--- a/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Fri Sep 28 15:23:32 2012 +0200
+++ b/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space.thy Fri Sep 28 17:06:07 2012 +0200
@@ -6,7 +6,8 @@
begin
lemma delta_mult_idempotent:
- "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)
+ "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)"
+ by (cases "k=a") auto
lemma setsum_Plus:
"\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
@@ -25,34 +26,42 @@
"setsum h {..<A * B :: nat} = (\<Sum>i\<in>{..<A}. \<Sum>j\<in>{..<B}. h (j + i * B))"
unfolding sumr_group[of h B A, unfolded atLeast0LessThan, symmetric]
proof (rule setsum_cong, simp, rule setsum_reindex_cong)
- fix i show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
+ fix i
+ show "inj_on (\<lambda>j. j + i * B) {..<B}" by (auto intro!: inj_onI)
show "{i * B..<i * B + B} = (\<lambda>j. j + i * B) ` {..<B}"
proof safe
fix j assume "j \<in> {i * B..<i * B + B}"
- thus "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
+ then show "j \<in> (\<lambda>j. j + i * B) ` {..<B}"
by (auto intro!: image_eqI[of _ _ "j - i * B"])
qed simp
qed simp
+
subsection{* Basic componentwise operations on vectors. *}
instantiation vec :: (times, finite) times
begin
- definition "op * \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))"
- instance ..
+
+definition "op * \<equiv> (\<lambda> x y. (\<chi> i. (x$i) * (y$i)))"
+instance ..
+
end
instantiation vec :: (one, finite) one
begin
- definition "1 \<equiv> (\<chi> i. 1)"
- instance ..
+
+definition "1 \<equiv> (\<chi> i. 1)"
+instance ..
+
end
instantiation vec :: (ord, finite) ord
begin
- definition "x \<le> y \<longleftrightarrow> (\<forall>i. x$i \<le> y$i)"
- definition "x < y \<longleftrightarrow> (\<forall>i. x$i < y$i)"
- instance ..
+
+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. *}
@@ -100,29 +109,30 @@
definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
where "c *s x = (\<chi> i. c * (x$i))"
+
subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
method_setup vector = {*
let
val ss1 = HOL_basic_ss addsimps [@{thm setsum_addf} RS sym,
- @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
- @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
+ @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
+ @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym]
val ss2 = @{simpset} addsimps
[@{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
+ 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 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
+ (* THEN' TRY o clarify_tac HOL_cs THEN' (TRY o rtac @{thm iffI}) *)
+ THEN' asm_full_simp_tac (ss2 addsimps ths)
+in
Attrib.thms >> (fn ths => K (SIMPLE_METHOD' (vector_arith_tac ths)))
- end
+end
*} "lift trivial vector statements to real arith statements"
lemma vec_0[simp]: "vec 0 = 0" by (vector zero_vec_def)
@@ -137,12 +147,17 @@
lemma vec_cmul: "vec(c * x) = c *s vec x " by (vector vec_def)
lemma vec_neg: "vec(- x) = - vec x " by (vector vec_def)
-lemma vec_setsum: assumes fS: "finite S"
+lemma vec_setsum:
+ assumes "finite S"
shows "vec(setsum f S) = setsum (vec o f) S"
- apply (induct rule: finite_induct[OF fS])
- apply (simp)
- apply (auto simp add: vec_add)
- done
+ using assms
+proof induct
+ case empty
+ then show ?case by simp
+next
+ case insert
+ then show ?case by (auto simp add: vec_add)
+qed
text{* Obvious "component-pushing". *}
@@ -162,6 +177,7 @@
vector_smult_component vector_minus_component vector_uminus_component
vector_scaleR_component cond_component
+
subsection {* Some frequently useful arithmetic lemmas over vectors. *}
instance vec :: (semigroup_mult, finite) semigroup_mult
@@ -200,21 +216,21 @@
instance vec :: (ring_1, finite) ring_1 ..
instance vec :: (real_algebra, finite) real_algebra
- apply intro_classes
- apply (simp_all add: vec_eq_iff)
- done
+ by default (simp_all add: vec_eq_iff)
instance vec :: (real_algebra_1, finite) real_algebra_1 ..
-lemma of_nat_index:
- "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
- apply (induct n)
- apply vector
- apply vector
- done
+lemma of_nat_index: "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
+proof (induct n)
+ case 0
+ then show ?case by vector
+next
+ case Suc
+ then show ?case by vector
+qed
-lemma one_index[simp]:
- "(1 :: 'a::one ^'n)$i = 1" by vector
+lemma one_index[simp]: "(1 :: 'a::one ^'n)$i = 1"
+ by vector
instance vec :: (semiring_char_0, finite) semiring_char_0
proof
@@ -227,7 +243,7 @@
instance vec :: (semiring_numeral, finite) semiring_numeral ..
lemma numeral_index [simp]: "numeral w $ i = numeral w"
- by (induct w, simp_all only: numeral.simps vector_add_component one_index)
+ by (induct w) (simp_all only: numeral.simps vector_add_component one_index)
lemma neg_numeral_index [simp]: "neg_numeral w $ i = neg_numeral w"
by (simp only: neg_numeral_def vector_uminus_component numeral_index)
@@ -291,7 +307,13 @@
lemma setsum_component [simp]:
fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
- by (cases "finite S", induct S set: finite, simp_all)
+proof (cases "finite S")
+ case True
+ then show ?thesis by induct simp_all
+next
+ case False
+ then show ?thesis by simp
+qed
lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
by (simp add: vec_eq_iff)
@@ -306,7 +328,7 @@
fixes f:: "'a \<Rightarrow> real ^'n"
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 CARD('n) * e"
-proof-
+proof -
let ?d = "real CARD('n)"
let ?nf = "\<lambda>x. norm (f x)"
let ?U = "UNIV :: 'n set"
@@ -314,7 +336,9 @@
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_cart)
+ apply (rule setsum_mono)
+ apply (rule norm_le_l1_cart)
+ done
also have "\<dots> \<le> 2 * ?d * e"
unfolding th0 th1
proof(rule setsum_bounded)
@@ -333,7 +357,8 @@
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
+ using fP thp0 apply auto
+ done
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
@@ -344,8 +369,10 @@
lemma split_dimensions'[consumes 1]:
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]] .
+ 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::euclidean_space)"
@@ -356,12 +383,14 @@
"(\<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)
- fix i :: 'b and j assume "j < DIM('a)"
+ fix i :: 'b and j
+ assume "j < DIM('a)"
note linear_less_than_times[OF pi'_range[of i] this]
moreover assume "?l"
ultimately show "P (j + \<pi>' i * DIM('a))" by auto
next
- fix i j assume "i < CARD('b)" "j < DIM('a)" and "?r"
+ fix i j
+ assume "i < CARD('b)" "j < DIM('a)" and "?r"
from `?r`[rule_format, OF `j < DIM('a)`, of "\<pi> i"] `i < CARD('b)`
show "P (j + i * DIM('a))" by simp
qed
@@ -408,9 +437,11 @@
have "i = j"
proof (cases rule: linorder_cases)
- assume "i < j" from eq[OF this `x < A` *] show "i = j" by simp
+ assume "i < j"
+ from eq[OF this `x < A` *] show "i = j" by simp
next
- assume "j < i" from eq[OF this `y < A` *[symmetric]] show "i = j" by simp
+ assume "j < i"
+ from eq[OF this `y < A` *[symmetric]] show "i = j" by simp
qed simp
thus "x = y \<and> i = j" using * by simp
qed simp
@@ -424,6 +455,7 @@
unfolding less_vec_def apply(subst eucl_less) by (simp add: cart_simps)
qed
+
subsection{* Basis vectors in coordinate directions. *}
definition "cart_basis k = (\<chi> i. if i = k then 1 else 0)"
@@ -435,7 +467,8 @@
shows "norm (cart_basis k :: real ^'n) = 1"
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
+ using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"] apply auto
+ done
lemma norm_basis_1: "norm(cart_basis 1 :: real ^'n::{finite,one}) = 1"
by (rule norm_basis)
@@ -443,10 +476,11 @@
lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
by (rule exI[where x="c *\<^sub>R cart_basis arbitrary"]) simp
-lemma vector_choose_dist: assumes e: "0 <= e"
+lemma vector_choose_dist:
+ assumes e: "0 <= e"
shows "\<exists>(y::real^'n). dist x y = e"
-proof-
- from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e"
+proof -
+ from vector_choose_size[OF e] obtain c:: "real ^'n" where "norm c = e"
by blast
then have "dist x (x - c) = e" by (simp add: dist_norm)
then show ?thesis by blast
@@ -455,23 +489,22 @@
lemma basis_inj[intro]: "inj (cart_basis :: 'n \<Rightarrow> real ^'n)"
by (simp add: inj_on_def vec_eq_iff)
-lemma basis_expansion:
- "setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
- by (auto simp add: vec_eq_iff if_distrib setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
+lemma basis_expansion: "setsum (\<lambda>i. (x$i) *s cart_basis i) UNIV = (x::('a::ring_1) ^'n)"
+ (is "?lhs = ?rhs" is "setsum ?f ?S = _")
+ by (auto simp add: 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 scaleR_vec_def by simp
-lemma basis_expansion':
- "setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"
+lemma basis_expansion': "setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) UNIV = x"
by (rule basis_expansion [where 'a=real, unfolded smult_conv_scaleR])
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$i)"
by (simp add: vec_eq_iff setsum_delta if_distrib cong del: if_weak_cong)
-lemma dot_basis:
- shows "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"
+lemma dot_basis: "cart_basis i \<bullet> x = x$i" "x \<bullet> (cart_basis i) = (x$i)"
by (auto simp add: inner_vec_def cart_basis_def cond_application_beta if_distrib setsum_delta
cong del: if_weak_cong)
@@ -485,8 +518,7 @@
lemma basis_eq_0: "cart_basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
by (auto simp add: vec_eq_iff)
-lemma basis_nonzero:
- shows "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
+lemma basis_nonzero: "cart_basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
by (simp add: basis_eq_0)
text {* some lemmas to map between Eucl and Cart *}
@@ -496,25 +528,22 @@
subsection {* Orthogonality on cartesian products *}
-lemma orthogonal_basis:
- shows "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"
+lemma orthogonal_basis: "orthogonal (cart_basis i) x \<longleftrightarrow> x$i = (0::real)"
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:
- shows "orthogonal (cart_basis i :: real^'n) (cart_basis j) \<longleftrightarrow> i \<noteq> j"
+lemma orthogonal_basis_basis: "orthogonal (cart_basis i :: real^'n) (cart_basis j) \<longleftrightarrow> i \<noteq> j"
unfolding orthogonal_basis[of i] basis_component[of j] by simp
subsection {* Linearity on cartesian products *}
lemma linear_vmul_component:
- assumes lf: "linear f"
+ assumes "linear f"
shows "linear (\<lambda>x. f x $ k *\<^sub>R v)"
- using lf
- by (auto simp add: linear_def algebra_simps)
+ using assms by (auto simp add: linear_def algebra_simps)
-subsection{* Adjoints on cartesian products *}
+subsection {* Adjoints on cartesian products *}
text {* TODO: The following lemmas about adjoints should hold for any
Hilbert space (i.e. complete inner product space).
