updated tagging for 9 theories: Cross3, Determinants, Tagged_Division, Change_of_Vars, Extended_Real_Limits, Fashoda, Finite_Cartesian_Product, Function_Topology, Finite_Product_Measure
--- a/NEWS Mon Jan 14 18:35:03 2019 +0000
+++ b/NEWS Wed Jan 16 21:27:33 2019 +0000
@@ -76,9 +76,10 @@
* Code generation: command 'export_code' without file keyword exports
code as regular theory export, which can be materialized using the
-command-line tool "isabelle export", or browsed in Isabelle/jEdit via
-the "isabelle-export:" file-system. To get generated code dumped into
-output, use "file \<open>\<close>". Minor INCOMPATIBILITY.
+command-line tool "isabelle export" or 'export_files' in the session
+ROOT, or browsed in Isabelle/jEdit via the "isabelle-export:"
+file-system. To get generated code dumped into output, use "file \<open>\<close>".
+Minor INCOMPATIBILITY.
* Simplified syntax setup for big operators under image. In rare
situations, type conversions are not inserted implicitly any longer
--- a/src/Doc/System/Sessions.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Doc/System/Sessions.thy Wed Jan 16 21:27:33 2019 +0000
@@ -76,7 +76,8 @@
;
document_files: @'document_files' ('(' dir ')')? (@{syntax embedded}+)
;
- export_files: @'export_files' ('(' dir ')')? (@{syntax embedded}+)
+ export_files: @'export_files' ('(' dir ')')? ('[' nat ']')? \<newline>
+ (@{syntax embedded}+)
\<close>
\<^descr> \isakeyword{session}~\<open>A = B + body\<close> defines a new session \<open>A\<close> based on
@@ -143,11 +144,14 @@
original directory hierarchy of \<open>base_dir\<close>. The default \<open>base_dir\<close> is
\<^verbatim>\<open>document\<close> within the session root directory.
- \<^descr> \isakeyword{export_files}~\<open>(\<close>\isakeyword{in}~\<open>target_dir) patterns\<close> writes
- theory exports to the file-system: the \<open>target_dir\<close> specification is
- relative to the session root directory; its default is \<^verbatim>\<open>export\<close>. Exports
- are selected via \<open>patterns\<close> as in @{tool_ref export}
- (\secref{sec:tool-export}).
+ \<^descr> \isakeyword{export_files}~\<open>(\<close>\isakeyword{in}~\<open>target_dir) [number]
+ patterns\<close> writes theory exports to the file-system: the \<open>target_dir\<close>
+ specification is relative to the session root directory; its default is
+ \<^verbatim>\<open>export\<close>. Exports are selected via \<open>patterns\<close> as in @{tool_ref export}
+ (\secref{sec:tool-export}). The number given in brackets (default: 0)
+ specifies elements that should be pruned from each name: it allows to reduce
+ the resulting directory hierarchy at the danger of overwriting files due to
+ loss of uniqueness.
\<close>
@@ -563,6 +567,7 @@
-l list exports
-n no build of session
-o OPTION override Isabelle system OPTION (via NAME=VAL or NAME)
+ -p NUM prune path of exported files by NUM elements
-s system build mode for session image
-x PATTERN extract files matching pattern (e.g.\ "*:**" for all)
@@ -594,6 +599,10 @@
Option \<^verbatim>\<open>-O\<close> specifies an alternative output directory for option \<^verbatim>\<open>-x\<close>: the
default is \<^verbatim>\<open>export\<close> within the current directory. Each theory creates its
own sub-directory hierarchy, using the session-qualified theory name.
+
+ Option \<^verbatim>\<open>-p\<close> specifies the number of elements that should be pruned from
+ each name: it allows to reduce the resulting directory hierarchy at the
+ danger of overwriting files due to loss of uniqueness.
\<close>
--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Wed Jan 16 21:27:33 2019 +0000
@@ -179,117 +179,18 @@
by force
qed
-lemma%unimportant matrix_mult_transpose_dot_column:
- shows "transpose A ** A = (\<chi> i j. inner (column i A) (column j A))"
- by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def column_def inner_vec_def)
-
-lemma%unimportant matrix_mult_transpose_dot_row:
- shows "A ** transpose A = (\<chi> i j. inner (row i A) (row j A))"
- by (simp add: matrix_matrix_mult_def vec_eq_iff transpose_def row_def inner_vec_def)
-
-text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
-
-lemma%important matrix_mult_dot: "A *v x = (\<chi> i. inner (A$i) x)"
- by (simp add: matrix_vector_mult_def inner_vec_def)
-
-lemma%unimportant 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
- sum_distrib_right sum_distrib_left)
- apply (subst sum.swap)
- apply (simp add: ac_simps)
- done
-
-lemma%important matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
- shows "matrix(adjoint f) = transpose(matrix f)"
-proof%unimportant -
- have "matrix(adjoint f) = matrix(adjoint ((*v) (matrix f)))"
- by (simp add: lf)
- also have "\<dots> = transpose(matrix f)"
- unfolding adjoint_matrix matrix_of_matrix_vector_mul
- apply rule
- done
- finally show ?thesis .
-qed
-
lemma%unimportant matrix_vector_mul_bounded_linear[intro, simp]: "bounded_linear ((*v) A)" for A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
using matrix_vector_mul_linear[of A]
by (simp add: linear_conv_bounded_linear linear_matrix_vector_mul_eq)
-lemma%unimportant (* FIX ME needs name*)
+lemma%unimportant
fixes A :: "'a::{euclidean_space,real_algebra_1}^'n^'m"
shows matrix_vector_mult_linear_continuous_at [continuous_intros]: "isCont ((*v) A) z"
and matrix_vector_mult_linear_continuous_on [continuous_intros]: "continuous_on S ((*v) A)"
by (simp_all add: linear_continuous_at linear_continuous_on)
-lemma%unimportant scalar_invertible:
- fixes A :: "('a::real_algebra_1)^'m^'n"
- assumes "k \<noteq> 0" and "invertible A"
- shows "invertible (k *\<^sub>R A)"
-proof -
- obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
- using assms unfolding invertible_def by auto
- with \<open>k \<noteq> 0\<close>
- have "(k *\<^sub>R A) ** ((1/k) *\<^sub>R A') = mat 1" "((1/k) *\<^sub>R A') ** (k *\<^sub>R A) = mat 1"
- by (simp_all add: assms matrix_scalar_ac)
- thus "invertible (k *\<^sub>R A)"
- unfolding invertible_def by auto
-qed
-lemma%unimportant scalar_invertible_iff:
- fixes A :: "('a::real_algebra_1)^'m^'n"
- assumes "k \<noteq> 0" and "invertible A"
- shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
- by (simp add: assms scalar_invertible)
-
-lemma%unimportant vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
- unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
- by simp
-
-lemma%unimportant transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
- unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
- by simp
-
-lemma%unimportant vector_scalar_commute:
- fixes A :: "'a::{field}^'m^'n"
- shows "A *v (c *s x) = c *s (A *v x)"
- by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
-
-lemma%unimportant scalar_vector_matrix_assoc:
- fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
- shows "(k *s x) v* A = k *s (x v* A)"
- by (metis transpose_matrix_vector vector_scalar_commute)
-
-lemma%unimportant vector_matrix_mult_0 [simp]: "0 v* A = 0"
- unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
-
-lemma%unimportant vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
- unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
-
-lemma%unimportant vector_matrix_mul_rid [simp]:
- fixes v :: "('a::semiring_1)^'n"
- shows "v v* mat 1 = v"
- by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
-
-lemma%unimportant scaleR_vector_matrix_assoc:
- fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
- shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
- by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
-
-lemma%important vector_scaleR_matrix_ac:
- fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
- shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
-proof%unimportant -
- have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
- unfolding vector_matrix_mult_def
- by (simp add: algebra_simps)
- with scaleR_vector_matrix_assoc
- show "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
- by auto
-qed
-
-
-subsection%important\<open>Some bounds on components etc. relative to operator norm\<close>
+subsection%important\<open>Bounds on components etc.\ relative to operator norm\<close>
lemma%important norm_column_le_onorm:
fixes A :: "real^'n^'m"
@@ -366,26 +267,6 @@
finally show "norm (A *v x) \<le> CARD('m) * real (CARD('n)) * B * norm x" .
qed
-subsection%important \<open>lambda skolemization on cartesian products\<close>
-
-lemma%important 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%unimportant -
- let ?S = "(UNIV :: 'n set)"
- { assume H: "?rhs"
- then have ?lhs by auto }
- moreover
- { 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
- from f have "P i (f i)" by metis
- then have "P i (?x $ i)" by auto
- }
- hence "\<forall>i. P i (?x$i)" by metis
- hence ?rhs by metis }
- ultimately show ?thesis by metis
-qed
lemma%unimportant rational_approximation:
assumes "e > 0"
@@ -416,36 +297,6 @@
lemma%unimportant vector_sub_project_orthogonal_cart: "(b::real^'n) \<bullet> (x - ((b \<bullet> x) / (b \<bullet> b)) *s b) = 0"
unfolding inner_simps scalar_mult_eq_scaleR by auto
-
-text \<open>The same result in terms of square matrices.\<close>
-
-
-text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
-
-definition%unimportant "rowvector v = (\<chi> i j. (v$j))"
-
-definition%unimportant "columnvector v = (\<chi> i j. (v$i))"
-
-lemma%unimportant transpose_columnvector: "transpose(columnvector v) = rowvector v"
- by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
-
-lemma%unimportant transpose_rowvector: "transpose(rowvector v) = columnvector v"
- by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
-
-lemma%unimportant dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
- by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
-
-lemma%unimportant 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%unimportant 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)$1"
- unfolding dot_matrix_product transpose_columnvector[symmetric]
- dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
-
lemma%unimportant infnorm_cart:"infnorm (x::real^'n) = Sup {\<bar>x$i\<bar> |i. i\<in>UNIV}"
by (simp add: infnorm_def inner_axis Basis_vec_def) (metis (lifting) inner_axis real_inner_1_right)
@@ -626,16 +477,16 @@
by (simp add: is_interval_def Ball_def Basis_vec_def inner_axis imp_ex)
lemma%unimportant closed_halfspace_component_le_cart: "closed {x::real^'n. x$i \<le> a}"
- by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
+ by (simp add: closed_Collect_le continuous_on_component)
lemma%unimportant closed_halfspace_component_ge_cart: "closed {x::real^'n. x$i \<ge> a}"
- by (simp add: closed_Collect_le continuous_on_const continuous_on_id continuous_on_component)
+ by (simp add: closed_Collect_le continuous_on_component)
lemma%unimportant open_halfspace_component_lt_cart: "open {x::real^'n. x$i < a}"
- by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
+ by (simp add: open_Collect_less continuous_on_component)
lemma%unimportant open_halfspace_component_gt_cart: "open {x::real^'n. x$i > a}"
- by (simp add: open_Collect_less continuous_on_const continuous_on_id continuous_on_component)
+ by (simp add: open_Collect_less continuous_on_component)
lemma%unimportant Lim_component_le_cart:
fixes f :: "'a \<Rightarrow> real^'n"
@@ -676,79 +527,6 @@
unfolding Collect_all_eq by (simp add: closed_INT)
qed
-lemma%important dim_substandard_cart: "vec.dim {x::'a::field^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
- (is "vec.dim ?A = _")
-proof%unimportant (rule vec.dim_unique)
- let ?B = "((\<lambda>x. axis x 1) ` d)"
- have subset_basis: "?B \<subseteq> cart_basis"
- by (auto simp: cart_basis_def)
- show "?B \<subseteq> ?A"
- by (auto simp: axis_def)
- show "vec.independent ((\<lambda>x. axis x 1) ` d)"
- using subset_basis
- by (rule vec.independent_mono[OF vec.independent_Basis])
- have "x \<in> vec.span ?B" if "\<forall>i. i \<notin> d \<longrightarrow> x $ i = 0" for x::"'a^'n"
- proof -
- have "finite ?B"
- using subset_basis finite_cart_basis
- by (rule finite_subset)
- have "x = (\<Sum>i\<in>UNIV. x $ i *s axis i 1)"
- by (rule basis_expansion[symmetric])
- also have "\<dots> = (\<Sum>i\<in>d. (x $ i) *s axis i 1)"
- by (rule sum.mono_neutral_cong_right) (auto simp: that)
- also have "\<dots> \<in> vec.span ?B"
- by (simp add: vec.span_sum vec.span_clauses)
- finally show "x \<in> vec.span ?B" .
