: ('a::euclidean_space) 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*\<^sub>R x) = c *\<^sub>R 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
-
-text {* 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:: ('a::euclidean_space) set)"
- and t: "subspace (T :: ('b::euclidean_space) 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
-
-text {* Linear functions are equal on a subspace if they are on a spanning set. *}
-
-lemma subspace_kernel:
- assumes lf: "linear f"
- 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"
-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"
- by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
-
-lemma linear_eq:
- assumes lf: "linear f" 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::euclidean_space \<Rightarrow> _)" and lg: "linear g"
- and fg: "\<forall>i<DIM('a::euclidean_space). f (basis i) = g(basis i)"
- shows "f = g"
-proof-
- let ?U = "{..<DIM('a)}"
- let ?I = "(basis::nat=>'a) ` {..<DIM('a)}"
- {fix x assume x: "x \<in> (UNIV :: 'a set)"
- from equalityD2[OF span_basis'[where 'a='a]]
- have IU: " (UNIV :: 'a set) \<subseteq> span ?I" by blast
- have "f x = g x" apply(rule linear_eq[OF lf lg IU,rule_format]) using fg x by auto }
- then show ?thesis by (auto intro: ext)
-qed
-
-text {* Similar results for bilinear functions. *}
-
-lemma bilinear_eq:
- assumes bf: "bilinear f"
- 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: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
- assumes bf: "bilinear f"
- and bg: "bilinear g"
- and fg: "\<forall>i<DIM('a). \<forall>j<DIM('b). f (basis i) (basis j) = g (basis i) (basis j)"
- shows "f = g"
-proof-
- from fg have th: "\<forall>x \<in> (basis ` {..<DIM('a)}). \<forall>y\<in> (basis ` {..<DIM('b)}). f x y = g x y" by blast
- from bilinear_eq[OF bf bg equalityD2[OF span_basis'] equalityD2[OF span_basis'] th]
- show ?thesis by (blast intro: ext)
-qed
-
-text {* Detailed theorems about left and right invertibility in general case. *}
-
-lemma linear_injective_left_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
- assumes lf: "linear f" 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_basis, OF lf fi]
- obtain h:: "'b => 'a" where h: "linear h"
- " \<forall>x \<in> f ` basis ` {..<DIM('a)}. h x = inv f x" by blast
- from h(2)
- have th: "\<forall>i<DIM('a). (h \<circ> f) (basis i) = id (basis i)"
- using inv_o_cancel[OF fi, unfolded fun_eq_iff 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: fixes f::"'a::euclidean_space => 'b::euclidean_space"
- assumes lf: "linear f" and sf: "surj f"
- shows "\<exists>g. linear g \<and> f o g = id"
-proof-
- from linear_independent_extend[OF independent_basis[where 'a='b],of "inv f"]
- obtain h:: "'b \<Rightarrow> 'a" where
- h: "linear h" "\<forall> x\<in> basis ` {..<DIM('b)}. h x = inv f x" by blast
- from h(2)
- have th: "\<forall>i<DIM('b). (f o h) (basis i) = id (basis i)"
- using sf by(auto simp add: surj_iff_all)
- 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
-
-text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
-
-lemma linear_injective_imp_surjective: fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and fi: "inj f"
- shows "surj f"
-proof-
- let ?U = "UNIV :: 'a 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
-
-text {* 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: fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and sf: "surj f"
- shows "inj f"
-proof-
- let ?U = "UNIV :: 'a 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
- 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
-
-text {* 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: fun_eq_iff o_def id_def)
-
-lemma linear_injective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" 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: fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" 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)
-
-text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
-
-lemma linear_inverse_left: fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" and lf': "linear f'"
- shows "f o f' = id \<longleftrightarrow> f' o f = id"
-proof-
- {fix f f':: "'a => 'a"
- assume lf: "linear f" "linear f'" and f: "f o f' = id"
- from f have sf: "surj f"
- apply (auto simp add: o_def id_def surj_def)
- by metis
- from linear_surjective_isomorphism[OF lf(1) sf] lf f
- have "f' o f = id" unfolding fun_eq_iff o_def id_def
- by metis}
- then show ?thesis using lf lf' by metis
-qed
-
-text {* Moreover, a one-sided inverse is automatically linear. *}
-
-lemma left_inverse_linear: fixes f::"'a::euclidean_space => 'a::euclidean_space"
- assumes lf: "linear f" 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 fun_eq_iff)
- by metis
- from linear_injective_isomorphism[OF lf fi]
- obtain h:: "'a \<Rightarrow> 'a" 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 fun_eq_iff)
- by metis
- with h(1) show ?thesis by blast
-qed
-
-subsection {* Infinity norm *}
-
-definition "infnorm (x::'a::euclidean_space) = Sup {abs(x$$i) |i. i<DIM('a)}"
-
-lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
- by auto
-
-lemma infnorm_set_image:
- "{abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)} =
- (\<lambda>i. abs(x$$i)) ` {..<DIM('a)}" by blast
-
-lemma infnorm_set_lemma:
- shows "finite {abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)}"
- and "{abs(x$$i) |i. i<DIM('a::euclidean_space)} \<noteq> {}"
- unfolding infnorm_set_image
- by auto
-
-lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
- unfolding infnorm_def
- unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
- unfolding infnorm_set_image
- by auto
-
-lemma infnorm_triangle: "infnorm ((x::'a::euclidean_space) + 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
- have *:"\<And>i. i \<in> {..<DIM('a)} \<longleftrightarrow> i <DIM('a)" by auto
- show ?thesis
- unfolding infnorm_def unfolding Sup_finite_le_iff[ OF infnorm_set_lemma[where 'a='a]]
- 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[where 'a='a]]
- unfolding infnorm_set_image ball_simps bex_simps
- unfolding euclidean_simps by (metis th2)
-qed
-
-lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 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
- apply(subst (1) euclidean_eq) unfolding euclidean_component.zero
- by auto
- 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 by(auto simp add: euclidean_simps)
-
-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_minus[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::'a::euclidean_space)"
-proof(cases "i<DIM('a)")
- case False thus ?thesis using infnorm_pos_le by auto
-next case True
- let ?U = "{..<DIM('a)}"
- let ?S = "{\<bar>x$$i\<bar> |i. i<DIM('a)}"
- have fS: "finite ?S" unfolding image_Collect[symmetric]
- apply (rule finite_imageI) by simp
- have S0: "?S \<noteq> {}" by blast
- have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
- show ?thesis unfolding infnorm_def
- apply(subst Sup_finite_ge_iff) using Sup_finite_in[OF fS S0]
- using infnorm_set_image using True by auto
-qed
-
-lemma infnorm_mul_lemma: "infnorm(a *\<^sub>R 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 euclidean_scaleR abs_mult
- using component_le_infnorm[of x] by(auto intro: mult_mono)
-
-lemma infnorm_mul: "infnorm(a *\<^sub>R 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) *\<^sub>R (a *\<^sub>R x) = x" by simp
- from a0 have ap: "\<bar>a\<bar> > 0" by arith
- from infnorm_mul_lemma[of "1/a" "a *\<^sub>R x"]
- have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*\<^sub>R x)"
- unfolding th by simp
- with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *\<^sub>R x))" by (simp add: field_simps)
- then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*\<^sub>R 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
-
-text {* 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 DIM('a)) * infnorm(x::'a::euclidean_space)"
-proof-
- let ?d = "DIM('a)"
- 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 apply(subst euclidean_inner)
- apply (subst power2_abs[symmetric])
- apply(rule order_trans[OF setsum_bounded[where K="\<bar>infnorm x\<bar>\<twosuperior>"]])
- 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 bexI) by auto
- from real_le_lsqrt[OF inner_ge_zero th th1]
- show ?thesis unfolding norm_eq_sqrt_inner id_def .
-qed
-
-text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
-
-lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R 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 *\<^sub>R x - norm x *\<^sub>R 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
- 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:
- shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
- norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm(x) *\<^sub>R y = - norm y *\<^sub>R 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 *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
- by simp
- 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_scaleR ..
- 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 :: "'a::real_inner"
- shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R 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 *\<^sub>R y = norm y *\<^sub>R 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
-
-subsection {* Collinearity *}
-
-definition
- collinear :: "'a::real_vector set \<Rightarrow> bool" where
- "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
-
-lemma collinear_empty: "collinear {}" by (simp add: collinear_def)
-
-lemma collinear_sing: "collinear {x}"
- by (simp add: collinear_def)
-
-lemma collinear_2: "collinear {x, y}"
- apply (simp add: collinear_def)
- apply (rule exI[where x="x - y"])
- apply auto
- apply (rule exI[where x=1], simp)
- apply (rule exI[where x="- 1"], simp)
- done
-
-lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R 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 *\<^sub>R 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 *\<^sub>R u" and cy: "y = cy *\<^sub>R 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 *\<^sub>R x"
- by simp
- hence ?rhs using x y by blast}
- moreover
- {assume h: "?rhs"
- then obtain c where c: "y = c *\<^sub>R 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)
- apply (rule exI[where x= "-c"], simp)
- apply (rule exI[where x=1], simp)
- apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
- apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
- done}
- ultimately have ?thesis by blast}
- ultimately show ?thesis by blast
-qed
-
-lemma norm_cauchy_schwarz_equal:
- shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0,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 *\<^sub>R y = norm y *\<^sub>R x")
-apply (rule exI[where x="(1/norm x) * norm y"])
-apply (drule sym)
-unfolding scaleR_scaleR[symmetric]
-apply (simp add: field_simps)
-apply (rule exI[where x="(1/norm x) * - norm y"])
-apply clarify
-apply (drule sym)
-unfolding scaleR_scaleR[symmetric]
-apply (simp add: field_simps)
-apply (erule exE)
-apply (erule ssubst)
-unfolding scaleR_scaleR
-unfolding norm_scaleR
-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
-
-subsection "Instantiate @{typ real} and @{typ complex} as typeclass @{text ordered_euclidean_space}."