@@ -525,26 +554,26 @@
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
-proof-
+proof -
let ?N = "UNIV :: 'n set"
let ?M = "UNIV :: 'm set"
have fN: "finite ?N" by simp
have fM: "finite ?M" by simp
- {fix y:: "real ^ 'm"
+ { fix y:: "real ^ 'm"
let ?w = "(\<chi> i. (f (cart_basis i) \<bullet> y)) :: real ^ 'n"
- {fix x
+ { fix x
have "f x \<bullet> y = f (setsum (\<lambda>i. (x$i) *\<^sub>R cart_basis i) ?N) \<bullet> y"
by (simp only: basis_expansion')
also have "\<dots> = (setsum (\<lambda>i. (x$i) *\<^sub>R f (cart_basis i)) ?N) \<bullet> y"
unfolding linear_setsum[OF lf fN]
by (simp add: linear_cmul[OF lf])
finally have "f x \<bullet> y = x \<bullet> ?w"
- apply (simp only: )
- apply (simp add: inner_vec_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
- done}
+ by (simp add: inner_vec_def setsum_left_distrib
+ setsum_right_distrib setsum_commute[of _ ?M ?N] field_simps)
+ }
}
- then show ?thesis unfolding adjoint_def
- some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
+ then show ?thesis
+ unfolding adjoint_def some_eq_ex[of "\<lambda>f'. \<forall>x y. f x \<bullet> y = x \<bullet> f' y"]
using choice_iff[of "\<lambda>a b. \<forall>x. f x \<bullet> a = x \<bullet> b "]
by metis
qed
@@ -566,7 +595,7 @@
fixes f:: "real ^'n \<Rightarrow> real ^'m"
assumes lf: "linear f"
shows "x \<bullet> adjoint f y = f x \<bullet> y"
- and "adjoint f y \<bullet> x = y \<bullet> f x"
+ and "adjoint f y \<bullet> x = y \<bullet> f x"
by (simp_all add: adjoint_works[OF lf] inner_commute)
lemma adjoint_adjoint:
@@ -580,13 +609,16 @@
text{* Matrix notation. NB: an MxN matrix is of type @{typ "'a^'n^'m"}, not @{typ "'a^'m^'n"} *}
-definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m" (infixl "**" 70)
+definition matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
+ (infixl "**" 70)
where "m ** m' == (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
-definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm" (infixl "*v" 70)
+definition matrix_vector_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"
+ (infixl "*v" 70)
where "m *v x \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
-definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n " (infixl "v*" 70)
+definition vector_matrix_mult :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "
+ (infixl "v*" 70)
where "v v* m == (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (v$i)) (UNIV :: 'm set)) :: 'a^'n"
definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
@@ -606,7 +638,9 @@
shows "mat 1 ** A = A"
apply (simp add: matrix_matrix_mult_def mat_def)
apply vector
- by (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite] mult_1_left mult_zero_left if_True UNIV_I)
+ apply (auto simp only: if_distrib cond_application_beta setsum_delta'[OF finite]
+ mult_1_left mult_zero_left if_True UNIV_I)
+ done
lemma matrix_mul_rid:
@@ -614,7 +648,9 @@
shows "A ** mat 1 = A"
apply (simp add: matrix_matrix_mult_def mat_def)
apply vector
- by (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite] mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
+ apply (auto simp only: if_distrib cond_application_beta setsum_delta[OF finite]
+ mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
+ done
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
@@ -623,17 +659,19 @@
done
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
- apply (vector matrix_matrix_mult_def matrix_vector_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
+ apply (vector matrix_matrix_mult_def matrix_vector_mult_def
+ setsum_right_distrib setsum_left_distrib mult_assoc)
apply (subst setsum_commute)
apply simp
done
lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
apply (vector matrix_vector_mult_def mat_def)
- by (simp add: if_distrib cond_application_beta
- setsum_delta' cong del: if_weak_cong)
+ apply (simp add: if_distrib cond_application_beta setsum_delta' cong del: if_weak_cong)
+ done
-lemma matrix_transpose_mul: "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
+lemma matrix_transpose_mul:
+ "transpose(A ** B) = transpose B ** transpose (A::'a::comm_semiring_1^_^_)"
by (simp add: matrix_matrix_mult_def transpose_def vec_eq_iff mult_commute)
lemma matrix_eq:
@@ -645,16 +683,18 @@
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)
+ apply (auto simp add: cart_basis_def if_distrib cond_application_beta
+ setsum_delta[OF finite] cong del: if_weak_cong)
+ done
-lemma matrix_vector_mul_component:
- shows "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
+lemma matrix_vector_mul_component: "((A::real^_^_) *v x)$k = (A$k) \<bullet> x"
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_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
apply (subst setsum_commute)
- by simp
+ apply simp
+ done
lemma transpose_mat: "transpose (mat n) = mat n"
by (vector transpose_def mat_def)
@@ -673,28 +713,31 @@
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)
+ by (auto simp add: rows_def columns_def row_transpose intro: set_eqI)
-lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A" by (metis transpose_transpose rows_transpose)
+lemma columns_transpose: "columns(transpose (A::'a::semiring_1^_^_)) = rows A"
+ by (metis transpose_transpose rows_transpose)
text{* Two sometimes fruitful ways of looking at matrix-vector multiplication. *}
lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
by (simp add: matrix_vector_mult_def inner_vec_def)
-lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
+lemma matrix_mult_vsum:
+ "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult_commute)
lemma vector_componentwise:
"(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (cart_basis i :: 'a^'n)$j) (UNIV :: 'n set))"
apply (subst basis_expansion[symmetric])
- by (vector vec_eq_iff setsum_component)
+ apply (vector vec_eq_iff setsum_component)
+ done
lemma linear_componentwise:
fixes f:: "real ^'m \<Rightarrow> real ^ _"
assumes lf: "linear f"
shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (cart_basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
-proof-
+proof -
let ?M = "(UNIV :: 'm set)"
let ?N = "(UNIV :: 'n set)"
have fM: "finite ?M" by simp
@@ -702,48 +745,57 @@
unfolding vector_smult_component[symmetric] smult_conv_scaleR
unfolding setsum_component[of "(\<lambda>i.(x$i) *\<^sub>R f (cart_basis i :: real^'m))" ?M]
..
- then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' ..
+ then show ?thesis
+ unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion' ..
qed
text{* Inverse matrices (not necessarily square) *}
-definition "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
+definition
+ "invertible(A::'a::semiring_1^'n^'m) \<longleftrightarrow> (\<exists>A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
-definition "matrix_inv(A:: 'a::semiring_1^'n^'m) =
- (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
+definition
+ "matrix_inv(A:: 'a::semiring_1^'n^'m) =
+ (SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
text{* Correspondence between matrices and linear operators. *}
-definition matrix:: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
-where "matrix f = (\<chi> i j. (f(cart_basis j))$i)"
+definition matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
+ where "matrix f = (\<chi> i j. (f(cart_basis j))$i)"
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
- by (simp add: linear_def matrix_vector_mult_def vec_eq_iff field_simps setsum_right_distrib setsum_addf)
+ by (simp add: linear_def matrix_vector_mult_def vec_eq_iff
+ field_simps setsum_right_distrib setsum_addf)
-lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::real ^ 'n)"
-apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
-apply clarify
-apply (rule linear_componentwise[OF lf, symmetric])
-done
+lemma matrix_works:
+ assumes lf: "linear f"
+ shows "matrix f *v x = f (x::real ^ 'n)"
+ apply (simp add: matrix_def matrix_vector_mult_def vec_eq_iff mult_commute)
+ apply clarify
+ apply (rule linear_componentwise[OF lf, symmetric])
+ done
-lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))" by (simp add: ext matrix_works)
+lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::real ^ 'n))"
+ by (simp add: ext matrix_works)
lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: real ^ 'n)) = A"
by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
lemma matrix_compose:
assumes lf: "linear (f::real^'n \<Rightarrow> real^'m)"
- and lg: "linear (g::real^'m \<Rightarrow> real^_)"
+ and lg: "linear (g::real^'m \<Rightarrow> real^_)"
shows "matrix (g o f) = matrix g ** matrix f"
using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
- by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
+ by (simp add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
-lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
+lemma matrix_vector_column:
+ "(A::'a::comm_semiring_1^'n^_) *v x = setsum (\<lambda>i. (x$i) *s ((transpose A)$i)) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def transpose_def 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_vec_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
@@ -751,7 +803,10 @@
lemma matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
shows "matrix(adjoint f) = transpose(matrix f)"
apply (subst matrix_vector_mul[OF lf])
- unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
+ unfolding adjoint_matrix matrix_of_matrix_vector_mul
+ apply rule
+ done
+
subsection {* lambda skolemization on cartesian products *}
@@ -759,15 +814,15 @@
lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
(\<exists>x::'a ^ 'n. \<forall>i. P i (x $ i))" (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
+proof -
let ?S = "(UNIV :: 'n set)"
- {assume H: "?rhs"
- then have ?lhs by auto}
+ { assume H: "?rhs"
+ then have ?lhs by auto }
moreover
- {assume H: "?lhs"
+ { assume H: "?lhs"
then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
- {fix i
+ { fix i
from f have "P i (f i)" by metis
then have "P i (?x $ i)" by auto
}
@@ -776,68 +831,69 @@
ultimately show ?thesis by metis
qed
+
subsection {* Standard bases are a spanning set, and obviously finite. *}
lemma span_stdbasis:"span {cart_basis i :: real^'n | i. i \<in> (UNIV :: 'n set)} = UNIV"
-apply (rule set_eqI)
-apply auto
-apply (subst basis_expansion'[symmetric])
-apply (rule span_setsum)
-apply simp
-apply auto
-apply (rule span_mul)
-apply (rule span_superset)
-apply auto
-done
+ apply (rule set_eqI)
+ apply auto
+ apply (subst basis_expansion'[symmetric])
+ apply (rule span_setsum)
+ apply simp
+ apply auto
+ apply (rule span_mul)
+ apply (rule span_superset)
+ apply auto
+ done
lemma finite_stdbasis: "finite {cart_basis i ::real^'n |i. i\<in> (UNIV:: 'n set)}" (is "finite ?S")
-proof-
- have eq: "?S = cart_basis ` UNIV" by blast
- show ?thesis unfolding eq by auto
+proof -
+ have "?S = cart_basis ` UNIV" by blast
+ then show ?thesis by auto
qed
lemma card_stdbasis: "card {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)} = CARD('n)" (is "card ?S = _")
-proof-
- have eq: "?S = cart_basis ` UNIV" by blast
- show ?thesis unfolding eq using card_image[OF basis_inj] by simp
+proof -
+ have "?S = cart_basis ` UNIV" by blast
+ then show ?thesis using card_image[OF basis_inj] by simp
qed
-
lemma independent_stdbasis_lemma:
assumes x: "(x::real ^ 'n) \<in> span (cart_basis ` S)"
- and iS: "i \<notin> S"
+ and iS: "i \<notin> S"
shows "(x$i) = 0"
-proof-
+proof -
let ?U = "UNIV :: 'n set"
let ?B = "cart_basis ` S"
let ?P = "{(x::real^_). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0}"
- {fix x::"real^_" assume xS: "x\<in> ?B"
- from xS have "x \<in> ?P" by auto}
- moreover
- have "subspace ?P"
- by (auto simp add: subspace_def)
- ultimately show ?thesis
- using x span_induct[of x ?B ?P] iS by blast
+ { fix x::"real^_" assume xS: "x\<in> ?B"
+ from xS have "x \<in> ?P" by auto }
+ moreover
+ have "subspace ?P"
+ by (auto simp add: subspace_def)
+ ultimately show ?thesis
+ using x span_induct[of x ?B ?P] iS by blast
qed
lemma independent_stdbasis: "independent {cart_basis i ::real^'n |i. i\<in> (UNIV :: 'n set)}"
-proof-
+proof -
let ?I = "UNIV :: 'n set"
let ?b = "cart_basis :: _ \<Rightarrow> real ^'n"
let ?B = "?b ` ?I"
- have eq: "{?b i|i. i \<in> ?I} = ?B"
- by auto
- {assume d: "dependent ?B"
+ have eq: "{?b i|i. i \<in> ?I} = ?B" by auto
+ { assume d: "dependent ?B"
then obtain k where k: "k \<in> ?I" "?b k \<in> span (?B - {?b k})"
unfolding dependent_def by auto
have eq1: "?B - {?b k} = ?B - ?b ` {k}" by simp
have eq2: "?B - {?b k} = ?b ` (?I - {k})"
unfolding eq1
apply (rule inj_on_image_set_diff[symmetric])
- apply (rule basis_inj) using k(1) by auto
+ apply (rule basis_inj) using k(1)
+ apply auto
+ done
from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
from independent_stdbasis_lemma[OF th0, of k, simplified]
- have False by simp}
+ have False by simp }
then show ?thesis unfolding eq dependent_def ..
