more uniform naming scheme for finite cartesian product type and related theorems
--- a/NEWS Wed Aug 10 10:13:16 2011 -0700
+++ b/NEWS Wed Aug 10 13:13:37 2011 -0700
@@ -183,6 +183,19 @@
* Limits.thy: Type "'a net" has been renamed to "'a filter", in
accordance with standard mathematical terminology. INCOMPATIBILITY.
+* Session Multivariate_Analysis: Type "('a, 'b) cart" has been renamed
+to "('a, 'b) vec" (the syntax "'a ^ 'b" remains unaffected). Constants
+"Cart_nth" and "Cart_lambda" have been respectively renamed to
+"vec_nth" and "vec_lambda"; theorems mentioning those names have
+changed to match. Definition theorems for overloaded constants now use
+the standard "foo_vec_def" naming scheme. A few other theorems have
+been renamed as follows (INCOMPATIBILITY):
+
+ Cart_eq ~> vec_eq_iff
+ dist_nth_le_cart ~> dist_vec_nth_le
+ tendsto_vector ~> vec_tendstoI
+ Cauchy_vector ~> vec_CauchyI
+
*** Document preparation ***
--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Wed Aug 10 10:13:16 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Wed Aug 10 13:13:37 2011 -0700
@@ -36,24 +36,23 @@
subsection{* Basic componentwise operations on vectors. *}
-instantiation cart :: (times,finite) times
+instantiation vec :: (times, finite) times
begin
- definition vector_mult_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 :: (one,finite) one
+instantiation vec :: (one, finite) one
begin
- definition vector_one_def : "1 \<equiv> (\<chi> i. 1)"
+ definition "1 \<equiv> (\<chi> i. 1)"
instance ..
end
-instantiation cart :: (ord,finite) ord
+instantiation vec :: (ord, finite) ord
begin
- definition vector_le_def:
- "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"
- definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"
- instance by (intro_classes)
+ definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
+ definition "x < y \<longleftrightarrow> (\<forall>i. x$i < y$i)"
+ instance ..
end
text{* The ordering on one-dimensional vectors is linear. *}
@@ -65,12 +64,12 @@
by (auto intro!: card_ge_0_finite) qed
end
-instantiation cart :: (linorder,cart_one) linorder begin
+instantiation vec :: (linorder,cart_one) linorder begin
instance proof
guess a B using UNIV_one[where 'a='b] unfolding card_Suc_eq apply- by(erule exE)+
hence *:"UNIV = {a}" by auto
have "\<And>P. (\<forall>i\<in>UNIV. P i) \<longleftrightarrow> P a" unfolding * by auto hence all:"\<And>P. (\<forall>i. P i) \<longleftrightarrow> P a" by auto
- fix x y z::"'a^'b::cart_one" note * = vector_le_def vector_less_def all Cart_eq
+ fix x y z::"'a^'b::cart_one" note * = less_eq_vec_def less_vec_def all vec_eq_iff
show "x\<le>x" "(x < y) = (x \<le> y \<and> \<not> y \<le> x)" "x\<le>y \<or> y\<le>x" unfolding * by(auto simp only:field_simps)
{ assume "x\<le>y" "y\<le>z" thus "x\<le>z" unfolding * by(auto simp only:field_simps) }
{ assume "x\<le>y" "y\<le>x" thus "x=y" unfolding * by(auto simp only:field_simps) }
@@ -93,16 +92,16 @@
@{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
@{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
val ss2 = @{simpset} addsimps
- [@{thm vector_add_def}, @{thm vector_mult_def},
- @{thm vector_minus_def}, @{thm vector_uminus_def},
- @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
- @{thm vector_scaleR_def},
- @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}]
+ [@{thm plus_vec_def}, @{thm times_vec_def},
+ @{thm minus_vec_def}, @{thm uminus_vec_def},
+ @{thm one_vec_def}, @{thm zero_vec_def}, @{thm vec_def},
+ @{thm scaleR_vec_def},
+ @{thm vec_lambda_beta}, @{thm vector_scalar_mult_def}]
fun vector_arith_tac ths =
simp_tac ss1
THEN' (fn i => rtac @{thm setsum_cong2} i
ORELSE rtac @{thm setsum_0'} i
- ORELSE simp_tac (HOL_basic_ss addsimps [@{thm "Cart_eq"}]) i)
+ ORELSE simp_tac (HOL_basic_ss addsimps [@{thm vec_eq_iff}]) i)
(* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *)
THEN' asm_full_simp_tac (ss2 addsimps ths)
in
@@ -110,8 +109,8 @@
end
*} "lift trivial vector statements to real arith statements"
-lemma vec_0[simp]: "vec 0 = 0" by (vector vector_zero_def)
-lemma vec_1[simp]: "vec 1 = 1" by (vector vector_one_def)
+lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
+lemma vec_1[simp]: "vec 1 = 1" by (vector one_vec_def)
lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
@@ -149,49 +148,47 @@
subsection {* Some frequently useful arithmetic lemmas over vectors. *}
-instance cart :: (semigroup_mult,finite) semigroup_mult
- apply (intro_classes) by (vector mult_assoc)
+instance vec :: (semigroup_mult, finite) semigroup_mult
+ by default (vector mult_assoc)
-instance cart :: (monoid_mult,finite) monoid_mult
- apply (intro_classes) by vector+
+instance vec :: (monoid_mult, finite) monoid_mult
+ by default vector+
-instance cart :: (ab_semigroup_mult,finite) ab_semigroup_mult
- apply (intro_classes) by (vector mult_commute)
+instance vec :: (ab_semigroup_mult, finite) ab_semigroup_mult
+ by default (vector mult_commute)
-instance cart :: (ab_semigroup_idem_mult,finite) ab_semigroup_idem_mult
- apply (intro_classes) by (vector mult_idem)
+instance vec :: (ab_semigroup_idem_mult, finite) ab_semigroup_idem_mult
+ by default (vector mult_idem)
-instance cart :: (comm_monoid_mult,finite) comm_monoid_mult
- apply (intro_classes) by vector
+instance vec :: (comm_monoid_mult, finite) comm_monoid_mult
+ by default vector
-instance cart :: (semiring,finite) semiring
- apply (intro_classes) by (vector field_simps)+
+instance vec :: (semiring, finite) semiring
+ by default (vector field_simps)+
-instance cart :: (semiring_0,finite) semiring_0
- apply (intro_classes) by (vector field_simps)+
-instance cart :: (semiring_1,finite) semiring_1
- apply (intro_classes) by vector
-instance cart :: (comm_semiring,finite) comm_semiring
- apply (intro_classes) by (vector field_simps)+
+instance vec :: (semiring_0, finite) semiring_0
+ by default (vector field_simps)+
+instance vec :: (semiring_1, finite) semiring_1
+ by default vector
+instance vec :: (comm_semiring, finite) comm_semiring
+ by default (vector field_simps)+
-instance cart :: (comm_semiring_0,finite) comm_semiring_0 by (intro_classes)
-instance cart :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
-instance cart :: (semiring_0_cancel,finite) semiring_0_cancel by (intro_classes)
-instance cart :: (comm_semiring_0_cancel,finite) comm_semiring_0_cancel by (intro_classes)
-instance cart :: (ring,finite) ring by (intro_classes)
-instance cart :: (semiring_1_cancel,finite) semiring_1_cancel by (intro_classes)
-instance cart :: (comm_semiring_1,finite) comm_semiring_1 by (intro_classes)
+instance vec :: (comm_semiring_0, finite) comm_semiring_0 ..