-
-instantiation real :: real_basis_with_inner
-begin
-definition [simp]: "basis i = (if i = 0 then (1::real) else 0)"
-
-lemma basis_real_range: "basis ` {..<1} = {1::real}" by auto
-
-instance proof
- let ?b = "basis::nat \<Rightarrow> real"
-
- from basis_real_range have "independent (?b ` {..<1})" by auto
- thus "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
- by (auto intro!: exI[of _ 1] inj_onI)
-
- { fix x::real
- have "x \<in> span (range ?b)"
- using span_mul[of 1 "range ?b" x] span_clauses(1)[of 1 "range ?b"]
- by auto }
- thus "span (range ?b) = UNIV" by auto
-qed
-end
-
-lemma DIM_real[simp]: "DIM(real) = 1"
- by (rule dimension_eq) (auto simp: basis_real_def)
-
-instance real::ordered_euclidean_space proof qed(auto simp add:euclidean_component_def)
-
-lemma Eucl_real_simps[simp]:
- "(x::real) $$ 0 = x"
- "(\<chi>\<chi> i. f i) = ((f 0)::real)"
- "\<And>i. i > 0 \<Longrightarrow> x $$ i = 0"
- defer apply(subst euclidean_eq) apply safe
- unfolding euclidean_lambda_beta'
- unfolding euclidean_component_def by auto
-
-instantiation complex :: real_basis_with_inner
-begin
-definition "basis i = (if i = 0 then 1 else if i = 1 then ii else 0)"
-
-lemma complex_basis[simp]:"basis 0 = (1::complex)" "basis 1 = ii" "basis (Suc 0) = ii"
- unfolding basis_complex_def by auto
-
-instance
-proof
- let ?b = "basis::nat \<Rightarrow> complex"
- have [simp]: "(range ?b) = {0, basis 0, basis 1}"
- by (auto simp: basis_complex_def split: split_if_asm)
- { fix z::complex
- have "z \<in> span (range ?b)"
- by (auto simp: span_finite complex_equality
- intro!: exI[of _ "\<lambda>i. if i = 1 then Re z else if i = ii then Im z else 0"]) }
- thus "span (range ?b) = UNIV" by auto
-
- have "{..<2} = {0, 1::nat}" by auto
- hence *: "?b ` {..<2} = {1, ii}"
- by (auto simp add: basis_complex_def)
- moreover have "1 \<notin> span {\<i>}"
- by (simp add: span_finite complex_equality complex_scaleR_def)
- hence "independent (?b ` {..<2})"
- by (simp add: * basis_complex_def independent_empty independent_insert)
- ultimately show "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
- by (auto intro!: exI[of _ 2] inj_onI simp: basis_complex_def split: split_if_asm)
-qed
end
lemma DIM_complex[simp]: "DIM(complex) = 2"
- by (rule dimension_eq) (auto simp: basis_complex_def)
+ by (rule dimension_complex_def)
-instance complex :: euclidean_space
- proof qed (auto simp add: basis_complex_def inner_complex_def)
+subsubsection {* Type @{typ "'a \<times> 'b"} *}
-section {* Products Spaces *}
-
-instantiation prod :: (real_basis, real_basis) real_basis
+instantiation prod :: (euclidean_space, euclidean_space) euclidean_space
begin
-definition "basis i = (if i < DIM('a) then (basis i, 0) else (0, basis (i - DIM('a))))"
-
-instance
-proof
- let ?b = "basis :: nat \<Rightarrow> 'a \<times> 'b"
- let ?b_a = "basis :: nat \<Rightarrow> 'a"
- let ?b_b = "basis :: nat \<Rightarrow> 'b"
-
- note image_range =
- image_add_atLeastLessThan[symmetric, of 0 "DIM('a)" "DIM('b)", simplified]
-
- have split_range:
- "{..<DIM('b) + DIM('a)} = {..<DIM('a)} \<union> {DIM('a)..<DIM('b) + DIM('a)}"
- by auto
- have *: "?b ` {DIM('a)..<DIM('b) + DIM('a)} = {0} \<times> (?b_b ` {..<DIM('b)})"
- "?b ` {..<DIM('a)} = (?b_a ` {..<DIM('a)}) \<times> {0}"
- unfolding image_range image_image basis_prod_def_raw range_basis
- by (auto simp: zero_prod_def basis_eq_0_iff)
- hence b_split:
- "?b ` {..<DIM('b) + DIM('a)} = (?b_a ` {..<DIM('a)}) \<times> {0} \<union> {0} \<times> (?b_b ` {..<DIM('b)})" (is "_ = ?prod")
- by (subst split_range) (simp add: image_Un)
-
- have b_0: "?b ` {DIM('b) + DIM('a)..} = {0}" unfolding basis_prod_def_raw
- by (auto simp: zero_prod_def image_iff basis_eq_0_iff elim!: ballE[of _ _ "DIM('a) + DIM('b)"])
-
- have split_UNIV:
- "UNIV = {..<DIM('b) + DIM('a)} \<union> {DIM('b)+DIM('a)..}"
- by auto
+definition
+ "dimension (t::('a \<times> 'b) itself) = DIM('a) + DIM('b)"
- have range_b: "range ?b = ?prod \<union> {0}"
- by (subst split_UNIV) (simp add: image_Un b_split b_0)
-
- have prod: "\<And>f A B. setsum f (A \<times> B) = (\<Sum>a\<in>A. \<Sum>b\<in>B. f (a, b))"
- by (simp add: setsum_cartesian_product)
+definition
+ "basis i = (if i < DIM('a) then (basis i, 0) else (0, basis (i - DIM('a))))"
- show "span (range ?b) = UNIV"
- unfolding span_explicit range_b
- proof safe
- fix a::'a and b::'b
- from in_span_basis[of a] in_span_basis[of b]
- obtain Sa ua Sb ub where span:
- "finite Sa" "Sa \<subseteq> basis ` {..<DIM('a)}" "a = (\<Sum>v\<in>Sa. ua v *\<^sub>R v)"
- "finite Sb" "Sb \<subseteq> basis ` {..<DIM('b)}" "b = (\<Sum>v\<in>Sb. ub v *\<^sub>R v)"
- unfolding span_explicit by auto
-
- let ?S = "((Sa - {0}) \<times> {0} \<union> {0} \<times> (Sb - {0}))"
- have *:
- "?S \<inter> {v. fst v = 0} \<inter> {v. snd v = 0} = {}"
- "?S \<inter> - {v. fst v = 0} \<inter> {v. snd v = 0} = (Sa - {0}) \<times> {0}"
- "?S \<inter> {v. fst v = 0} \<inter> - {v. snd v = 0} = {0} \<times> (Sb - {0})"
- by (auto simp: zero_prod_def)
- show "\<exists>S u. finite S \<and> S \<subseteq> ?prod \<union> {0} \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = (a, b)"
- apply (rule exI[of _ ?S])
- apply (rule exI[of _ "\<lambda>(v, w). (if w = 0 then ua v else 0) + (if v = 0 then ub w else 0)"])
- using span
- apply (simp add: prod_case_unfold setsum_addf if_distrib cond_application_beta setsum_cases prod *)
- by (auto simp add: setsum_prod intro!: setsum_mono_zero_cong_left)
- qed simp
+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)
- show "\<exists>d>0. ?b ` {d..} = {0} \<and> independent (?b ` {..<d}) \<and> inj_on ?b {..<d}"
- apply (rule exI[of _ "DIM('b) + DIM('a)"]) unfolding b_0
- proof (safe intro!: DIM_positive del: notI)
- show inj_on: "inj_on ?b {..<DIM('b) + DIM('a)}" unfolding split_range
- using inj_on_iff[OF basis_inj[where 'a='a]] inj_on_iff[OF basis_inj[where 'a='b]]
- by (auto intro!: inj_onI simp: basis_prod_def basis_eq_0_iff)
+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
- show "independent (?b ` {..<DIM('b) + DIM('a)})"
- unfolding independent_eq_inj_on[OF inj_on]
- proof safe
- fix i u assume i_upper: "i < DIM('b) + DIM('a)" and
- "(\<Sum>j\<in>{..<DIM('b) + DIM('a)} - {i}. u (?b j) *\<^sub>R ?b j) = ?b i" (is "?SUM = _")
- let ?left = "{..<DIM('a)}" and ?right = "{DIM('a)..<DIM('b) + DIM('a)}"
- show False
- proof cases
- assume "i < DIM('a)"
- hence "(basis i, 0) = ?SUM" unfolding `?SUM = ?b i` unfolding basis_prod_def by auto
- also have "\<dots> = (\<Sum>j\<in>?left - {i}. u (?b j) *\<^sub>R ?b j) +
- (\<Sum>j\<in>?right. u (?b j) *\<^sub>R ?b j)"
- using `i < DIM('a)` by (subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
- also have "\<dots> = (\<Sum>j\<in>?left - {i}. u (?b_a j, 0) *\<^sub>R (?b_a j, 0)) +
- (\<Sum>j\<in>?right. u (0, ?b_b (j-DIM('a))) *\<^sub>R (0, ?b_b (j-DIM('a))))"
- unfolding basis_prod_def by auto
- finally have "basis i = (\<Sum>j\<in>?left - {i}. u (?b_a j, 0) *\<^sub>R ?b_a j)"
- by (simp add: setsum_prod)
- moreover
- note independent_basis[where 'a='a, unfolded independent_eq_inj_on[OF basis_inj]]
- note this[rule_format, of i "\<lambda>v. u (v, 0)"]
- ultimately show False using `i < DIM('a)` by auto
- next
- let ?i = "i - DIM('a)"
- assume not: "\<not> i < DIM('a)" hence "DIM('a) \<le> i" by auto
- hence "?i < DIM('b)" using `i < DIM('b) + DIM('a)` by auto
-
- have inj_on: "inj_on (\<lambda>j. j - DIM('a)) {DIM('a)..<DIM('b) + DIM('a)}"
- by (auto intro!: inj_onI)
- with i_upper not have *: "{..<DIM('b)} - {?i} = (\<lambda>j. j-DIM('a))`(?right - {i})"
- by (auto simp: inj_on_image_set_diff image_minus_const_atLeastLessThan_nat)
-
- have "(0, basis ?i) = ?SUM" unfolding `?SUM = ?b i`
- unfolding basis_prod_def using not `?i < DIM('b)` by auto
- also have "\<dots> = (\<Sum>j\<in>?left. u (?b j) *\<^sub>R ?b j) +
- (\<Sum>j\<in>?right - {i}. u (?b j) *\<^sub>R ?b j)"
- using not by (subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
- also have "\<dots> = (\<Sum>j\<in>?left. u (?b_a j, 0) *\<^sub>R (?b_a j, 0)) +
- (\<Sum>j\<in>?right - {i}. u (0, ?b_b (j-DIM('a))) *\<^sub>R (0, ?b_b (j-DIM('a))))"
- unfolding basis_prod_def by auto
- finally have "basis ?i = (\<Sum>j\<in>{..<DIM('b)} - {?i}. u (0, ?b_b j) *\<^sub>R ?b_b j)"
- unfolding *
- by (subst setsum_reindex[OF inj_on[THEN subset_inj_on]])
- (auto simp: setsum_prod)
- moreover
- note independent_basis[where 'a='b, unfolded independent_eq_inj_on[OF basis_inj]]
- note this[rule_format, of ?i "\<lambda>v. u (0, v)"]
- ultimately show False using `?i < DIM('b)` by auto
- qed
- qed
- qed
-qed
end
-lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('b::real_basis) + DIM('a::real_basis)"
- by (rule dimension_eq) (auto simp: basis_prod_def zero_prod_def basis_eq_0_iff)
-
-instance prod :: (euclidean_space, euclidean_space) euclidean_space
-proof (default, safe)
- let ?b = "basis :: nat \<Rightarrow> 'a \<times> 'b"
- fix i j assume "i < DIM('a \<times> 'b)" "j < DIM('a \<times> 'b)"
- thus "?b i \<bullet> ?b j = (if i = j then 1 else 0)"
- unfolding basis_prod_def by (auto simp: dot_basis)
-qed
-
-instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
-begin
-
-definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i \<le> y $$ i)"
-definition "x < (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i < y $$ i)"
-
-instance proof qed (auto simp: less_prod_def less_eq_prod_def)
end
-
-
-end
--- a/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy Wed Aug 10 21:24:26 2011 +0200
@@ -248,7 +248,7 @@
show "eventually (\<lambda>x. a * X x \<in> S) net"
by (rule eventually_mono[OF _ *]) auto
qed
-qed auto
+qed (auto intro: tendsto_const)
lemma ereal_lim_uminus:
fixes X :: "'a \<Rightarrow> ereal" shows "((\<lambda>i. - X i) ---> -L) net \<longleftrightarrow> (X ---> L) net"
@@ -460,12 +460,12 @@
assumes inc: "incseq X" and lim: "X ----> L"
shows "X N \<le> L"
using inc
- by (intro ereal_lim_mono[of N, OF _ Lim_const lim]) (simp add: incseq_def)
+ by (intro ereal_lim_mono[of N, OF _ tendsto_const lim]) (simp add: incseq_def)
lemma decseq_ge_ereal: assumes dec: "decseq X"
and lim: "X ----> (L::ereal)" shows "X N >= L"
using dec
- by (intro ereal_lim_mono[of N, OF _ lim Lim_const]) (simp add: decseq_def)
+ by (intro ereal_lim_mono[of N, OF _ lim tendsto_const]) (simp add: decseq_def)
lemma liminf_bounded_open:
fixes x :: "nat \<Rightarrow> ereal"
@@ -519,7 +519,7 @@
lemma lim_ereal_increasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n >= f m"
obtains l where "f ----> (l::ereal)"
proof(cases "f = (\<lambda>x. - \<infinity>)")
- case True then show thesis using Lim_const[of "- \<infinity>" sequentially] by (intro that[of "-\<infinity>"]) auto
+ case True then show thesis using tendsto_const[of "- \<infinity>" sequentially] by (intro that[of "-\<infinity>"]) auto
next
case False
from this obtain N where N_def: "f N > (-\<infinity>)" by (auto simp: fun_eq_iff)
@@ -1138,7 +1138,7 @@
by (induct i) (insert assms, auto) }
note this[simp]
show ?thesis unfolding sums_def
- by (rule LIMSEQ_offset[of _ n]) (auto simp add: atLeast0LessThan)
+ by (rule LIMSEQ_offset[of _ n]) (auto simp add: atLeast0LessThan intro: tendsto_const)
qed
lemma suminf_finite:
@@ -1298,4 +1298,4 @@
apply (subst SUP_commute) ..