qed
@@ -846,27 +902,31 @@
lemma linear_eq_stdbasis_cart:
assumes lf: "linear (f::real^'m \<Rightarrow> _)" and lg: "linear g"
- and fg: "\<forall>i. f (cart_basis i) = g(cart_basis i)"
+ and fg: "\<forall>i. f (cart_basis i) = g(cart_basis i)"
shows "f = g"
-proof-
+proof -
let ?U = "UNIV :: 'm set"
let ?I = "{cart_basis i:: real^'m|i. i \<in> ?U}"
- {fix x assume x: "x \<in> (UNIV :: (real^'m) set)"
+ { fix x assume x: "x \<in> (UNIV :: (real^'m) set)"
from equalityD2[OF span_stdbasis]
have IU: " (UNIV :: (real^'m) set) \<subseteq> span ?I" by blast
from linear_eq[OF lf lg IU] fg x
- have "f x = g x" unfolding Ball_def mem_Collect_eq by metis}
+ have "f x = g x" unfolding Ball_def mem_Collect_eq by metis
+ }
then show ?thesis by auto
qed
lemma bilinear_eq_stdbasis_cart:
assumes bf: "bilinear (f:: real^'m \<Rightarrow> real^'n \<Rightarrow> _)"
- and bg: "bilinear g"
- and fg: "\<forall>i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)"
+ and bg: "bilinear g"
+ and fg: "\<forall>i j. f (cart_basis i) (cart_basis j) = g (cart_basis i) (cart_basis j)"
shows "f = g"
-proof-
- from fg have th: "\<forall>x \<in> {cart_basis i| i. i\<in> (UNIV :: 'm set)}. \<forall>y\<in> {cart_basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y" by blast
- from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by blast
+proof -
+ from fg have th: "\<forall>x \<in> {cart_basis i| i. i\<in> (UNIV :: 'm set)}.
+ \<forall>y\<in> {cart_basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y"
+ by blast
+ from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th]
+ show ?thesis by blast
qed
lemma left_invertible_transpose:
@@ -878,21 +938,21 @@
by (metis matrix_transpose_mul transpose_mat transpose_transpose)
lemma matrix_left_invertible_injective:
-"(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
-proof-
- {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
+ "(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
+proof -
+ { fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
from xy have "B*v (A *v x) = B *v (A*v y)" by simp
hence "x = y"
- unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid .}
+ unfolding matrix_vector_mul_assoc B matrix_vector_mul_lid . }
moreover
- {assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
+ { assume A: "\<forall>x y. A *v x = A *v y \<longrightarrow> x = y"
hence i: "inj (op *v A)" unfolding inj_on_def by auto
from linear_injective_left_inverse[OF matrix_vector_mul_linear i]
obtain g where g: "linear g" "g o op *v A = id" by blast
have "matrix g ** A = mat 1"
unfolding matrix_eq matrix_vector_mul_lid matrix_vector_mul_assoc[symmetric] matrix_works[OF g(1)]
using g(2) by (simp add: fun_eq_iff)
- then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast}
+ then have "\<exists>B. (B::real ^'m^'n) ** A = mat 1" by blast }
ultimately show ?thesis by blast
qed
@@ -903,15 +963,16 @@
by (simp add: inj_on_def)
lemma matrix_right_invertible_surjective:
-"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
-proof-
- {fix B :: "real ^'m^'n" assume AB: "A ** B = mat 1"
- {fix x :: "real ^ 'm"
+ "(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
+proof -
+ { fix B :: "real ^'m^'n"
+ assume AB: "A ** B = mat 1"
+ { fix x :: "real ^ 'm"
have "A *v (B *v x) = x"
- by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB)}
+ by (simp add: matrix_vector_mul_lid matrix_vector_mul_assoc AB) }
hence "surj (op *v A)" unfolding surj_def by metis }
moreover
- {assume sf: "surj (op *v A)"
+ { assume sf: "surj (op *v A)"
from linear_surjective_right_inverse[OF matrix_vector_mul_linear sf]
obtain g:: "real ^'m \<Rightarrow> real ^'n" where g: "linear g" "op *v A o g = id"
by blast
@@ -928,13 +989,14 @@
lemma matrix_left_invertible_independent_columns:
fixes A :: "real^'n^'m"
- shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof-
+ shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow>
+ (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof -
let ?U = "UNIV :: 'n set"
- {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
- {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
- and i: "i \<in> ?U"
+ { assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
+ { fix c i
+ assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0" and i: "i \<in> ?U"
let ?x = "\<chi> i. c i"
have th0:"A *v ?x = 0"
using c
@@ -942,82 +1004,95 @@
by auto
from k[rule_format, OF th0] i
have "c i = 0" by (vector vec_eq_iff)}
- hence ?rhs by blast}
+ hence ?rhs by blast }
moreover
- {assume H: ?rhs
- {fix x assume x: "A *v x = 0"
+ { assume H: ?rhs
+ { fix x assume x: "A *v x = 0"
let ?c = "\<lambda>i. ((x$i ):: real)"
from H[rule_format, of ?c, unfolded matrix_mult_vsum[symmetric], OF x]
- have "x = 0" by vector}}
+ have "x = 0" by vector }
+ }
ultimately show ?thesis unfolding matrix_left_invertible_ker by blast
qed
lemma matrix_right_invertible_independent_rows:
fixes A :: "real^'n^'m"
- shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
+ shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow>
+ (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
unfolding left_invertible_transpose[symmetric]
matrix_left_invertible_independent_columns
by (simp add: column_transpose)
lemma matrix_right_invertible_span_columns:
- "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
-proof-
+ "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow>
+ span (columns A) = UNIV" (is "?lhs = ?rhs")
+proof -
let ?U = "UNIV :: 'm set"
have fU: "finite ?U" by simp
have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
- apply (subst eq_commute) ..
+ apply (subst eq_commute)
+ apply rule
+ done
have rhseq: "?rhs \<longleftrightarrow> (\<forall>x. x \<in> span (columns A))" by blast
- {assume h: ?lhs
- {fix x:: "real ^'n"
- from h[unfolded lhseq, rule_format, of x] obtain y:: "real ^'m"
- where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
- have "x \<in> span (columns A)"
- unfolding y[symmetric]
- apply (rule span_setsum[OF fU])
- apply clarify
- unfolding smult_conv_scaleR
- apply (rule span_mul)
- apply (rule span_superset)
- unfolding columns_def
- by blast}
- then have ?rhs unfolding rhseq by blast}
+ { assume h: ?lhs
+ { fix x:: "real ^'n"
+ from h[unfolded lhseq, rule_format, of x] obtain y :: "real ^'m"
+ where y: "setsum (\<lambda>i. (y$i) *s column i A) ?U = x" by blast
+ have "x \<in> span (columns A)"
+ unfolding y[symmetric]
+ apply (rule span_setsum[OF fU])
+ apply clarify
+ unfolding smult_conv_scaleR
+ apply (rule span_mul)
+ apply (rule span_superset)
+ unfolding columns_def
+ apply blast
+ done
+ }
+ then have ?rhs unfolding rhseq by blast }
moreover
- {assume h:?rhs
+ { assume h:?rhs
let ?P = "\<lambda>(y::real ^'n). \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y"
- {fix y have "?P y"
- proof(rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR])
+ { fix y
+ have "?P y"
+ proof (rule span_induct_alt[of ?P "columns A", folded smult_conv_scaleR])
show "\<exists>x\<Colon>real ^ 'm. setsum (\<lambda>i. (x$i) *s column i A) ?U = 0"
by (rule exI[where x=0], simp)
next
- fix c y1 y2 assume y1: "y1 \<in> columns A" and y2: "?P y2"
+ fix c y1 y2
+ assume y1: "y1 \<in> columns A" and y2: "?P y2"
from y1 obtain i where i: "i \<in> ?U" "y1 = column i A"
unfolding columns_def by blast
from y2 obtain x:: "real ^'m" where
x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
show "?P (c*s y1 + y2)"
- proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib right_distrib cond_application_beta cong del: if_weak_cong)
- fix j
- have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
- else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" using i(1)
- by (simp add: field_simps)
- have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
- else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
- apply (rule setsum_cong[OF refl])
- using th by blast
- also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
- by (simp add: setsum_addf)
- also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
- unfolding setsum_delta[OF fU]
- using i(1) by simp
- finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
- else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
- qed
- next
- show "y \<in> span (columns A)" unfolding h by blast
- qed}
- then have ?lhs unfolding lhseq ..}
+ proof (rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] if_distrib right_distrib cond_application_beta cong del: if_weak_cong)
+ fix j
+ have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
+ else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))"
+ using i(1) by (simp add: field_simps)
+ have "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
+ else (x$xa) * ((column xa A$j))) ?U = setsum (\<lambda>xa. (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))) ?U"
+ apply (rule setsum_cong[OF refl])
+ using th apply blast
+ done
+ also have "\<dots> = setsum (\<lambda>xa. if xa = i then c * ((column i A)$j) else 0) ?U + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
+ by (simp add: setsum_addf)
+ also have "\<dots> = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U"
+ unfolding setsum_delta[OF fU]
+ using i(1) by simp
+ finally show "setsum (\<lambda>xa. if xa = i then (c + (x$i)) * ((column xa A)$j)
+ else (x$xa) * ((column xa A$j))) ?U = c * ((column i A)$j) + setsum (\<lambda>xa. ((x$xa) * ((column xa A)$j))) ?U" .