- qed
- then show "?A \<subseteq> vec.span ?B" by auto
-qed (simp add: card_image inj_on_def axis_eq_axis)
-
-lemma%unimportant dim_subset_UNIV_cart_gen:
- fixes S :: "('a::field^'n) set"
- shows "vec.dim S \<le> CARD('n)"
- by (metis vec.dim_eq_full vec.dim_subset_UNIV vec.span_UNIV vec_dim_card)
-
-lemma%unimportant dim_subset_UNIV_cart:
- fixes S :: "(real^'n) set"
- shows "dim S \<le> CARD('n)"
- using dim_subset_UNIV_cart_gen[of S] by (simp add: dim_vec_eq)
-
-lemma%unimportant affinity_inverses:
- assumes m0: "m \<noteq> (0::'a::field)"
- shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
- "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
- using m0
- by (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
-
-lemma%important vector_affinity_eq:
- assumes m0: "(m::'a::field) \<noteq> 0"
- shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
-proof%unimportant
- assume h: "m *s x + c = y"
- hence "m *s x = y - c" by (simp add: field_simps)
- hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
- then show "x = inverse m *s y + - (inverse m *s c)"
- using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
-next
- assume h: "x = inverse m *s y + - (inverse m *s c)"
- show "m *s x + c = y" unfolding h
- using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
-qed
-
-lemma%unimportant vector_eq_affinity:
- "(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%unimportant vector_cart:
- fixes f :: "real^'n \<Rightarrow> real"
- shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
- unfolding euclidean_eq_iff[where 'a="real^'n"]
- by simp (simp add: Basis_vec_def inner_axis)
-
-lemma%unimportant const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
- by (rule vector_cart)
-
subsection%important "Convex Euclidean Space"
lemma%unimportant Cart_1:"(1::real^'n) = \<Sum>Basis"
@@ -801,7 +579,7 @@
qed
-subsection%important \<open>Component of the differential must be zero if it exists at a local
+text%important \<open>Component of the differential must be zero if it exists at a local
maximum or minimum for that corresponding component\<close>
lemma%important differential_zero_maxmin_cart:
@@ -813,282 +591,8 @@
vector_cart[of "\<lambda>j. frechet_derivative f (at x) j $ k"]
by (simp add: Basis_vec_def axis_eq_axis inner_axis jacobian_def matrix_def)
-subsection%unimportant \<open>Lemmas for working on \<^typ>\<open>real^1\<close>\<close>
-
-lemma forall_1[simp]: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
- by (metis (full_types) num1_eq_iff)
-
-lemma ex_1[simp]: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
- by auto (metis (full_types) num1_eq_iff)
-
-lemma exhaust_2:
- fixes x :: 2
- shows "x = 1 \<or> x = 2"
-proof (induct x)
- case (of_int z)
- then have "0 \<le> z" and "z < 2" by simp_all
- then have "z = 0 | z = 1" by arith
- then show ?case by auto
-qed
-
-lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
- by (metis exhaust_2)
-
-lemma exhaust_3:
- fixes x :: 3
- shows "x = 1 \<or> x = 2 \<or> x = 3"
-proof (induct x)
- case (of_int z)
- then have "0 \<le> z" and "z < 3" by simp_all
- then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
- then show ?case by auto
-qed
-
-lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
- by (metis exhaust_3)
-
-lemma UNIV_1 [simp]: "UNIV = {1::1}"
- by (auto simp add: num1_eq_iff)
-
-lemma UNIV_2: "UNIV = {1::2, 2::2}"
- using exhaust_2 by auto
-
-lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
- using exhaust_3 by auto
-
-lemma sum_1: "sum f (UNIV::1 set) = f 1"
- unfolding UNIV_1 by simp
-
-lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
- unfolding UNIV_2 by simp
-
-lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
- unfolding UNIV_3 by (simp add: ac_simps)
-
-lemma num1_eqI:
- fixes a::num1 shows "a = b"
- by (metis (full_types) UNIV_1 UNIV_I empty_iff insert_iff)
-
-lemma num1_eq1 [simp]:
- fixes a::num1 shows "a = 1"
- by (rule num1_eqI)
-
-instantiation num1 :: cart_one
-begin
-
-instance
-proof
- show "CARD(1) = Suc 0" by auto
-qed
-
-end
-
-instantiation num1 :: linorder begin
-definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b"
-definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b"
-instance
- by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI)
-end
-
-instance num1 :: wellorder
- by intro_classes (auto simp: less_eq_num1_def less_num1_def)
-
-subsection%unimportant\<open>The collapse of the general concepts to dimension one\<close>
-
-lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
- by (simp add: vec_eq_iff)
-
-lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
- apply auto
- apply (erule_tac x= "x$1" in allE)
- apply (simp only: vector_one[symmetric])
- done
-
-lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
- by (simp add: norm_vec_def)
-
-lemma dist_vector_1:
- fixes x :: "'a::real_normed_vector^1"
- shows "dist x y = dist (x$1) (y$1)"
- by (simp add: dist_norm norm_vector_1)
-
-lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"
- by (simp add: norm_vector_1)
-
-lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
- by (auto simp add: norm_real dist_norm)
-
-subsection%important\<open> Rank of a matrix\<close>
-
-text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close>
-
-lemma%unimportant matrix_vector_mult_in_columnspace_gen:
- fixes A :: "'a::field^'n^'m"
- shows "(A *v x) \<in> vec.span(columns A)"
- apply (simp add: matrix_vector_column columns_def transpose_def column_def)
- apply (intro vec.span_sum vec.span_scale)
- apply (force intro: vec.span_base)
- done
-
-lemma%unimportant matrix_vector_mult_in_columnspace:
- fixes A :: "real^'n^'m"
- shows "(A *v x) \<in> span(columns A)"
- using matrix_vector_mult_in_columnspace_gen[of A x] by (simp add: span_vec_eq)
-
-lemma%important orthogonal_nullspace_rowspace:
- fixes A :: "real^'n^'m"
- assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)"
- shows "orthogonal x y"
- using y
-proof%unimportant (induction rule: span_induct)
- case base
- then show ?case
- by (simp add: subspace_orthogonal_to_vector)
-next
- case (step v)
- then obtain i where "v = row i A"
- by (auto simp: rows_def)
- with 0 show ?case
- unfolding orthogonal_def inner_vec_def matrix_vector_mult_def row_def
- by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
-qed
-
-lemma%unimportant nullspace_inter_rowspace:
- fixes A :: "real^'n^'m"
- shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0"
- using orthogonal_nullspace_rowspace orthogonal_self span_zero matrix_vector_mult_0_right
- by blast
-
-lemma%unimportant matrix_vector_mul_injective_on_rowspace:
- fixes A :: "real^'n^'m"
- shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y"
- using nullspace_inter_rowspace [of A "x-y"]
- by (metis diff_eq_diff_eq diff_self matrix_vector_mult_diff_distrib span_diff)
-
-definition%important rank :: "'a::field^'n^'m=>nat"
- where row_rank_def_gen: "rank A \<equiv> vec.dim(rows A)"
-
-lemma%important row_rank_def: "rank A = dim (rows A)" for A::"real^'n^'m"
- by%unimportant (auto simp: row_rank_def_gen dim_vec_eq)
-
-lemma%important dim_rows_le_dim_columns:
- fixes A :: "real^'n^'m"
- shows "dim(rows A) \<le> dim(columns A)"
-proof%unimportant -
- have "dim (span (rows A)) \<le> dim (span (columns A))"
- proof -
- obtain B where "independent B" "span(rows A) \<subseteq> span B"
- and B: "B \<subseteq> span(rows A)""card B = dim (span(rows A))"
- using basis_exists [of "span(rows A)"] by metis
- with span_subspace have eq: "span B = span(rows A)"
- by auto
- then have inj: "inj_on ((*v) A) (span B)"
- by (simp add: inj_on_def matrix_vector_mul_injective_on_rowspace)
- then have ind: "independent ((*v) A ` B)"
- by (rule linear_independent_injective_image [OF Finite_Cartesian_Product.matrix_vector_mul_linear \<open>independent B\<close>])
- have "dim (span (rows A)) \<le> card ((*v) A ` B)"
- unfolding B(2)[symmetric]
- using inj
- by (auto simp: card_image inj_on_subset span_superset)
- also have "\<dots> \<le> dim (span (columns A))"
- using _ ind
- by (rule independent_card_le_dim) (auto intro!: matrix_vector_mult_in_columnspace)
- finally show ?thesis .
- qed
- then show ?thesis
- by (simp add: dim_span)
-qed
-
-lemma%unimportant column_rank_def:
- fixes A :: "real^'n^'m"
- shows "rank A = dim(columns A)"
- unfolding row_rank_def
- by (metis columns_transpose dim_rows_le_dim_columns le_antisym rows_transpose)
-
-lemma%unimportant rank_transpose:
- fixes A :: "real^'n^'m"
- shows "rank(transpose A) = rank A"
- by (metis column_rank_def row_rank_def rows_transpose)
-
-lemma%unimportant matrix_vector_mult_basis:
- fixes A :: "real^'n^'m"
- shows "A *v (axis k 1) = column k A"
- by (simp add: cart_eq_inner_axis column_def matrix_mult_dot)
-
-lemma%unimportant columns_image_basis:
- fixes A :: "real^'n^'m"
- shows "columns A = (*v) A ` (range (\<lambda>i. axis i 1))"
- by (force simp: columns_def matrix_vector_mult_basis [symmetric])
-
-lemma%important rank_dim_range:
- fixes A :: "real^'n^'m"
- shows "rank A = dim(range (\<lambda>x. A *v x))"
- unfolding column_rank_def
-proof%unimportant (rule span_eq_dim)
- have "span (columns A) \<subseteq> span (range ((*v) A))" (is "?l \<subseteq> ?r")
- by (simp add: columns_image_basis image_subsetI span_mono)
- then show "?l = ?r"
- by (metis (no_types, lifting) image_subset_iff matrix_vector_mult_in_columnspace
- span_eq span_span)
-qed
-
-lemma%unimportant rank_bound:
- fixes A :: "real^'n^'m"
- shows "rank A \<le> min CARD('m) (CARD('n))"
- by (metis (mono_tags, lifting) dim_subset_UNIV_cart min.bounded_iff
- column_rank_def row_rank_def)
-
-lemma%unimportant full_rank_injective:
- fixes A :: "real^'n^'m"
- shows "rank A = CARD('n) \<longleftrightarrow> inj ((*v) A)"
- by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows row_rank_def
- dim_eq_full [symmetric] card_cart_basis vec.dimension_def)
-
-lemma%unimportant full_rank_surjective:
- fixes A :: "real^'n^'m"
- shows "rank A = CARD('m) \<longleftrightarrow> surj ((*v) A)"
- by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric]
- matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose)
-
-lemma%unimportant rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
- by (simp add: full_rank_injective inj_on_def)
-
-lemma%unimportant less_rank_noninjective:
- fixes A :: "real^'n^'m"
- shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj ((*v) A)"
-using less_le rank_bound by (auto simp: full_rank_injective [symmetric])
-
-lemma%unimportant matrix_nonfull_linear_equations_eq:
- fixes A :: "real^'n^'m"
- shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> rank A \<noteq> CARD('n)"
- by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker)
-
-lemma%unimportant rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank (0::real^'n^'m) = 0"
- for A :: "real^'n^'m"
- by (auto simp: rank_dim_range matrix_eq)
-
-lemma%important rank_mul_le_right:
- fixes A :: "real^'n^'m" and B :: "real^'p^'n"
- shows "rank(A ** B) \<le> rank B"
-proof%unimportant -
- have "rank(A ** B) \<le> dim ((*v) A ` range ((*v) B))"
- by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc)
- also have "\<dots> \<le> rank B"
- by (simp add: rank_dim_range dim_image_le)
- finally show ?thesis .