+instance vec :: (cancel_comm_monoid_add, finite) cancel_comm_monoid_add ..
+instance vec :: (semiring_0_cancel, finite) semiring_0_cancel ..
+instance vec :: (comm_semiring_0_cancel, finite) comm_semiring_0_cancel ..
+instance vec :: (ring, finite) ring ..
+instance vec :: (semiring_1_cancel, finite) semiring_1_cancel ..
+instance vec :: (comm_semiring_1, finite) comm_semiring_1 ..
-instance cart :: (ring_1,finite) ring_1 ..
+instance vec :: (ring_1, finite) ring_1 ..
-instance cart :: (real_algebra,finite) real_algebra
+instance vec :: (real_algebra, finite) real_algebra
apply intro_classes
- apply (simp_all add: vector_scaleR_def field_simps)
- apply vector
- apply vector
+ apply (simp_all add: vec_eq_iff)
done
-instance cart :: (real_algebra_1,finite) real_algebra_1 ..
+instance vec :: (real_algebra_1, finite) real_algebra_1 ..
lemma of_nat_index:
"(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
@@ -203,15 +200,15 @@
lemma one_index[simp]:
"(1 :: 'a::one ^'n)$i = 1" by vector
-instance cart :: (semiring_char_0, finite) semiring_char_0
+instance vec :: (semiring_char_0, finite) semiring_char_0
proof
fix m n :: nat
show "inj (of_nat :: nat \<Rightarrow> 'a ^ 'b)"
- by (auto intro!: injI simp add: Cart_eq of_nat_index)
+ by (auto intro!: injI simp add: vec_eq_iff of_nat_index)
qed
-instance cart :: (comm_ring_1,finite) comm_ring_1 ..
-instance cart :: (ring_char_0,finite) ring_char_0 ..
+instance vec :: (comm_ring_1, finite) comm_ring_1 ..
+instance vec :: (ring_char_0, finite) ring_char_0 ..
lemma vector_smult_assoc: "a *s (b *s x) = ((a::'a::semigroup_mult) * b) *s x"
by (vector mult_assoc)
@@ -231,7 +228,7 @@
by (vector field_simps)
lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
- by (simp add: Cart_eq)
+ by (simp add: vec_eq_iff)
lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
@@ -246,7 +243,7 @@
by (metis vector_mul_rcancel)
lemma component_le_norm_cart: "\<bar>x$i\<bar> <= norm x"
- apply (simp add: norm_vector_def)
+ apply (simp add: norm_vec_def)
apply (rule member_le_setL2, simp_all)
done
@@ -257,10 +254,10 @@
by (metis component_le_norm_cart basic_trans_rules(21))
lemma norm_le_l1_cart: "norm x <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
- by (simp add: norm_vector_def setL2_le_setsum)
+ by (simp add: norm_vec_def setL2_le_setsum)
lemma scalar_mult_eq_scaleR: "c *s x = c *\<^sub>R x"
- unfolding vector_scaleR_def vector_scalar_mult_def by simp
+ unfolding scaleR_vec_def vector_scalar_mult_def by simp
lemma dist_mul[simp]: "dist (c *s x) (c *s y) = \<bar>c\<bar> * dist x y"
unfolding dist_norm scalar_mult_eq_scaleR
@@ -272,12 +269,12 @@
by (cases "finite S", induct S set: finite, simp_all)
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
- by (simp add: Cart_eq)
+ by (simp add: vec_eq_iff)
lemma setsum_cmul:
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
- by (simp add: Cart_eq setsum_right_distrib)
+ by (simp add: vec_eq_iff setsum_right_distrib)
(* TODO: use setsum_norm_allsubsets_bound *)
lemma setsum_norm_allsubsets_bound_cart:
@@ -359,10 +356,10 @@
lemmas cart_simps = forall_CARD_DIM exists_CARD_DIM forall_CARD exists_CARD
lemma cart_euclidean_nth[simp]:
- fixes x :: "('a::euclidean_space, 'b::finite) cart"
+ fixes x :: "('a::euclidean_space, 'b::finite) vec"
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
+ unfolding euclidean_component_def inner_vec_def basis_eq_pi'[OF j] if_distrib cond_application_beta
by (simp add: setsum_cases)
lemma real_euclidean_nth:
@@ -393,13 +390,13 @@
thus "x = y \<and> i = j" using * by simp
qed simp
-instance cart :: (ordered_euclidean_space,finite) ordered_euclidean_space
+instance vec :: (ordered_euclidean_space, finite) ordered_euclidean_space
proof
fix x y::"'a^'b"
- show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) cart). x $$ i \<le> y $$ i)"
- unfolding vector_le_def apply(subst eucl_le) by (simp add: cart_simps)
- show"(x < y) = (\<forall>i<DIM(('a, 'b) cart). x $$ i < y $$ i)"
- unfolding vector_less_def apply(subst eucl_less) by (simp add: cart_simps)
+ show "(x \<le> y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i \<le> y $$ i)"
+ unfolding less_eq_vec_def apply(subst eucl_le) by (simp add: cart_simps)
+ show"(x < y) = (\<forall>i<DIM(('a, 'b) vec). x $$ i < y $$ i)"
+ unfolding less_vec_def apply(subst eucl_less) by (simp add: cart_simps)
qed
subsection{* Basis vectors in coordinate directions. *}
@@ -411,7 +408,7 @@
lemma norm_basis[simp]:
shows "norm (cart_basis k :: real ^'n) = 1"
- apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vector_def
+ apply (simp add: cart_basis_def norm_eq_sqrt_inner) unfolding inner_vec_def
apply (vector delta_mult_idempotent)
using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] by auto
@@ -431,14 +428,14 @@
qed
lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)"
- by (simp add: inj_on_def Cart_eq)
+ by (simp add: inj_on_def vec_eq_iff)
lemma basis_expansion:
"setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
- by (auto simp add: Cart_eq if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
+ by (auto simp add: vec_eq_iff if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
lemma smult_conv_scaleR: "c *s x = scaleR c x"
- unfolding vector_scalar_mult_def vector_scaleR_def by simp
+ unfolding vector_scalar_mult_def scaleR_vec_def by simp
lemma basis_expansion':
"setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"
@@ -446,22 +443,22 @@
lemma basis_expansion_unique:
"setsum (\<lambda>i. f i *s cart_basis i) UNIV = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i. f i = x$i)"
- by (simp add: Cart_eq setsum_delta if_distrib cong del: if_weak_cong)
+ by (simp add: vec_eq_iff setsum_delta if_distrib cong del: if_weak_cong)
lemma dot_basis:
shows "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"
- by (auto simp add: inner_vector_def cart_basis_def cond_application_beta if_distrib setsum_delta
+ by (auto simp add: inner_vec_def cart_basis_def cond_application_beta if_distrib setsum_delta
cong del: if_weak_cong)
lemma inner_basis:
fixes x :: "'a::{real_inner, real_algebra_1} ^ 'n"
shows "inner (cart_basis i) x = inner 1 (x $ i)"
and "inner x (cart_basis i) = inner (x $ i) 1"
- unfolding inner_vector_def cart_basis_def
+ unfolding inner_vec_def cart_basis_def