+ qed
+ next
+ show "y \<in> span (columns A)"
+ unfolding h by blast
+ qed
+ }
+ then have ?lhs unfolding lhseq ..
+ }
ultimately show ?thesis by blast
qed
@@ -1026,29 +1101,34 @@
unfolding right_invertible_transpose[symmetric]
unfolding columns_transpose[symmetric]
unfolding matrix_right_invertible_span_columns
- ..
+ ..
text {* The same result in terms of square matrices. *}
lemma matrix_left_right_inverse:
fixes A A' :: "real ^'n^'n"
shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
-proof-
- {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
+proof -
+ { fix A A' :: "real ^'n^'n"
+ assume AA': "A ** A' = mat 1"
have sA: "surj (op *v A)"
unfolding surj_def
apply clarify
apply (rule_tac x="(A' *v y)" in exI)
- by (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
+ apply (simp add: matrix_vector_mul_assoc AA' matrix_vector_mul_lid)
+ done
from linear_surjective_isomorphism[OF matrix_vector_mul_linear sA]
obtain f' :: "real ^'n \<Rightarrow> real ^'n"
where f': "linear f'" "\<forall>x. f' (A *v x) = x" "\<forall>x. A *v f' x = x" by blast
have th: "matrix f' ** A = mat 1"
- by (simp add: matrix_eq matrix_works[OF f'(1)] matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
+ by (simp add: matrix_eq matrix_works[OF f'(1)]
+ matrix_vector_mul_assoc[symmetric] matrix_vector_mul_lid f'(2)[rule_format])
hence "(matrix f' ** A) ** A' = mat 1 ** A'" by simp
- hence "matrix f' = A'" by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
+ hence "matrix f' = A'"
+ by (simp add: matrix_mul_assoc[symmetric] AA' matrix_mul_rid matrix_mul_lid)
hence "matrix f' ** A = A' ** A" by simp
- hence "A' ** A = mat 1" by (simp add: th)}
+ hence "A' ** A = mat 1" by (simp add: th)
+ }
then show ?thesis by blast
qed
@@ -1058,67 +1138,64 @@
definition "columnvector v = (\<chi> i j. (v$i))"
-lemma transpose_columnvector:
- "transpose(columnvector v) = rowvector v"
+lemma transpose_columnvector: "transpose(columnvector v) = rowvector v"
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 vec_eq_iff)
-lemma dot_rowvector_columnvector:
- "columnvector (A *v v) = A ** columnvector v"
+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"
+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_vec_def)
lemma dot_matrix_vector_mul:
fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
shows "(A *v x) \<bullet> (B *v y) =
(((rowvector x :: real^'n^1) ** ((transpose A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
-unfolding dot_matrix_product transpose_columnvector[symmetric]
- dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
+ unfolding dot_matrix_product transpose_columnvector[symmetric]
+ dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
lemma infnorm_cart:"infnorm (x::real^'n) = Sup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
unfolding infnorm_def apply(rule arg_cong[where f=Sup]) apply safe
apply(rule_tac x="\<pi> i" in exI) defer
- apply(rule_tac x="\<pi>' i" in exI) unfolding nth_conv_component using pi'_range by auto
+ apply(rule_tac x="\<pi>' i" in exI)
+ unfolding nth_conv_component
+ using pi'_range apply auto
+ done
-lemma infnorm_set_image_cart:
- "{abs(x$i) |i. i\<in> (UNIV :: _ set)} =
+lemma infnorm_set_image_cart: "{abs(x$i) |i. i\<in> (UNIV :: _ set)} =
(\<lambda>i. abs(x$i)) ` (UNIV)" by blast
lemma infnorm_set_lemma_cart:
- shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"
- and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
- unfolding infnorm_set_image_cart
- by auto
+ "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> (UNIV :: 'n set)}"
+ "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
+ unfolding infnorm_set_image_cart by auto
-lemma component_le_infnorm_cart:
- shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
+lemma component_le_infnorm_cart: "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
unfolding nth_conv_component
using component_le_infnorm[of x] .
-lemma continuous_component:
- shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
+lemma continuous_component: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x $ i)"
unfolding continuous_def by (rule tendsto_vec_nth)
-lemma continuous_on_component:
- shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
+lemma continuous_on_component: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x $ i)"
unfolding continuous_on_def by (fast intro: tendsto_vec_nth)
lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i. 0 \<le>x$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le)
lemma bounded_component_cart: "bounded s \<Longrightarrow> bounded ((\<lambda>x. x $ i) ` s)"
-unfolding bounded_def
-apply clarify
-apply (rule_tac x="x $ i" in exI)
-apply (rule_tac x="e" in exI)
-apply clarify
-apply (rule order_trans [OF dist_vec_nth_le], simp)
-done
+ unfolding bounded_def
+ apply clarify
+ apply (rule_tac x="x $ i" in exI)
+ apply (rule_tac x="e" in exI)
+ apply clarify
+ apply (rule order_trans [OF dist_vec_nth_le], simp)
+ done
lemma compact_lemma_cart:
fixes f :: "nat \<Rightarrow> 'a::heine_borel ^ 'n"
@@ -1127,24 +1204,35 @@
\<exists>l r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
proof
- fix d::"'n set" have "finite d" by simp
+ fix d :: "'n set"
+ have "finite d" by simp
thus "\<exists>l::'a ^ 'n. \<exists>r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) $ i) (l $ i) < e) sequentially)"
- proof(induct d) case empty thus ?case unfolding subseq_def by auto
- next case (insert k d)
- have s': "bounded ((\<lambda>x. x $ k) ` s)" using `bounded s` by (rule bounded_component_cart)
- obtain l1::"'a^'n" and r1 where r1:"subseq r1" and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
+ proof (induct d)
+ case empty
+ thus ?case unfolding subseq_def by auto
+ next
+ case (insert k d)
+ have s': "bounded ((\<lambda>x. x $ k) ` s)"
+ using `bounded s` by (rule bounded_component_cart)
+ obtain l1::"'a^'n" and r1 where r1:"subseq r1"
+ and lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
using insert(3) by auto
- have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s" using `\<forall>n. f n \<in> s` by simp
- obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
+ have f': "\<forall>n. f (r1 n) $ k \<in> (\<lambda>x. x $ k) ` s"
+ using `\<forall>n. f n \<in> s` by simp
+ obtain l2 r2 where r2: "subseq r2"
+ and lr2: "((\<lambda>i. f (r1 (r2 i)) $ k) ---> l2) sequentially"
using bounded_imp_convergent_subsequence[OF s' f'] unfolding o_def by auto
- def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
+ def r \<equiv> "r1 \<circ> r2"
+ have r: "subseq r"
using r1 and r2 unfolding r_def o_def subseq_def by auto
moreover
def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::'a^'n"
- { fix e::real assume "e>0"
- from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially" by blast
- from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially" by (rule tendstoD)
+ { fix e :: real assume "e > 0"
+ from lr1 `e>0` have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) $ i) (l1 $ i) < e) sequentially"
+ by blast
+ from lr2 `e>0` have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) $ k) l2 < e) sequentially"
+ by (rule tendstoD)
from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) $ i) (l1 $ i) < e) sequentially"
by (rule eventually_subseq)
have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) $ i) (l $ i) < e) sequentially"
@@ -1159,16 +1247,17 @@
fix s :: "('a ^ 'b) set" and f :: "nat \<Rightarrow> 'a ^ 'b"
assume s: "bounded s" and f: "\<forall>n. f n \<in> s"
then obtain l r where r: "subseq r"
- and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
+ and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>UNIV. dist (f (r n) $ i) (l $ i) < e) sequentially"
using compact_lemma_cart [OF s f] by blast
let ?d = "UNIV::'b set"
{ fix e::real assume "e>0"
hence "0 < e / (real_of_nat (card ?d))"
- using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
+ using zero_less_card_finite divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
with l have "eventually (\<lambda>n. \<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))) sequentially"
by simp
moreover
- { fix n assume n: "\<forall>i. dist (f (r n) $ i) (l $ i) < e / (real_of_nat (card ?d))"
+ { 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_vec_def using zero_le_dist by (rule setL2_le_setsum)
also have "\<dots> < (\<Sum>i\<in>?d. e / (real_of_nat (card ?d)))"
@@ -1178,28 +1267,31 @@
ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
by (rule eventually_elim1)
}
- hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
+ hence "((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
qed
-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}"
+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 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 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)"
+ and "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 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
- "({a .. b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
-proof-
+lemma interval_eq_empty_cart:
+ fixes a :: "real^'n"
+ shows "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1)
+ and "({a .. b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
+proof -
{ fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval_cart by auto
hence "a$i < b$i" by auto
- hence False using as by auto }
+ hence False using as by auto }
moreover
{ assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
@@ -1207,14 +1299,14 @@
have "a$i < b$i" using as[THEN spec[where x=i]] by auto
hence "a$i < ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i < b$i"
unfolding vector_smult_component and vector_add_component
- by auto }
- hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
+ by auto }
+ hence "{a <..< b} \<noteq> {}" using mem_interval_cart(1)[of "?x" a b] by auto }
ultimately show ?th1 by blast
{ fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval_cart by auto
hence "a$i \<le> b$i" by auto
- hence False using as by auto }
+ hence False using as by auto }
moreover
{ assume as:"\<forall>i. \<not> (b$i < a$i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
@@ -1222,37 +1314,41 @@
have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
hence "a$i \<le> ((1/2) *\<^sub>R (a+b)) $ i" "((1/2) *\<^sub>R (a+b)) $ i \<le> b$i"
unfolding vector_smult_component and vector_add_component
- by auto }
+ by auto }
hence "{a .. b} \<noteq> {}" using mem_interval_cart(2)[of "?x" a b] by auto }
ultimately show ?th2 by blast
qed
-lemma interval_ne_empty_cart: fixes a :: "real^'n" shows
- "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and
- "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
+lemma interval_ne_empty_cart:
+ fixes a :: "real^'n"
+ shows "{a .. b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)"
+ and "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
unfolding interval_eq_empty_cart[of a b] by (auto simp add: not_less not_le)
(* BH: Why doesn't just "auto" work here? *)
-lemma subset_interval_imp_cart: fixes a :: "real^'n" shows
- "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
- "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
- "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
- "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
+lemma subset_interval_imp_cart:
+ fixes a :: "real^'n"
+ shows "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}"