-qed
-
-lemma%unimportant rank_mul_le_left:
- fixes A :: "real^'n^'m" and B :: "real^'p^'n"
- shows "rank(A ** B) \<le> rank A"
- by (metis matrix_transpose_mul rank_mul_le_right rank_transpose)
-
subsection%unimportant\<open>Routine results connecting the types \<^typ>\<open>real^1\<close> and \<^typ>\<open>real\<close>\<close>
-lemma vector_one_nth [simp]:
- fixes x :: "'a^1" shows "vec (x $ 1) = x"
- by (metis vec_def vector_one)
-
lemma vec_cbox_1_eq [simp]:
shows "vec ` cbox u v = cbox (vec u) (vec v ::real^1)"
by (force simp: Basis_vec_def cart_eq_inner_axis [symmetric] mem_box)
@@ -1119,76 +623,6 @@
qed
qed simp
-lemma tendsto_at_within_vector_1:
- fixes S :: "'a :: metric_space set"
- assumes "(f \<longlongrightarrow> fx) (at x within S)"
- shows "((\<lambda>y::'a^1. \<chi> i. f (y $ 1)) \<longlongrightarrow> (vec fx::'a^1)) (at (vec x) within vec ` S)"
-proof (rule topological_tendstoI)
- fix T :: "('a^1) set"
- assume "open T" "vec fx \<in> T"
- have "\<forall>\<^sub>F x in at x within S. f x \<in> (\<lambda>x. x $ 1) ` T"
- using \<open>open T\<close> \<open>vec fx \<in> T\<close> assms open_image_vec_nth tendsto_def by fastforce
- then show "\<forall>\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\<chi> i. f (x $ 1)) \<in> T"
- unfolding eventually_at dist_norm [symmetric]
- by (rule ex_forward)
- (use \<open>open T\<close> in
- \<open>fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\<close>)
-qed
-
-lemma has_derivative_vector_1:
- assumes der_g: "(g has_derivative (\<lambda>x. x * g' a)) (at a within S)"
- shows "((\<lambda>x. vec (g (x $ 1))) has_derivative (*\<^sub>R) (g' a))
- (at ((vec a)::real^1) within vec ` S)"
- using der_g
- apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1)
- apply (drule tendsto_at_within_vector_1, vector)
- apply (auto simp: algebra_simps eventually_at tendsto_def)
- done
-
-
-subsection%unimportant\<open>Explicit vector construction from lists\<close>
-
-definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
-
-lemma vector_1 [simp]: "(vector[x]) $1 = x"
- unfolding vector_def by simp
-
-lemma vector_2 [simp]: "(vector[x,y]) $1 = x" "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
- unfolding vector_def by simp_all
-
-lemma vector_3 [simp]:
- "(vector [x,y,z] ::('a::zero)^3)$1 = x"
- "(vector [x,y,z] ::('a::zero)^3)$2 = y"
- "(vector [x,y,z] ::('a::zero)^3)$3 = z"
- unfolding vector_def by simp_all
-
-lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
- by (metis vector_1 vector_one)
-
-lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
- apply auto
- apply (erule_tac x="v$1" in allE)
- apply (erule_tac x="v$2" in allE)
- apply (subgoal_tac "vector [v$1, v$2] = v")
- apply simp
- apply (vector vector_def)
- apply (simp add: forall_2)
- done
-
-lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
- apply auto
- apply (erule_tac x="v$1" in allE)
- apply (erule_tac x="v$2" in allE)
- apply (erule_tac x="v$3" in allE)
- apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
- apply simp
- apply (vector vector_def)
- apply (simp add: forall_3)
- done
-
-lemma bounded_linear_component_cart[intro]: "bounded_linear (\<lambda>x::real^'n. x $ k)"
- apply (rule bounded_linear_intro[where K=1])
- using component_le_norm_cart[of _ k] unfolding real_norm_def by auto
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)}"
@@ -1201,6 +635,5 @@
lemmas cartesian_euclidean_space_uniform_limit_intros[uniform_limit_intros] =
bounded_linear.uniform_limit[OF blinfun.bounded_linear_right]
bounded_linear.uniform_limit[OF bounded_linear_vec_nth]
- bounded_linear.uniform_limit[OF bounded_linear_component_cart]
end
--- a/src/HOL/Analysis/Cartesian_Space.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Cartesian_Space.thy Wed Jan 16 21:27:33 2019 +0000
@@ -6,11 +6,15 @@
Author: Fabian Immler, TUM
*)
+section "Linear Algebra on Finite Cartesian Products"
+
theory Cartesian_Space
imports
Finite_Cartesian_Product Linear_Algebra
begin
+subsection \<open>Type @{typ \<open>'a ^ 'n\<close>} and fields as vector spaces\<close>
+
definition%unimportant "cart_basis = {axis i 1 | i. i\<in>UNIV}"
lemma%unimportant finite_cart_basis: "finite (cart_basis)" unfolding cart_basis_def
@@ -20,7 +24,7 @@
unfolding cart_basis_def Setcompr_eq_image
by (rule card_image) (auto simp: inj_on_def axis_eq_axis)
-interpretation vec: vector_space "(*s) "
+interpretation%important vec: vector_space "(*s) "
by unfold_locales (vector algebra_simps)+
lemma%unimportant independent_cart_basis:
@@ -448,8 +452,8 @@
finally show ?thesis .
qed
-interpretation vector_space_over_itself: vector_space "(*) :: 'a::field => 'a => 'a"
- by unfold_locales (simp_all add: algebra_simps)
+interpretation%important vector_space_over_itself: vector_space "(*) :: 'a::field \<Rightarrow> 'a \<Rightarrow> 'a"
+by unfold_locales (simp_all add: algebra_simps)
lemmas [simp del] = vector_space_over_itself.scale_scale
@@ -460,4 +464,470 @@
lemma dimension_eq_1[code_unfold]: "vector_space_over_itself.dimension TYPE('a::field)= 1"
unfolding vector_space_over_itself.dimension_def by simp
+
+lemma%unimportant dim_subset_UNIV_cart_gen:
+ fixes S :: "('a::field^'n) set"
+ shows "vec.dim S \<le> CARD('n)"
+ by (metis vec.dim_eq_full vec.dim_subset_UNIV vec.span_UNIV vec_dim_card)
+
+lemma%unimportant dim_subset_UNIV_cart:
+ fixes S :: "(real^'n) set"
+ shows "dim S \<le> CARD('n)"
+ using dim_subset_UNIV_cart_gen[of S] by (simp add: dim_vec_eq)
+
+text\<open>Two sometimes fruitful ways of looking at matrix-vector multiplication.\<close>
+
+lemma%important matrix_mult_dot: "A *v x = (\<chi> i. inner (A$i) x)"
+ by (simp add: matrix_vector_mult_def inner_vec_def)
+
+lemma%unimportant 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
+ sum_distrib_right sum_distrib_left)
+ apply (subst sum.swap)
+ apply (simp add: ac_simps)
+ done
+
+lemma%important matrix_adjoint: assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'m)"
+ shows "matrix(adjoint f) = transpose(matrix f)"
+proof%unimportant -
+ have "matrix(adjoint f) = matrix(adjoint ((*v) (matrix f)))"
+ by (simp add: lf)
+ also have "\<dots> = transpose(matrix f)"
+ unfolding adjoint_matrix matrix_of_matrix_vector_mul
+ apply rule
+ done
+ finally show ?thesis .
+qed
+
+
+subsection%important\<open> Rank of a matrix\<close>
+
+text\<open>Equivalence of row and column rank is taken from George Mackiw's paper, Mathematics Magazine 1995, p. 285.\<close>
+
+lemma%unimportant matrix_vector_mult_in_columnspace_gen:
+ fixes A :: "'a::field^'n^'m"
+ shows "(A *v x) \<in> vec.span(columns A)"
+ apply (simp add: matrix_vector_column columns_def transpose_def column_def)
+ apply (intro vec.span_sum vec.span_scale)
+ apply (force intro: vec.span_base)
+ done
+
+lemma%unimportant matrix_vector_mult_in_columnspace:
+ fixes A :: "real^'n^'m"
+ shows "(A *v x) \<in> span(columns A)"
+ using matrix_vector_mult_in_columnspace_gen[of A x] by (simp add: span_vec_eq)
+
+lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
+ by (simp add: subspace_def orthogonal_clauses)
+
+lemma%important orthogonal_nullspace_rowspace:
+ fixes A :: "real^'n^'m"
+ assumes 0: "A *v x = 0" and y: "y \<in> span(rows A)"
+ shows "orthogonal x y"
+ using y
+proof%unimportant (induction rule: span_induct)
+ case base
+ then show ?case
+ by (simp add: subspace_orthogonal_to_vector)
+next
+ case (step v)
+ then obtain i where "v = row i A"
+ by (auto simp: rows_def)
+ with 0 show ?case
+ unfolding orthogonal_def inner_vec_def matrix_vector_mult_def row_def
+ by (simp add: mult.commute) (metis (no_types) vec_lambda_beta zero_index)
+qed
+
+lemma%unimportant nullspace_inter_rowspace:
+ fixes A :: "real^'n^'m"
+ shows "A *v x = 0 \<and> x \<in> span(rows A) \<longleftrightarrow> x = 0"
+ using orthogonal_nullspace_rowspace orthogonal_self span_zero matrix_vector_mult_0_right
+ by blast
+
+lemma%unimportant matrix_vector_mul_injective_on_rowspace:
+ fixes A :: "real^'n^'m"
+ shows "\<lbrakk>A *v x = A *v y; x \<in> span(rows A); y \<in> span(rows A)\<rbrakk> \<Longrightarrow> x = y"
+ using nullspace_inter_rowspace [of A "x-y"]
+ by (metis diff_eq_diff_eq diff_self matrix_vector_mult_diff_distrib span_diff)
+
+definition%important rank :: "'a::field^'n^'m=>nat"
+ where row_rank_def_gen: "rank A \<equiv> vec.dim(rows A)"
+
+lemma%important row_rank_def: "rank A = dim (rows A)" for A::"real^'n^'m"
+ by%unimportant (auto simp: row_rank_def_gen dim_vec_eq)
+
+lemma%important dim_rows_le_dim_columns:
+ fixes A :: "real^'n^'m"
+ shows "dim(rows A) \<le> dim(columns A)"
+proof%unimportant -
+ have "dim (span (rows A)) \<le> dim (span (columns A))"
+ proof -
+ obtain B where "independent B" "span(rows A) \<subseteq> span B"
+ and B: "B \<subseteq> span(rows A)""card B = dim (span(rows A))"
+ using basis_exists [of "span(rows A)"] by metis
+ with span_subspace have eq: "span B = span(rows A)"
+ by auto
+ then have inj: "inj_on ((*v) A) (span B)"
+ by (simp add: inj_on_def matrix_vector_mul_injective_on_rowspace)
+ then have ind: "independent ((*v) A ` B)"
+ by (rule linear_independent_injective_image [OF Finite_Cartesian_Product.matrix_vector_mul_linear \<open>independent B\<close>])
+ have "dim (span (rows A)) \<le> card ((*v) A ` B)"
+ unfolding B(2)[symmetric]
+ using inj
+ by (auto simp: card_image inj_on_subset span_superset)
+ also have "\<dots> \<le> dim (span (columns A))"
+ using _ ind
+ by (rule independent_card_le_dim) (auto intro!: matrix_vector_mult_in_columnspace)
+ finally show ?thesis .