by (auto simp add: cond_application_beta if_distrib setsum_delta cong del: if_weak_cong)
lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
- by (auto simp add: Cart_eq)
+ by (auto simp add: vec_eq_iff)
lemma basis_nonzero:
shows "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
@@ -469,14 +466,14 @@
text {* some lemmas to map between Eucl and Cart *}
lemma basis_real_n[simp]:"(basis (\<pi>' i)::real^'a) = cart_basis i"
- unfolding basis_cart_def using pi'_range[where 'n='a]
- by (auto simp: Cart_eq Cart_lambda_beta)
+ unfolding basis_vec_def using pi'_range[where 'n='a]
+ by (auto simp: vec_eq_iff)
subsection {* Orthogonality on cartesian products *}
lemma orthogonal_basis:
shows "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"
- by (auto simp add: orthogonal_def inner_vector_def cart_basis_def if_distrib
+ by (auto simp add: orthogonal_def inner_vec_def cart_basis_def if_distrib
cond_application_beta setsum_delta cong del: if_weak_cong)
lemma orthogonal_basis_basis:
@@ -518,7 +515,7 @@
by (simp add: linear_cmul[OF lf])
finally have "f x \<bullet> y = x \<bullet> ?w"
apply (simp only: )
- apply (simp add: inner_vector_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
+ apply (simp add: inner_vec_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
done}
}
then show ?thesis unfolding adjoint_def
@@ -612,25 +609,25 @@
setsum_delta' cong del: if_weak_cong)
lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
- by (simp add: matrix_matrix_mult_def transpose_def Cart_eq mult_commute)
+ by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
lemma matrix_eq:
fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
apply auto
- apply (subst Cart_eq)
+ apply (subst vec_eq_iff)
apply clarify
- apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta Cart_eq cong del: if_weak_cong)
+ apply (clarsimp simp add: matrix_vector_mult_def cart_basis_def if_distrib cond_application_beta vec_eq_iff cong del: if_weak_cong)
apply (erule_tac x="cart_basis ia" in allE)
apply (erule_tac x="i" in allE)
by (auto simp add: cart_basis_def if_distrib cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
lemma matrix_vector_mul_component:
shows "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
- by (simp add: matrix_vector_mult_def inner_vector_def)
+ by (simp add: matrix_vector_mult_def inner_vec_def)
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
- apply (simp add: inner_vector_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
+ apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
apply (subst setsum_commute)
by simp
@@ -643,12 +640,12 @@
lemma row_transpose:
fixes A:: "'a::semiring_1^_^_"
shows "row i (transpose A) = column i A"
- by (simp add: row_def column_def transpose_def Cart_eq)
+ by (simp add: row_def column_def transpose_def vec_eq_iff)
lemma column_transpose:
fixes A:: "'a::semiring_1^_^_"
shows "column i (transpose A) = row i A"
- by (simp add: row_def column_def transpose_def Cart_eq)
+ by (simp add: row_def column_def transpose_def vec_eq_iff)
lemma rows_transpose: "rows(transpose (A::'a::semiring_1^_^_)) = columns A"
by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
@@ -658,15 +655,15 @@
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
- by (simp add: matrix_vector_mult_def inner_vector_def)
+ by (simp add: matrix_vector_mult_def inner_vec_def)
lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
- by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
+ by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
lemma vector_componentwise:
"(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (cart_basis i :: 'a^'n)$j) (UNIV :: 'n set))"
apply (subst basis_expansion[symmetric])
- by (vector Cart_eq setsum_component)
+ by (vector vec_eq_iff setsum_component)
lemma linear_componentwise:
fixes f:: "real ^'m \<Rightarrow> real ^ _"
@@ -696,10 +693,10 @@
where "matrix f = (\<chi> i j. (f(cart_basis j))$i)"
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
- by (simp add: linear_def matrix_vector_mult_def Cart_eq field_simps setsum_right_distrib setsum_addf)
+ by (simp add: linear_def matrix_vector_mult_def vec_eq_iff field_simps setsum_right_distrib setsum_addf)
lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::real ^ 'n)"
-apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
+apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
apply clarify
apply (rule linear_componentwise[OF lf, symmetric])
done
@@ -717,11 +714,11 @@
by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
- by (simp add: matrix_vector_mult_def transpose_def Cart_eq mult_commute)
+ by (simp add: matrix_vector_mult_def transpose_def vec_eq_iff mult_commute)
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
apply (rule adjoint_unique)
- apply (simp add: transpose_def inner_vector_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
+ apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
apply (subst setsum_commute)
apply (auto simp add: mult_ac)
done
@@ -916,10 +913,10 @@
let ?x = "\<chi> i. c i"
have th0:"A *v ?x = 0"
using c
- unfolding matrix_mult_vsum Cart_eq
+ unfolding matrix_mult_vsum vec_eq_iff
by auto
from k[rule_format, OF th0] i
- have "c i = 0" by (vector Cart_eq)}
+ have "c i = 0" by (vector vec_eq_iff)}
hence ?rhs by blast}
moreover
{assume H: ?rhs
@@ -1038,17 +1035,17 @@
lemma transpose_columnvector:
"transpose(columnvector v) = rowvector v"
- by (simp add: transpose_def rowvector_def columnvector_def Cart_eq)
+ by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
lemma transpose_rowvector: "transpose(rowvector v) = columnvector v"
- by (simp add: transpose_def columnvector_def rowvector_def Cart_eq)
+ by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
lemma dot_rowvector_columnvector:
"columnvector (A *v v) = A ** columnvector v"
by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
lemma dot_matrix_product: "(x::real^'n) \<bullet> y = (((rowvector x ::real^'n^1) ** (columnvector y :: real^1^'n))$1)$1"
- by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vector_def)
+ by (vector matrix_matrix_mult_def rowvector_def columnvector_def inner_vec_def)
lemma dot_matrix_vector_mul:
fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
@@ -1078,18 +1075,18 @@
unfolding nth_conv_component
using component_le_infnorm[of x] .