+ and "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}"
+ and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}"
+ and "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
unfolding subset_eq[unfolded Ball_def] unfolding mem_interval_cart
by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
-lemma interval_sing: fixes a :: "'a::linorder^'n" shows
- "{a .. a} = {a} \<and> {a<..<a} = {}"
-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
+lemma interval_sing:
+ fixes a :: "'a::linorder^'n"
+ shows "{a .. a} = {a} \<and> {a<..<a} = {}"
+ 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
-lemma interval_open_subset_closed_cart: fixes a :: "'a::preorder^'n" shows
- "{a<..<b} \<subseteq> {a .. b}"
-proof(simp add: subset_eq, rule)
+lemma interval_open_subset_closed_cart:
+ fixes a :: "'a::preorder^'n"
+ shows "{a<..<b} \<subseteq> {a .. b}"
+proof (simp add: subset_eq, rule)
fix x
- assume x:"x \<in>{a<..<b}"
+ assume x: "x \<in>{a<..<b}"
{ fix i
have "a $ i \<le> x $ i"
using x order_less_imp_le[of "a$i" "x$i"]
@@ -1269,105 +1365,123 @@
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
- "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1) and
- "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2) and
- "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3) and
- "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
+lemma subset_interval_cart:
+ fixes a :: "real^'n"
+ shows "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1)
+ and "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2)
+ and "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3)
+ and "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
using subset_interval[of c d a b] by (simp_all add: cart_simps real_euclidean_nth)
-lemma disjoint_interval_cart: fixes a::"real^'n" shows
- "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1) and
- "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2) and
- "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3) and
- "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
+lemma disjoint_interval_cart:
+ fixes a::"real^'n"
+ shows "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1)
+ and "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2)
+ and "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3)
+ and "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
using disjoint_interval[of a b c d] by (simp_all add: cart_simps real_euclidean_nth)
-lemma inter_interval_cart: fixes a :: "'a::linorder^'n" shows
- "{a .. b} \<inter> {c .. d} = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
+lemma inter_interval_cart:
+ fixes a :: "'a::linorder^'n"
+ shows "{a .. b} \<inter> {c .. d} = {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
unfolding set_eq_iff and Int_iff and mem_interval_cart
by auto
-lemma closed_interval_left_cart: fixes b::"real^'n"
+lemma closed_interval_left_cart:
+ fixes b :: "real^'n"
shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le)
-lemma closed_interval_right_cart: fixes a::"real^'n"
+lemma closed_interval_right_cart:
+ fixes a::"real^'n"
shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
by (simp add: Collect_all_eq closed_INT closed_Collect_le)
-lemma is_interval_cart:"is_interval (s::(real^'n) set) \<longleftrightarrow>
- (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
- unfolding is_interval_def Ball_def by (simp add: cart_simps real_euclidean_nth)
+lemma is_interval_cart:
+ "is_interval (s::(real^'n) set) \<longleftrightarrow>
+ (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i. ((a$i \<le> x$i \<and> x$i \<le> b$i) \<or> (b$i \<le> x$i \<and> x$i \<le> a$i))) \<longrightarrow> x \<in> s)"
+ by (simp add: is_interval_def Ball_def cart_simps real_euclidean_nth)
-lemma closed_halfspace_component_le_cart:
- shows "closed {x::real^'n. x$i \<le> a}"
+lemma closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
by (simp add: closed_Collect_le)
-lemma closed_halfspace_component_ge_cart:
- shows "closed {x::real^'n. x$i \<ge> a}"
+lemma closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
by (simp add: closed_Collect_le)
-lemma open_halfspace_component_lt_cart:
- shows "open {x::real^'n. x$i < a}"
+lemma open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
+ by (simp add: open_Collect_less)
+
+lemma open_halfspace_component_gt_cart: "open {x::real^'n. x$i > a}"
by (simp add: open_Collect_less)
-lemma open_halfspace_component_gt_cart:
- shows "open {x::real^'n. x$i > a}"
- by (simp add: open_Collect_less)
-
-lemma Lim_component_le_cart: fixes f :: "'a \<Rightarrow> real^'n"
+lemma Lim_component_le_cart:
+ fixes f :: "'a \<Rightarrow> real^'n"
assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. f(x)$i \<le> b) net"
shows "l$i \<le> b"
-proof-
- { fix x have "x \<in> {x::real^'n. inner (cart_basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding inner_basis by auto } note * = this
- show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<le> b}" f net l] unfolding *
+proof -
+ { fix x
+ have "x \<in> {x::real^'n. inner (cart_basis i) x \<le> b} \<longleftrightarrow> x$i \<le> b"
+ unfolding inner_basis by auto }
+ then show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<le> b}" f net l]
using closed_halfspace_le[of "(cart_basis i)::real^'n" b] and assms(1,2,3) by auto
qed
-lemma Lim_component_ge_cart: fixes f :: "'a \<Rightarrow> real^'n"
+lemma Lim_component_ge_cart:
+ fixes f :: "'a \<Rightarrow> real^'n"
assumes "(f ---> l) net" "\<not> (trivial_limit net)" "eventually (\<lambda>x. b \<le> (f x)$i) net"
shows "b \<le> l$i"
-proof-
- { fix x have "x \<in> {x::real^'n. inner (cart_basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding inner_basis by auto } note * = this
- show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<ge> b}" f net l] unfolding *
+proof -
+ { fix x
+ have "x \<in> {x::real^'n. inner (cart_basis i) x \<ge> b} \<longleftrightarrow> x$i \<ge> b"
+ unfolding inner_basis by auto }
+ then show ?thesis using Lim_in_closed_set[of "{x. inner (cart_basis i) x \<ge> b}" f net l]
using closed_halfspace_ge[of b "(cart_basis i)::real^'n"] and assms(1,2,3) by auto
qed
-lemma Lim_component_eq_cart: fixes f :: "'a \<Rightarrow> real^'n"
- assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
+lemma Lim_component_eq_cart:
+ fixes f :: "'a \<Rightarrow> real^'n"
+ assumes net: "(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
shows "l$i = b"
- using ev[unfolded order_eq_iff eventually_conj_iff] using Lim_component_ge_cart[OF net, of b i] and
+ using ev[unfolded order_eq_iff eventually_conj_iff] and
+ Lim_component_ge_cart[OF net, of b i] and
Lim_component_le_cart[OF net, of i b] by auto
-lemma connected_ivt_component_cart: fixes x::"real^'n" shows
- "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s. z$k = a)"
- using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a] by (auto simp add: inner_basis)
+lemma connected_ivt_component_cart:
+ fixes x :: "real^'n"
+ shows "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s. z$k = a)"
+ using connected_ivt_hyperplane[of s x y "(cart_basis k)::real^'n" a]
+ by (auto simp add: inner_basis)
-lemma subspace_substandard_cart:
- "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
+lemma subspace_substandard_cart: "subspace {x::real^_. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
unfolding subspace_def by auto
lemma closed_substandard_cart:
"closed {x::'a::real_normed_vector ^ 'n. \<forall>i. P i \<longrightarrow> x$i = 0}"
-proof-
+proof -
{ fix i::'n
have "closed {x::'a ^ 'n. P i \<longrightarrow> x$i = 0}"
- by (cases "P i", simp_all add: closed_Collect_eq) }
+ by (cases "P i") (simp_all add: closed_Collect_eq) }
thus ?thesis
unfolding Collect_all_eq by (simp add: closed_INT)
qed
-lemma dim_substandard_cart:
- shows "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
-proof- have *:"{x. \<forall>i<DIM((real, 'n) vec). i \<notin> \<pi>' ` d \<longrightarrow> x $$ i = 0} =
- {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"apply safe
- apply(erule_tac x="\<pi>' i" in allE) defer
- apply(erule_tac x="\<pi> i" in allE)
- unfolding image_iff real_euclidean_nth[symmetric] by (auto simp: pi'_inj[THEN inj_eq])
- have " \<pi>' ` d \<subseteq> {..<DIM((real, 'n) vec)}" using pi'_range[where 'n='n] by auto
- thus ?thesis using dim_substandard[of "\<pi>' ` d", where 'a="real^'n"]
- unfolding * using card_image[of "\<pi>'" d] using pi'_inj unfolding inj_on_def by auto
+lemma dim_substandard_cart: "dim {x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
+ (is "dim ?A = _")
+proof -
+ have *: "{x. \<forall>i<DIM((real, 'n) vec). i \<notin> \<pi>' ` d \<longrightarrow> x $$ 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]
+ apply (auto simp: pi'_inj[THEN inj_eq])
+ done
+ 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
lemma affinity_inverses:
@@ -1375,8 +1489,9 @@
shows "(\<lambda>x. m *s x + c) o (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
"(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) o (\<lambda>x. m *s x + c) = id"
using m0
-apply (auto simp add: fun_eq_iff vector_add_ldistrib)
-by (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
+ apply (auto simp add: fun_eq_iff vector_add_ldistrib)
+ apply (simp_all add: vector_smult_lneg[symmetric] vector_smult_assoc vector_sneg_minus1[symmetric])
+ done
lemma vector_affinity_eq:
assumes m0: "(m::'a::field) \<noteq> 0"
@@ -1394,25 +1509,30 @@
qed
lemma vector_eq_affinity:
- "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
+ "(m::'a::field) \<noteq> 0 ==> (y = m *s x + c \<longleftrightarrow> inverse(m) *s y + -(inverse(m) *s c) = x)"
using vector_affinity_eq[where m=m and x=x and y=y and c=c]
by metis
lemma const_vector_cart:"((\<chi> i. d)::real^'n) = (\<chi>\<chi> i. d)"
apply(subst euclidean_eq)
-proof safe case goal1
- hence *:"(basis i::real^'n) = cart_basis (\<pi> i)"
- unfolding basis_real_n[THEN sym] by auto
+proof safe
+ case goal1
+ hence *: "(basis i::real^'n) = cart_basis (\<pi> i)"
+ unfolding basis_real_n[symmetric] by auto
have "((\<chi> i. d)::real^'n) $$ i = d" unfolding euclidean_component_def *
unfolding dot_basis by auto
thus ?case using goal1 by auto
qed
+
subsection "Convex Euclidean Space"
lemma Cart_1:"(1::real^'n) = (\<chi>\<chi> i. 1)"
apply(subst euclidean_eq)
-proof safe case goal1 thus ?case using nth_conv_component[THEN sym,where i1="\<pi> i" and x1="1::real^'n"] by auto
+proof safe
+ case goal1
+ thus ?case
+ using nth_conv_component[THEN sym,where i1="\<pi> i" and x1="1::real^'n"] by auto
qed
declare vector_add_ldistrib[simp] vector_ssub_ldistrib[simp] vector_smult_assoc[simp] vector_smult_rneg[simp]
@@ -1433,21 +1553,32 @@
unfolding Cart_1 unit_interval_convex_hull[where 'a="real^'n"]
apply(rule arg_cong[where f="\<lambda>x. convex hull x"]) apply(rule set_eqI) unfolding mem_Collect_eq
apply safe apply(erule_tac x="\<pi>' i" in allE) unfolding nth_conv_component defer
- apply(erule_tac x="\<pi> i" in allE) by auto
+ apply(erule_tac x="\<pi> i" in allE)
+ apply auto
+ done
lemma cube_convex_hull_cart:
- assumes "0 < d" obtains s::"(real^'n) set" where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s"
-proof- from cube_convex_hull[OF assms, where 'a="real^'n" and x=x] guess s . note s=this
- show thesis apply(rule that[OF s(1)]) unfolding s(2)[THEN sym] const_vector_cart ..