+ qed
+ then show ?thesis
+ by (simp add: dim_span)
+qed
+
+lemma%unimportant column_rank_def:
+ fixes A :: "real^'n^'m"
+ shows "rank A = dim(columns A)"
+ unfolding row_rank_def
+ by (metis columns_transpose dim_rows_le_dim_columns le_antisym rows_transpose)
+
+lemma%unimportant rank_transpose:
+ fixes A :: "real^'n^'m"
+ shows "rank(transpose A) = rank A"
+ by (metis column_rank_def row_rank_def rows_transpose)
+
+lemma%unimportant matrix_vector_mult_basis:
+ fixes A :: "real^'n^'m"
+ shows "A *v (axis k 1) = column k A"
+ by (simp add: cart_eq_inner_axis column_def matrix_mult_dot)
+
+lemma%unimportant columns_image_basis:
+ fixes A :: "real^'n^'m"
+ shows "columns A = (*v) A ` (range (\<lambda>i. axis i 1))"
+ by (force simp: columns_def matrix_vector_mult_basis [symmetric])
+
+lemma%important rank_dim_range:
+ fixes A :: "real^'n^'m"
+ shows "rank A = dim(range (\<lambda>x. A *v x))"
+ unfolding column_rank_def
+proof%unimportant (rule span_eq_dim)
+ have "span (columns A) \<subseteq> span (range ((*v) A))" (is "?l \<subseteq> ?r")
+ by (simp add: columns_image_basis image_subsetI span_mono)
+ then show "?l = ?r"
+ by (metis (no_types, lifting) image_subset_iff matrix_vector_mult_in_columnspace
+ span_eq span_span)
+qed
+
+lemma%unimportant rank_bound:
+ fixes A :: "real^'n^'m"
+ shows "rank A \<le> min CARD('m) (CARD('n))"
+ by (metis (mono_tags, lifting) dim_subset_UNIV_cart min.bounded_iff
+ column_rank_def row_rank_def)
+
+lemma%unimportant full_rank_injective:
+ fixes A :: "real^'n^'m"
+ shows "rank A = CARD('n) \<longleftrightarrow> inj ((*v) A)"
+ by (simp add: matrix_left_invertible_injective [symmetric] matrix_left_invertible_span_rows row_rank_def
+ dim_eq_full [symmetric] card_cart_basis vec.dimension_def)
+
+lemma%unimportant full_rank_surjective:
+ fixes A :: "real^'n^'m"
+ shows "rank A = CARD('m) \<longleftrightarrow> surj ((*v) A)"
+ by (simp add: matrix_right_invertible_surjective [symmetric] left_invertible_transpose [symmetric]
+ matrix_left_invertible_injective full_rank_injective [symmetric] rank_transpose)
+
+lemma%unimportant rank_I: "rank(mat 1::real^'n^'n) = CARD('n)"
+ by (simp add: full_rank_injective inj_on_def)
+
+lemma%unimportant less_rank_noninjective:
+ fixes A :: "real^'n^'m"
+ shows "rank A < CARD('n) \<longleftrightarrow> \<not> inj ((*v) A)"
+using less_le rank_bound by (auto simp: full_rank_injective [symmetric])
+
+lemma%unimportant matrix_nonfull_linear_equations_eq:
+ fixes A :: "real^'n^'m"
+ shows "(\<exists>x. (x \<noteq> 0) \<and> A *v x = 0) \<longleftrightarrow> rank A \<noteq> CARD('n)"
+ by (meson matrix_left_invertible_injective full_rank_injective matrix_left_invertible_ker)
+
+lemma%unimportant rank_eq_0: "rank A = 0 \<longleftrightarrow> A = 0" and rank_0 [simp]: "rank (0::real^'n^'m) = 0"
+ for A :: "real^'n^'m"
+ by (auto simp: rank_dim_range matrix_eq)
+
+lemma%important rank_mul_le_right:
+ fixes A :: "real^'n^'m" and B :: "real^'p^'n"
+ shows "rank(A ** B) \<le> rank B"
+proof%unimportant -
+ have "rank(A ** B) \<le> dim ((*v) A ` range ((*v) B))"
+ by (auto simp: rank_dim_range image_comp o_def matrix_vector_mul_assoc)
+ also have "\<dots> \<le> rank B"
+ by (simp add: rank_dim_range dim_image_le)
+ finally show ?thesis .
+qed
+
+lemma%unimportant rank_mul_le_left:
+ fixes A :: "real^'n^'m" and B :: "real^'p^'n"
+ shows "rank(A ** B) \<le> rank A"
+ by (metis matrix_transpose_mul rank_mul_le_right rank_transpose)
+
+
+subsection%unimportant \<open>Lemmas for working on \<open>real^1/2/3\<close>\<close>
+
+lemma exhaust_2:
+ fixes x :: 2
+ shows "x = 1 \<or> x = 2"
+proof (induct x)
+ case (of_int z)
+ then have "0 \<le> z" and "z < 2" by simp_all
+ then have "z = 0 | z = 1" by arith
+ then show ?case by auto
+qed
+
+lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
+ by (metis exhaust_2)
+
+lemma exhaust_3:
+ fixes x :: 3
+ shows "x = 1 \<or> x = 2 \<or> x = 3"
+proof (induct x)
+ case (of_int z)
+ then have "0 \<le> z" and "z < 3" by simp_all
+ then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
+ then show ?case by auto
+qed
+
+lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
+ by (metis exhaust_3)
+
+lemma UNIV_1 [simp]: "UNIV = {1::1}"
+ by (auto simp add: num1_eq_iff)
+
+lemma UNIV_2: "UNIV = {1::2, 2::2}"
+ using exhaust_2 by auto
+
+lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
+ using exhaust_3 by auto
+
+lemma sum_1: "sum f (UNIV::1 set) = f 1"
+ unfolding UNIV_1 by simp
+
+lemma sum_2: "sum f (UNIV::2 set) = f 1 + f 2"
+ unfolding UNIV_2 by simp
+
+lemma sum_3: "sum f (UNIV::3 set) = f 1 + f 2 + f 3"
+ unfolding UNIV_3 by (simp add: ac_simps)
+
+subsection%unimportant\<open>The collapse of the general concepts to dimension one\<close>
+
+lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
+ by (simp add: vec_eq_iff)
+
+lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
+ apply auto
+ apply (erule_tac x= "x$1" in allE)
+ apply (simp only: vector_one[symmetric])
+ done
+
+lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
+ by (simp add: norm_vec_def)
+
+lemma dist_vector_1:
+ fixes x :: "'a::real_normed_vector^1"
+ shows "dist x y = dist (x$1) (y$1)"
+ by (simp add: dist_norm norm_vector_1)
+
+lemma norm_real: "norm(x::real ^ 1) = \<bar>x$1\<bar>"
+ by (simp add: norm_vector_1)
+
+lemma dist_real: "dist(x::real ^ 1) y = \<bar>(x$1) - (y$1)\<bar>"
+ by (auto simp add: norm_real dist_norm)
+
+
+subsection%unimportant\<open>Routine results connecting the types \<^typ>\<open>real^1\<close> and \<^typ>\<open>real\<close>\<close>
+
+lemma vector_one_nth [simp]:
+ fixes x :: "'a^1" shows "vec (x $ 1) = x"
+ by (metis vec_def vector_one)
+
+lemma tendsto_at_within_vector_1:
+ fixes S :: "'a :: metric_space set"
+ assumes "(f \<longlongrightarrow> fx) (at x within S)"
+ shows "((\<lambda>y::'a^1. \<chi> i. f (y $ 1)) \<longlongrightarrow> (vec fx::'a^1)) (at (vec x) within vec ` S)"
+proof (rule topological_tendstoI)
+ fix T :: "('a^1) set"
+ assume "open T" "vec fx \<in> T"
+ have "\<forall>\<^sub>F x in at x within S. f x \<in> (\<lambda>x. x $ 1) ` T"
+ using \<open>open T\<close> \<open>vec fx \<in> T\<close> assms open_image_vec_nth tendsto_def by fastforce
+ then show "\<forall>\<^sub>F x::'a^1 in at (vec x) within vec ` S. (\<chi> i. f (x $ 1)) \<in> T"
+ unfolding eventually_at dist_norm [symmetric]
+ by (rule ex_forward)
+ (use \<open>open T\<close> in
+ \<open>fastforce simp: dist_norm dist_vec_def L2_set_def image_iff vector_one open_vec_def\<close>)
+qed
+
+lemma has_derivative_vector_1:
+ assumes der_g: "(g has_derivative (\<lambda>x. x * g' a)) (at a within S)"
+ shows "((\<lambda>x. vec (g (x $ 1))) has_derivative (*\<^sub>R) (g' a))
+ (at ((vec a)::real^1) within vec ` S)"
+ using der_g
+ apply (auto simp: Deriv.has_derivative_within bounded_linear_scaleR_right norm_vector_1)
+ apply (drule tendsto_at_within_vector_1, vector)
+ apply (auto simp: algebra_simps eventually_at tendsto_def)
+ done
+
+
+subsection%unimportant\<open>Explicit vector construction from lists\<close>
+
+definition "vector l = (\<chi> i. foldr (\<lambda>x f n. fun_upd (f (n+1)) n x) l (\<lambda>n x. 0) 1 i)"
+
+lemma vector_1 [simp]: "(vector[x]) $1 = x"
+ unfolding vector_def by simp
+
+lemma vector_2 [simp]: "(vector[x,y]) $1 = x" "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
+ unfolding vector_def by simp_all
+
+lemma vector_3 [simp]:
+ "(vector [x,y,z] ::('a::zero)^3)$1 = x"
+ "(vector [x,y,z] ::('a::zero)^3)$2 = y"
+ "(vector [x,y,z] ::('a::zero)^3)$3 = z"
+ unfolding vector_def by simp_all
+
+lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
+ by (metis vector_1 vector_one)
+
+lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
+ apply auto
+ apply (erule_tac x="v$1" in allE)
+ apply (erule_tac x="v$2" in allE)
+ apply (subgoal_tac "vector [v$1, v$2] = v")
+ apply simp
+ apply (vector vector_def)
+ apply (simp add: forall_2)
+ done
+
+lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
+ apply auto
+ apply (erule_tac x="v$1" in allE)
+ apply (erule_tac x="v$2" in allE)
+ apply (erule_tac x="v$3" in allE)
+ apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
+ apply simp
+ apply (vector vector_def)
+ apply (simp add: forall_3)
+ done
+
+subsection%unimportant \<open>lambda skolemization on cartesian products\<close>
+
+lemma%important 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%unimportant -
+ let ?S = "(UNIV :: 'n set)"
+ { assume H: "?rhs"
+ then have ?lhs by auto }
+ moreover
+ { 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
+ from f have "P i (f i)" by metis
+ then have "P i (?x $ i)" by auto
+ }
+ hence "\<forall>i. P i (?x$i)" by metis
+ hence ?rhs by metis }
+ ultimately show ?thesis by metis
+qed
+
+
+text \<open>The same result in terms of square matrices.\<close>
+
+
+text \<open>Considering an n-element vector as an n-by-1 or 1-by-n matrix.\<close>
+
+definition%unimportant "rowvector v = (\<chi> i j. (v$j))"
+
+definition%unimportant "columnvector v = (\<chi> i j. (v$i))"
+
+lemma%unimportant transpose_columnvector: "transpose(columnvector v) = rowvector v"
+ by (simp add: transpose_def rowvector_def columnvector_def vec_eq_iff)
+
+lemma%unimportant transpose_rowvector: "transpose(rowvector v) = columnvector v"
+ by (simp add: transpose_def columnvector_def rowvector_def vec_eq_iff)
+
+lemma%unimportant dot_rowvector_columnvector: "columnvector (A *v v) = A ** columnvector v"
+ by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
+
+lemma%unimportant 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%unimportant 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)$1"
+ unfolding dot_matrix_product transpose_columnvector[symmetric]
+ dot_rowvector_columnvector matrix_transpose_mul matrix_mul_assoc ..
