--- a/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy Wed Aug 10 10:13:16 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Finite_Cartesian_Product.thy Wed Aug 10 13:13:37 2011 -0700
@@ -13,148 +13,148 @@
subsection {* Finite Cartesian products, with indexing and lambdas. *}
-typedef (open Cart)
- ('a, 'b) cart = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
- morphisms Cart_nth Cart_lambda ..
+typedef (open)
+ ('a, 'b) vec = "UNIV :: (('b::finite) \<Rightarrow> 'a) set"
+ morphisms vec_nth vec_lambda ..
notation
- Cart_nth (infixl "$" 90) and
- Cart_lambda (binder "\<chi>" 10)
+ vec_nth (infixl "$" 90) and
+ vec_lambda (binder "\<chi>" 10)
(*
Translate "'b ^ 'n" into "'b ^ ('n :: finite)". When 'n has already more than
- the finite type class write "cart 'b 'n"
+ the finite type class write "vec 'b 'n"
*)
-syntax "_finite_cart" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
+syntax "_finite_vec" :: "type \<Rightarrow> type \<Rightarrow> type" ("(_ ^/ _)" [15, 16] 15)
parse_translation {*
let
- fun cart t u = Syntax.const @{type_syntax cart} $ t $ u;
- fun finite_cart_tr [t, u as Free (x, _)] =
+ fun vec t u = Syntax.const @{type_syntax vec} $ t $ u;
+ fun finite_vec_tr [t, u as Free (x, _)] =
if Lexicon.is_tid x then
- cart t (Syntax.const @{syntax_const "_ofsort"} $ u $ Syntax.const @{class_syntax finite})
- else cart t u
- | finite_cart_tr [t, u] = cart t u
+ vec t (Syntax.const @{syntax_const "_ofsort"} $ u $ Syntax.const @{class_syntax finite})
+ else vec t u
+ | finite_vec_tr [t, u] = vec t u
in
- [(@{syntax_const "_finite_cart"}, finite_cart_tr)]
+ [(@{syntax_const "_finite_vec"}, finite_vec_tr)]
end
*}
lemma stupid_ext: "(\<forall>x. f x = g x) \<longleftrightarrow> (f = g)"
by (auto intro: ext)
-lemma Cart_eq: "(x = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
- by (simp add: Cart_nth_inject [symmetric] fun_eq_iff)
+lemma vec_eq_iff: "(x = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
+ by (simp add: vec_nth_inject [symmetric] fun_eq_iff)
-lemma Cart_lambda_beta [simp]: "Cart_lambda g $ i = g i"
- by (simp add: Cart_lambda_inverse)
+lemma vec_lambda_beta [simp]: "vec_lambda g $ i = g i"
+ by (simp add: vec_lambda_inverse)
-lemma Cart_lambda_unique: "(\<forall>i. f$i = g i) \<longleftrightarrow> Cart_lambda g = f"
- by (auto simp add: Cart_eq)
+lemma vec_lambda_unique: "(\<forall>i. f$i = g i) \<longleftrightarrow> vec_lambda g = f"
+ by (auto simp add: vec_eq_iff)
-lemma Cart_lambda_eta: "(\<chi> i. (g$i)) = g"
- by (simp add: Cart_eq)
+lemma vec_lambda_eta: "(\<chi> i. (g$i)) = g"
+ by (simp add: vec_eq_iff)
subsection {* Group operations and class instances *}
-instantiation cart :: (zero,finite) zero
+instantiation vec :: (zero, finite) zero
begin
- definition vector_zero_def : "0 \<equiv> (\<chi> i. 0)"
+ definition "0 \<equiv> (\<chi> i. 0)"
instance ..
end
-instantiation cart :: (plus,finite) plus
+instantiation vec :: (plus, finite) plus
begin
- definition vector_add_def : "op + \<equiv> (\<lambda> x y. (\<chi> i. (x$i) + (y$i)))"
+ definition "op + \<equiv> (\<lambda> x y. (\<chi> i. x$i + y$i))"
instance ..
end
-instantiation cart :: (minus,finite) minus
+instantiation vec :: (minus, finite) minus
begin
- definition vector_minus_def : "op - \<equiv> (\<lambda> x y. (\<chi> i. (x$i) - (y$i)))"
+ definition "op - \<equiv> (\<lambda> x y. (\<chi> i. x$i - y$i))"
instance ..
end
-instantiation cart :: (uminus,finite) uminus
+instantiation vec :: (uminus, finite) uminus
begin
- definition vector_uminus_def : "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x$i)))"
+ definition "uminus \<equiv> (\<lambda> x. (\<chi> i. - (x$i)))"
instance ..