+ assumes "0 < d"
+ obtains s::"(real^'n) set"
+ where "finite s" "{x - (\<chi> i. d) .. x + (\<chi> i. d)} = convex hull s"
+proof -
+ obtain s where s: "finite s" "{x - (\<chi>\<chi> i. d)..x + (\<chi>\<chi> i. d)} = convex hull s"
+ by (rule cube_convex_hull [OF assms])
+ show thesis
+ apply(rule that[OF s(1)]) unfolding s(2)[symmetric] const_vector_cart ..
qed
lemma std_simplex_cart:
"(insert (0::real^'n) { cart_basis i | i. i\<in>UNIV}) =
- (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
- apply(rule_tac x="\<pi> i" in exI) using pi'_range[where 'n='n] by auto
+ (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[symmetric]
+ apply safe
+ apply (rule_tac x="\<pi>' i" in exI) defer
+ apply (rule_tac x="\<pi> i" in exI) using pi'_range[where 'n='n]
+ apply auto
+ done
+
subsection "Brouwer Fixpoint"
@@ -1457,89 +1588,142 @@
(\<forall>x i. P x \<and> Q i \<and> (x$i = 0) \<longrightarrow> (l x i = 0)) \<and>
(\<forall>x i. P x \<and> Q i \<and> (x$i = 1) \<longrightarrow> (l x i = 1)) \<and>
(\<forall>x i. P x \<and> Q i \<and> (l x i = 0) \<longrightarrow> x$i \<le> f(x)$i) \<and>
- (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)" proof-
- have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)" by auto
- have *:"\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)" by auto
- show ?thesis unfolding and_forall_thm apply(subst choice_iff[THEN sym])+ proof(rule,rule) case goal1
+ (\<forall>x i. P x \<and> Q i \<and> (l x i = 1) \<longrightarrow> f(x)$i \<le> x$i)"
+proof -
+ have and_forall_thm:"\<And>P Q. (\<forall>x. P x) \<and> (\<forall>x. Q x) \<longleftrightarrow> (\<forall>x. P x \<and> Q x)"
+ by auto
+ have *: "\<forall>x y::real. 0 \<le> x \<and> x \<le> 1 \<and> 0 \<le> y \<and> y \<le> 1 \<longrightarrow> (x \<noteq> 1 \<and> x \<le> y \<or> x \<noteq> 0 \<and> y \<le> x)"
+ by auto
+ show ?thesis
+ unfolding and_forall_thm apply(subst choice_iff[symmetric])+
+ proof (rule, rule)
+ case goal1
let ?R = "\<lambda>y. (P x \<and> Q xa \<and> x $ xa = 0 \<longrightarrow> y = (0::nat)) \<and>
- (P x \<and> Q xa \<and> x $ xa = 1 \<longrightarrow> y = 1) \<and> (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $ xa \<le> f x $ xa) \<and> (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $ xa \<le> x $ xa)"
- { assume "P x" "Q xa" hence "0 \<le> f x $ xa \<and> f x $ xa \<le> 1" using assms(2)[rule_format,of "f x" xa]
- apply(drule_tac assms(1)[rule_format]) by auto }
- hence "?R 0 \<or> ?R 1" by auto thus ?case by auto qed qed
+ (P x \<and> Q xa \<and> x $ xa = 1 \<longrightarrow> y = 1) \<and>
+ (P x \<and> Q xa \<and> y = 0 \<longrightarrow> x $ xa \<le> f x $ xa) \<and>
+ (P x \<and> Q xa \<and> y = 1 \<longrightarrow> f x $ xa \<le> x $ xa)"
+ { assume "P x" "Q xa"
+ hence "0 \<le> f x $ xa \<and> f x $ xa \<le> 1"
+ using assms(2)[rule_format,of "f x" xa]
+ apply (drule_tac assms(1)[rule_format])
+ apply auto
+ done
+ }
+ hence "?R 0 \<or> ?R 1" by auto
+ thus ?case by auto
+ qed
+qed
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$i))::real^'n)"
unfolding interval_bij_def apply(rule ext)+ apply safe
unfolding vec_eq_iff vec_lambda_beta unfolding nth_conv_component
- apply rule apply(subst euclidean_lambda_beta) using pi'_range by auto
+ apply rule
+ apply (subst euclidean_lambda_beta)
+ using pi'_range apply auto
+ done
lemma interval_bij_affine_cart:
"interval_bij (a,b) (u,v) = (\<lambda>x. (\<chi> i. (v$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 vec_eq_iff interval_bij_cart vector_component_simps
- by(auto simp add: field_simps add_divide_distrib[THEN sym])
+ apply rule
+ unfolding vec_eq_iff interval_bij_cart vector_component_simps
+ apply (auto simp add: field_simps add_divide_distrib[symmetric])
+ done
+
subsection "Derivative"
-lemma has_derivative_vmul_component_cart: fixes c::"real^'a \<Rightarrow> real^'b" and v::"real^'c"
+lemma has_derivative_vmul_component_cart:
+ fixes c :: "real^'a \<Rightarrow> real^'b" and v :: "real^'c"
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"
- unfolding nth_conv_component
- by (intro has_derivative_intros assms)
+ unfolding nth_conv_component by (intro has_derivative_intros assms)
-lemma differentiable_at_imp_differentiable_on: "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
+lemma differentiable_at_imp_differentiable_on:
+ "(\<forall>x\<in>(s::(real^'n) set). f differentiable at x) \<Longrightarrow> f differentiable_on s"
unfolding differentiable_on_def by(auto intro!: differentiable_at_withinI)
definition "jacobian f net = matrix(frechet_derivative f net)"
-lemma jacobian_works: "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow> (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
- apply rule unfolding jacobian_def apply(simp only: matrix_works[OF linear_frechet_derivative]) defer
- apply(rule differentiableI) apply assumption unfolding frechet_derivative_works by assumption
+lemma jacobian_works:
+ "(f::(real^'a) \<Rightarrow> (real^'b)) differentiable net \<longleftrightarrow>
+ (f has_derivative (\<lambda>h. (jacobian f net) *v h)) net"
+ apply rule
+ unfolding jacobian_def
+ apply (simp only: matrix_works[OF linear_frechet_derivative]) defer
+ apply (rule differentiableI)
+ apply assumption
+ unfolding frechet_derivative_works
+ apply assumption
+ done
-subsection {* Component of the differential must be zero if it exists at a local *)
-(* maximum or minimum for that corresponding component. *}
-lemma differential_zero_maxmin_component: fixes f::"real^'a \<Rightarrow> real^'b"
- assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
- "f differentiable (at x)" shows "jacobian f (at x) $ k = 0"
-(* FIXME: reuse proof of generic differential_zero_maxmin_component*)
+subsection {* Component of the differential must be zero if it exists at a local
+ maximum or minimum for that corresponding component. *}
-proof(rule ccontr)
- def D \<equiv> "jacobian f (at x)" assume "jacobian f (at x) $ k \<noteq> 0"
+lemma differential_zero_maxmin_component:
+ fixes f::"real^'a \<Rightarrow> real^'b"
+ assumes "0 < e" "((\<forall>y \<in> ball x e. (f y)$k \<le> (f x)$k) \<or> (\<forall>y\<in>ball x e. (f x)$k \<le> (f y)$k))"
+ "f differentiable (at x)" shows "jacobian f (at x) $ k = 0"
+(* FIXME: reuse proof of generic differential_zero_maxmin_component*)
+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 vec_eq_iff D_def by auto
- hence *:"abs (jacobian f (at x) $ k $ j) / 2 > 0" unfolding 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
guess d using real_lbound_gt_zero[OF assms(1) e'[THEN conjunct1]] .. note d = this
- { fix c assume "abs c \<le> d"
- hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'" using norm_basis[of j] d by auto
- have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le> norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))"
- by(rule component_le_norm_cart)
- also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j] unfolding D_def[symmetric] by auto
- finally have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
- hence "\<bar>f (x + c *\<^sub>R cart_basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
- unfolding vector_component_simps matrix_vector_mul_component unfolding smult_conv_scaleR[symmetric]
- unfolding inner_simps dot_basis smult_conv_scaleR by simp } note * = this
+ { fix c
+ assume "abs c \<le> d"
+ hence *:"norm (x + c *\<^sub>R cart_basis j - x) < e'"
+ using norm_basis[of j] d by auto
+ have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le>
+ norm (f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j))"
+ by (rule component_le_norm_cart)
+ also have "\<dots> \<le> \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
+ using e'[THEN conjunct2,rule_format,OF *] and norm_basis[of j]
+ unfolding D_def[symmetric] by auto
+ finally
+ have "\<bar>(f (x + c *\<^sub>R cart_basis j) - f x - D *v (c *\<^sub>R cart_basis j)) $ k\<bar> \<le>
+ \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>" by simp
+ hence "\<bar>f (x + c *\<^sub>R cart_basis j) $ k - f x $ k - c * D $ k $ j\<bar> \<le>
+ \<bar>D $ k $ j\<bar> / 2 * \<bar>c\<bar>"
+ unfolding vector_component_simps matrix_vector_mul_component
+ unfolding smult_conv_scaleR[symmetric]
+ unfolding inner_simps dot_basis smult_conv_scaleR by simp
+ } note * = this
have "x + d *\<^sub>R cart_basis j \<in> ball x e" "x - d *\<^sub>R cart_basis j \<in> ball x e"
unfolding mem_ball dist_norm using norm_basis[of j] d by auto
- hence **:"((f (x - d *\<^sub>R cart_basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<le> (f x)$k) \<or>
- ((f (x - d *\<^sub>R cart_basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<ge> (f x)$k)" using assms(2) by auto
- 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)
+ hence **: "((f (x - d *\<^sub>R cart_basis j))$k \<le> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<le> (f x)$k) \<or>
+ ((f (x - d *\<^sub>R cart_basis j))$k \<ge> (f x)$k \<and> (f (x + d *\<^sub>R cart_basis j))$k \<ge> (f x)$k)"
+ using assms(2) by auto
+ 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
+ apply (auto intro: mult_pos_pos)
+ done
qed
+
subsection {* Lemmas for working on @{typ "real^1"} *}
lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
- by (metis num1_eq_iff)
+ by (metis (full_types) num1_eq_iff)
lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
- by auto (metis num1_eq_iff)
+ by auto (metis (full_types) num1_eq_iff)
lemma exhaust_2:
- fixes x :: 2 shows "x = 1 \<or> x = 2"
+ fixes x :: 2
+ shows "x = 1 \<or> x = 2"
proof (induct x)
case (of_int z)
then have "0 <= z" and "z < 2" by simp_all
@@ -1551,7 +1735,8 @@
by (metis exhaust_2)
lemma exhaust_3:
- fixes x :: 3 shows "x = 1 \<or> x = 2 \<or> x = 3"
+ fixes x :: 3
+ shows "x = 1 \<or> x = 2 \<or> x = 3"
proof (induct x)
case (of_int z)
then have "0 <= z" and "z < 3" by simp_all
@@ -1580,17 +1765,21 @@
lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
unfolding UNIV_3 by (simp add: add_ac)
-instantiation num1 :: cart_one begin
-instance proof
+instantiation num1 :: cart_one
+begin
+
+instance
+proof
show "CARD(1) = Suc 0" by auto
-qed end
+qed
+
+end
(* "lift" from 'a to 'a^1 and "drop" from 'a^1 to 'a -- FIXME: potential use of transfer *)
abbreviation vec1:: "'a \<Rightarrow> 'a ^ 1" where "vec1 x \<equiv> vec x"
-abbreviation dest_vec1:: "'a ^1 \<Rightarrow> 'a"
- where "dest_vec1 x \<equiv> (x$1)"
+abbreviation dest_vec1:: "'a ^1 \<Rightarrow> 'a" where "dest_vec1 x \<equiv> (x$1)"
lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x"
by (simp add: vec_eq_iff)
@@ -1604,6 +1793,7 @@
lemma dest_vec1_eq[simp]: "dest_vec1 x = dest_vec1 y \<longleftrightarrow> x = y"
by (metis vec1_dest_vec1(1))
+
subsection{* The collapse of the general concepts to dimension one. *}
lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
@@ -1624,6 +1814,7 @@
lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
by (auto simp add: norm_real dist_norm)
+
subsection{* Explicit vector construction from lists. *}
definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
@@ -1672,14 +1863,17 @@
apply (simp add: forall_3)
done
-lemma range_vec1[simp]:"range vec1 = UNIV" apply(rule set_eqI,rule) unfolding image_iff defer
- apply(rule_tac x="dest_vec1 x" in bexI) by auto
+lemma range_vec1[simp]:"range vec1 = UNIV"
+ apply (rule set_eqI,rule) unfolding image_iff defer
+ apply (rule_tac x="dest_vec1 x" in bexI)
+ apply auto
+ done
lemma dest_vec1_lambda: "dest_vec1(\<chi> i. x i) = x 1"
- by (simp)
+ by simp
lemma dest_vec1_vec: "dest_vec1(vec x) = x"
- by (simp)
+ by simp
lemma dest_vec1_sum: assumes fS: "finite S"
shows "dest_vec1(setsum f S) = setsum (dest_vec1 o f) S"
@@ -1696,13 +1890,19 @@
lemma abs_dest_vec1: "norm x = \<bar>dest_vec1 x\<bar>"
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
- vec_inj[where 'b=1] vec_cmul[THEN sym] smult_conv_scaleR[THEN sym] o_def dist_real_def real_norm_def
+lemmas vec1_dest_vec1_simps =
+ forall_vec1 vec_add[symmetric] dist_vec1 vec_sub[symmetric] vec1_dest_vec1 norm_vec1 vector_smult_component
+ vec_inj[where 'b=1] vec_cmul[symmetric] smult_conv_scaleR[symmetric] o_def dist_real_def real_norm_def