-instance cart :: (perfect_space, finite) perfect_space
+instance vec :: (perfect_space, finite) perfect_space
proof
fix x :: "'a ^ 'b"
show "x islimpt UNIV"
apply (rule islimptI)
- apply (unfold open_vector_def)
+ apply (unfold open_vec_def)
apply (drule (1) bspec)
apply clarsimp
apply (subgoal_tac "\<forall>i\<in>UNIV. \<exists>y. y \<in> A i \<and> y \<noteq> x $ i")
apply (drule finite_choice [OF finite_UNIV], erule exE)
- apply (rule_tac x="Cart_lambda f" in exI)
- apply (simp add: Cart_eq)
+ apply (rule_tac x="vec_lambda f" in exI)
+ apply (simp add: vec_eq_iff)
apply (rule ballI, drule_tac x=i in spec, clarify)
apply (cut_tac x="x $ i" in islimpt_UNIV)
apply (simp add: islimpt_def)
@@ -1122,7 +1119,7 @@
apply (clarify)
apply (drule spec, drule (1) mp)
apply (erule eventually_elim1)
- apply (erule le_less_trans [OF dist_nth_le_cart])
+ apply (erule le_less_trans [OF dist_vec_nth_le])
done
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
@@ -1131,7 +1128,7 @@
apply (rule_tac x="x $ i" in exI)
apply (rule_tac x="e" in exI)
apply clarify
-apply (rule order_trans [OF dist_nth_le_cart], simp)
+apply (rule order_trans [OF dist_vec_nth_le], simp)
done
lemma compact_lemma_cart:
@@ -1168,7 +1165,7 @@
qed
qed
-instance cart :: (heine_borel, finite) heine_borel
+instance vec :: (heine_borel, finite) heine_borel
proof
fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
@@ -1184,7 +1181,7 @@
moreover
{ fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
have "dist (f (r n)) l \<le> (\<Sum>i\<in>?d. dist (f (r n) $ i) (l $ i))"
- unfolding dist_vector_def using zero_le_dist by (rule setL2_le_setsum)
+ unfolding dist_vec_def using zero_le_dist by (rule setL2_le_setsum)
also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
by (rule setsum_strict_mono) (simp_all add: n)
finally have "dist (f (r n)) l < e" by simp
@@ -1205,12 +1202,12 @@
lemma interval_cart: fixes a :: "'a::ord^'n" shows
"{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and
"{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
- by (auto simp add: set_eq_iff vector_less_def vector_le_def)
+ by (auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
lemma mem_interval_cart: fixes a :: "'a::ord^'n" shows
"x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
"x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
- using interval_cart[of a b] by(auto simp add: set_eq_iff vector_less_def vector_le_def)
+ using interval_cart[of a b] by(auto simp add: set_eq_iff less_vec_def less_eq_vec_def)
lemma interval_eq_empty_cart: fixes a :: "real^'n" shows
"({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and
@@ -1263,7 +1260,7 @@
lemma interval_sing: fixes a :: "'a::linorder^'n" shows
"{a .. a} = {a} \<and> {a<..<a} = {}"
-apply(auto simp add: set_eq_iff vector_less_def vector_le_def Cart_eq)
+apply(auto simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
apply (simp add: order_eq_iff)
apply (auto simp add: not_less less_imp_le)
done
@@ -1276,17 +1273,17 @@
{ fix i
have "a $ i \<le> x $ i"
using x order_less_imp_le[of "a$i" "x$i"]
- by(simp add: set_eq_iff vector_less_def vector_le_def Cart_eq)
+ by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
}
moreover
{ fix i
have "x $ i \<le> b $ i"
using x order_less_imp_le[of "x$i" "b$i"]
- by(simp add: set_eq_iff vector_less_def vector_le_def Cart_eq)
+ by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
}
ultimately
show "a \<le> x \<and> x \<le> b"
- by(simp add: set_eq_iff vector_less_def vector_le_def Cart_eq)
+ by(simp add: set_eq_iff less_vec_def less_eq_vec_def vec_eq_iff)
qed
lemma subset_interval_cart: fixes a :: "real^'n" shows
@@ -1406,12 +1403,12 @@
lemma dim_substandard_cart:
shows "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
-proof- have *:"{x. \<forall>i<DIM((real, 'n) cart). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} =
+proof- have *:"{x. \<forall>i<DIM((real, 'n) vec). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} =
{x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"apply safe
apply(erule_tac x="\<pi>' i" in allE) defer
apply(erule_tac x="\<pi> i" in allE)
unfolding image_iff real_euclidean_nth[symmetric] by (auto simp: pi'_inj[THEN inj_eq])
- have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) cart)}" using pi'_range[where 'n='n] by auto
+ have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) vec)}" using pi'_range[where 'n='n] by auto
thus ?thesis using dim_substandard[of "\<pi>' ` d", where 'a="real^'n"]
unfolding * using card_image[of "\<pi>'" d] using pi'_inj unfolding inj_on_def by auto
qed
@@ -1464,7 +1461,7 @@
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
declare vector_sadd_rdistrib[simp] vector_sub_rdistrib[simp]
-lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component vector_le_def Cart_lambda_beta basis_component vector_uminus_component
+lemmas vector_component_simps = vector_minus_component vector_smult_component vector_add_component less_eq_vec_def vec_lambda_beta basis_component vector_uminus_component
lemma convex_box_cart:
assumes "\<And>i. convex {x. P i x}"
@@ -1489,7 +1486,7 @@
lemma std_simplex_cart:
"(insert (0::real^'n) { cart_basis i | i. i\<in>UNIV}) =
- (insert 0 { basis i | i. i<DIM((real,'n) cart)})"
+ (insert 0 { basis i | i. i<DIM((real,'n) vec)})"
apply(rule arg_cong[where f="\<lambda>s. (insert 0 s)"])
unfolding basis_real_n[THEN sym] apply safe
apply(rule_tac x="\<pi>' i" in exI) defer
@@ -1516,13 +1513,13 @@
lemma interval_bij_cart:"interval_bij = (\<lambda> (a,b) (u,v) (x::real^'n).