end
lemma zero_index [simp]: "0 $ i = 0"
- unfolding vector_zero_def by simp
+ unfolding zero_vec_def by simp
lemma vector_add_component [simp]: "(x + y)$i = x$i + y$i"
- unfolding vector_add_def by simp
+ unfolding plus_vec_def by simp
lemma vector_minus_component [simp]: "(x - y)$i = x$i - y$i"
- unfolding vector_minus_def by simp
+ unfolding minus_vec_def by simp
lemma vector_uminus_component [simp]: "(- x)$i = - (x$i)"
- unfolding vector_uminus_def by simp
+ unfolding uminus_vec_def by simp
-instance cart :: (semigroup_add, finite) semigroup_add
- by default (simp add: Cart_eq add_assoc)
+instance vec :: (semigroup_add, finite) semigroup_add
+ by default (simp add: vec_eq_iff add_assoc)
-instance cart :: (ab_semigroup_add, finite) ab_semigroup_add
- by default (simp add: Cart_eq add_commute)
+instance vec :: (ab_semigroup_add, finite) ab_semigroup_add
+ by default (simp add: vec_eq_iff add_commute)
-instance cart :: (monoid_add, finite) monoid_add
- by default (simp_all add: Cart_eq)
+instance vec :: (monoid_add, finite) monoid_add
+ by default (simp_all add: vec_eq_iff)
-instance cart :: (comm_monoid_add, finite) comm_monoid_add
- by default (simp add: Cart_eq)
+instance vec :: (comm_monoid_add, finite) comm_monoid_add
+ by default (simp add: vec_eq_iff)
-instance cart :: (cancel_semigroup_add, finite) cancel_semigroup_add
- by default (simp_all add: Cart_eq)
+instance vec :: (cancel_semigroup_add, finite) cancel_semigroup_add
+ by default (simp_all add: vec_eq_iff)
-instance cart :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
- by default (simp add: Cart_eq)
+instance vec :: (cancel_ab_semigroup_add, finite) cancel_ab_semigroup_add
+ by default (simp add: vec_eq_iff)
-instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
+instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
-instance cart :: (group_add, finite) group_add
- by default (simp_all add: Cart_eq diff_minus)
+instance vec :: (group_add, finite) group_add
+ by default (simp_all add: vec_eq_iff diff_minus)
-instance cart :: (ab_group_add, finite) ab_group_add
- by default (simp_all add: Cart_eq)
+instance vec :: (ab_group_add, finite) ab_group_add
+ by default (simp_all add: vec_eq_iff)
subsection {* Real vector space *}
-instantiation cart :: (real_vector, finite) real_vector
+instantiation vec :: (real_vector, finite) real_vector
begin
-definition vector_scaleR_def: "scaleR = (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
+definition "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
- unfolding vector_scaleR_def by simp
+ unfolding scaleR_vec_def by simp
instance
- by default (simp_all add: Cart_eq scaleR_left_distrib scaleR_right_distrib)
+ by default (simp_all add: vec_eq_iff scaleR_left_distrib scaleR_right_distrib)
end
subsection {* Topological space *}
-instantiation cart :: (topological_space, finite) topological_space
+instantiation vec :: (topological_space, finite) topological_space
begin
-definition open_vector_def:
+definition
"open (S :: ('a ^ 'b) set) \<longleftrightarrow>
(\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>
(\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> S))"
instance proof
show "open (UNIV :: ('a ^ 'b) set)"
- unfolding open_vector_def by auto
+ unfolding open_vec_def by auto
next
fix S T :: "('a ^ 'b) set"
assume "open S" "open T" thus "open (S \<inter> T)"
- unfolding open_vector_def
+ unfolding open_vec_def
apply clarify
apply (drule (1) bspec)+
apply (clarify, rename_tac Sa Ta)
@@ -164,7 +164,7 @@
next
fix K :: "('a ^ 'b) set set"
assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
- unfolding open_vector_def
+ unfolding open_vec_def
apply clarify
apply (drule (1) bspec)
apply (drule (1) bspec)
@@ -177,32 +177,32 @@
end
lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
-unfolding open_vector_def by auto
+ unfolding open_vec_def by auto
-lemma open_vimage_Cart_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
-unfolding open_vector_def
-apply clarify
-apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
-done
+lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
+ unfolding open_vec_def
+ apply clarify
+ apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
+ done
-lemma closed_vimage_Cart_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
-unfolding closed_open vimage_Compl [symmetric]
-by (rule open_vimage_Cart_nth)
+lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
+ unfolding closed_open vimage_Compl [symmetric]