+
+
+lemma%important dim_substandard_cart: "vec.dim {x::'a::field^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d"
+ (is "vec.dim ?A = _")
+proof%unimportant (rule vec.dim_unique)
+ let ?B = "((\<lambda>x. axis x 1) ` d)"
+ have subset_basis: "?B \<subseteq> cart_basis"
+ by (auto simp: cart_basis_def)
+ show "?B \<subseteq> ?A"
+ by (auto simp: axis_def)
+ show "vec.independent ((\<lambda>x. axis x 1) ` d)"
+ using subset_basis
+ by (rule vec.independent_mono[OF vec.independent_Basis])
+ have "x \<in> vec.span ?B" if "\<forall>i. i \<notin> d \<longrightarrow> x $ i = 0" for x::"'a^'n"
+ proof -
+ have "finite ?B"
+ using subset_basis finite_cart_basis
+ by (rule finite_subset)
+ have "x = (\<Sum>i\<in>UNIV. x $ i *s axis i 1)"
+ by (rule basis_expansion[symmetric])
+ also have "\<dots> = (\<Sum>i\<in>d. (x $ i) *s axis i 1)"
+ by (rule sum.mono_neutral_cong_right) (auto simp: that)
+ also have "\<dots> \<in> vec.span ?B"
+ by (simp add: vec.span_sum vec.span_clauses)
+ finally show "x \<in> vec.span ?B" .
+ qed
+ then show "?A \<subseteq> vec.span ?B" by auto
+qed (simp add: card_image inj_on_def axis_eq_axis)
+
+lemma%unimportant affinity_inverses:
+ assumes m0: "m \<noteq> (0::'a::field)"
+ shows "(\<lambda>x. m *s x + c) \<circ> (\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) = id"
+ "(\<lambda>x. inverse(m) *s x + (-(inverse(m) *s c))) \<circ> (\<lambda>x. m *s x + c) = id"
+ using m0
+ by (auto simp add: fun_eq_iff vector_add_ldistrib diff_conv_add_uminus simp del: add_uminus_conv_diff)
+
+lemma%important vector_affinity_eq:
+ assumes m0: "(m::'a::field) \<noteq> 0"
+ shows "m *s x + c = y \<longleftrightarrow> x = inverse m *s y + -(inverse m *s c)"
+proof%unimportant
+ assume h: "m *s x + c = y"
+ hence "m *s x = y - c" by (simp add: field_simps)
+ hence "inverse m *s (m *s x) = inverse m *s (y - c)" by simp
+ then show "x = inverse m *s y + - (inverse m *s c)"
+ using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
+next
+ assume h: "x = inverse m *s y + - (inverse m *s c)"
+ show "m *s x + c = y" unfolding h
+ using m0 by (simp add: vector_smult_assoc vector_ssub_ldistrib)
+qed
+
+lemma%unimportant vector_eq_affinity:
+ "(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%unimportant vector_cart:
+ fixes f :: "real^'n \<Rightarrow> real"
+ shows "(\<chi> i. f (axis i 1)) = (\<Sum>i\<in>Basis. f i *\<^sub>R i)"
+ unfolding euclidean_eq_iff[where 'a="real^'n"]
+ by simp (simp add: Basis_vec_def inner_axis)
+
+lemma%unimportant const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
+ by (rule vector_cart)
+
end
\ No newline at end of file
--- a/src/HOL/Analysis/Change_Of_Vars.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Change_Of_Vars.thy Wed Jan 16 21:27:33 2019 +0000
@@ -10,8 +10,11 @@
begin
subsection%important\<open>Induction on matrix row operations\<close>
-
-lemma%unimportant induct_matrix_row_operations:
+(*FIX move first 3 lemmas of subsection to linear algebra, contain technical tools on matrix operations.
+Keep the rest of the subsection contents in this theory but rename the subsection, they refer
+to lebesgue measure
+*)
+lemma induct_matrix_row_operations:
fixes P :: "real^'n^'n \<Rightarrow> bool"
assumes zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0) \<Longrightarrow> P A"
@@ -120,7 +123,7 @@
by blast
qed
-lemma%unimportant induct_matrix_elementary:
+lemma induct_matrix_elementary:
fixes P :: "real^'n^'n \<Rightarrow> bool"
assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
@@ -151,7 +154,7 @@
by (rule induct_matrix_row_operations [OF zero_row diagonal swap row])
qed
-lemma%unimportant induct_matrix_elementary_alt:
+lemma induct_matrix_elementary_alt:
fixes P :: "real^'n^'n \<Rightarrow> bool"
assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
@@ -185,11 +188,11 @@
by (rule induct_matrix_elementary) (auto intro: assms *)
qed
-lemma%unimportant matrix_vector_mult_matrix_matrix_mult_compose:
+lemma matrix_vector_mult_matrix_matrix_mult_compose:
"(*v) (A ** B) = (*v) A \<circ> (*v) B"
by (auto simp: matrix_vector_mul_assoc)
-lemma%unimportant induct_linear_elementary:
+lemma induct_linear_elementary:
fixes f :: "real^'n \<Rightarrow> real^'n"
assumes "linear f"
and comp: "\<And>f g. \<lbrakk>linear f; linear g; P f; P g\<rbrakk> \<Longrightarrow> P(f \<circ> g)"
@@ -250,14 +253,14 @@
qed
-proposition%important
+proposition (*FIX needs name *)
fixes a :: "real^'n"
assumes "m \<noteq> n" and ab_ne: "cbox a b \<noteq> {}" and an: "0 \<le> a$n"
shows measurable_shear_interval: "(\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i) ` (cbox a b) \<in> lmeasurable"
(is "?f ` _ \<in> _")
and measure_shear_interval: "measure lebesgue ((\<lambda>x. \<chi> i. if i = m then x$m + x$n else x$i) ` cbox a b)
= measure lebesgue (cbox a b)" (is "?Q")
-proof%unimportant -
+proof -
have lin: "linear ?f"
by (rule linearI) (auto simp: plus_vec_def scaleR_vec_def algebra_simps)
show fab: "?f ` cbox a b \<in> lmeasurable"
@@ -359,13 +362,13 @@
qed
-proposition%important
+proposition (*FIX needs name *)
fixes S :: "(real^'n) set"
assumes "S \<in> lmeasurable"
shows measurable_stretch: "((\<lambda>x. \<chi> k. m k * x$k) ` S) \<in> lmeasurable" (is "?f ` S \<in> _")
and measure_stretch: "measure lebesgue ((\<lambda>x. \<chi> k. m k * x$k) ` S) = \<bar>prod m UNIV\<bar> * measure lebesgue S"
(is "?MEQ")
-proof%unimportant -
+proof -
have "(?f ` S) \<in> lmeasurable \<and> ?MEQ"
proof (cases "\<forall>k. m k \<noteq> 0")
case True
@@ -463,12 +466,12 @@
qed
-proposition%important
+proposition (*FIX needs name *)
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes "linear f" "S \<in> lmeasurable"
shows measurable_linear_image: "(f ` S) \<in> lmeasurable"
and measure_linear_image: "measure lebesgue (f ` S) = \<bar>det (matrix f)\<bar> * measure lebesgue S" (is "?Q f S")
-proof%unimportant -
+proof -
have "\<forall>S \<in> lmeasurable. (f ` S) \<in> lmeasurable \<and> ?Q f S"
proof (rule induct_linear_elementary [OF \<open>linear f\<close>]; intro ballI)
fix f g and S :: "(real,'n) vec set"
@@ -611,7 +614,7 @@
qed
-lemma%unimportant
+lemma (* needs name *)
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes f: "orthogonal_transformation f" and S: "S \<in> lmeasurable"
shows measurable_orthogonal_image: "f ` S \<in> lmeasurable"
@@ -634,10 +637,10 @@
inductive%important gdelta :: "'a::topological_space set \<Rightarrow> bool" where
"(\<And>n::nat. open (F n)) \<Longrightarrow> gdelta (\<Inter>(F ` UNIV))"
-lemma%important fsigma_Union_compact:
+proposition fsigma_Union_compact:
fixes S :: "'a::{real_normed_vector,heine_borel} set"
shows "fsigma S \<longleftrightarrow> (\<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> Collect compact \<and> S = \<Union>(F ` UNIV))"
-proof%unimportant safe
+proof safe
assume "fsigma S"
then obtain F :: "nat \<Rightarrow> 'a set" where F: "range F \<subseteq> Collect closed" "S = \<Union>(F ` UNIV)"
by (meson fsigma.cases image_subsetI mem_Collect_eq)
@@ -668,7 +671,7 @@
by (simp add: compact_imp_closed fsigma.intros image_subset_iff)
qed
-lemma%unimportant gdelta_imp_fsigma: "gdelta S \<Longrightarrow> fsigma (- S)"
+lemma gdelta_imp_fsigma: "gdelta S \<Longrightarrow> fsigma (- S)"
proof (induction rule: gdelta.induct)
case (1 F)
have "- \<Inter>(F ` UNIV) = (\<Union>i. -(F i))"
@@ -677,7 +680,7 @@
by (simp add: fsigma.intros closed_Compl 1)
qed
-lemma%unimportant fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"
+lemma fsigma_imp_gdelta: "fsigma S \<Longrightarrow> gdelta (- S)"
proof (induction rule: fsigma.induct)
case (1 F)
have "- \<Union>(F ` UNIV) = (\<Inter>i. -(F i))"
@@ -686,11 +689,11 @@
by (simp add: 1 gdelta.intros open_closed)
qed
-lemma%unimportant gdelta_complement: "gdelta(- S) \<longleftrightarrow> fsigma S"
+lemma gdelta_complement: "gdelta(- S) \<longleftrightarrow> fsigma S"
using fsigma_imp_gdelta gdelta_imp_fsigma by force
text\<open>A Lebesgue set is almost an \<open>F_sigma\<close> or \<open>G_delta\<close>.\<close>
-lemma%unimportant lebesgue_set_almost_fsigma:
+lemma lebesgue_set_almost_fsigma:
assumes "S \<in> sets lebesgue"
obtains C T where "fsigma C" "negligible T" "C \<union> T = S" "disjnt C T"
proof -
@@ -725,7 +728,7 @@
qed
qed
-lemma%unimportant lebesgue_set_almost_gdelta:
+lemma lebesgue_set_almost_gdelta:
assumes "S \<in> sets lebesgue"
obtains C T where "gdelta C" "negligible T" "S \<union> T = C" "disjnt S T"
proof -
@@ -743,12 +746,12 @@
qed
-proposition%important measure_semicontinuous_with_hausdist_explicit:
+proposition measure_semicontinuous_with_hausdist_explicit:
assumes "bounded S" and neg: "negligible(frontier S)" and "e > 0"
obtains d where "d > 0"
"\<And>T. \<lbrakk>T \<in> lmeasurable; \<And>y. y \<in> T \<Longrightarrow> \<exists>x. x \<in> S \<and> dist x y < d\<rbrakk>
\<Longrightarrow> measure lebesgue T < measure lebesgue S + e"
-proof%unimportant (cases "S = {}")
+proof (cases "S = {}")
case True
with that \<open>e > 0\<close> show ?thesis by force
next
@@ -812,10 +815,10 @@
qed
qed
-proposition%important lebesgue_regular_inner:
+proposition lebesgue_regular_inner:
assumes "S \<in> sets lebesgue"
obtains K C where "negligible K" "\<And>n::nat. compact(C n)" "S = (\<Union>n. C n) \<union> K"
-proof%unimportant -
+proof -
have "\<exists>T. closed T \<and> T \<subseteq> S \<and> (S - T) \<in> lmeasurable \<and> measure lebesgue (S - T) < (1/2)^n" for n
using sets_lebesgue_inner_closed assms
by (metis sets_lebesgue_inner_closed zero_less_divide_1_iff zero_less_numeral zero_less_power)
@@ -865,7 +868,7 @@
qed
-lemma%unimportant sets_lebesgue_continuous_image:
+lemma sets_lebesgue_continuous_image:
assumes T: "T \<in> sets lebesgue" and contf: "continuous_on S f"
and negim: "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible(f ` T)" and "T \<subseteq> S"
shows "f ` T \<in> sets lebesgue"
@@ -890,7 +893,7 @@
by (simp add: Teq image_Un image_Union)