qed
-end
\ No newline at end of file
+end
--- a/src/HOL/Multivariate_Analysis/Integration.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Integration.thy Wed Aug 10 21:24:26 2011 +0200
@@ -4476,7 +4476,7 @@
"bounded {integral {a..b} (f k) | k . k \<in> UNIV}"
shows "g integrable_on {a..b} \<and> ((\<lambda>k. integral ({a..b}) (f k)) ---> integral ({a..b}) g) sequentially"
proof(case_tac[!] "content {a..b} = 0") assume as:"content {a..b} = 0"
- show ?thesis using integrable_on_null[OF as] unfolding integral_null[OF as] using Lim_const by auto
+ show ?thesis using integrable_on_null[OF as] unfolding integral_null[OF as] using tendsto_const by auto
next assume ab:"content {a..b} \<noteq> 0"
have fg:"\<forall>x\<in>{a..b}. \<forall> k. (f k x) $$ 0 \<le> (g x) $$ 0"
proof safe case goal1 note assms(3)[rule_format,OF this]
@@ -4631,7 +4631,8 @@
proof(rule monotone_convergence_interval,safe)
case goal1 show ?case using int .
next case goal2 thus ?case apply-apply(cases "x\<in>s") using assms(3) by auto
- next case goal3 thus ?case apply-apply(cases "x\<in>s") using assms(4) by auto
+ next case goal3 thus ?case apply-apply(cases "x\<in>s")
+ using assms(4) by (auto intro: tendsto_const)
next case goal4 note * = integral_nonneg
have "\<And>k. norm (integral {a..b} (\<lambda>x. if x \<in> s then f k x else 0)) \<le> norm (integral s (f k))"
unfolding real_norm_def apply(subst abs_of_nonneg) apply(rule *[OF int])
@@ -4681,13 +4682,13 @@
proof- case goal1 thus ?case using *[of x 0 "Suc k"] by auto
next case goal2 thus ?case apply(rule integrable_sub) using assms(1) by auto
next case goal3 thus ?case using *[of x "Suc k" "Suc (Suc k)"] by auto
- next case goal4 thus ?case apply-apply(rule Lim_sub)
- using seq_offset[OF assms(3)[rule_format],of x 1] by auto
+ next case goal4 thus ?case apply-apply(rule tendsto_diff)
+ using seq_offset[OF assms(3)[rule_format],of x 1] by (auto intro: tendsto_const)
next case goal5 thus ?case using assms(4) unfolding bounded_iff
apply safe apply(rule_tac x="a + norm (integral s (\<lambda>x. f 0 x))" in exI)
apply safe apply(erule_tac x="integral s (\<lambda>x. f (Suc k) x)" in ballE) unfolding sub
apply(rule order_trans[OF norm_triangle_ineq4]) by auto qed
- note conjunctD2[OF this] note Lim_add[OF this(2) Lim_const[of "integral s (f 0)"]]
+ note conjunctD2[OF this] note tendsto_add[OF this(2) tendsto_const[of "integral s (f 0)"]]
integrable_add[OF this(1) assms(1)[rule_format,of 0]]
thus ?thesis unfolding sub apply-apply rule defer apply(subst(asm) integral_sub)
using assms(1) apply auto apply(rule seq_offset_rev[where k=1]) by auto qed
@@ -4702,11 +4703,11 @@
apply(rule_tac x=k in exI) unfolding integral_neg[OF assm(1)] by auto
have "(\<lambda>x. - g x) integrable_on s \<and> ((\<lambda>k. integral s (\<lambda>x. - f k x))
---> integral s (\<lambda>x. - g x)) sequentially" apply(rule monotone_convergence_increasing)
- apply(safe,rule integrable_neg) apply(rule assm) defer apply(rule Lim_neg)
+ apply(safe,rule integrable_neg) apply(rule assm) defer apply(rule tendsto_minus)
apply(rule assm,assumption) unfolding * apply(rule bounded_scaling) using assm by auto
note * = conjunctD2[OF this]
show ?thesis apply rule using integrable_neg[OF *(1)] defer
- using Lim_neg[OF *(2)] apply- unfolding integral_neg[OF assm(1)]
+ using tendsto_minus[OF *(2)] apply- unfolding integral_neg[OF assm(1)]
unfolding integral_neg[OF *(1),THEN sym] by auto qed
subsection {* absolute integrability (this is the same as Lebesgue integrability). *}
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Linear_Algebra.thy Wed Aug 10 21:24:26 2011 +0200
@@ -0,0 +1,3181 @@
+(* Title: HOL/Multivariate_Analysis/Linear_Algebra.thy
+ Author: Amine Chaieb, University of Cambridge
+*)
+
+header {* Elementary linear algebra on Euclidean spaces *}
+
+theory Linear_Algebra
+imports
+ Euclidean_Space
+ "~~/src/HOL/Library/Infinite_Set"
+ L2_Norm
+ "~~/src/HOL/Library/Convex"
+uses
+ "~~/src/HOL/Library/positivstellensatz.ML" (* FIXME duplicate use!? *)
+ ("normarith.ML")
+begin
+
+lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
+ by auto
+
+notation inner (infix "\<bullet>" 70)
+
+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 "\<And>d::real. 0 < d \<Longrightarrow> 0 < d/2 \<and> d/2 < d" by simp
+ then have th3: "\<And>d::real. d > 0 \<Longrightarrow> \<exists>e > 0. e < d" by blast
+ {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
+ let ?d' = "min (d/2) ((l - a)/2)"
+ have "?d' < d \<and> 0 < ?d' \<and> ?d' < l - a" using lap d(1)
+ by (simp add: min_max.less_infI2)
+ then have "\<exists>d'. d' < d \<and> d' >0 \<and> l - d' > a" by auto
+ 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 "\<And>d::real. 0 < d \<Longrightarrow> d/2 < d \<and> 0 < d/2" by simp
+ then have "\<exists>d'. d' < d \<and> d' >0" using d(1) by blast
+ 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. *}
+
+(* FIXME: same as norm_scaleR
+lemma norm_mul[simp]: "norm(a *\<^sub>R 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: setL2_def power2_eq_square)
+
+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 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_left_mono)
+ 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)
+*} "prove simple linear statements about vector norms"
+
+
+text{* Hence more metric properties. *}
+
+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_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_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_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_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 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)
+
+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. *}
+
+context real_inner
+begin
+
+definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
+
+lemma orthogonal_clauses:
+ "orthogonal a 0"
+ "orthogonal a x \<Longrightarrow> orthogonal a (c *\<^sub>R x)"
+ "orthogonal a x \<Longrightarrow> orthogonal a (-x)"
+ "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x + y)"
+ "orthogonal a x \<Longrightarrow> orthogonal a y \<Longrightarrow> orthogonal a (x - y)"
+ "orthogonal 0 a"
+ "orthogonal x a \<Longrightarrow> orthogonal (c *\<^sub>R x) a"
+ "orthogonal x a \<Longrightarrow> orthogonal (-x) a"
+ "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x + y) a"
+ "orthogonal x a \<Longrightarrow> orthogonal y a \<Longrightarrow> orthogonal (x - y) a"
+ unfolding orthogonal_def inner_simps inner_add_left inner_add_right inner_diff_left inner_diff_right (*FIXME*) by auto
+
+end
+
+lemma orthogonal_commute: "orthogonal x y \<longleftrightarrow> orthogonal y x"
+ by (simp add: orthogonal_def inner_commute)
+
+subsection{* Linear functions. *}
+
+definition
+ linear :: "('a::real_vector \<Rightarrow> 'b::real_vector) \<Rightarrow> bool" where
+ "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *\<^sub>R x) = c *\<^sub>R f x)"
+
+lemma linearI: assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+ shows "linear f" using assms unfolding linear_def by auto
+
+lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. c *\<^sub>R f x)"
+ by (simp add: linear_def algebra_simps)
+
+lemma linear_compose_neg: "linear f ==> linear (\<lambda>x. -(f(x)))"
+ by (simp add: linear_def)
+
+lemma linear_compose_add: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f(x) + g(x))"
+ by (simp add: linear_def algebra_simps)
+
+lemma linear_compose_sub: "linear f \<Longrightarrow> linear g ==> linear (\<lambda>x. f x - g x)"
+ by (simp add: linear_def algebra_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)" by (simp add: linear_def)
+
+lemma linear_compose_setsum:
+ assumes fS: "finite S" and lS: "\<forall>a \<in> S. linear (f a)"
+ shows "linear(\<lambda>x. setsum (\<lambda>a. f a x) S)"
+ using lS
+ apply (induct rule: finite_induct[OF fS])
+ by (auto simp add: linear_zero intro: linear_compose_add)
+
+lemma linear_0: "linear f \<Longrightarrow> f 0 = 0"
+ unfolding linear_def
+ apply clarsimp
+ apply (erule allE[where x="0::'a"])
+ apply simp
+ done
+
+lemma linear_cmul: "linear f ==> f(c *\<^sub>R x) = c *\<^sub>R f x" by (simp add: linear_def)
+
+lemma linear_neg: "linear f ==> f (-x) = - f x"
+ using linear_cmul [where c="-1"] by simp
+
+lemma linear_add: "linear f ==> f(x + y) = f x + f y" by (metis linear_def)
+
+lemma linear_sub: "linear f ==> f(x - y) = f x - f y"
+ by (simp add: diff_minus linear_add linear_neg)
+
+lemma linear_setsum:
+ 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:
+ assumes lf: "linear f" and fS: "finite S"
+ shows "f (setsum (\<lambda>i. c i *\<^sub>R v i) S) = setsum (\<lambda>i. c i *\<^sub>R f (v i)) S"
+ using linear_setsum[OF lf fS, of "\<lambda>i. c i *\<^sub>R v i" , unfolded o_def]
+ linear_cmul[OF lf] by simp
+
+lemma linear_injective_0:
+ assumes lf: "linear f"
+ 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
+
+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 *\<^sub>R x) y = c *\<^sub>R (h x y)"
+ by (simp add: bilinear_def linear_def)
+
+lemma bilinear_rmul: "bilinear h ==> h x (c *\<^sub>R y) = c *\<^sub>R (h x y)"
+ by (simp add: bilinear_def linear_def)
+
+lemma bilinear_lneg: "bilinear h ==> h (- x) y = -(h x y)"
+ by (simp only: scaleR_minus1_left [symmetric] bilinear_lmul)
+
+lemma bilinear_rneg: "bilinear h ==> h x (- y) = - h x y"
+ by (simp only: scaleR_minus1_left [symmetric] 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:
+ 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:
+ 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) z = h x z - h y z"
+ by (simp add: diff_minus bilinear_ladd bilinear_lneg)
+
+lemma bilinear_rsub: "bilinear h ==> h z (x - y) = h z x - h z y"
+ by (simp add: diff_minus bilinear_radd bilinear_rneg)
+
+lemma bilinear_setsum:
+ 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
+
+subsection{* Adjoints. *}
+
+definition "adjoint f = (SOME f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y)"
+
+lemma adjoint_unique:
+ assumes "\<forall>x y. inner (f x) y = inner x (g y)"
+ shows "adjoint f = g"
+unfolding adjoint_def
+proof (rule some_equality)
+ show "\<forall>x y. inner (f x) y = inner x (g y)" using assms .