+ by (rule open_vimage_vec_nth)
lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
proof -
have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
- by (simp add: closed_INT closed_vimage_Cart_nth)
+ by (simp add: closed_INT closed_vimage_vec_nth)
qed
-lemma tendsto_Cart_nth [tendsto_intros]:
+lemma tendsto_vec_nth [tendsto_intros]:
assumes "((\<lambda>x. f x) ---> a) net"
shows "((\<lambda>x. f x $ i) ---> a $ i) net"
proof (rule topological_tendstoI)
fix S assume "open S" "a $ i \<in> S"
then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
- by (simp_all add: open_vimage_Cart_nth)
+ by (simp_all add: open_vimage_vec_nth)
with assms have "eventually (\<lambda>x. f x \<in> (\<lambda>y. y $ i) -` S) net"
by (rule topological_tendstoD)
then show "eventually (\<lambda>x. f x $ i \<in> S) net"
@@ -220,14 +220,14 @@
shows "eventually (\<lambda>x. \<forall>y. P x y) net"
using eventually_Ball_finite [of UNIV P] assms by simp
-lemma tendsto_vector:
+lemma vec_tendstoI:
assumes "\<And>i. ((\<lambda>x. f x $ i) ---> a $ i) net"
shows "((\<lambda>x. f x) ---> a) net"
proof (rule topological_tendstoI)
fix S assume "open S" and "a \<in> S"
then obtain A where A: "\<And>i. open (A i)" "\<And>i. a $ i \<in> A i"
and S: "\<And>y. \<forall>i. y $ i \<in> A i \<Longrightarrow> y \<in> S"
- unfolding open_vector_def by metis
+ unfolding open_vec_def by metis
have "\<And>i. eventually (\<lambda>x. f x $ i \<in> A i) net"
using assms A by (rule topological_tendstoD)
hence "eventually (\<lambda>x. \<forall>i. f x $ i \<in> A i) net"
@@ -236,10 +236,10 @@
by (rule eventually_elim1, simp add: S)
qed
-lemma tendsto_Cart_lambda [tendsto_intros]:
+lemma tendsto_vec_lambda [tendsto_intros]:
assumes "\<And>i. ((\<lambda>x. f x i) ---> a i) net"
shows "((\<lambda>x. \<chi> i. f x i) ---> (\<chi> i. a i)) net"
-using assms by (simp add: tendsto_vector)
+ using assms by (simp add: vec_tendstoI)
subsection {* Metric *}
@@ -251,25 +251,24 @@
apply (rule_tac x="f(x:=y)" in exI, simp)
done
-instantiation cart :: (metric_space, finite) metric_space
+instantiation vec :: (metric_space, finite) metric_space
begin
-definition dist_vector_def:
+definition
"dist x y = setL2 (\<lambda>i. dist (x$i) (y$i)) UNIV"
-lemma dist_nth_le_cart: "dist (x $ i) (y $ i) \<le> dist x y"
-unfolding dist_vector_def
-by (rule member_le_setL2) simp_all
+lemma dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
+ unfolding dist_vec_def by (rule member_le_setL2) simp_all
instance proof
fix x y :: "'a ^ 'b"
show "dist x y = 0 \<longleftrightarrow> x = y"
- unfolding dist_vector_def
- by (simp add: setL2_eq_0_iff Cart_eq)
+ unfolding dist_vec_def
+ by (simp add: setL2_eq_0_iff vec_eq_iff)
next
fix x y z :: "'a ^ 'b"
show "dist x y \<le> dist x z + dist y z"
- unfolding dist_vector_def
+ unfolding dist_vec_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
apply (simp add: setL2_mono dist_triangle2)
done
@@ -277,7 +276,7 @@
(* FIXME: long proof! *)
fix S :: "('a ^ 'b) set"
show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
- unfolding open_vector_def open_dist
+ unfolding open_vec_def open_dist
apply safe
apply (drule (1) bspec)
apply clarify
@@ -286,7 +285,7 @@
apply (rule_tac x=e in exI, clarify)
apply (drule spec, erule mp, clarify)
apply (drule spec, drule spec, erule mp)
- apply (erule le_less_trans [OF dist_nth_le_cart])
+ apply (erule le_less_trans [OF dist_vec_nth_le])
apply (subgoal_tac "\<forall>i\<in>UNIV. \<exists>e>0. \<forall>y. dist y (x$i) < e \<longrightarrow> y \<in> A i")
apply (drule finite_choice [OF finite], clarify)
apply (rule_tac x="Min (range f)" in exI, simp)
@@ -308,7 +307,7 @@
apply simp
apply clarify
apply (drule spec, erule mp)
- apply (simp add: dist_vector_def setL2_strict_mono)
+ apply (simp add: dist_vec_def setL2_strict_mono)
apply (rule_tac x="\<lambda>i. e / sqrt (of_nat CARD('b))" in exI)
apply (simp add: divide_pos_pos setL2_constant)
done
@@ -316,11 +315,11 @@
end
-lemma Cauchy_Cart_nth:
+lemma Cauchy_vec_nth:
"Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
-unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_nth_le_cart])
+ unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
-lemma Cauchy_vector:
+lemma vec_CauchyI:
fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
shows "Cauchy (\<lambda>n. X n)"
@@ -340,7 +339,7 @@
fix m n :: nat
assume "M \<le> m" "M \<le> n"
have "dist (X m) (X n) = setL2 (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
- unfolding dist_vector_def ..