(\<chi> i. u$i + (x$i - a$i) / (b$i - a$i) * (v$i - u$i))::real^'n)"
unfolding interval_bij_def apply(rule ext)+ apply safe
- unfolding Cart_eq Cart_lambda_beta unfolding nth_conv_component
+ unfolding vec_eq_iff vec_lambda_beta unfolding nth_conv_component
apply rule apply(subst euclidean_lambda_beta) using pi'_range by auto
lemma interval_bij_affine_cart:
"interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$i - u$i) / (b$i - a$i) * x$i) +
(\<chi> i. u$i - (v$i - u$i) / (b$i - a$i) * a$i)::real^'n)"
- apply rule unfolding Cart_eq interval_bij_cart vector_component_simps
+ apply rule unfolding vec_eq_iff interval_bij_cart vector_component_simps
by(auto simp add: field_simps add_divide_distrib[THEN sym])
subsection "Derivative"
@@ -1552,7 +1549,7 @@
proof(rule ccontr)
def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
- then obtain j where j:"D$k$j \<noteq> 0" unfolding Cart_eq D_def by auto
+ then obtain j where j:"D$k$j \<noteq> 0" unfolding vec_eq_iff D_def by auto
hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding D_def by auto
note as = assms(3)[unfolded jacobian_works has_derivative_at_alt]
guess e' using as[THEN conjunct2,rule_format,OF *] .. note e' = this
@@ -1639,10 +1636,10 @@
where "dest_vec1 x \<equiv> (x$1)"
lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
- by (simp_all add: Cart_eq)
+ by (simp_all add: vec_eq_iff)
lemma vec1_component[simp]: "(vec1 x)$1 = x"
- by (simp_all add: Cart_eq)
+ by (simp_all add: vec_eq_iff)
declare vec1_dest_vec1(1) [simp]
@@ -1661,7 +1658,7 @@
subsection{* The collapse of the general concepts to dimension one. *}
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
- by (simp add: Cart_eq)
+ by (simp add: vec_eq_iff)
lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
apply auto
@@ -1670,7 +1667,7 @@
done
lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
- by (simp add: norm_vector_def)
+ by (simp add: norm_vec_def)
lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
by (simp add: norm_vector_1)
@@ -1751,7 +1748,7 @@
by (metis vec1_dest_vec1(1) norm_vec1)
lemmas vec1_dest_vec1_simps = forall_vec1 vec_add[THEN sym] dist_vec1 vec_sub[THEN sym] vec1_dest_vec1 norm_vec1 vector_smult_component
- vec1_eq vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def norm_vec1 real_norm_def
+ vec1_eq vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def real_norm_def
lemma bounded_linear_vec1:"bounded_linear (vec1::real\<Rightarrow>real^1)"
unfolding bounded_linear_def additive_def bounded_linear_axioms_def
@@ -1770,14 +1767,14 @@
unfolding smult_conv_scaleR
apply (rule ext)
apply (subst matrix_works[OF lf, symmetric])
- apply (auto simp add: Cart_eq matrix_vector_mult_def column_def mult_commute)
+ apply (auto simp add: vec_eq_iff matrix_vector_mult_def column_def mult_commute)
done
lemma linear_to_scalars: assumes lf: "linear (f::real ^'n \<Rightarrow> real^1)"
shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
apply (rule ext)
apply (subst matrix_works[OF lf, symmetric])
- apply (simp add: Cart_eq matrix_vector_mult_def row_def inner_vector_def mult_commute)
+ apply (simp add: vec_eq_iff matrix_vector_mult_def row_def inner_vec_def mult_commute)
done
lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
@@ -1801,14 +1798,14 @@
using assms unfolding continuous_on_iff apply safe
apply(erule_tac x="x$1" in ballE,erule_tac x=e in allE) apply safe
apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real
- apply(erule_tac x="dest_vec1 x'" in ballE) by(auto simp add:vector_le_def)
+ apply(erule_tac x="dest_vec1 x'" in ballE) by(auto simp add:less_eq_vec_def)
lemma continuous_on_o_vec1: fixes f::"real^1 \<Rightarrow> 'a::real_normed_vector"
assumes "continuous_on {a..b} f" shows "continuous_on {dest_vec1 a..dest_vec1 b} (f o vec1)"
using assms unfolding continuous_on_iff apply safe
apply(erule_tac x="vec x" in ballE,erule_tac x=e in allE) apply safe
apply(rule_tac x=d in exI) apply safe unfolding o_def dist_real_def dist_real
- apply(erule_tac x="vec1 x'" in ballE) by(auto simp add:vector_le_def)
+ apply(erule_tac x="vec1 x'" in ballE) by(auto simp add:less_eq_vec_def)
lemma continuous_on_vec1:"continuous_on A (vec1::real\<Rightarrow>real^1)"
by(rule linear_continuous_on[OF bounded_linear_vec1])
@@ -1816,12 +1813,12 @@
lemma mem_interval_1: fixes x :: "real^1" shows
"(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b)"
"(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-by(simp_all add: Cart_eq vector_less_def vector_le_def)
+by(simp_all add: vec_eq_iff less_vec_def less_eq_vec_def)
lemma vec1_interval:fixes a::"real" shows
"vec1 ` {a .. b} = {vec1 a .. vec1 b}"
"vec1 ` {a<..<b} = {vec1 a<..<vec1 b}"
- apply(rule_tac[!] set_eqI) unfolding image_iff vector_less_def unfolding mem_interval_cart
+ apply(rule_tac[!] set_eqI) unfolding image_iff less_vec_def unfolding mem_interval_cart
unfolding forall_1 unfolding vec1_dest_vec1_simps
apply rule defer apply(rule_tac x="dest_vec1 x" in bexI) prefer 3 apply rule defer
apply(rule_tac x="dest_vec1 x" in bexI) by auto
@@ -1830,12 +1827,12 @@
lemma interval_cases_1: fixes x :: "real^1" shows
"x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
- unfolding Cart_eq vector_less_def vector_le_def mem_interval_cart by(auto simp del:dest_vec1_eq)
+ unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart by(auto simp del:dest_vec1_eq)
lemma in_interval_1: fixes x :: "real^1" shows