+next
+ fix h assume "\<forall>x y. inner (f x) y = inner x (h y)"
+ hence "\<forall>x y. inner x (g y) = inner x (h y)" using assms by simp
+ hence "\<forall>x y. inner x (g y - h y) = 0" by (simp add: inner_diff_right)
+ hence "\<forall>y. inner (g y - h y) (g y - h y) = 0" by simp
+ hence "\<forall>y. h y = g y" by simp
+ thus "h = g" by (simp add: ext)
+qed
+
+lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
+
+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 assms apply auto
+ apply (erule_tac x="n" in allE)
+ apply (erule_tac x="n" in allE)
+ apply auto
+ done
+
+
+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
+
+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
+
+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
+
+subsection {* 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 (in real_vector)
+ subspace :: "'a set \<Rightarrow> bool" where
+ "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 *\<^sub>R x \<in>S )"
+
+definition (in real_vector) "span S = (subspace hull S)"
+definition (in real_vector) "dependent S \<longleftrightarrow> (\<exists>a \<in> S. a \<in> span(S - {a}))"
+abbreviation (in real_vector) "independent s == ~(dependent s)"
+
+text {* Closure properties of subspaces. *}
+
+lemma subspace_UNIV[simp]: "subspace(UNIV)" by (simp add: subspace_def)
+
+lemma (in real_vector) subspace_0: "subspace S ==> 0 \<in> S" by (metis subspace_def)
+
+lemma (in real_vector) subspace_add: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S ==> x + y \<in> S"
+ by (metis subspace_def)
+
+lemma (in real_vector) subspace_mul: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> c *\<^sub>R x \<in> S"
+ by (metis subspace_def)
+
+lemma subspace_neg: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> - x \<in> S"
+ by (metis scaleR_minus1_left subspace_mul)
+
+lemma subspace_sub: "subspace S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x - y \<in> S"
+ by (metis diff_minus subspace_add subspace_neg)
+
+lemma (in real_vector) 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" 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 *\<^sub>R x" in bexI, auto)
+ done
+
+lemma subspace_linear_preimage: "linear f ==> 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}"
+ by (simp add: subspace_def)
+
+lemma (in real_vector) subspace_inter: "subspace A \<Longrightarrow> subspace B ==> subspace (A \<inter> B)"
+ by (simp add: subspace_def)
+
+lemma (in real_vector) span_mono: "A \<subseteq> B ==> span A \<subseteq> span B"
+ by (metis span_def hull_mono)
+
+lemma (in real_vector) 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 (in real_vector) 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 *\<^sub>R x \<in> span S"
+ by (metis span_def hull_subset subset_eq)
+ (metis subspace_span subspace_def)+
+
+lemma (in real_vector) 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[simp]: "span {} = {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 (in real_vector) independent_empty[intro]: "independent {}"
+ by (simp add: dependent_def)
+
+lemma dependent_single[simp]:
+ "dependent {x} \<longleftrightarrow> x = 0"
+ unfolding dependent_def by auto
+
+lemma (in real_vector) 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 (in real_vector) 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 (in real_vector) 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 (in real_vector) span_induct_alt_help for S:: "'a \<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 *\<^sub>R x + z)"
+
+lemma span_induct_alt':
+ assumes h0: "h 0" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" shows "\<forall>x \<in> span S. h x"
+proof-
+ {fix x:: "'a" 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 *\<^sub>R x)"
+ apply (induct rule: span_induct_alt_help.induct)
+ apply (simp add: span_induct_alt_help_0)
+ apply (simp add: scaleR_right_distrib)
+ 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" and hS: "\<And>c x y. x \<in> S \<Longrightarrow> h y \<Longrightarrow> h (c *\<^sub>R x + y)" and x: "x \<in> span S"
+ shows "h x"
+using span_induct_alt'[of h S] h0 hS x by blast
+
+text {* Individual closure properties. *}
+
+lemma span_span: "span (span A) = span A"
+ unfolding span_def hull_hull ..
+
+lemma (in real_vector) span_superset: "x \<in> S ==> x \<in> span S" by (metis span_clauses(1))
+
+lemma (in real_vector) span_0: "0 \<in> span S" by (metis subspace_span subspace_0)
+
+lemma span_inc: "S \<subseteq> span S"
+ by (metis subset_eq span_superset)
+
+lemma (in real_vector) dependent_0: assumes "0\<in>A" shows "dependent A"
+ unfolding dependent_def apply(rule_tac x=0 in bexI)
+ using assms span_0 by auto
+
+lemma (in real_vector) span_add: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x + y \<in> span S"
+ by (metis subspace_add subspace_span)
+
+lemma (in real_vector) span_mul: "x \<in> span S ==> (c *\<^sub>R x) \<in> span S"
+ by (metis subspace_span subspace_mul)
+
+lemma span_neg: "x \<in> span S ==> - x \<in> span S"
+ by (metis subspace_neg subspace_span)
+
+lemma span_sub: "x \<in> span S \<Longrightarrow> y \<in> span S ==> x - y \<in> span S"
+ by (metis subspace_span subspace_sub)
+
+lemma (in real_vector) 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 \<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)
+
+text {* Mapping under linear image. *}
+
+lemma span_linear_image: assumes lf: "linear f"
+ 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_eqI)
+ 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
+
+text {* The key breakdown property. *}
+
+lemma span_breakdown:
+ assumes bS: "b \<in> S" and aS: "a \<in> span S"
+ shows "\<exists>k. a - k *\<^sub>R 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) *\<^sub>R b = (x - k*\<^sub>R b) + (y - ka *\<^sub>R b)")
+ apply (simp only: )
+ apply (rule span_add[unfolded mem_def])
+ apply assumption+
+ apply (simp add: algebra_simps)
+ apply (clarsimp simp add: mem_def)
+ apply (rule_tac x= "c*k" in exI)
+ apply (subgoal_tac "c *\<^sub>R x - (c * k) *\<^sub>R b = c*\<^sub>R (x - k*\<^sub>R b)")
+ apply (simp only: )
+ apply (rule span_mul[unfolded mem_def])
+ apply assumption
+ by (simp add: algebra_simps)
+ ultimately show "?P a" using aS span_induct[where S=S and P= "?P"] by metis
+qed
+
+lemma span_breakdown_eq:
+ "x \<in> span (insert a S) \<longleftrightarrow> (\<exists>k. (x - k *\<^sub>R 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 *\<^sub>R a \<in> span S"
+ have eq: "x = (x - k *\<^sub>R a) + k *\<^sub>R a" by simp
+ have "(x - k *\<^sub>R a) + k *\<^sub>R 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
+
+text {* Hence some "reversal" results. *}
+
+lemma in_span_insert:
+ assumes a: "a \<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*\<^sub>R 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) *\<^sub>R a - ((1/k) *\<^sub>R a - b)" by simp
+ from k0 have eq': "(1/k) *\<^sub>R (a - k*\<^sub>R b) = (1/k) *\<^sub>R a - b"
+ by (simp add: algebra_simps)
+ from k have "(1/k) *\<^sub>R (a - k*\<^sub>R b) \<in> span (S - {b})"
+ by (rule span_mul)
+ hence th: "(1/k) *\<^sub>R 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 \<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
+
+text {* Transitivity property. *}
+
+lemma span_trans:
+ assumes x: "x \<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*\<^sub>R x \<in> span (S - {x})" by auto
+ have eq: "y = (y - k *\<^sub>R x) + k *\<^sub>R x" by simp
+ 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
+
+lemma span_insert_0[simp]: "span (insert 0 S) = span S"
+ using span_mono[of S "insert 0 S"] by (auto intro: span_trans span_0)
+
+text {* An explicit expansion is sometimes needed. *}
+
+lemma span_explicit:
+ "span P = {y. \<exists>S u. finite S \<and> S \<subseteq> P \<and> setsum (\<lambda>v. u v *\<^sub>R 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 *\<^sub>R 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 *\<^sub>R 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 *\<^sub>R v) ?S =(\<Sum>v\<in>S - {x}. u v *\<^sub>R v) + (u x + c) *\<^sub>R 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 *\<^sub>R v) + c *\<^sub>R x"
+ apply (simp add: setsum_Un_disjoint[OF Sss, unfolded S1[symmetric]])
+ by (simp add: algebra_simps)
+ also have "\<dots> = c*\<^sub>R x + y"
+ by (simp add: add_commute u)
+ finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = c*\<^sub>R x + y" .
+ then have "?Q ?S ?u (c*\<^sub>R 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) *\<^sub>R v) = y"
+ unfolding u[symmetric]
+ apply (rule setsum_cong2)
+ using xS by auto
+ have "?Q ?S ?u (c*\<^sub>R 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*\<^sub>R x + y)"
+ by (cases "x \<in> S", simp, simp)
+ then show "?h (c*\<^sub>R 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\<in>S. u v \<noteq> 0 \<and> setsum (\<lambda>v. u v *\<^sub>R 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 *\<^sub>R 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 *\<^sub>R v) ?S = 0"
+ using fS aS
+ apply (simp add: 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 *\<^sub>R 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 *\<^sub>R v) ?S = setsum (\<lambda>v. (- (inverse (u ?a))) *\<^sub>R (u v *\<^sub>R v)) S - ?u v *\<^sub>R v"
+ using fS vS uv
+ by (simp add: setsum_diff1 divide_inverse field_simps)
+ also have "\<dots> = ?a"
+ unfolding scaleR_right.setsum [symmetric] u
+ using uv by simp
+ finally have "setsum (\<lambda>v. ?u v *\<^sub>R v) ?S = ?a" .
+ with th0 have ?lhs
+ unfolding dependent_def span_explicit
+ apply -
+ apply (rule bexI[where x= "?a"])
+ apply (simp_all del: scaleR_minus_left)
+ apply (rule exI[where x= "?S"])
+ by (auto simp del: scaleR_minus_left)}
+ ultimately show ?thesis by blast
+qed
+
+
+lemma span_finite:
+ assumes fS: "finite S"
+ shows "span S = {y. \<exists>u. setsum (\<lambda>v. u v *\<^sub>R 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 *\<^sub>R v) S' = y" unfolding span_explicit by blast
+ let ?u = "\<lambda>x. if x \<in> S' then u x else 0"
+ have "setsum (\<lambda>v. ?u v *\<^sub>R v) S = setsum (\<lambda>v. u v *\<^sub>R v) S'"
+ using SS' fS by (auto intro!: setsum_mono_zero_cong_right)
+ hence "setsum (\<lambda>v. ?u v *\<^sub>R v) S = y" by (metis u)
+ hence "y \<in> ?rhs" by auto}
+ moreover
+ {fix y u assume u: "setsum (\<lambda>v. u v *\<^sub>R v) S = y"
+ then have "y \<in> span S" using fS unfolding span_explicit by auto}
+ ultimately show ?thesis by blast
+qed
+
+lemma Int_Un_cancel: "(A \<union> B) \<inter> A = A" "(A \<union> B) \<inter> B = B" by auto
+
+lemma span_union: "span (A \<union> B) = (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
+proof safe
+ fix x assume "x \<in> span (A \<union> B)"
+ then obtain S u where S: "finite S" "S \<subseteq> A \<union> B" and x: "x = (\<Sum>v\<in>S. u v *\<^sub>R v)"
+ unfolding span_explicit by auto
+
+ let ?Sa = "\<Sum>v\<in>S\<inter>A. u v *\<^sub>R v"
+ let ?Sb = "(\<Sum>v\<in>S\<inter>(B - A). u v *\<^sub>R v)"
+ show "x \<in> (\<lambda>(a, b). a + b) ` (span A \<times> span B)"
+ proof
+ show "x = (case (?Sa, ?Sb) of (a, b) \<Rightarrow> a + b)"
+ unfolding x using S
+ by (simp, subst setsum_Un_disjoint[symmetric]) (auto intro!: setsum_cong)
+
+ from S have "?Sa \<in> span A" unfolding span_explicit
+ by (auto intro!: exI[of _ "S \<inter> A"])
+ moreover from S have "?Sb \<in> span B" unfolding span_explicit
+ by (auto intro!: exI[of _ "S \<inter> (B - A)"])
+ ultimately show "(?Sa, ?Sb) \<in> span A \<times> span B" by simp
+ qed
+next
+ fix a b assume "a \<in> span A" and "b \<in> span B"
+ then obtain Sa ua Sb ub where span:
+ "finite Sa" "Sa \<subseteq> A" "a = (\<Sum>v\<in>Sa. ua v *\<^sub>R v)"
+ "finite Sb" "Sb \<subseteq> B" "b = (\<Sum>v\<in>Sb. ub v *\<^sub>R v)"
+ unfolding span_explicit by auto
+ let "?u v" = "(if v \<in> Sa then ua v else 0) + (if v \<in> Sb then ub v else 0)"
+ from span have "finite (Sa \<union> Sb)" "Sa \<union> Sb \<subseteq> A \<union> B"
+ and "a + b = (\<Sum>v\<in>(Sa\<union>Sb). ?u v *\<^sub>R v)"
+ unfolding setsum_addf scaleR_left_distrib
+ by (auto simp add: if_distrib cond_application_beta setsum_cases Int_Un_cancel)
+ thus "a + b \<in> span (A \<union> B)"
+ unfolding span_explicit by (auto intro!: exI[of _ ?u])
+qed
+
+text {* This is useful for building a basis step-by-step. *}
+
+lemma independent_insert:
+ "independent(insert a 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
+
+text {* 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 spanning_subset_independent:
+ assumes BA: "B \<subseteq> A" and iA: "independent A"
+ 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
+
+text {* The general case of the Exchange Lemma, the key to what follows. *}
+
+lemma exchange_lemma:
+ assumes f:"finite t" 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
+
+text {* This implies corresponding size bounds. *}
+
+lemma independent_span_bound:
+ assumes f: "finite t" and i: "independent s" 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
+
+subsection{* Euclidean Spaces as Typeclass*}
+
+lemma (in euclidean_space) basis_inj[simp, intro]: "inj_on basis {..<DIM('a)}"
+ by (rule inj_onI, rule ccontr, cut_tac i=x and j=y in dot_basis, simp)
+
+lemma (in euclidean_space) basis_finite: "basis ` {DIM('a)..} = {0}"
+ by (auto intro: image_eqI [where x="DIM('a)"])
+
+lemma independent_eq_inj_on:
+ fixes D :: nat and f :: "nat \<Rightarrow> 'c::real_vector" assumes *: "inj_on f {..<D}"
+ shows "independent (f ` {..<D}) \<longleftrightarrow> (\<forall>a u. a < D \<longrightarrow> (\<Sum>i\<in>{..<D}-{a}. u (f i) *\<^sub>R f i) \<noteq> f a)"
+proof -
+ from * have eq: "\<And>i. i < D \<Longrightarrow> f ` {..<D} - {f i} = f`({..<D} - {i})"
+ and inj: "\<And>i. inj_on f ({..<D} - {i})"
+ by (auto simp: inj_on_def)
+ have *: "\<And>i. finite (f ` {..<D} - {i})" by simp
+ show ?thesis unfolding dependent_def span_finite[OF *]
+ by (auto simp: eq setsum_reindex[OF inj])
+qed
+
+lemma independent_basis:
+ "independent (basis ` {..<DIM('a)} :: 'a::euclidean_space set)"
+ unfolding independent_eq_inj_on [OF basis_inj]
+ apply clarify
+ apply (drule_tac f="inner (basis a)" in arg_cong)
+ apply (simp add: inner_right.setsum dot_basis)
+ done
+
+lemma dimensionI:
+ assumes "\<And>d. \<lbrakk> 0 < d; basis ` {d..} = {0::'a::euclidean_space};
+ independent (basis ` {..<d} :: 'a set);
+ inj_on (basis :: nat \<Rightarrow> 'a) {..<d} \<rbrakk> \<Longrightarrow> P d"
+ shows "P DIM('a::euclidean_space)"
+ using DIM_positive basis_finite independent_basis basis_inj
+ by (rule assms)
+
+lemma (in euclidean_space) dimension_eq:
+ assumes "\<And>i. i < d \<Longrightarrow> basis i \<noteq> 0"
+ assumes "\<And>i. d \<le> i \<Longrightarrow> basis i = 0"
+ shows "DIM('a) = d"
+proof (rule linorder_cases [of "DIM('a)" d])
+ assume "DIM('a) < d"
+ hence "basis DIM('a) \<noteq> 0" by (rule assms)
+ thus ?thesis by simp
+next
+ assume "d < DIM('a)"
+ hence "basis d \<noteq> 0" by simp
+ thus ?thesis by (simp add: assms)
+next
+ assume "DIM('a) = d" thus ?thesis .