qed
-lemma%unimportant differentiable_image_in_sets_lebesgue:
+lemma differentiable_image_in_sets_lebesgue:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes S: "S \<in> sets lebesgue" and dim: "DIM('m) \<le> DIM('n)" and f: "f differentiable_on S"
shows "f`S \<in> sets lebesgue"
@@ -902,7 +905,7 @@
by (auto simp: intro!: negligible_differentiable_image_negligible [OF dim])
qed auto
-lemma%unimportant sets_lebesgue_on_continuous_image:
+lemma sets_lebesgue_on_continuous_image:
assumes S: "S \<in> sets lebesgue" and X: "X \<in> sets (lebesgue_on S)" and contf: "continuous_on S f"
and negim: "\<And>T. \<lbrakk>negligible T; T \<subseteq> S\<rbrakk> \<Longrightarrow> negligible(f ` T)"
shows "f ` X \<in> sets (lebesgue_on (f ` S))"
@@ -917,7 +920,7 @@
by (auto simp: sets_restrict_space_iff)
qed
-lemma%unimportant differentiable_image_in_sets_lebesgue_on:
+lemma differentiable_image_in_sets_lebesgue_on:
fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
assumes S: "S \<in> sets lebesgue" and X: "X \<in> sets (lebesgue_on S)" and dim: "DIM('m) \<le> DIM('n)"
and f: "f differentiable_on S"
@@ -931,7 +934,7 @@
qed
-proposition%important
+proposition (*FIX needs name *)
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes S: "S \<in> lmeasurable"
and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
@@ -941,7 +944,7 @@
"f ` S \<in> lmeasurable"
and measure_bounded_differentiable_image:
"measure lebesgue (f ` S) \<le> B * measure lebesgue S" (is "?M")
-proof%unimportant -
+proof -
have "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) \<le> B * measure lebesgue S"
proof (cases "B < 0")
case True
@@ -1167,13 +1170,13 @@
then show "f ` S \<in> lmeasurable" ?M by blast+
qed
-lemma%important
+proposition (*FIX needs name *)
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes S: "S \<in> lmeasurable"
and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
and int: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
shows m_diff_image_weak: "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
-proof%unimportant -
+proof -
let ?\<mu> = "measure lebesgue"
have aint_S: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
using int unfolding absolutely_integrable_on_def by auto
@@ -1365,7 +1368,7 @@
qed
-theorem%important
+theorem (*FIX needs name *)
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes S: "S \<in> sets lebesgue"
and deriv: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
@@ -1373,7 +1376,7 @@
shows measurable_differentiable_image: "f ` S \<in> lmeasurable"
and measure_differentiable_image:
"measure lebesgue (f ` S) \<le> integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)" (is "?M")
-proof%unimportant -
+proof -
let ?I = "\<lambda>n::nat. cbox (vec (-n)) (vec n) \<inter> S"
let ?\<mu> = "measure lebesgue"
have "x \<in> cbox (vec (- real (nat \<lceil>norm x\<rceil>))) (vec (real (nat \<lceil>norm x\<rceil>)))" for x :: "real^'n::_"
@@ -1418,7 +1421,7 @@
qed
-lemma%unimportant borel_measurable_simple_function_limit_increasing:
+lemma borel_measurable_simple_function_limit_increasing:
fixes f :: "'a::euclidean_space \<Rightarrow> real"
shows "(f \<in> borel_measurable lebesgue \<and> (\<forall>x. 0 \<le> f x)) \<longleftrightarrow>
(\<exists>g. (\<forall>n x. 0 \<le> g n x \<and> g n x \<le> f x) \<and> (\<forall>n x. g n x \<le> (g(Suc n) x)) \<and>
@@ -1617,7 +1620,7 @@
subsection%important\<open>Borel measurable Jacobian determinant\<close>
-lemma%unimportant lemma_partial_derivatives0:
+lemma lemma_partial_derivatives0:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" and lim0: "((\<lambda>x. f x /\<^sub>R norm x) \<longlongrightarrow> 0) (at 0 within S)"
and lb: "\<And>v. v \<noteq> 0 \<Longrightarrow> (\<exists>k>0. \<forall>e>0. \<exists>x. x \<in> S - {0} \<and> norm x < e \<and> k * norm x \<le> \<bar>v \<bullet> x\<bar>)"
@@ -1690,7 +1693,7 @@
mem_Collect_eq module_hom_zero span_base span_raw_def)
qed
-lemma%unimportant lemma_partial_derivatives:
+lemma lemma_partial_derivatives:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
assumes "linear f" and lim: "((\<lambda>x. f (x - a) /\<^sub>R norm (x - a)) \<longlongrightarrow> 0) (at a within S)"
and lb: "\<And>v. v \<noteq> 0 \<Longrightarrow> (\<exists>k>0. \<forall>e>0. \<exists>x \<in> S - {a}. norm(a - x) < e \<and> k * norm(a - x) \<le> \<bar>v \<bullet> (x - a)\<bar>)"
@@ -1708,12 +1711,12 @@
qed
-proposition%important borel_measurable_partial_derivatives:
+proposition borel_measurable_partial_derivatives:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n"
assumes S: "S \<in> sets lebesgue"
and f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
shows "(\<lambda>x. (matrix(f' x)$m$n)) \<in> borel_measurable (lebesgue_on S)"
-proof%unimportant -
+proof -
have contf: "continuous_on S f"
using continuous_on_eq_continuous_within f has_derivative_continuous by blast
have "{x \<in> S. (matrix (f' x)$m$n) \<le> b} \<in> sets lebesgue" for b
@@ -2299,7 +2302,7 @@
qed
-theorem%important borel_measurable_det_Jacobian:
+theorem borel_measurable_det_Jacobian:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes S: "S \<in> sets lebesgue" and f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
shows "(\<lambda>x. det(matrix(f' x))) \<in> borel_measurable (lebesgue_on S)"
@@ -2313,7 +2316,7 @@
assumes "S \<in> sets lebesgue"
shows "f \<in> borel_measurable (lebesgue_on S) \<longleftrightarrow>
(\<forall>T. T \<in> sets borel \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue)"
-proof%unimportant -
+proof -
have "{x. (if x \<in> S then f x else 0) \<in> T} \<in> sets lebesgue \<longleftrightarrow> {x \<in> S. f x \<in> T} \<in> sets lebesgue"
if "T \<in> sets borel" for T
proof (cases "0 \<in> T")
@@ -2335,7 +2338,7 @@
by blast
qed
-lemma%unimportant sets_lebesgue_almost_borel:
+lemma sets_lebesgue_almost_borel:
assumes "S \<in> sets lebesgue"
obtains B N where "B \<in> sets borel" "negligible N" "B \<union> N = S"
proof -
@@ -2345,7 +2348,7 @@
by (metis negligible_iff_null_sets negligible_subset null_sets_completionI that)
qed
-lemma%unimportant double_lebesgue_sets:
+lemma double_lebesgue_sets:
assumes S: "S \<in> sets lebesgue" and T: "T \<in> sets lebesgue" and fim: "f ` S \<subseteq> T"
shows "(\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue) \<longleftrightarrow>
f \<in> borel_measurable (lebesgue_on S) \<and>
@@ -2390,7 +2393,7 @@
subsection%important\<open>Simplest case of Sard's theorem (we don't need continuity of derivative)\<close>
-lemma%important Sard_lemma00:
+lemma Sard_lemma00:
fixes P :: "'b::euclidean_space set"
assumes "a \<ge> 0" and a: "a *\<^sub>R i \<noteq> 0" and i: "i \<in> Basis"
and P: "P \<subseteq> {x. a *\<^sub>R i \<bullet> x = 0}"
@@ -2398,7 +2401,7 @@
obtains S where "S \<in> lmeasurable"
and "{z. norm z \<le> m \<and> (\<exists>t \<in> P. norm(z - t) \<le> e)} \<subseteq> S"
and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (DIM('b) - 1)"
-proof%unimportant -
+proof -
have "a > 0"
using assms by simp
let ?v = "(\<Sum>j\<in>Basis. (if j = i then e else m) *\<^sub>R j)"
@@ -2426,7 +2429,7 @@
qed
text\<open>As above, but reorienting the vector (HOL Light's @text{GEOM\_BASIS\_MULTIPLE\_TAC})\<close>
-lemma%unimportant Sard_lemma0:
+lemma Sard_lemma0:
fixes P :: "(real^'n::{finite,wellorder}) set"
assumes "a \<noteq> 0"
and P: "P \<subseteq> {x. a \<bullet> x = 0}" and "0 \<le> m" "0 \<le> e"
@@ -2486,13 +2489,13 @@
qed
text\<open>As above, but translating the sets (HOL Light's @text{GEN\_GEOM\_ORIGIN\_TAC})\<close>
-lemma%important Sard_lemma1:
+lemma Sard_lemma1:
fixes P :: "(real^'n::{finite,wellorder}) set"
assumes P: "dim P < CARD('n)" and "0 \<le> m" "0 \<le> e"
obtains S where "S \<in> lmeasurable"
and "{z. norm(z - w) \<le> m \<and> (\<exists>t \<in> P. norm(z - w - t) \<le> e)} \<subseteq> S"
and "measure lebesgue S \<le> (2 * e) * (2 * m) ^ (CARD('n) - 1)"
-proof%unimportant -
+proof -
obtain a where "a \<noteq> 0" "P \<subseteq> {x. a \<bullet> x = 0}"
using lowdim_subset_hyperplane [of P] P span_base by auto
then obtain S where S: "S \<in> lmeasurable"
@@ -2510,7 +2513,7 @@
qed
qed
-lemma%important Sard_lemma2:
+lemma Sard_lemma2:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n::{finite,wellorder}"
assumes mlen: "CARD('m) \<le> CARD('n)" (is "?m \<le> ?n")
and "B > 0" "bounded S"
@@ -2518,7 +2521,7 @@
and rank: "\<And>x. x \<in> S \<Longrightarrow> rank(matrix(f' x)) < CARD('n)"
and B: "\<And>x. x \<in> S \<Longrightarrow> onorm(f' x) \<le> B"
shows "negligible(f ` S)"
-proof%unimportant -
+proof -
have lin_f': "\<And>x. x \<in> S \<Longrightarrow> linear(f' x)"
using derS has_derivative_linear by blast
show ?thesis
@@ -2725,13 +2728,13 @@
qed
-theorem%important baby_Sard:
+theorem baby_Sard:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n::{finite,wellorder}"
assumes mlen: "CARD('m) \<le> CARD('n)"
and der: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
and rank: "\<And>x. x \<in> S \<Longrightarrow> rank(matrix(f' x)) < CARD('n)"
shows "negligible(f ` S)"
-proof%unimportant -
+proof -
let ?U = "\<lambda>n. {x \<in> S. norm(x) \<le> n \<and> onorm(f' x) \<le> real n}"
have "\<And>x. x \<in> S \<Longrightarrow> \<exists>n. norm x \<le> real n \<and> onorm (f' x) \<le> real n"
by (meson linear order_trans real_arch_simple)
@@ -2753,7 +2756,7 @@
subsection%important\<open>A one-way version of change-of-variables not assuming injectivity. \<close>
-lemma%important integral_on_image_ubound_weak:
+lemma integral_on_image_ubound_weak:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real"
assumes S: "S \<in> sets lebesgue"
and f: "f \<in> borel_measurable (lebesgue_on (g ` S))"
@@ -2764,7 +2767,7 @@
shows "f integrable_on (g ` S) \<and>
integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
(is "_ \<and> _ \<le> ?b")
-proof%unimportant -
+proof -
let ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar>"