"(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
- unfolding Cart_eq vector_less_def vector_le_def mem_interval_cart by(auto simp del:dest_vec1_eq)
+ unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart by(auto simp del:dest_vec1_eq)
lemma interval_eq_empty_1: fixes a :: "real^1" shows
"{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
@@ -1871,10 +1868,10 @@
lemma open_closed_interval_1: fixes a :: "real^1" shows
"{a<..<b} = {a .. b} - {a, b}"
- unfolding set_eq_iff apply simp unfolding vector_less_def and vector_le_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
+ unfolding set_eq_iff apply simp unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
- unfolding set_eq_iff apply simp unfolding vector_less_def and vector_le_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
+ unfolding set_eq_iff apply simp unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[THEN sym] by(auto simp del:dest_vec1_eq)
lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows
"(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
@@ -1903,7 +1900,7 @@
lemma dest_vec1_simps[simp]: fixes a::"real^1"
shows "a$1 = 0 \<longleftrightarrow> a = 0" (*"a \<le> 1 \<longleftrightarrow> dest_vec1 a \<le> 1" "0 \<le> a \<longleftrightarrow> 0 \<le> dest_vec1 a"*)
"a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
- by(auto simp add: vector_le_def Cart_eq)
+ by(auto simp add: less_eq_vec_def vec_eq_iff)
lemma dest_vec1_inverval:
"dest_vec1 ` {a .. b} = {dest_vec1 a .. dest_vec1 b}"
@@ -1915,18 +1912,18 @@
apply(rule_tac [!] allI)apply(rule_tac [!] impI)
apply(rule_tac[2] x="vec1 x" in exI)apply(rule_tac[4] x="vec1 x" in exI)
apply(rule_tac[6] x="vec1 x" in exI)apply(rule_tac[8] x="vec1 x" in exI)
- by (auto simp add: vector_less_def vector_le_def)
+ by (auto simp add: less_vec_def less_eq_vec_def)
lemma dest_vec1_setsum: assumes "finite S"
shows " dest_vec1 (setsum f S) = setsum (\<lambda>x. dest_vec1 (f x)) S"
using dest_vec1_sum[OF assms] by auto
lemma open_dest_vec1_vimage: "open S \<Longrightarrow> open (dest_vec1 -` S)"
-unfolding open_vector_def forall_1 by auto
+unfolding open_vec_def forall_1 by auto
lemma tendsto_dest_vec1 [tendsto_intros]:
"(f ---> l) net \<Longrightarrow> ((\<lambda>x. dest_vec1 (f x)) ---> dest_vec1 l) net"
-by(rule tendsto_Cart_nth)
+by(rule tendsto_vec_nth)
lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
unfolding continuous_def by (rule tendsto_dest_vec1)
@@ -1952,9 +1949,9 @@
unfolding vec1_dest_vec1_simps by auto qed
lemma vec1_le[simp]:fixes a::real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
- unfolding vector_le_def by auto
+ unfolding less_eq_vec_def by auto
lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
- unfolding vector_less_def by auto
+ unfolding less_vec_def by auto
subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}
@@ -1998,7 +1995,7 @@
lemma onorm_vec1: fixes f::"real \<Rightarrow> real"
shows "onorm (\<lambda>x. vec1 (f (dest_vec1 x))) = onorm f" proof-
- have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:Cart_eq)
+ have "\<forall>x::real^1. norm x = 1 \<longleftrightarrow> x\<in>{vec1 -1, vec1 (1::real)}" unfolding forall_vec1 by(auto simp add:vec_eq_iff)
hence 1:"{x. norm x = 1} = {vec1 -1, vec1 (1::real)}" by auto
have 2:"{norm (vec1 (f (dest_vec1 x))) |x. norm x = 1} = (\<lambda>x. norm (vec1 (f (dest_vec1 x)))) ` {x. norm x=1}" by auto
have "\<forall>x::real. norm x = 1 \<longleftrightarrow> x\<in>{-1, 1}" by auto hence 3:"{x. norm x = 1} = {-1, (1::real)}" by auto
@@ -2037,11 +2034,11 @@
"{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"
"{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"
apply(rule_tac[!] set_eqI) unfolding Int_iff mem_interval_cart mem_Collect_eq
- unfolding Cart_lambda_beta by auto
+ unfolding vec_lambda_beta by auto
(*lemma content_split_cart:
"content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"
-proof- note simps = interval_split_cart content_closed_interval_cases_cart Cart_lambda_beta vector_le_def
+proof- note simps = interval_split_cart content_closed_interval_cases_cart vec_lambda_beta less_eq_vec_def
{ presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }
have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto
have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))"
@@ -2051,7 +2048,7 @@
\<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"
by (auto simp add:field_simps)
moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"
- unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto
+ unfolding not_le using as[unfolded less_eq_vec_def,rule_format,of k] by auto
ultimately show ?thesis
unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)
qed*)
@@ -2059,7 +2056,7 @@
lemma has_integral_vec1: assumes "(f has_integral k) {a..b}"
shows "((\<lambda>x. vec1 (f x)) has_integral (vec1 k)) {a..b}"
proof- have *:"\<And>p. (\<Sum>(x, k)\<in>p. content k *\<^sub>R vec1 (f x)) - vec1 k = vec1 ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k)"
- unfolding vec_sub Cart_eq by(auto simp add: split_beta)
+ unfolding vec_sub vec_eq_iff by(auto simp add: split_beta)
show ?thesis using assms unfolding has_integral apply safe
apply(erule_tac x=e in allE,safe) apply(rule_tac x=d in exI,safe)
apply(erule_tac x=p in allE,safe) unfolding * norm_vector_1 by auto qed