+ unfolding dist_vec_def ..
also have "\<dots> \<le> setsum (\<lambda>i. dist (X m $ i) (X n $ i)) UNIV"
by (rule setL2_le_setsum [OF zero_le_dist])
also have "\<dots> < setsum (\<lambda>i::'n. ?s) UNIV"
@@ -354,14 +353,14 @@
then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < r" ..
qed
-instance cart :: (complete_space, finite) complete_space
+instance vec :: (complete_space, finite) complete_space
proof
fix X :: "nat \<Rightarrow> 'a ^ 'b" assume "Cauchy X"
have "\<And>i. (\<lambda>n. X n $ i) ----> lim (\<lambda>n. X n $ i)"
- using Cauchy_Cart_nth [OF `Cauchy X`]
+ using Cauchy_vec_nth [OF `Cauchy X`]
by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
- hence "X ----> Cart_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
- by (simp add: tendsto_vector)
+ hence "X ----> vec_lambda (\<lambda>i. lim (\<lambda>n. X n $ i))"
+ by (simp add: vec_tendstoI)
then show "convergent X"
by (rule convergentI)
qed
@@ -369,11 +368,10 @@
subsection {* Normed vector space *}
-instantiation cart :: (real_normed_vector, finite) real_normed_vector
+instantiation vec :: (real_normed_vector, finite) real_normed_vector
begin
-definition norm_vector_def:
- "norm x = setL2 (\<lambda>i. norm (x$i)) UNIV"
+definition "norm x = setL2 (\<lambda>i. norm (x$i)) UNIV"
definition vector_sgn_def:
"sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
@@ -381,69 +379,68 @@
instance proof
fix a :: real and x y :: "'a ^ 'b"
show "0 \<le> norm x"
- unfolding norm_vector_def
+ unfolding norm_vec_def
by (rule setL2_nonneg)
show "norm x = 0 \<longleftrightarrow> x = 0"
- unfolding norm_vector_def
- by (simp add: setL2_eq_0_iff Cart_eq)
+ unfolding norm_vec_def
+ by (simp add: setL2_eq_0_iff vec_eq_iff)
show "norm (x + y) \<le> norm x + norm y"
- unfolding norm_vector_def
+ unfolding norm_vec_def
apply (rule order_trans [OF _ setL2_triangle_ineq])
apply (simp add: setL2_mono norm_triangle_ineq)
done
show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
- unfolding norm_vector_def
+ unfolding norm_vec_def
by (simp add: setL2_right_distrib)
show "sgn x = scaleR (inverse (norm x)) x"
by (rule vector_sgn_def)
show "dist x y = norm (x - y)"
- unfolding dist_vector_def norm_vector_def
+ unfolding dist_vec_def norm_vec_def
by (simp add: dist_norm)
qed
end
lemma norm_nth_le: "norm (x $ i) \<le> norm x"
-unfolding norm_vector_def
+unfolding norm_vec_def
by (rule member_le_setL2) simp_all
-interpretation Cart_nth: bounded_linear "\<lambda>x. x $ i"
+interpretation vec_nth: bounded_linear "\<lambda>x. x $ i"
apply default
apply (rule vector_add_component)
apply (rule vector_scaleR_component)
apply (rule_tac x="1" in exI, simp add: norm_nth_le)
done
-instance cart :: (banach, finite) banach ..