have cont_g: "continuous_on S g"
using der_g has_derivative_continuous_on by blast
@@ -2942,14 +2945,14 @@
qed
-lemma%important integral_on_image_ubound_nonneg:
+lemma integral_on_image_ubound_nonneg:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real"
assumes nonneg_fg: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) integrable_on S"
shows "f integrable_on (g ` S) \<and> integral (g ` S) f \<le> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x))"
(is "_ \<and> _ \<le> ?b")
-proof%unimportant -
+proof -
let ?D = "\<lambda>x. det (matrix (g' x))"
define S' where "S' \<equiv> {x \<in> S. ?D x * f(g x) \<noteq> 0}"
then have der_gS': "\<And>x. x \<in> S' \<Longrightarrow> (g has_derivative g' x) (at x within S')"
@@ -3072,7 +3075,7 @@
qed
-lemma%unimportant absolutely_integrable_on_image_real:
+proposition absolutely_integrable_on_image_real:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f(g x)) absolutely_integrable_on S"
@@ -3147,7 +3150,7 @@
qed
-proposition%important absolutely_integrable_on_image:
+proposition absolutely_integrable_on_image:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and intS: "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S"
@@ -3155,7 +3158,7 @@
apply (rule absolutely_integrable_componentwise [OF absolutely_integrable_on_image_real [OF der_g]])
using%unimportant absolutely_integrable_component [OF intS] by%unimportant auto
-proposition%important integral_on_image_ubound:
+proposition integral_on_image_ubound:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
assumes"\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f(g x)"
and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
@@ -3169,7 +3172,7 @@
text\<open>The classic change-of-variables theorem. We have two versions with quite general hypotheses,
the first that the transforming function has a continuous inverse, the second that the base set is
Lebesgue measurable.\<close>
-lemma%unimportant cov_invertible_nonneg_le:
+lemma cov_invertible_nonneg_le:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
@@ -3240,7 +3243,7 @@
qed
-lemma%unimportant cov_invertible_nonneg_eq:
+lemma cov_invertible_nonneg_eq:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
@@ -3253,7 +3256,7 @@
by (simp add: has_integral_iff) (meson eq_iff)
-lemma%unimportant cov_invertible_real:
+lemma cov_invertible_real:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real" and g :: "real^'n::_ \<Rightarrow> real^'n::_"
assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
@@ -3419,7 +3422,7 @@
qed
-lemma%unimportant cv_inv_version3:
+lemma cv_inv_version3:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and der_h: "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"
@@ -3445,7 +3448,7 @@
qed
-lemma%unimportant cv_inv_version4:
+lemma cv_inv_version4:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S) \<and> invertible(matrix(g' x))"
and hg: "\<And>x. x \<in> S \<Longrightarrow> continuous_on (g ` S) h \<and> h(g x) = x"
@@ -3470,7 +3473,7 @@
qed
-proposition%important has_absolute_integral_change_of_variables_invertible:
+proposition has_absolute_integral_change_of_variables_invertible:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and hg: "\<And>x. x \<in> S \<Longrightarrow> h(g x) = x"
@@ -3478,7 +3481,7 @@
shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b \<longleftrightarrow>
f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
(is "?lhs = ?rhs")
-proof%unimportant -
+proof -
let ?S = "{x \<in> S. invertible (matrix (g' x))}" and ?D = "\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)"
have *: "?D absolutely_integrable_on ?S \<and> integral ?S ?D = b
\<longleftrightarrow> f absolutely_integrable_on (g ` ?S) \<and> integral (g ` ?S) f = b"
@@ -3514,7 +3517,7 @@
-lemma%unimportant has_absolute_integral_change_of_variables_compact:
+proposition has_absolute_integral_change_of_variables_compact:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes "compact S"
and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
@@ -3532,7 +3535,7 @@
qed
-lemma%unimportant has_absolute_integral_change_of_variables_compact_family:
+lemma has_absolute_integral_change_of_variables_compact_family:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes compact: "\<And>n::nat. compact (F n)"
and der_g: "\<And>x. x \<in> (\<Union>n. F n) \<Longrightarrow> (g has_derivative g' x) (at x within (\<Union>n. F n))"
@@ -3711,7 +3714,7 @@
qed
-proposition%important has_absolute_integral_change_of_variables:
+proposition has_absolute_integral_change_of_variables:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes S: "S \<in> sets lebesgue"
and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
@@ -3719,7 +3722,7 @@
shows "(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) = b
\<longleftrightarrow> f absolutely_integrable_on (g ` S) \<and> integral (g ` S) f = b"
-proof%unimportant -
+proof -
obtain C N where "fsigma C" "negligible N" and CNS: "C \<union> N = S" and "disjnt C N"
using lebesgue_set_almost_fsigma [OF S] .
then obtain F :: "nat \<Rightarrow> (real^'m::_) set"
@@ -3773,16 +3776,16 @@
qed
-corollary%important absolutely_integrable_change_of_variables:
+corollary absolutely_integrable_change_of_variables:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes "S \<in> sets lebesgue"
and "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
and "inj_on g S"
shows "f absolutely_integrable_on (g ` S)
\<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S"
- using%unimportant assms has_absolute_integral_change_of_variables by%unimportant blast
+ using assms has_absolute_integral_change_of_variables by blast
-corollary%important integral_change_of_variables:
+corollary integral_change_of_variables:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes S: "S \<in> sets lebesgue"
and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"
@@ -3790,10 +3793,10 @@
and disj: "(f absolutely_integrable_on (g ` S) \<or>
(\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S)"
shows "integral (g ` S) f = integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f(g x))"
- using%unimportant has_absolute_integral_change_of_variables [OF S der_g inj] disj
- by%unimportant blast
+ using has_absolute_integral_change_of_variables [OF S der_g inj] disj
+ by blast
-lemma%unimportant has_absolute_integral_change_of_variables_1:
+lemma has_absolute_integral_change_of_variables_1:
fixes f :: "real \<Rightarrow> real^'n::{finite,wellorder}" and g :: "real \<Rightarrow> real"
assumes S: "S \<in> sets lebesgue"
and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_vector_derivative g' x) (at x within S)"
@@ -3832,19 +3835,19 @@
qed
-corollary%important absolutely_integrable_change_of_variables_1:
+corollary absolutely_integrable_change_of_variables_1:
fixes f :: "real \<Rightarrow> real^'n::{finite,wellorder}" and g :: "real \<Rightarrow> real"
assumes S: "S \<in> sets lebesgue"
and der_g: "\<And>x. x \<in> S \<Longrightarrow> (g has_vector_derivative g' x) (at x within S)"
and inj: "inj_on g S"
shows "(f absolutely_integrable_on g ` S \<longleftrightarrow>
(\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S)"
- using%unimportant has_absolute_integral_change_of_variables_1 [OF assms] by%unimportant auto
+ using has_absolute_integral_change_of_variables_1 [OF assms] by auto
subsection%important\<open>Change of variables for integrals: special case of linear function\<close>
-lemma%unimportant has_absolute_integral_change_of_variables_linear:
+lemma has_absolute_integral_change_of_variables_linear:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes "linear g"
shows "(\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S \<and>
@@ -3869,14 +3872,14 @@
qed (use h in auto)
qed
-lemma%unimportant absolutely_integrable_change_of_variables_linear:
+lemma absolutely_integrable_change_of_variables_linear:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes "linear g"
shows "(\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f(g x)) absolutely_integrable_on S
\<longleftrightarrow> f absolutely_integrable_on (g ` S)"
using assms has_absolute_integral_change_of_variables_linear by blast
-lemma%unimportant absolutely_integrable_on_linear_image:
+lemma absolutely_integrable_on_linear_image:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes "linear g"
shows "f absolutely_integrable_on (g ` S)
@@ -3884,12 +3887,12 @@
unfolding assms absolutely_integrable_change_of_variables_linear [OF assms, symmetric] absolutely_integrable_on_scaleR_iff
by (auto simp: set_integrable_def)
-lemma%important integral_change_of_variables_linear:
+lemma integral_change_of_variables_linear:
fixes f :: "real^'m::{finite,wellorder} \<Rightarrow> real^'n" and g :: "real^'m::_ \<Rightarrow> real^'m::_"
assumes "linear g"
and "f absolutely_integrable_on (g ` S) \<or> (f \<circ> g) absolutely_integrable_on S"
shows "integral (g ` S) f = \<bar>det (matrix g)\<bar> *\<^sub>R integral S (f \<circ> g)"
-proof%unimportant -
+proof -
have "((\<lambda>x. \<bar>det (matrix g)\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S) \<or> (f absolutely_integrable_on g ` S)"
using absolutely_integrable_on_linear_image assms by blast
moreover
@@ -3903,18 +3906,18 @@
subsection%important\<open>Change of variable for measure\<close>
-lemma%unimportant has_measure_differentiable_image:
+lemma has_measure_differentiable_image:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes "S \<in> sets lebesgue"
and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
and "inj_on f S"
shows "f ` S \<in> lmeasurable \<and> measure lebesgue (f ` S) = m
\<longleftrightarrow> ((\<lambda>x. \<bar>det (matrix (f' x))\<bar>) has_integral m) S"
- using%unimportant has_absolute_integral_change_of_variables [OF assms, of "\<lambda>x. (1::real^1)" "vec m"]
- unfolding%unimportant absolutely_integrable_on_1_iff integral_on_1_eq integrable_on_1_iff absolutely_integrable_on_def