+qed
+
+lemma (in euclidean_space) range_basis:
+ "range basis = insert 0 (basis ` {..<DIM('a)})"
+proof -
+ have *: "UNIV = {..<DIM('a)} \<union> {DIM('a)..}" by auto
+ show ?thesis unfolding * image_Un basis_finite by auto
+qed
+
+lemma (in euclidean_space) range_basis_finite[intro]:
+ "finite (range basis)"
+ unfolding range_basis by auto
+
+lemma span_basis: "span (range basis) = (UNIV :: 'a::euclidean_space set)"
+proof -
+ { fix x :: 'a
+ have "(\<Sum>i<DIM('a). (x $$ i) *\<^sub>R basis i) \<in> span (range basis :: 'a set)"
+ by (simp add: span_setsum span_mul span_superset)
+ hence "x \<in> span (range basis)"
+ by (simp only: euclidean_representation [symmetric])
+ } thus ?thesis by auto
+qed
+
+lemma basis_representation:
+ "\<exists>u. x = (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R (v\<Colon>'a\<Colon>euclidean_space))"
+proof -
+ have "x\<in>UNIV" by auto from this[unfolded span_basis[THEN sym]]
+ have "\<exists>u. (\<Sum>v\<in>basis ` {..<DIM('a)}. u v *\<^sub>R v) = x"
+ unfolding range_basis span_insert_0 apply(subst (asm) span_finite) by auto
+ thus ?thesis by fastsimp
+qed
+
+lemma span_basis'[simp]:"span ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = UNIV"
+ apply(subst span_basis[symmetric]) unfolding range_basis by auto
+
+lemma card_basis[simp]:"card ((basis::nat=>'a) ` {..<DIM('a::euclidean_space)}) = DIM('a)"
+ apply(subst card_image) using basis_inj by auto
+
+lemma in_span_basis: "(x::'a::euclidean_space) \<in> span (basis ` {..<DIM('a)})"
+ unfolding span_basis' ..
+
+lemma component_le_norm: "\<bar>x$$i\<bar> \<le> norm (x::'a::euclidean_space)"
+ unfolding euclidean_component_def
+ apply(rule order_trans[OF real_inner_class.Cauchy_Schwarz_ineq2]) by auto
+
+lemma norm_bound_component_le: "norm (x::'a::euclidean_space) \<le> e \<Longrightarrow> \<bar>x$$i\<bar> <= e"
+ by (metis component_le_norm order_trans)
+
+lemma norm_bound_component_lt: "norm (x::'a::euclidean_space) < e \<Longrightarrow> \<bar>x$$i\<bar> < e"
+ by (metis component_le_norm basic_trans_rules(21))
+
+lemma norm_le_l1: "norm (x::'a::euclidean_space) \<le> (\<Sum>i<DIM('a). \<bar>x $$ i\<bar>)"
+ apply (subst euclidean_representation[of x])
+ apply (rule order_trans[OF setsum_norm])
+ by (auto intro!: setsum_mono)
+
+lemma setsum_norm_allsubsets_bound:
+ fixes f:: "'a \<Rightarrow> 'n::euclidean_space"
+ 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 DIM('n) * e"
+proof-
+ let ?d = "real DIM('n)"
+ let ?nf = "\<lambda>x. norm (f x)"
+ let ?U = "{..<DIM('n)}"
+ 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]
+ unfolding euclidean_component.setsum by(auto intro: abs_le_D1)
+ have Pne: "setsum (\<lambda>x. \<bar>f x $$ i\<bar>) ?Pn \<le> e"
+ using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i] fPs[OF PnP]
+ unfolding euclidean_component.setsum euclidean_component.minus
+ by(auto simp add: setsum_negf intro: abs_le_D1)
+ have "setsum (\<lambda>x. \<bar>f x $$ 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 $$ i\<bar>) P \<le> 2*e" .
+ qed
+ finally show ?thesis .
+qed
+
+lemma choice_iff': "(\<forall>x<d. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x<d. P x (f x))" by metis
+
+lemma lambda_skolem': "(\<forall>i<DIM('a::euclidean_space). \<exists>x. P i x) \<longleftrightarrow>
+ (\<exists>x::'a. \<forall>i<DIM('a). P i (x$$i))" (is "?lhs \<longleftrightarrow> ?rhs")
+proof-
+ let ?S = "{..<DIM('a)}"
+ {assume H: "?rhs"
+ then have ?lhs by auto}
+ moreover
+ {assume H: "?lhs"
+ then obtain f where f:"\<forall>i<DIM('a). P i (f i)" unfolding choice_iff' by metis
+ let ?x = "(\<chi>\<chi> i. (f i)) :: 'a"
+ {fix i assume i:"i<DIM('a)"
+ with f have "P i (f i)" by metis
+ then have "P i (?x$$i)" using i by auto
+ }
+ hence "\<forall>i<DIM('a). P i (?x$$i)" by metis
+ hence ?rhs by metis }
+ ultimately show ?thesis by metis
+qed
+
+subsection {* An ordering on euclidean spaces that will allow us to talk about intervals *}
+
+class ordered_euclidean_space = ord + euclidean_space +
+ assumes eucl_le: "x \<le> y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i \<le> y $$ i)"
+ and eucl_less: "x < y \<longleftrightarrow> (\<forall>i < DIM('a). x $$ i < y $$ i)"
+
+lemma eucl_less_not_refl[simp, intro!]: "\<not> x < (x::'a::ordered_euclidean_space)"
+ unfolding eucl_less[where 'a='a] by auto
+
+lemma euclidean_trans[trans]:
+ fixes x y z :: "'a::ordered_euclidean_space"
+ shows "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
+ and "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
+ and "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
+ by (force simp: eucl_less[where 'a='a] eucl_le[where 'a='a])+
+
+subsection {* Linearity and Bilinearity continued *}
+
+lemma linear_bounded:
+ fixes f:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ assumes lf: "linear f"
+ shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
+proof-
+ let ?S = "{..<DIM('a)}"
+ let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
+ have fS: "finite ?S" by simp
+ {fix x:: "'a"
+ let ?g = "(\<lambda> i. (x$$i) *\<^sub>R (basis i) :: 'a)"
+ have "norm (f x) = norm (f (setsum (\<lambda>i. (x$$i) *\<^sub>R (basis i)) ?S))"
+ apply(subst euclidean_representation[of x]) ..
+ also have "\<dots> = norm (setsum (\<lambda> i. (x$$i) *\<^sub>R f (basis i)) ?S)"
+ using linear_setsum[OF lf fS, of ?g, unfolded o_def] linear_cmul[OF lf] by auto
+ finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$$i) *\<^sub>R f (basis i))?S)" .
+ {fix i assume i: "i \<in> ?S"
+ from component_le_norm[of x i]
+ have "norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x"
+ unfolding norm_scaleR
+ apply (simp only: mult_commute)
+ apply (rule mult_mono)
+ by (auto simp add: field_simps) }
+ then have th: "\<forall>i\<in> ?S. norm ((x$$i) *\<^sub>R f (basis i :: 'a)) \<le> norm (f (basis i)) * norm x" by metis
+ from setsum_norm_le[OF fS, of "\<lambda>i. (x$$i) *\<^sub>R (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:: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ 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 "((\<chi>\<chi> i. 1)::'a) \<noteq> 0" unfolding euclidean_eq[where 'a='a]
+ by(auto intro!:exI[where x=0] simp add:euclidean_component.zero)
+ hence "norm ((\<chi>\<chi> i. 1)::'a) > 0" by auto
+ with C have "B * norm ((\<chi>\<chi> i. 1)::'a) < 0"
+ by (simp add: mult_less_0_iff)
+ with B[rule_format, of "(\<chi>\<chi> i. 1)::'a"] norm_ge_zero[of "f ((\<chi>\<chi> i. 1)::'a)"] have False by simp
+ }
+ then have Bp: "B \<ge> 0" by (metis not_leE)
+ {fix x::"'a"
+ 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 linear_conv_bounded_linear:
+ fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ 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
+ 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"
+ by (simp add: f.add f.scaleR linear_def)
+qed
+
+lemma bounded_linearI': fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "\<And>x y. f (x + y) = f x + f y" "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
+ shows "bounded_linear f" unfolding linear_conv_bounded_linear[THEN sym]
+ by(rule linearI[OF assms])
+
+
+lemma bilinear_bounded:
+ fixes h:: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space \<Rightarrow> 'k::real_normed_vector"
+ assumes bh: "bilinear h"
+ shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
+proof-
+ let ?M = "{..<DIM('m)}"
+ let ?N = "{..<DIM('n)}"
+ 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:: "'m" and y :: "'n"
+ have "norm (h x y) = norm (h (setsum (\<lambda>i. (x$$i) *\<^sub>R basis i) ?M) (setsum (\<lambda>i. (y$$i) *\<^sub>R basis i) ?N))"
+ apply(subst euclidean_representation[where 'a='m])
+ apply(subst euclidean_representation[where 'a='n]) ..