+instance vec :: (banach, finite) banach ..
subsection {* Inner product space *}
-instantiation cart :: (real_inner, finite) real_inner
+instantiation vec :: (real_inner, finite) real_inner
begin
-definition inner_vector_def:
- "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
+definition "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
instance proof
fix r :: real and x y z :: "'a ^ 'b"
show "inner x y = inner y x"
- unfolding inner_vector_def
+ unfolding inner_vec_def
by (simp add: inner_commute)
show "inner (x + y) z = inner x z + inner y z"
- unfolding inner_vector_def
+ unfolding inner_vec_def
by (simp add: inner_add_left setsum_addf)
show "inner (scaleR r x) y = r * inner x y"
- unfolding inner_vector_def
+ unfolding inner_vec_def
by (simp add: setsum_right_distrib)
show "0 \<le> inner x x"
- unfolding inner_vector_def
+ unfolding inner_vec_def
by (simp add: setsum_nonneg)
show "inner x x = 0 \<longleftrightarrow> x = 0"
- unfolding inner_vector_def
- by (simp add: Cart_eq setsum_nonneg_eq_0_iff)
+ unfolding inner_vec_def
+ by (simp add: vec_eq_iff setsum_nonneg_eq_0_iff)
show "norm x = sqrt (inner x x)"
- unfolding inner_vector_def norm_vector_def setL2_def
+ unfolding inner_vec_def norm_vec_def setL2_def
by (simp add: power2_norm_eq_inner)
qed
@@ -453,16 +450,16 @@
text {* A bijection between @{text "'n::finite"} and @{text "{..<CARD('n)}"} *}
-definition cart_bij_nat :: "nat \<Rightarrow> ('n::finite)" where
- "cart_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"
+definition vec_bij_nat :: "nat \<Rightarrow> ('n::finite)" where
+ "vec_bij_nat = (SOME p. bij_betw p {..<CARD('n)} (UNIV::'n set) )"
-abbreviation "\<pi> \<equiv> cart_bij_nat"
+abbreviation "\<pi> \<equiv> vec_bij_nat"
definition "\<pi>' = inv_into {..<CARD('n)} (\<pi>::nat \<Rightarrow> ('n::finite))"
lemma bij_betw_pi:
"bij_betw \<pi> {..<CARD('n::finite)} (UNIV::('n::finite) set)"
using ex_bij_betw_nat_finite[of "UNIV::'n set"]
- by (auto simp: cart_bij_nat_def atLeast0LessThan
+ by (auto simp: vec_bij_nat_def atLeast0LessThan
intro!: someI_ex[of "\<lambda>x. bij_betw x {..<CARD('n)} (UNIV::'n set)"])
lemma bij_betw_pi'[intro]: "bij_betw \<pi>' (UNIV::'n set) {..<CARD('n::finite)}"
@@ -486,7 +483,7 @@
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 :: (euclidean_space, finite) euclidean_space
+instantiation vec :: (euclidean_space, finite) euclidean_space
begin
definition "dimension (t :: ('a ^ 'b) itself) = CARD('b) * DIM('a)"
@@ -503,7 +500,7 @@
have "j + i * DIM('a) < DIM('a) * (i + 1)" using assms by (auto simp: field_simps)
also have "\<dots> \<le> DIM('a) * CARD('b)" using assms unfolding mult_le_cancel1 by auto
finally show ?thesis
- unfolding basis_cart_def using assms by (auto simp: Cart_eq not_less field_simps)
+ unfolding basis_vec_def using assms by (auto simp: vec_eq_iff not_less field_simps)
qed
lemma basis_eq_pi':
@@ -551,7 +548,7 @@
qed
lemma DIM_cart[simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
- by (rule dimension_cart_def)
+ by (rule dimension_vec_def)
lemma all_less_DIM_cart:
fixes m n :: nat
@@ -582,17 +579,17 @@
instance proof
show "0 < DIM('a ^ 'b)"
- unfolding dimension_cart_def
+ unfolding dimension_vec_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
+ unfolding dimension_vec_def basis_vec_def
by simp
next
show "\<forall>i<DIM('a ^ 'b). \<forall>j<DIM('a ^ 'b).
inner (basis i :: 'a ^ 'b) (basis j) = (if i = j then 1 else 0)"
- apply (simp add: inner_vector_def)
+ apply (simp add: inner_vec_def)
apply safe
apply (erule split_CARD_DIM, simp add: basis_eq_pi')
apply (simp add: inner_if setsum_delta cong: if_cong)
@@ -605,11 +602,11 @@
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
+ unfolding inner_vec_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)
+ apply (simp add: vec_eq_iff)
done
qed