--- a/src/HOL/Multivariate_Analysis/Fashoda.thy Wed Aug 10 10:13:16 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Fashoda.thy Wed Aug 10 13:13:37 2011 -0700
@@ -46,7 +46,7 @@
apply(rule assms)+ apply(rule continuous_on_compose,subst sqprojection_def)
apply(rule continuous_on_mul ) apply(rule continuous_at_imp_continuous_on,rule) apply(rule continuous_at_inv[unfolded o_def])
apply(rule continuous_at_infnorm) unfolding infnorm_eq_0 defer apply(rule continuous_on_id) apply(rule linear_continuous_on) proof-
- show "bounded_linear negatex" apply(rule bounded_linearI') unfolding Cart_eq proof(rule_tac[!] allI) fix i::2 and x y::"real^2" and c::real
+ show "bounded_linear negatex" apply(rule bounded_linearI') unfolding vec_eq_iff proof(rule_tac[!] allI) fix i::2 and x y::"real^2" and c::real
show "negatex (x + y) $ i = (negatex x + negatex y) $ i" "negatex (c *\<^sub>R x) $ i = (c *\<^sub>R negatex x) $ i"
apply-apply(case_tac[!] "i\<noteq>1") prefer 3 apply(drule_tac[1-2] 21)
unfolding negatex_def by(auto simp add:vector_2 ) qed qed(insert x0, auto)
@@ -66,7 +66,7 @@
apply- apply(rule_tac[!] allI impI)+ proof- fix x::"real^2" and i::2 assume x:"x\<noteq>0"
have "inverse (infnorm x) > 0" using x[unfolded infnorm_pos_lt[THEN sym]] by auto
thus "(0 < sqprojection x $ i) = (0 < x $ i)" "(sqprojection x $ i < 0) = (x $ i < 0)"
- unfolding sqprojection_def vector_component_simps Cart_nth.scaleR real_scaleR_def
+ unfolding sqprojection_def vector_component_simps vec_nth.scaleR real_scaleR_def
unfolding zero_less_mult_iff mult_less_0_iff by(auto simp add:field_simps) qed
note lem3 = this[rule_format]
have x1:"x $ 1 \<in> {- 1..1::real}" "x $ 2 \<in> {- 1..1::real}" using x(1) unfolding mem_interval_cart by auto
@@ -77,7 +77,7 @@
next assume as:"x$1 = 1"
hence *:"f (x $ 1) $ 1 = 1" using assms(6) by auto
have "sqprojection (f (x$1) - g (x$2)) $ 1 < 0"
- using x(2)[unfolded o_def Cart_eq,THEN spec[where x=1]]
+ using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
unfolding as negatex_def vector_2 by auto moreover
from x1 have "g (x $ 2) \<in> {- 1..1}" apply-apply(rule assms(2)[unfolded subset_eq,rule_format]) by auto
ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart
@@ -85,7 +85,7 @@
next assume as:"x$1 = -1"
hence *:"f (x $ 1) $ 1 = - 1" using assms(5) by auto
have "sqprojection (f (x$1) - g (x$2)) $ 1 > 0"
- using x(2)[unfolded o_def Cart_eq,THEN spec[where x=1]]
+ using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=1]]
unfolding as negatex_def vector_2 by auto moreover
from x1 have "g (x $ 2) \<in> {- 1..1}" apply-apply(rule assms(2)[unfolded subset_eq,rule_format]) by auto
ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart
@@ -93,7 +93,7 @@
next assume as:"x$2 = 1"
hence *:"g (x $ 2) $ 2 = 1" using assms(8) by auto
have "sqprojection (f (x$1) - g (x$2)) $ 2 > 0"
- using x(2)[unfolded o_def Cart_eq,THEN spec[where x=2]]
+ using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
unfolding as negatex_def vector_2 by auto moreover
from x1 have "f (x $ 1) \<in> {- 1..1}" apply-apply(rule assms(1)[unfolded subset_eq,rule_format]) by auto
ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart
@@ -101,7 +101,7 @@
next assume as:"x$2 = -1"
hence *:"g (x $ 2) $ 2 = - 1" using assms(7) by auto
have "sqprojection (f (x$1) - g (x$2)) $ 2 < 0"
- using x(2)[unfolded o_def Cart_eq,THEN spec[where x=2]]
+ using x(2)[unfolded o_def vec_eq_iff,THEN spec[where x=2]]
unfolding as negatex_def vector_2 by auto moreover
from x1 have "f (x $ 1) \<in> {- 1..1}" apply-apply(rule assms(1)[unfolded subset_eq,rule_format]) by auto
ultimately show False unfolding lem3[OF nz] vector_component_simps * mem_interval_cart
@@ -120,7 +120,7 @@
have *:"continuous_on {- 1..1} iscale" unfolding iscale_def by(rule continuous_on_intros)+
show "continuous_on {- 1..1} (f \<circ> iscale)" "continuous_on {- 1..1} (g \<circ> iscale)"
apply-apply(rule_tac[!] continuous_on_compose[OF *]) apply(rule_tac[!] continuous_on_subset[OF _ isc])
- by(rule assms)+ have *:"(1 / 2) *\<^sub>R (1 + (1::real^1)) = 1" unfolding Cart_eq by auto
+ by(rule assms)+ have *:"(1 / 2) *\<^sub>R (1 + (1::real^1)) = 1" unfolding vec_eq_iff by auto
show "(f \<circ> iscale) (- 1) $ 1 = - 1" "(f \<circ> iscale) 1 $ 1 = 1" "(g \<circ> iscale) (- 1) $ 2 = -1" "(g \<circ> iscale) 1 $ 2 = 1"
unfolding o_def iscale_def using assms by(auto simp add:*) qed
then guess s .. from this(2) guess t .. note st=this
@@ -132,7 +132,7 @@
(* move *)
lemma interval_bij_bij_cart: fixes x::"real^'n" assumes "\<forall>i. a$i < b$i \<and> u$i < v$i"
shows "interval_bij (a,b) (u,v) (interval_bij (u,v) (a,b) x) = x"
- unfolding interval_bij_cart split_conv Cart_eq Cart_lambda_beta
+ unfolding interval_bij_cart split_conv vec_eq_iff vec_lambda_beta
apply(rule,insert assms,erule_tac x=i in allE) by auto
lemma fashoda: fixes b::"real^2"
@@ -142,23 +142,23 @@
obtains z where "z \<in> path_image f" "z \<in> path_image g" proof-
fix P Q S presume "P \<or> Q \<or> S" "P \<Longrightarrow> thesis" "Q \<Longrightarrow> thesis" "S \<Longrightarrow> thesis" thus thesis by auto
next have "{a..b} \<noteq> {}" using assms(3) using path_image_nonempty by auto
- hence "a \<le> b" unfolding interval_eq_empty_cart vector_le_def by(auto simp add: not_less)
- thus "a$1 = b$1 \<or> a$2 = b$2 \<or> (a$1 < b$1 \<and> a$2 < b$2)" unfolding vector_le_def forall_2 by auto
+ hence "a \<le> b" unfolding interval_eq_empty_cart less_eq_vec_def by(auto simp add: not_less)