+ also have "\<dots> = norm (setsum (\<lambda> (i,j). h ((x$$i) *\<^sub>R basis i) ((y$$j) *\<^sub>R 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 simp del: scaleR_scaleR)
+ 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:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
+ 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::'a and y::'b
+ 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 :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::real_normed_vector"
+ 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
+ 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
+ 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 {* We continue. *}
+
+lemma independent_bound:
+ fixes S:: "('a::euclidean_space) set"
+ shows "independent S \<Longrightarrow> finite S \<and> card S <= DIM('a::euclidean_space)"
+ using independent_span_bound[of "(basis::nat=>'a) ` {..<DIM('a)}" S] by auto
+
+lemma dependent_biggerset: "(finite (S::('a::euclidean_space) set) ==> card S > DIM('a)) ==> dependent S"
+ by (metis independent_bound not_less)
+
+text {* Hence we can create a maximal independent subset. *}
+
+lemma maximal_independent_subset_extend:
+ assumes sv: "(S::('a::euclidean_space) 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 "DIM('a) - 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 = "DIM('a)"
+ {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:: ('a::euclidean_space) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
+ by (metis maximal_independent_subset_extend[of "{}:: ('a::euclidean_space) set"] empty_subsetI independent_empty)
+
+
+text {* 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 :: ('a::euclidean_space) 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
+
+text {* Consequences of independence or spanning for cardinality. *}
+
+lemma independent_card_le_dim:
+ assumes "(B::('a::euclidean_space) 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::('a::euclidean_space) 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:: ('a::euclidean_space) 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::('a::euclidean_space) 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)
+
+text {* More lemmas about dimension. *}
+
+lemma dim_UNIV: "dim (UNIV :: ('a::euclidean_space) set) = DIM('a)"
+ apply (rule dim_unique[of "(basis::nat=>'a) ` {..<DIM('a)}"])
+ using independent_basis by auto
+
+lemma dim_subset:
+ "(S:: ('a::euclidean_space) 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:: ('a::euclidean_space) set) \<le> DIM('a)"
+ by (metis dim_subset subset_UNIV dim_UNIV)
+
+text {* Converses to those. *}
+
+lemma card_ge_dim_independent:
+ assumes BV:"(B::('a::euclidean_space) 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:: ('a::euclidean_space) 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:: ('a::euclidean_space) 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)
+
+text {* More general size bound lemmas. *}
+
+lemma independent_bound_general:
+ "independent (S:: ('a::euclidean_space) 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:: ('a::euclidean_space) 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:: ('a::euclidean_space) 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:: ('a::euclidean_space) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
+ by (metis dim_span dim_subset)
+
+lemma span_eq_dim: "span (S:: ('a::euclidean_space) set) = span T ==> dim S = dim T"
+ by (metis dim_span)
+
+lemma spans_image:
+ assumes lf: "linear f" 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 :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+ assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S)"
+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
+
+text {* Relation between bases and injectivity/surjectivity of map. *}
+
+lemma spanning_surjective_image:
+ assumes us: "UNIV \<subseteq> span S"
+ 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" 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
+
+text {* 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::'a::euclidean_space) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *\<^sub>R b) = 0"
+ unfolding inner_simps by auto
+
+lemma basis_orthogonal:
+ fixes B :: "('a::euclidean_space) 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)) *\<^sub>R 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::'a) b c. a - (b - c) = c + (a - b)" by (simp add: field_simps)
+ have "x - k *\<^sub>R (a - (\<Sum>x\<in>C. (x \<bullet> a / (x \<bullet> x)) *\<^sub>R x)) \<in> span C \<longleftrightarrow> x - k *\<^sub>R a \<in> span C"
+ apply (simp only: scaleR_right_diff_distrib 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 set_eq_iff 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)
+ 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)
+ 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 :: "('a::euclidean_space) 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)
+
+text {* Low-dimensional subset is in a hyperplane (weak orthogonal complement). *}
+
+lemma span_not_univ_orthogonal: fixes S::"('a::euclidean_space) set"
+ assumes sU: "span S \<noteq> UNIV"
+ shows "\<exists>(a::'a). 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)) *\<^sub>R b) B"
+ have "setsum (\<lambda>b. (a \<bullet> b / (b \<bullet> b)) *\<^sub>R 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)
+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
+ apply (clarsimp simp add: inner_simps 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 ::('a::euclidean_space) 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: fixes S::"('a::euclidean_space) set"
+ assumes d: "dim S < DIM('a)"
+ shows "\<exists>(a::'a). a \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
+proof-
+ {assume "span S = UNIV"
+ hence "dim (span S) = dim (UNIV :: ('a) set)" by simp
+ hence "dim S = DIM('a)" 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
+
+text {* 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 = 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*\<^sub>R a \<in> span (b -{a})" by blast
+ have "f (x - k*\<^sub>R 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*\<^sub>R f a \<in> span (f ` b)"
+ by (simp add: linear_sub[OF lf] linear_cmul[OF lf])
+ hence th: "-k *\<^sub>R f a \<in> span (f ` b)"
+ using "2.prems"(5) by simp
+ {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
+ 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(8), 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
+
+text {* We can extend a linear mapping from basis. *}
+
+lemma linear_independent_extend_lemma:
+ fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
+ assumes fi: "finite B" and ib: "independent B"
+ shows "\<exists>g. (\<forall>x\<in> span B. \<forall>y\<in> span B. g (x + y) = g x + g y)
+ \<and> (\<forall>x\<in> span B. \<forall>c. g (c*\<^sub>R x) = c *\<^sub>R 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 *\<^sub>R x) = c *\<^sub>R g x" "\<forall>x\<in>b. g x = f x" by blast
+ let ?h = "\<lambda>z. SOME k. (z - k *\<^sub>R a) \<in> span b"
+ {fix z assume z: "z \<in> span (insert a b)"
+ have th0: "z - ?h z *\<^sub>R a \<in> span b"
+ apply (rule someI_ex)
+ unfolding span_breakdown_eq[symmetric]
+ using z .
+ {fix k assume k: "z - k *\<^sub>R a \<in> span b"
+ have eq: "z - ?h z *\<^sub>R a - (z - k*\<^sub>R a) = (k - ?h z) *\<^sub>R a"
+ by (simp add: field_simps scaleR_left_distrib [symmetric])
+ from span_sub[OF th0 k]
+ have khz: "(k - ?h z) *\<^sub>R 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
+ 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 *\<^sub>R a \<in> span b \<and> (\<forall>k. z - k *\<^sub>R a \<in> span b \<longrightarrow> k = ?h z)" by blast}
+ note h = this
+ let ?g = "\<lambda>z. ?h z *\<^sub>R f a + g (z - ?h z *\<^sub>R 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) y a k l. (x + y) - (k + l) *\<^sub>R a = (x - k *\<^sub>R a) + (y - l *\<^sub>R a)"
+ by (simp add: algebra_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: scaleR_left_distrib)}
+ moreover
+ {fix x:: "'a" and c:: real assume x: "x \<in> span (insert a b)"
+ have tha: "\<And>(x::'a) c k a. c *\<^sub>R x - (c * k) *\<^sub>R a = c *\<^sub>R (x - k *\<^sub>R a)"
+ by (simp add: algebra_simps)
+ have hc: "?h (c *\<^sub>R 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 *\<^sub>R x) = c*\<^sub>R ?g x"
+ unfolding hc tha g(2)[rule_format, OF conjunct1[OF h, OF x]]
+ by (simp add: algebra_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:: ('a::euclidean_space) 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*\<^sub>R x) = c *\<^sub>R 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
+
+text {* 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:: ('a::euclidean_space) set)"
+ and t: "subspace (T :: ('b::euclidean_space) 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
+
+text {* Linear functions are equal on a subspace if they are on a spanning set. *}
+
+lemma subspace_kernel:
+ assumes lf: "linear f"
+ 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"
+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"
+ by (metis linear_eq_0_span[OF lf] subset_eq SB f0)
+
+lemma linear_eq:
+ assumes lf: "linear f" 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::euclidean_space \<Rightarrow> _)" and lg: "linear g"
+ and fg: "\<forall>i<DIM('a::euclidean_space). f (basis i) = g(basis i)"
+ shows "f = g"
+proof-
+ let ?U = "{..<DIM('a)}"
+ let ?I = "(basis::nat=>'a) ` {..<DIM('a)}"
+ {fix x assume x: "x \<in> (UNIV :: 'a set)"
+ from equalityD2[OF span_basis'[where 'a='a]]
+ have IU: " (UNIV :: 'a set) \<subseteq> span ?I" by blast
+ have "f x = g x" apply(rule linear_eq[OF lf lg IU,rule_format]) using fg x by auto }
+ then show ?thesis by (auto intro: ext)
+qed
+
+text {* Similar results for bilinear functions. *}
+
+lemma bilinear_eq:
+ assumes bf: "bilinear f"
+ 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: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> _"
+ assumes bf: "bilinear f"
+ and bg: "bilinear g"
+ and fg: "\<forall>i<DIM('a). \<forall>j<DIM('b). f (basis i) (basis j) = g (basis i) (basis j)"
+ shows "f = g"
+proof-
+ from fg have th: "\<forall>x \<in> (basis ` {..<DIM('a)}). \<forall>y\<in> (basis ` {..<DIM('b)}). f x y = g x y" by blast
+ from bilinear_eq[OF bf bg equalityD2[OF span_basis'] equalityD2[OF span_basis'] th]
+ show ?thesis by (blast intro: ext)
+qed
+
+text {* Detailed theorems about left and right invertibility in general case. *}
+
+lemma linear_injective_left_inverse: fixes f::"'a::euclidean_space => 'b::euclidean_space"
+ assumes lf: "linear f" 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_basis, OF lf fi]
+ obtain h:: "'b => 'a" where h: "linear h"
+ " \<forall>x \<in> f ` basis ` {..<DIM('a)}. h x = inv f x" by blast
+ from h(2)
+ have th: "\<forall>i<DIM('a). (h \<circ> f) (basis i) = id (basis i)"
+ using inv_o_cancel[OF fi, unfolded fun_eq_iff 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: fixes f::"'a::euclidean_space => 'b::euclidean_space"
+ assumes lf: "linear f" and sf: "surj f"
+ shows "\<exists>g. linear g \<and> f o g = id"
+proof-
+ from linear_independent_extend[OF independent_basis[where 'a='b],of "inv f"]
+ obtain h:: "'b \<Rightarrow> 'a" where
+ h: "linear h" "\<forall> x\<in> basis ` {..<DIM('b)}. h x = inv f x" by blast
+ from h(2)
+ have th: "\<forall>i<DIM('b). (f o h) (basis i) = id (basis i)"
+ using sf by(auto simp add: surj_iff_all)
+ 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
+
+text {* An injective map @{typ "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"} is also surjective. *}
+
+lemma linear_injective_imp_surjective: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+ assumes lf: "linear f" and fi: "inj f"
+ shows "surj f"
+proof-
+ let ?U = "UNIV :: 'a 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
+