-lemma bounded_linear_vec1:"bounded_linear (vec1::real\<Rightarrow>real^1)"
+lemma bounded_linear_vec1: "bounded_linear (vec1::real\<Rightarrow>real^1)"
unfolding bounded_linear_def additive_def bounded_linear_axioms_def
- unfolding smult_conv_scaleR[THEN sym] unfolding vec1_dest_vec1_simps
- apply(rule conjI) defer apply(rule conjI) defer apply(rule_tac x=1 in exI) by auto
+ unfolding smult_conv_scaleR[symmetric]
+ unfolding vec1_dest_vec1_simps
+ apply (rule conjI) defer
+ apply (rule conjI) defer
+ apply (rule_tac x=1 in exI)
+ apply auto
+ done
lemma linear_vmul_dest_vec1:
fixes f:: "real^_ \<Rightarrow> real^1"
@@ -1719,7 +1919,8 @@
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)"
+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])
@@ -1729,127 +1930,190 @@
lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
by (simp add: dest_vec1_eq[symmetric])
-lemma setsum_scalars: assumes fS: "finite S"
+lemma setsum_scalars:
+ assumes fS: "finite S"
shows "setsum f S = vec1 (setsum (dest_vec1 o f) S)"
unfolding vec_setsum[OF fS] by simp
-lemma dest_vec1_wlog_le: "(\<And>(x::'a::linorder ^ 1) y. P x y \<longleftrightarrow> P y x) \<Longrightarrow> (\<And>x y. dest_vec1 x <= dest_vec1 y ==> P x y) \<Longrightarrow> P x y"
+lemma dest_vec1_wlog_le:
+ "(\<And>(x::'a::linorder ^ 1) y. P x y \<longleftrightarrow> P y x)
+ \<Longrightarrow> (\<And>x y. dest_vec1 x <= dest_vec1 y ==> P x y) \<Longrightarrow> P x y"
apply (cases "dest_vec1 x \<le> dest_vec1 y")
apply simp
apply (subgoal_tac "dest_vec1 y \<le> dest_vec1 x")
- apply (auto)
+ apply auto
done
text{* Lifting and dropping *}
-lemma continuous_on_o_dest_vec1: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
- assumes "continuous_on {a..b::real} f" shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)"
+lemma continuous_on_o_dest_vec1:
+ fixes f::"real \<Rightarrow> 'a::real_normed_vector"
+ assumes "continuous_on {a..b::real} f"
+ shows "continuous_on {vec1 a..vec1 b} (f o dest_vec1)"
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:less_eq_vec_def)
+ 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)
+ apply (auto simp add:less_eq_vec_def)
+ done
-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: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)
+ apply (auto simp add:less_eq_vec_def)
+ done
lemma continuous_on_vec1:"continuous_on A (vec1::real\<Rightarrow>real^1)"
- by(rule linear_continuous_on[OF bounded_linear_vec1])
+ by (rule linear_continuous_on[OF bounded_linear_vec1])
-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: vec_eq_iff less_vec_def less_eq_vec_def)
+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)"
+ and "(x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
+ 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 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
+lemma vec1_interval:
+ fixes a::"real"
+ shows "vec1 ` {a .. b} = {vec1 a .. vec1 b}"
+ and "vec1 ` {a<..<b} = {vec1 a<..<vec1 b}"
+ apply (rule_tac[!] set_eqI)
+ unfolding image_iff less_vec_def
+ unfolding mem_interval_cart
+ unfolding forall_1 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)
+ apply auto
+ done
(* Some special cases for intervals in R^1. *)
-lemma interval_cases_1: fixes x :: "real^1" shows
- "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
- unfolding vec_eq_iff less_vec_def less_eq_vec_def mem_interval_cart by(auto simp del:dest_vec1_eq)
+lemma interval_cases_1:
+ fixes x :: "real^1"
+ shows "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
+ 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 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 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"
- "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
+lemma interval_eq_empty_1:
+ fixes a :: "real^1"
+ shows "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
+ and "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
unfolding interval_eq_empty_cart and ex_1 by auto
-lemma subset_interval_1: fixes a :: "real^1" shows
- "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or>
- dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
- "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or>
- dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
- "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
- dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
- "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
- dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
+lemma subset_interval_1:
+ fixes a :: "real^1"
+ shows "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or>
+ dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
+ "({a .. b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or>
+ dest_vec1 c < dest_vec1 a \<and> dest_vec1 a \<le> dest_vec1 b \<and> dest_vec1 b < dest_vec1 d)"
+ "({a<..<b} \<subseteq> {c .. d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
+ dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
+ "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
+ dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
unfolding subset_interval_cart[of a b c d] unfolding forall_1 by auto
-lemma eq_interval_1: fixes a :: "real^1" shows
- "{a .. b} = {c .. d} \<longleftrightarrow>
+lemma eq_interval_1:
+ fixes a :: "real^1"
+ shows "{a .. b} = {c .. d} \<longleftrightarrow>
dest_vec1 b < dest_vec1 a \<and> dest_vec1 d < dest_vec1 c \<or>
dest_vec1 a = dest_vec1 c \<and> dest_vec1 b = dest_vec1 d"
-unfolding set_eq_subset[of "{a .. b}" "{c .. d}"]
-unfolding subset_interval_1(1)[of a b c d]
-unfolding subset_interval_1(1)[of c d a b]
-by auto
+ unfolding set_eq_subset[of "{a .. b}" "{c .. d}"]
+ unfolding subset_interval_1(1)[of a b c d]
+ unfolding subset_interval_1(1)[of c d a b]
+ by auto
-lemma disjoint_interval_1: fixes a :: "real^1" shows
- "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d < dest_vec1 c \<or> dest_vec1 b < dest_vec1 c \<or> dest_vec1 d < dest_vec1 a"
- "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
- "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
- "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
+lemma disjoint_interval_1:
+ fixes a :: "real^1"
+ shows
+ "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow>
+ dest_vec1 b < dest_vec1 a \<or> dest_vec1 d < dest_vec1 c \<or> dest_vec1 b < dest_vec1 c \<or> dest_vec1 d < dest_vec1 a"
+ "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow>
+ dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
+ "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow>
+ dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
+ "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow>
+ dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c \<or> dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
unfolding disjoint_interval_cart and ex_1 by auto
-lemma open_closed_interval_1: fixes a :: "real^1" shows
- "{a<..<b} = {a .. b} - {a, b}"
- 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 open_closed_interval_1:
+ fixes a :: "real^1"
+ shows "{a<..<b} = {a .. b} - {a, b}"
+ unfolding set_eq_iff apply simp
+ unfolding less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[symmetric]
+ apply (auto simp del:dest_vec1_eq)
+ done
-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 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 less_vec_def and less_eq_vec_def and forall_1 and dest_vec1_eq[symmetric]
+ apply (auto simp del:dest_vec1_eq)
+ done
-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"
+lemma Lim_drop_le:
+ fixes f :: "'a \<Rightarrow> real^1"
+ shows "(f ---> l) net \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow>
+ eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
using Lim_component_le_cart[of f l net 1 b] by auto
-lemma Lim_drop_ge: fixes f :: "'a \<Rightarrow> real^1" shows
- "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
+lemma Lim_drop_ge:
+ fixes f :: "'a \<Rightarrow> real^1"
+ shows "(f ---> l) net \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow>
+ eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
using Lim_component_ge_cart[of f l net b 1] by auto
+
text{* Also more convenient formulations of monotone convergence. *}
-lemma bounded_increasing_convergent: fixes s::"nat \<Rightarrow> real^1"
+lemma bounded_increasing_convergent:
+ fixes s :: "nat \<Rightarrow> real^1"
assumes "bounded {s n| n::nat. True}" "\<forall>n. dest_vec1(s n) \<le> dest_vec1(s(Suc n))"
shows "\<exists>l. (s ---> l) sequentially"
-proof-
- obtain a where a:"\<forall>n. \<bar>dest_vec1 (s n)\<bar> \<le> a" using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto
+proof -
+ obtain a where a:"\<forall>n. \<bar>dest_vec1 (s n)\<bar> \<le> a"
+ using assms(1)[unfolded bounded_iff abs_dest_vec1] by auto
{ fix m::nat
have "\<And> n. n\<ge>m \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
- apply(induct_tac n) apply simp using assms(2) apply(erule_tac x="na" in allE) by(auto simp add: not_less_eq_eq) }
- hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)" by auto
- then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e" using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
+ apply (induct_tac n)
+ apply simp
+ using assms(2) apply (erule_tac x="na" in allE)
+ apply (auto simp add: not_less_eq_eq)
+ done
+ }
+ hence "\<forall>m n. m \<le> n \<longrightarrow> dest_vec1 (s m) \<le> dest_vec1 (s n)"
+ by auto
+ then obtain l where "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>dest_vec1 (s n) - l\<bar> < e"
+ using convergent_bounded_monotone[OF a] unfolding monoseq_def by auto
thus ?thesis unfolding LIMSEQ_def apply(rule_tac x="vec1 l" in exI)
- unfolding dist_norm unfolding abs_dest_vec1 by auto
+ unfolding dist_norm unfolding abs_dest_vec1 by auto
qed
-lemma dest_vec1_simps[simp]: fixes a::"real^1"
+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: less_eq_vec_def vec_eq_iff)
+ "a \<le> b \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 b" "dest_vec1 (1::real^1) = 1"
+ 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}"
@@ -1861,106 +2125,162 @@
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: less_vec_def less_eq_vec_def)
+ apply (auto simp add: less_vec_def less_eq_vec_def)
+ done
-lemma dest_vec1_setsum: assumes "finite S"
+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_vec_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_vec_nth)
+ by (rule tendsto_vec_nth)
-lemma continuous_dest_vec1: "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
+lemma continuous_dest_vec1:
+ "continuous net f \<Longrightarrow> continuous net (\<lambda>x. dest_vec1 (f x))"
unfolding continuous_def by (rule tendsto_dest_vec1)
lemma forall_dest_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P(dest_vec1 x))"
- apply safe defer apply(erule_tac x="vec1 x" in allE) by auto
+ apply safe defer
+ apply (erule_tac x="vec1 x" in allE)
+ apply auto
+ done
lemma forall_of_dest_vec1: "(\<forall>v. P (\<lambda>x. dest_vec1 (v x))) \<longleftrightarrow> (\<forall>x. P x)"
- apply rule apply rule apply(erule_tac x="(vec1 \<circ> x)" in allE) unfolding o_def vec1_dest_vec1 by auto
+ apply rule
+ apply rule
+ apply (erule_tac x="vec1 \<circ> x" in allE)
+ unfolding o_def vec1_dest_vec1
+ apply auto
+ done
lemma forall_of_dest_vec1': "(\<forall>v. P (dest_vec1 v)) \<longleftrightarrow> (\<forall>x. P x)"
- apply rule apply rule apply(erule_tac x="(vec1 x)" in allE) defer apply rule
- apply(erule_tac x="dest_vec1 v" in allE) unfolding o_def vec1_dest_vec1 by auto
+ apply rule
+ apply rule
+ apply (erule_tac x="(vec1 x)" in allE) defer
+ apply rule
+ apply (erule_tac x="dest_vec1 v" in allE)
+ unfolding o_def vec1_dest_vec1
+ apply auto
+ done
-lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x" unfolding dist_norm by auto
+lemma dist_vec1_0[simp]: "dist(vec1 (x::real)) 0 = norm x"
+ unfolding dist_norm by auto
-lemma bounded_linear_vec1_dest_vec1: fixes f::"real \<Rightarrow> real"
- shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r") proof-
- { assume ?l guess K using linear_bounded[OF `?l`] ..
- hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K" apply(rule_tac x=K in exI)
- unfolding vec1_dest_vec1_simps by (auto simp add:field_simps) }
- thus ?thesis unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
- unfolding vec1_dest_vec1_simps by auto qed
+lemma bounded_linear_vec1_dest_vec1:
+ fixes f :: "real \<Rightarrow> real"
+ shows "linear (vec1 \<circ> f \<circ> dest_vec1) = bounded_linear f" (is "?l = ?r")
+proof -
+ { assume ?l
+ then have "\<exists>K. \<forall>x. norm ((vec1 \<circ> f \<circ> dest_vec1) x) \<le> K * norm x" by (rule linear_bounded)
+ then guess K ..
+ hence "\<exists>K. \<forall>x. \<bar>f x\<bar> \<le> \<bar>x\<bar> * K"
+ apply(rule_tac x=K in exI)
+ unfolding vec1_dest_vec1_simps by (auto simp add:field_simps)
+ }
+ thus ?thesis
+ unfolding linear_def bounded_linear_def additive_def bounded_linear_axioms_def o_def
+ 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"
+lemma vec1_le[simp]: fixes a :: real shows "vec1 a \<le> vec1 b \<longleftrightarrow> a \<le> b"
unfolding less_eq_vec_def by auto
-lemma vec1_less[simp]:fixes a::real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
+lemma vec1_less[simp]: fixes a :: real shows "vec1 a < vec1 b \<longleftrightarrow> a < b"
unfolding less_vec_def by auto
subsection {* Derivatives on real = Derivatives on @{typ "real^1"} *}
-lemma has_derivative_within_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
- "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
- = (f has_derivative f') (at x within s)"
- unfolding has_derivative_within unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
+lemma has_derivative_within_vec1_dest_vec1:
+ fixes f :: "real \<Rightarrow> real"
+ shows "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s)
+ = (f has_derivative f') (at x within s)"
+ unfolding has_derivative_within
+ unfolding bounded_linear_vec1_dest_vec1[unfolded linear_conv_bounded_linear]
unfolding o_def Lim_within Ball_def unfolding forall_vec1
- unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
+ unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff
+ by auto
-lemma has_derivative_at_vec1_dest_vec1: fixes f::"real\<Rightarrow>real" shows
- "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
- using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV] by auto
+lemma has_derivative_at_vec1_dest_vec1:
+ fixes f :: "real \<Rightarrow> real"
+ shows "((vec1 \<circ> f \<circ> dest_vec1) has_derivative (vec1 \<circ> f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
+ using has_derivative_within_vec1_dest_vec1[where s=UNIV, unfolded range_vec1 within_UNIV]
+ by auto
-lemma bounded_linear_vec1': fixes f::"'a::real_normed_vector\<Rightarrow>real"
+lemma bounded_linear_vec1':
+ fixes f :: "'a::real_normed_vector\<Rightarrow>real"
shows "bounded_linear f = bounded_linear (vec1 \<circ> f)"
unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
unfolding vec1_dest_vec1_simps by auto
-lemma bounded_linear_dest_vec1: fixes f::"real\<Rightarrow>'a::real_normed_vector"
+lemma bounded_linear_dest_vec1:
+ fixes f :: "real\<Rightarrow>'a::real_normed_vector"
shows "bounded_linear f = bounded_linear (f \<circ> dest_vec1)"
unfolding bounded_linear_def additive_def bounded_linear_axioms_def o_def
- unfolding vec1_dest_vec1_simps by auto
+ unfolding vec1_dest_vec1_simps
+ by auto
-lemma has_derivative_at_vec1: fixes f::"'a::real_normed_vector\<Rightarrow>real" shows
- "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
- unfolding has_derivative_at unfolding bounded_linear_vec1'[unfolded linear_conv_bounded_linear]
- unfolding o_def Lim_at unfolding vec1_dest_vec1_simps dist_vec1_0 by auto
+lemma has_derivative_at_vec1:
+ fixes f :: "'a::real_normed_vector\<Rightarrow>real"
+ shows "(f has_derivative f') (at x) = ((vec1 \<circ> f) has_derivative (vec1 \<circ> f')) (at x)"
+ unfolding has_derivative_at
+ unfolding bounded_linear_vec1'[unfolded linear_conv_bounded_linear]
+ unfolding o_def Lim_at
+ unfolding vec1_dest_vec1_simps dist_vec1_0
+ by auto
-lemma has_derivative_within_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
- "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) = (f has_derivative f') (at x within s)"
- unfolding has_derivative_within bounded_linear_dest_vec1 unfolding o_def Lim_within Ball_def
- unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff by auto
+lemma has_derivative_within_dest_vec1:
+ fixes f :: "real\<Rightarrow>'a::real_normed_vector"
+ shows "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x) within vec1 ` s) =
+ (f has_derivative f') (at x within s)"
+ unfolding has_derivative_within bounded_linear_dest_vec1
+ unfolding o_def Lim_within Ball_def
+ unfolding vec1_dest_vec1_simps dist_vec1_0 image_iff
+ by auto
-lemma has_derivative_at_dest_vec1:fixes f::"real\<Rightarrow>'a::real_normed_vector" shows
- "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) = (f has_derivative f') (at x)"
+lemma has_derivative_at_dest_vec1:
+ fixes f :: "real\<Rightarrow>'a::real_normed_vector"
+ shows "((f \<circ> dest_vec1) has_derivative (f' \<circ> dest_vec1)) (at (vec1 x)) =
+ (f has_derivative f') (at x)"
using has_derivative_within_dest_vec1[where s=UNIV] by simp
+
subsection {* In particular if we have a mapping into @{typ "real^1"}. *}
-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: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
+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: 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
have 4:"{norm (f x) |x. norm x = 1} = (\<lambda>x. norm (f x)) ` {x. norm x=1}" by auto
- show ?thesis unfolding onorm_def 1 2 3 4 by(simp add:Sup_finite_Max) qed
+ show ?thesis
+ unfolding onorm_def 1 2 3 4 by (simp add:Sup_finite_Max)
+qed
lemma convex_vec1:"convex (vec1 ` s) = convex (s::real set)"
- unfolding convex_def Ball_def forall_vec1 unfolding vec1_dest_vec1_simps image_iff by auto
+ unfolding convex_def Ball_def forall_vec1
+ unfolding vec1_dest_vec1_simps image_iff
+ by auto
lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
- apply(rule bounded_linearI[where K=1])
+ apply (rule bounded_linearI[where K=1])
using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
-lemma bounded_vec1[intro]: "bounded s \<Longrightarrow> bounded (vec1 ` (s::real set))"
+lemma bounded_vec1[intro]: "bounded s \<Longrightarrow> bounded (vec1 ` (s::real set))"
unfolding bounded_def apply safe apply(rule_tac x="vec1 x" in exI,rule_tac x=e in exI)
- by(auto simp add: dist_real dist_real_def)
+ apply (auto simp add: dist_real dist_real_def)
+ done
(*lemma content_closed_interval_cases_cart:
"content {a..b::real^'n} =
@@ -1975,15 +2295,19 @@
sorry*)
-lemma integral_component_eq_cart[simp]: fixes f::"'n::ordered_euclidean_space \<Rightarrow> real^'m"
- assumes "f integrable_on s" shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
+lemma integral_component_eq_cart[simp]:
+ fixes f :: "'n::ordered_euclidean_space \<Rightarrow> real^'m"
+ assumes "f integrable_on s"
+ shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
using integral_linear[OF assms(1) bounded_linear_component_cart,unfolded o_def] .
lemma interval_split_cart:
"{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 vec_lambda_beta by auto
+ apply (rule_tac[!] set_eqI)
+ unfolding Int_iff mem_interval_cart mem_Collect_eq
+ 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})"
@@ -2006,7 +2330,8 @@
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)"
+ 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 vec_eq_iff by (auto simp add: split_beta)
show ?thesis
using assms unfolding has_integral