+ thus "a$1 = b$1 \<or> a$2 = b$2 \<or> (a$1 < b$1 \<and> a$2 < b$2)" unfolding less_eq_vec_def forall_2 by auto
next assume as:"a$1 = b$1" have "\<exists>z\<in>path_image g. z$2 = (pathstart f)$2" apply(rule connected_ivt_component_cart)
apply(rule connected_path_image assms)+apply(rule pathstart_in_path_image,rule pathfinish_in_path_image)
unfolding assms using assms(3)[unfolded path_image_def subset_eq,rule_format,of "f 0"]
- unfolding pathstart_def by(auto simp add: vector_le_def) then guess z .. note z=this
+ unfolding pathstart_def by(auto simp add: less_eq_vec_def) then guess z .. note z=this
have "z \<in> {a..b}" using z(1) assms(4) unfolding path_image_def by blast
- hence "z = f 0" unfolding Cart_eq forall_2 unfolding z(2) pathstart_def
+ hence "z = f 0" unfolding vec_eq_iff forall_2 unfolding z(2) pathstart_def
using assms(3)[unfolded path_image_def subset_eq mem_interval_cart,rule_format,of "f 0" 1]
unfolding mem_interval_cart apply(erule_tac x=1 in allE) using as by auto
thus thesis apply-apply(rule that[OF _ z(1)]) unfolding path_image_def by auto
next assume as:"a$2 = b$2" have "\<exists>z\<in>path_image f. z$1 = (pathstart g)$1" apply(rule connected_ivt_component_cart)
apply(rule connected_path_image assms)+apply(rule pathstart_in_path_image,rule pathfinish_in_path_image)
unfolding assms using assms(4)[unfolded path_image_def subset_eq,rule_format,of "g 0"]
- unfolding pathstart_def by(auto simp add: vector_le_def) then guess z .. note z=this
+ unfolding pathstart_def by(auto simp add: less_eq_vec_def) then guess z .. note z=this
have "z \<in> {a..b}" using z(1) assms(3) unfolding path_image_def by blast
- hence "z = g 0" unfolding Cart_eq forall_2 unfolding z(2) pathstart_def
+ hence "z = g 0" unfolding vec_eq_iff forall_2 unfolding z(2) pathstart_def
using assms(4)[unfolded path_image_def subset_eq mem_interval_cart,rule_format,of "g 0" 2]
unfolding mem_interval_cart apply(erule_tac x=2 in allE) using as by auto
thus thesis apply-apply(rule that[OF z(1)]) unfolding path_image_def by auto
@@ -180,7 +180,7 @@
"(interval_bij (a, b) (- 1, 1) \<circ> f) 1 $ 1 = 1"
"(interval_bij (a, b) (- 1, 1) \<circ> g) 0 $ 2 = -1"
"(interval_bij (a, b) (- 1, 1) \<circ> g) 1 $ 2 = 1"
- unfolding interval_bij_cart Cart_lambda_beta vector_component_simps o_def split_conv
+ unfolding interval_bij_cart vec_lambda_beta vector_component_simps o_def split_conv
unfolding assms[unfolded pathstart_def pathfinish_def] using as by auto qed note z=this
from z(1) guess zf unfolding image_iff .. note zf=this
from z(2) guess zg unfolding image_iff .. note zg=this
@@ -197,7 +197,7 @@
proof-
let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
{ presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
- unfolding Cart_eq forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
+ unfolding vec_eq_iff forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
{ assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
{ fix b a assume "b + u * a > a + u * b"
hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
@@ -221,7 +221,7 @@
proof-
let ?L = "\<exists>u. (x $ 1 = (1 - u) * a $ 1 + u * b $ 1 \<and> x $ 2 = (1 - u) * a $ 2 + u * b $ 2) \<and> 0 \<le> u \<and> u \<le> 1"
{ presume "?L \<Longrightarrow> ?R" "?R \<Longrightarrow> ?L" thus ?thesis unfolding closed_segment_def mem_Collect_eq
- unfolding Cart_eq forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
+ unfolding vec_eq_iff forall_2 smult_conv_scaleR[THEN sym] vector_component_simps by blast }
{ assume ?L then guess u apply-apply(erule exE)apply(erule conjE)+ . note u=this
{ fix b a assume "b + u * a > a + u * b"
hence "(1 - u) * b > (1 - u) * a" by(auto simp add:field_simps)
@@ -274,7 +274,7 @@
path_image(linepath(vector[(pathfinish g)$1,a$2 - 1])(vector[b$1 + 1,a$2 - 1])) \<union>
path_image(linepath(vector[b$1 + 1,a$2 - 1])(vector[b$1 + 1,b$2 + 3]))" using assms(1-2)
by(auto simp add: path_image_join path_linepath)
- have abab: "{a..b} \<subseteq> {?a..?b}" by(auto simp add:vector_le_def forall_2 vector_2)
+ have abab: "{a..b} \<subseteq> {?a..?b}" by(auto simp add:less_eq_vec_def forall_2 vector_2)
guess z apply(rule fashoda[of ?P1 ?P2 ?a ?b])
unfolding pathstart_join pathfinish_join pathstart_linepath pathfinish_linepath vector_2 proof-
show "path ?P1" "path ?P2" using assms by auto
@@ -318,11 +318,11 @@
qed hence "z \<in> path_image f \<or> z \<in> path_image g" using z unfolding Un_iff by blast
hence z':"z\<in>{a..b}" using assms(3-4) by auto
have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart f $ 1 \<or> z $ 1 = pathfinish f $ 1) \<Longrightarrow> (z = pathstart f \<or> z = pathfinish f)"
- unfolding Cart_eq forall_2 assms by auto
+ unfolding vec_eq_iff forall_2 assms by auto
with z' show "z\<in>path_image f" using z(1) unfolding Un_iff mem_interval_cart forall_2 apply-
apply(simp only: segment_vertical segment_horizontal vector_2) unfolding assms by auto
have "a $ 2 = z $ 2 \<Longrightarrow> (z $ 1 = pathstart g $ 1 \<or> z $ 1 = pathfinish g $ 1) \<Longrightarrow> (z = pathstart g \<or> z = pathfinish g)"
- unfolding Cart_eq forall_2 assms by auto
+ unfolding vec_eq_iff forall_2 assms by auto
with z' show "z\<in>path_image g" using z(2) unfolding Un_iff mem_interval_cart forall_2 apply-
apply(simp only: segment_vertical segment_horizontal vector_2) unfolding assms by auto
qed qed
--- 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