+text {* 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: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+ assumes lf: "linear f" and sf: "surj f"
+ shows "inj f"
+proof-
+ let ?U = "UNIV :: 'a 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
+ 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
+
+text {* 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: fun_eq_iff o_def id_def)
+
+lemma linear_injective_isomorphism: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+ assumes lf: "linear f" 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: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+ assumes lf: "linear f" 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)
+
+text {* Left and right inverses are the same for @{typ "'a::euclidean_space => 'a::euclidean_space"}. *}
+
+lemma linear_inverse_left: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+ assumes lf: "linear f" and lf': "linear f'"
+ shows "f o f' = id \<longleftrightarrow> f' o f = id"
+proof-
+ {fix f f':: "'a => 'a"
+ assume lf: "linear f" "linear f'" and f: "f o f' = id"
+ from f have sf: "surj f"
+ apply (auto simp add: o_def id_def surj_def)
+ by metis
+ from linear_surjective_isomorphism[OF lf(1) sf] lf f
+ have "f' o f = id" unfolding fun_eq_iff o_def id_def
+ by metis}
+ then show ?thesis using lf lf' by metis
+qed
+
+text {* Moreover, a one-sided inverse is automatically linear. *}
+
+lemma left_inverse_linear: fixes f::"'a::euclidean_space => 'a::euclidean_space"
+ assumes lf: "linear f" 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 fun_eq_iff)
+ by metis
+ from linear_injective_isomorphism[OF lf fi]
+ obtain h:: "'a \<Rightarrow> 'a" 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 fun_eq_iff)
+ by metis
+ with h(1) show ?thesis by blast
+qed
+
+subsection {* Infinity norm *}
+
+definition "infnorm (x::'a::euclidean_space) = Sup {abs(x$$i) |i. i<DIM('a)}"
+
+lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
+ by auto
+
+lemma infnorm_set_image:
+ "{abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)} =
+ (\<lambda>i. abs(x$$i)) ` {..<DIM('a)}" by blast
+
+lemma infnorm_set_lemma:
+ shows "finite {abs((x::'a::euclidean_space)$$i) |i. i<DIM('a)}"
+ and "{abs(x$$i) |i. i<DIM('a::euclidean_space)} \<noteq> {}"
+ unfolding infnorm_set_image
+ by auto
+
+lemma infnorm_pos_le: "0 \<le> infnorm (x::'a::euclidean_space)"
+ unfolding infnorm_def
+ unfolding Sup_finite_ge_iff[ OF infnorm_set_lemma]
+ unfolding infnorm_set_image
+ by auto
+
+lemma infnorm_triangle: "infnorm ((x::'a::euclidean_space) + 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
+ have *:"\<And>i. i \<in> {..<DIM('a)} \<longleftrightarrow> i <DIM('a)" by auto
+ show ?thesis
+ unfolding infnorm_def unfolding Sup_finite_le_iff[ OF infnorm_set_lemma[where 'a='a]]
+ 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[where 'a='a]]
+ unfolding infnorm_set_image ball_simps bex_simps
+ unfolding euclidean_simps by (metis th2)
+qed
+
+lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::_::euclidean_space) = 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
+ apply(subst (1) euclidean_eq) unfolding euclidean_component.zero
+ by auto
+ 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 by(auto simp add: euclidean_simps)
+
+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_minus[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$$i\<bar> \<le> infnorm (x::'a::euclidean_space)"
+proof(cases "i<DIM('a)")
+ case False thus ?thesis using infnorm_pos_le by auto
+next case True
+ let ?U = "{..<DIM('a)}"
+ let ?S = "{\<bar>x$$i\<bar> |i. i<DIM('a)}"
+ have fS: "finite ?S" unfolding image_Collect[symmetric]
+ apply (rule finite_imageI) by simp
+ have S0: "?S \<noteq> {}" by blast
+ have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
+ show ?thesis unfolding infnorm_def
+ apply(subst Sup_finite_ge_iff) using Sup_finite_in[OF fS S0]
+ using infnorm_set_image using True by auto
+qed
+
+lemma infnorm_mul_lemma: "infnorm(a *\<^sub>R 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 euclidean_scaleR abs_mult
+ using component_le_infnorm[of x] by(auto intro: mult_mono)
+
+lemma infnorm_mul: "infnorm(a *\<^sub>R 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) *\<^sub>R (a *\<^sub>R x) = x" by simp
+ from a0 have ap: "\<bar>a\<bar> > 0" by arith
+ from infnorm_mul_lemma[of "1/a" "a *\<^sub>R x"]
+ have "infnorm x \<le> 1/\<bar>a\<bar> * infnorm (a*\<^sub>R x)"
+ unfolding th by simp
+ with ap have "\<bar>a\<bar> * infnorm x \<le> \<bar>a\<bar> * (1/\<bar>a\<bar> * infnorm (a *\<^sub>R x))" by (simp add: field_simps)
+ then have "\<bar>a\<bar> * infnorm x \<le> infnorm (a*\<^sub>R 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
+
+text {* 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 DIM('a)) * infnorm(x::'a::euclidean_space)"
+proof-
+ let ?d = "DIM('a)"
+ 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 apply(subst euclidean_inner)
+ apply (subst power2_abs[symmetric])
+ apply(rule order_trans[OF setsum_bounded[where K="\<bar>infnorm x\<bar>\<twosuperior>"]])
+ 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 bexI) by auto
+ from real_le_lsqrt[OF inner_ge_zero th th1]
+ show ?thesis unfolding norm_eq_sqrt_inner id_def .
+qed
+
+text {* Equality in Cauchy-Schwarz and triangle inequalities. *}
+
+lemma norm_cauchy_schwarz_eq: "x \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R 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 *\<^sub>R x - norm x *\<^sub>R 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
+ 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:
+ shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
+ norm x *\<^sub>R y = norm y *\<^sub>R x \<or> norm(x) *\<^sub>R y = - norm y *\<^sub>R 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 *\<^sub>R y = norm y *\<^sub>R x \<or> norm (- x) *\<^sub>R y = norm y *\<^sub>R (- x)"
+ by simp
+ 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_scaleR ..
+ 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 :: "'a::real_inner"
+ shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *\<^sub>R y = norm y *\<^sub>R 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 *\<^sub>R y = norm y *\<^sub>R 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
+
+subsection {* Collinearity *}
+
+definition
+ collinear :: "'a::real_vector set \<Rightarrow> bool" where
+ "collinear S \<longleftrightarrow> (\<exists>u. \<forall>x \<in> S. \<forall> y \<in> S. \<exists>c. x - y = c *\<^sub>R u)"
+
+lemma collinear_empty: "collinear {}" by (simp add: collinear_def)
+
+lemma collinear_sing: "collinear {x}"
+ by (simp add: collinear_def)
+
+lemma collinear_2: "collinear {x, y}"
+ apply (simp add: collinear_def)
+ apply (rule exI[where x="x - y"])
+ apply auto
+ apply (rule exI[where x=1], simp)
+ apply (rule exI[where x="- 1"], simp)
+ done
+
+lemma collinear_lemma: "collinear {0,x,y} \<longleftrightarrow> x = 0 \<or> y = 0 \<or> (\<exists>c. y = c *\<^sub>R 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 *\<^sub>R 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 *\<^sub>R u" and cy: "y = cy *\<^sub>R 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 *\<^sub>R x"
+ by simp
+ hence ?rhs using x y by blast}
+ moreover
+ {assume h: "?rhs"
+ then obtain c where c: "y = c *\<^sub>R 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)
+ apply (rule exI[where x= "-c"], simp)
+ apply (rule exI[where x=1], simp)
+ apply (rule exI[where x="1 - c"], simp add: scaleR_left_diff_distrib)
+ apply (rule exI[where x="c - 1"], simp add: scaleR_left_diff_distrib)
+ done}
+ ultimately have ?thesis by blast}
+ ultimately show ?thesis by blast
+qed
+
+lemma norm_cauchy_schwarz_equal:
+ shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {0,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 *\<^sub>R y = norm y *\<^sub>R x")
+apply (rule exI[where x="(1/norm x) * norm y"])
+apply (drule sym)
+unfolding scaleR_scaleR[symmetric]
+apply (simp add: field_simps)
+apply (rule exI[where x="(1/norm x) * - norm y"])
+apply clarify
+apply (drule sym)
+unfolding scaleR_scaleR[symmetric]
+apply (simp add: field_simps)
+apply (erule exE)
+apply (erule ssubst)
+unfolding scaleR_scaleR
+unfolding norm_scaleR
+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
+
+subsection "Instantiate @{typ real} and @{typ complex} as typeclass @{text ordered_euclidean_space}."
+
+lemma basis_real_range: "basis ` {..<1} = {1::real}" by auto
+
+instance real::ordered_euclidean_space
+ by default (auto simp add: euclidean_component_def)
+
+lemma Eucl_real_simps[simp]:
+ "(x::real) $$ 0 = x"
+ "(\<chi>\<chi> i. f i) = ((f 0)::real)"
+ "\<And>i. i > 0 \<Longrightarrow> x $$ i = 0"
+ defer apply(subst euclidean_eq) apply safe
+ unfolding euclidean_lambda_beta'
+ unfolding euclidean_component_def by auto
+
+lemma complex_basis[simp]:
+ shows "basis 0 = (1::complex)" and "basis 1 = ii" and "basis (Suc 0) = ii"
+ unfolding basis_complex_def by auto
+
+section {* Products Spaces *}
+
+lemma DIM_prod[simp]: "DIM('a \<times> 'b) = DIM('b::euclidean_space) + DIM('a::euclidean_space)"
+ (* FIXME: why this orientation? Why not "DIM('a) + DIM('b)" ? *)
+ unfolding dimension_prod_def by (rule add_commute)
+
+instantiation prod :: (ordered_euclidean_space, ordered_euclidean_space) ordered_euclidean_space
+begin
+
+definition "x \<le> (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i \<le> y $$ i)"
+definition "x < (y::('a\<times>'b)) \<longleftrightarrow> (\<forall>i<DIM('a\<times>'b). x $$ i < y $$ i)"
+
+instance proof qed (auto simp: less_prod_def less_eq_prod_def)
+end
+
+
+end
--- a/src/HOL/Multivariate_Analysis/Operator_Norm.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Operator_Norm.thy Wed Aug 10 21:24:26 2011 +0200
@@ -5,7 +5,7 @@
header {* Operator Norm *}
theory Operator_Norm
-imports Euclidean_Space
+imports Linear_Algebra
begin
definition "onorm f = Sup {norm (f x)| x. norm x = 1}"
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Wed Aug 10 21:24:26 2011 +0200
@@ -7,7 +7,7 @@
header {* Elementary topology in Euclidean space. *}
theory Topology_Euclidean_Space
-imports SEQ Euclidean_Space "~~/src/HOL/Library/Glbs"
+imports SEQ Linear_Algebra "~~/src/HOL/Library/Glbs"
begin
(* to be moved elsewhere *)
@@ -473,29 +473,15 @@
using islimpt_UNIV [of x]
by (simp add: islimpt_approachable)
-instance real :: perfect_space
-apply default
-apply (rule islimpt_approachable [THEN iffD2])
-apply (clarify, rule_tac x="x + e/2" in bexI)
-apply (auto simp add: dist_norm)
-done
-
instance euclidean_space \<subseteq> perfect_space
-proof fix x::'a
+proof
+ fix x :: 'a
{ fix e :: real assume "0 < e"
- def a \<equiv> "x $$ 0"
- have "a islimpt UNIV" by (rule islimpt_UNIV)
- with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
- unfolding islimpt_approachable by auto
- def y \<equiv> "\<chi>\<chi> i. if i = 0 then b else x$$i :: 'a"
- from `b \<noteq> a` have "y \<noteq> x" unfolding a_def y_def apply(subst euclidean_eq) apply safe
- apply(erule_tac x=0 in allE) using DIM_positive[where 'a='a] by auto
-
- have *:"(\<Sum>i<DIM('a). (dist (y $$ i) (x $$ i))\<twosuperior>) = (\<Sum>i\<in>{0}. (dist (y $$ i) (x $$ i))\<twosuperior>)"
- apply(rule setsum_mono_zero_right) unfolding y_def by auto
- from `dist b a < e` have "dist y x < e"
- apply(subst euclidean_dist_l2)
- unfolding setL2_def * unfolding y_def a_def using `0 < e` by auto
+ def y \<equiv> "x + scaleR (e/2) (sgn (basis 0))"
+ from `0 < e` have "y \<noteq> x"
+ unfolding y_def by (simp add: sgn_zero_iff basis_eq_0_iff DIM_positive)
+ from `0 < e` have "dist y x < e"
+ unfolding y_def by (simp add: dist_norm norm_sgn)
from `y \<noteq> x` and `dist y x < e`
have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
}
@@ -1237,62 +1223,15 @@
thus ?lhs unfolding islimpt_approachable by auto
qed
-text{* Basic arithmetical combining theorems for limits. *}
-
-lemma Lim_linear:
- assumes "(f ---> l) net" "bounded_linear h"
- shows "((\<lambda>x. h (f x)) ---> h l) net"
-using `bounded_linear h` `(f ---> l) net`
-by (rule bounded_linear.tendsto)
-
-lemma Lim_ident_at: "((\<lambda>x. x) ---> a) (at a)"
- unfolding tendsto_def Limits.eventually_at_topological by fast
-
-lemma Lim_const[intro]: "((\<lambda>x. a) ---> a) net" by (rule tendsto_const)
-
-lemma Lim_cmul[intro]:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f ---> l) net ==> ((\<lambda>x. c *\<^sub>R f x) ---> c *\<^sub>R l) net"
- by (intro tendsto_intros)
-
-lemma Lim_neg:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f ---> l) net ==> ((\<lambda>x. -(f x)) ---> -l) net"
- by (rule tendsto_minus)
-
-lemma Lim_add: fixes f :: "'a \<Rightarrow> 'b::real_normed_vector" shows
- "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
- by (rule tendsto_add)
-
-lemma Lim_sub:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) ---> l - m) net"
- by (rule tendsto_diff)
-
-lemma Lim_mul:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "(c ---> d) net" "(f ---> l) net"
- shows "((\<lambda>x. c(x) *\<^sub>R f x) ---> (d *\<^sub>R l)) net"
- using assms by (rule scaleR.tendsto)
-
-lemma Lim_inv:
+lemma Lim_inv: (* TODO: delete *)
fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
assumes "(f ---> l) A" and "l \<noteq> 0"
shows "((inverse o f) ---> inverse l) A"
unfolding o_def using assms by (rule tendsto_inverse)
-lemma Lim_vmul:
- fixes c :: "'a \<Rightarrow> real" and v :: "'b::real_normed_vector"
- shows "(c ---> d) net ==> ((\<lambda>x. c(x) *\<^sub>R v) ---> d *\<^sub>R v) net"
- by (intro tendsto_intros)
-
lemma Lim_null:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net" by (simp add: Lim dist_norm)
-
-lemma Lim_null_norm:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f ---> 0) net \<longleftrightarrow> ((\<lambda>x. norm(f x)) ---> 0) net"
+ shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
by (simp add: Lim dist_norm)
lemma Lim_null_comparison:
@@ -1311,16 +1250,6 @@
using assms `e>0` unfolding tendsto_iff by auto
qed
-lemma Lim_component:
- fixes f :: "'a \<Rightarrow> ('a::euclidean_space)"
- shows "(f ---> l) net \<Longrightarrow> ((\<lambda>a. f a $$i) ---> l$$i) net"
- unfolding tendsto_iff
- apply (clarify)
- apply (drule spec, drule (1) mp)
- apply (erule eventually_elim1)
- apply (erule le_less_trans [OF dist_nth_le])
- done
-
lemma Lim_transform_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
@@ -1436,8 +1365,6 @@
unfolding tendsto_def Limits.eventually_within eventually_at_topological
by auto
-lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id
-
lemma Lim_at_id: "(id ---> a) (at a)"
apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
@@ -1492,10 +1419,10 @@
unfolding netlimit_def
apply (rule some_equality)
apply (rule Lim_at_within)
-apply (rule Lim_ident_at)
+apply (rule LIM_ident)
apply (erule tendsto_unique [OF assms])
apply (rule Lim_at_within)
-apply (rule Lim_ident_at)
+apply (rule LIM_ident)
done
lemma netlimit_at:
@@ -1512,8 +1439,8 @@
assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
shows "(g ---> l) net"
proof-
- from assms have "((\<lambda>x. f x - g x - f x) ---> 0 - l) net" using Lim_sub[of "\<lambda>x. f x - g x" 0 net f l] by auto
- thus "?thesis" using Lim_neg [of "\<lambda> x. - g x" "-l" net] by auto
+ from assms have "((\<lambda>x. f x - g x - f x) ---> 0 - l) net" using tendsto_diff[of "\<lambda>x. f x - g x" 0 net f l] by auto
+ thus "?thesis" using tendsto_minus [of "\<lambda> x. - g x" "-l" net] by auto
qed
lemma Lim_transform_eventually:
@@ -1606,7 +1533,7 @@
proof
assume "?lhs" moreover
{ assume "l \<in> S"
- hence "?rhs" using Lim_const[of l sequentially] by auto
+ hence "?rhs" using tendsto_const[of l sequentially] by auto
} moreover
{ assume "l islimpt S"
hence "?rhs" unfolding islimpt_sequential by auto
@@ -2823,7 +2750,7 @@
by (rule infinite_enumerate)
then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = l" by auto
hence "subseq r \<and> ((f \<circ> r) ---> l) sequentially"
- unfolding o_def by (simp add: fr Lim_const)
+ unfolding o_def by (simp add: fr tendsto_const)
hence "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
by - (rule exI)
from f have "\<forall>n. f (r n) \<in> s" by simp
@@ -3611,7 +3538,7 @@
\<longrightarrow> ((\<lambda>n. f(x n) - f(y n)) ---> 0) sequentially)" (is "?lhs = ?rhs")
(* BH: maybe the previous lemma should replace this one? *)
unfolding uniformly_continuous_on_sequentially'
-unfolding dist_norm Lim_null_norm [symmetric] ..
+unfolding dist_norm tendsto_norm_zero_iff ..
text{* The usual transformation theorems. *}
@@ -3642,34 +3569,34 @@
text{* Combination results for pointwise continuity. *}
lemma continuous_const: "continuous net (\<lambda>x. c)"
- by (auto simp add: continuous_def Lim_const)
+ by (auto simp add: continuous_def tendsto_const)
lemma continuous_cmul:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "continuous net f ==> continuous net (\<lambda>x. c *\<^sub>R f x)"
- by (auto simp add: continuous_def Lim_cmul)
+ by (auto simp add: continuous_def intro: tendsto_intros)
lemma continuous_neg:
fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "continuous net f ==> continuous net (\<lambda>x. -(f x))"
- by (auto simp add: continuous_def Lim_neg)
+ by (auto simp add: continuous_def tendsto_minus)
lemma continuous_add:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x + g x)"
- by (auto simp add: continuous_def Lim_add)
+ by (auto simp add: continuous_def tendsto_add)
lemma continuous_sub:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
shows "continuous net f \<Longrightarrow> continuous net g \<Longrightarrow> continuous net (\<lambda>x. f x - g x)"
- by (auto simp add: continuous_def Lim_sub)
+ by (auto simp add: continuous_def tendsto_diff)
text{* Same thing for setwise continuity. *}
lemma continuous_on_const:
"continuous_on s (\<lambda>x. c)"
- unfolding continuous_on_def by auto
+ unfolding continuous_on_def by (auto intro: tendsto_intros)
lemma continuous_on_cmul:
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
@@ -3706,11 +3633,11 @@
proof-
{ fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
hence "((\<lambda>n. c *\<^sub>R f (x n) - c *\<^sub>R f (y n)) ---> 0) sequentially"
- using Lim_cmul[of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
+ using scaleR.tendsto [OF tendsto_const, of "(\<lambda>n. f (x n) - f (y n))" 0 sequentially c]
unfolding scaleR_zero_right scaleR_right_diff_distrib by auto
}
thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
- unfolding dist_norm Lim_null_norm [symmetric] by auto
+ unfolding dist_norm tendsto_norm_zero_iff by auto
qed
lemma dist_minus:
@@ -3732,10 +3659,10 @@
{ fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
"((\<lambda>n. g (x n) - g (y n)) ---> 0) sequentially"
hence "((\<lambda>xa. f (x xa) - f (y xa) + (g (x xa) - g (y xa))) ---> 0 + 0) sequentially"
- using Lim_add[of "\<lambda> n. f (x n) - f (y n)" 0 sequentially "\<lambda> n. g (x n) - g (y n)" 0] by auto
+ using tendsto_add[of "\<lambda> n. f (x n) - f (y n)" 0 sequentially "\<lambda> n. g (x n) - g (y n)" 0] by auto
hence "((\<lambda>n. f (x n) + g (x n) - (f (y n) + g (y n))) ---> 0) sequentially" unfolding Lim_sequentially and add_diff_add [symmetric] by auto }
thus ?thesis using assms unfolding uniformly_continuous_on_sequentially'
- unfolding dist_norm Lim_null_norm [symmetric] by auto
+ unfolding dist_norm tendsto_norm_zero_iff by auto
qed
lemma uniformly_continuous_on_sub:
@@ -3750,11 +3677,11 @@
lemma continuous_within_id:
"continuous (at a within s) (\<lambda>x. x)"
- unfolding continuous_within by (rule Lim_at_within [OF Lim_ident_at])
+ unfolding continuous_within by (rule Lim_at_within [OF LIM_ident])
lemma continuous_at_id:
"continuous (at a) (\<lambda>x. x)"
- unfolding continuous_at by (rule Lim_ident_at)
+ unfolding continuous_at by (rule LIM_ident)
lemma continuous_on_id:
"continuous_on s (\<lambda>x. x)"
@@ -4117,7 +4044,7 @@
lemma continuous_vmul:
fixes c :: "'a::metric_space \<Rightarrow> real" and v :: "'b::real_normed_vector"
shows "continuous net c ==> continuous net (\<lambda>x. c(x) *\<^sub>R v)"
- unfolding continuous_def using Lim_vmul[of c] by auto
+ unfolding continuous_def by (intro tendsto_intros)
lemma continuous_mul:
fixes c :: "'a::metric_space \<Rightarrow> real"
@@ -4448,7 +4375,7 @@
proof (rule continuous_attains_sup [OF assms])
{ fix x assume "x\<in>s"
have "(dist a ---> dist a x) (at x within s)"
- by (intro tendsto_dist tendsto_const Lim_at_within Lim_ident_at)
+ by (intro tendsto_dist tendsto_const Lim_at_within LIM_ident)
}
thus "continuous_on s (dist a)"
unfolding continuous_on ..
@@ -4695,7 +4622,7 @@
obtain l' r where "l'\<in>s" and r:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> l') sequentially"
using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
have "((\<lambda>n. snd (f (r n))) ---> l - l') sequentially"
- using Lim_sub[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
+ using tendsto_diff[OF lim_subseq[OF r as(2)] lr] and f(1) unfolding o_def by auto
hence "l - l' \<in> t"
using assms(2)[unfolded closed_sequential_limits, THEN spec[where x="\<lambda> n. snd (f (r n))"], THEN spec[where x="l - l'"]]
using f(3) by auto
@@ -5140,8 +5067,8 @@
hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
unfolding Lim_sequentially by(auto simp add: dist_norm)
hence "(f ---> x) sequentially" unfolding f_def
- using Lim_add[OF Lim_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
- using Lim_vmul[of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
+ using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
+ using scaleR.tendsto [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto }
ultimately have "x \<in> closure {a<..<b}"
using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto }
thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
@@ -6171,4 +6098,21 @@
(** TODO move this someplace else within this theory **)
instance euclidean_space \<subseteq> banach ..
+declare tendsto_const [intro] (* FIXME: move *)
+
+text {* Legacy theorem names *}
+
+lemmas Lim_ident_at = LIM_ident
+lemmas Lim_const = tendsto_const
+lemmas Lim_cmul = scaleR.tendsto [OF tendsto_const]
+lemmas Lim_neg = tendsto_minus
+lemmas Lim_add = tendsto_add
+lemmas Lim_sub = tendsto_diff
+lemmas Lim_mul = scaleR.tendsto
+lemmas Lim_vmul = scaleR.tendsto [OF _ tendsto_const]
+lemmas Lim_null_norm = tendsto_norm_zero_iff [symmetric]
+lemmas Lim_linear = bounded_linear.tendsto [COMP swap_prems_rl]
+lemmas Lim_component = euclidean_component.tendsto
+lemmas Lim_intros = Lim_add Lim_const Lim_sub Lim_cmul Lim_vmul Lim_within_id
+
end
--- a/src/HOL/RealVector.thy Wed Aug 10 20:53:43 2011 +0200
+++ b/src/HOL/RealVector.thy Wed Aug 10 21:24:26 2011 +0200
@@ -974,6 +974,13 @@
end
+lemma bounded_linear_intro:
+ assumes "\<And>x y. f (x + y) = f x + f y"
+ assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
+ assumes "\<And>x. norm (f x) \<le> norm x * K"
+ shows "bounded_linear f"
+ by default (fast intro: assms)+
+
locale bounded_bilinear =
fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
\<Rightarrow> 'c::real_normed_vector"
@@ -1030,21 +1037,19 @@
lemma bounded_linear_left:
"bounded_linear (\<lambda>a. a ** b)"
-apply (unfold_locales)
+apply (cut_tac bounded, safe)
+apply (rule_tac K="norm b * K" in bounded_linear_intro)
apply (rule add_left)
apply (rule scaleR_left)
-apply (cut_tac bounded, safe)
-apply (rule_tac x="norm b * K" in exI)
apply (simp add: mult_ac)
done
lemma bounded_linear_right:
"bounded_linear (\<lambda>b. a ** b)"
-apply (unfold_locales)
+apply (cut_tac bounded, safe)
+apply (rule_tac K="norm a * K" in bounded_linear_intro)
apply (rule add_right)
apply (rule scaleR_right)
-apply (cut_tac bounded, safe)
-apply (rule_tac x="norm a * K" in exI)
apply (simp add: mult_ac)
done