- by%unimportant (auto simp: has_integral_iff lmeasurable_iff_integrable_on lmeasure_integral)
+ using has_absolute_integral_change_of_variables [OF assms, of "\<lambda>x. (1::real^1)" "vec m"]
+ unfolding absolutely_integrable_on_1_iff integral_on_1_eq integrable_on_1_iff absolutely_integrable_on_def
+ by (auto simp: has_integral_iff lmeasurable_iff_integrable_on lmeasure_integral)
-lemma%unimportant measurable_differentiable_image_eq:
+lemma measurable_differentiable_image_eq:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes "S \<in> sets lebesgue"
and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
@@ -3923,23 +3926,23 @@
using has_measure_differentiable_image [OF assms]
by blast
-lemma%important measurable_differentiable_image_alt:
+lemma measurable_differentiable_image_alt:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes "S \<in> sets lebesgue"
and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
and "inj_on f S"
shows "f ` S \<in> lmeasurable \<longleftrightarrow> (\<lambda>x. \<bar>det (matrix (f' x))\<bar>) absolutely_integrable_on S"
- using%unimportant measurable_differentiable_image_eq [OF assms]
- by%unimportant (simp only: absolutely_integrable_on_iff_nonneg)
+ using measurable_differentiable_image_eq [OF assms]
+ by (simp only: absolutely_integrable_on_iff_nonneg)
-lemma%important measure_differentiable_image_eq:
+lemma measure_differentiable_image_eq:
fixes f :: "real^'n::{finite,wellorder} \<Rightarrow> real^'n::_"
assumes S: "S \<in> sets lebesgue"
and der_f: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
and inj: "inj_on f S"
and intS: "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"
shows "measure lebesgue (f ` S) = integral S (\<lambda>x. \<bar>det (matrix (f' x))\<bar>)"
- using%unimportant measurable_differentiable_image_eq [OF S der_f inj]
+ using measurable_differentiable_image_eq [OF S der_f inj]
assms has_measure_differentiable_image by%unimportant blast
end
--- a/src/HOL/Analysis/Cross3.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Cross3.thy Wed Jan 16 21:27:33 2019 +0000
@@ -79,14 +79,14 @@
hide_fact (open) left_diff_distrib right_diff_distrib
-lemma%important Jacobi: "x \<times> (y \<times> z) + y \<times> (z \<times> x) + z \<times> (x \<times> y) = 0" for x::"real^3"
+proposition Jacobi: "x \<times> (y \<times> z) + y \<times> (z \<times> x) + z \<times> (x \<times> y) = 0" for x::"real^3"
by%unimportant (simp add: cross3_simps)
-lemma%important Lagrange: "x \<times> (y \<times> z) = (x \<bullet> z) *\<^sub>R y - (x \<bullet> y) *\<^sub>R z"
+proposition Lagrange: "x \<times> (y \<times> z) = (x \<bullet> z) *\<^sub>R y - (x \<bullet> y) *\<^sub>R z"
by%unimportant (simp add: cross3_simps) (metis (full_types) exhaust_3)
-lemma cross_triple: "(x \<times> y) \<bullet> z = (y \<times> z) \<bullet> x"
- by (simp add: cross3_def inner_vec_def sum_3 vec_eq_iff algebra_simps)
+proposition cross_triple: "(x \<times> y) \<bullet> z = (y \<times> z) \<bullet> x"
+ by%unimportant (simp add: cross3_def inner_vec_def sum_3 vec_eq_iff algebra_simps)
lemma cross_components:
"(x \<times> y)$1 = x$2 * y$3 - y$2 * x$3" "(x \<times> y)$2 = x$3 * y$1 - y$3 * x$1" "(x \<times> y)$3 = x$1 * y$2 - y$1 * x$2"
@@ -126,15 +126,15 @@
lemma cross_cross_det: "(w \<times> x) \<times> (y \<times> z) = det(vector[w,x,z]) *\<^sub>R y - det(vector[w,x,y]) *\<^sub>R z"
using exhaust_3 by (force simp add: cross3_simps)
-lemma%important dot_cross: "(w \<times> x) \<bullet> (y \<times> z) = (w \<bullet> y) * (x \<bullet> z) - (w \<bullet> z) * (x \<bullet> y)"
+proposition dot_cross: "(w \<times> x) \<bullet> (y \<times> z) = (w \<bullet> y) * (x \<bullet> z) - (w \<bullet> z) * (x \<bullet> y)"
by%unimportant (force simp add: cross3_simps)
-lemma norm_cross: "(norm (x \<times> y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \<bullet> y)\<^sup>2"
+proposition norm_cross: "(norm (x \<times> y))\<^sup>2 = (norm x)\<^sup>2 * (norm y)\<^sup>2 - (x \<bullet> y)\<^sup>2"
unfolding power2_norm_eq_inner power_mult_distrib
by (simp add: cross3_simps power2_eq_square)
-lemma%important cross_eq_0: "x \<times> y = 0 \<longleftrightarrow> collinear{0,x,y}"
-proof%unimportant -
+lemma cross_eq_0: "x \<times> y = 0 \<longleftrightarrow> collinear{0,x,y}"
+proof -
have "x \<times> y = 0 \<longleftrightarrow> norm (x \<times> y) = 0"
by simp
also have "... \<longleftrightarrow> (norm x * norm y)\<^sup>2 = (x \<bullet> y)\<^sup>2"
@@ -175,10 +175,10 @@
apply (simp add: vector_matrix_mult_def matrix_vector_mult_def forall_3 cross3_simps)
done
-lemma%important cross_orthogonal_matrix:
+lemma cross_orthogonal_matrix:
assumes "orthogonal_matrix A"
shows "(A *v x) \<times> (A *v y) = det A *\<^sub>R (A *v (x \<times> y))"
-proof%unimportant -
+proof -
have "mat 1 = transpose (A ** transpose A)"
by (metis (no_types) assms orthogonal_matrix_def transpose_mat)
then show ?thesis
@@ -191,10 +191,10 @@
lemma cross_rotoinversion_matrix: "rotoinversion_matrix A \<Longrightarrow> (A *v x) \<times> (A *v y) = - A *v (x \<times> y)"
by (simp add: rotoinversion_matrix_def cross_orthogonal_matrix scaleR_matrix_vector_assoc)
-lemma%important cross_orthogonal_transformation:
+lemma cross_orthogonal_transformation:
assumes "orthogonal_transformation f"
shows "(f x) \<times> (f y) = det(matrix f) *\<^sub>R f(x \<times> y)"
-proof%unimportant -
+proof -
have orth: "orthogonal_matrix (matrix f)"
using assms orthogonal_transformation_matrix by blast
have "matrix f *v z = f z" for z
@@ -208,7 +208,7 @@
\<Longrightarrow> (f x) \<times> (f y) = f(x \<times> y)"
by (simp add: cross_orthogonal_transformation orthogonal_transformation)
-subsection%unimportant \<open>Continuity\<close>
+subsection%important \<open>Continuity\<close>
lemma continuous_cross: "\<lbrakk>continuous F f; continuous F g\<rbrakk> \<Longrightarrow> continuous F (\<lambda>x. (f x) \<times> (g x))"
apply (subst continuous_componentwise)
--- a/src/HOL/Analysis/Determinants.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Determinants.thy Wed Jan 16 21:27:33 2019 +0000
@@ -15,19 +15,19 @@
definition%important trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a"
where "trace A = sum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
-lemma%unimportant trace_0: "trace (mat 0) = 0"
+lemma trace_0: "trace (mat 0) = 0"
by (simp add: trace_def mat_def)
-lemma%unimportant trace_I: "trace (mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
+lemma trace_I: "trace (mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
by (simp add: trace_def mat_def)
-lemma%unimportant trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
+lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
by (simp add: trace_def sum.distrib)
-lemma%unimportant trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
+lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
by (simp add: trace_def sum_subtractf)
-lemma%important trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
+lemma trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
apply (simp add: trace_def matrix_matrix_mult_def)
apply (subst sum.swap)
apply (simp add: mult.commute)
@@ -42,8 +42,8 @@
text \<open>Basic determinant properties\<close>
-lemma%important det_transpose [simp]: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
-proof%unimportant -
+proposition det_transpose [simp]: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
+proof -
let ?di = "\<lambda>A i j. A$i$j"
let ?U = "(UNIV :: 'n set)"
have fU: "finite ?U" by simp
@@ -80,7 +80,7 @@
by (subst sum_permutations_inverse) (blast intro: sum.cong)
qed
-lemma%unimportant det_lowerdiagonal:
+lemma det_lowerdiagonal:
fixes A :: "'a::comm_ring_1^('n::{finite,wellorder})^('n::{finite,wellorder})"
assumes ld: "\<And>i j. i < j \<Longrightarrow> A$i$j = 0"
shows "det A = prod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
@@ -107,11 +107,11 @@
unfolding det_def by (simp add: sign_id)
qed
-lemma%important det_upperdiagonal:
+lemma det_upperdiagonal:
fixes A :: "'a::comm_ring_1^'n::{finite,wellorder}^'n::{finite,wellorder}"
assumes ld: "\<And>i j. i > j \<Longrightarrow> A$i$j = 0"
shows "det A = prod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
-proof%unimportant -
+proof -
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "(\<lambda>p. of_int (sign p) * prod (\<lambda>i. A$i$p i) (UNIV :: 'n set))"
@@ -134,11 +134,11 @@
unfolding det_def by (simp add: sign_id)
qed
-lemma%important det_diagonal:
+proposition det_diagonal:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0"
shows "det A = prod (\<lambda>i. A$i$i) (UNIV::'n set)"
-proof%unimportant -
+proof -
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "\<lambda>p. of_int (sign p) * prod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
@@ -161,13 +161,13 @@
unfolding det_def by (simp add: sign_id)
qed
-lemma%unimportant det_I [simp]: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
+lemma det_I [simp]: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
by (simp add: det_diagonal mat_def)
-lemma%unimportant det_0 [simp]: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
+lemma det_0 [simp]: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
by (simp add: det_def prod_zero power_0_left)
-lemma%unimportant det_permute_rows:
+lemma det_permute_rows:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes p: "p permutes (UNIV :: 'n::finite set)"
shows "det (\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
@@ -204,11 +204,11 @@
done
qed
-lemma%important det_permute_columns:
+lemma det_permute_columns:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes p: "p permutes (UNIV :: 'n set)"
shows "det(\<chi> i j. A$i$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
-proof%unimportant -
+proof -
let ?Ap = "\<chi> i j. A$i$ p j :: 'a^'n^'n"
let ?At = "transpose A"
have "of_int (sign p) * det A = det (transpose (\<chi> i. transpose A $ p i))"
@@ -220,7 +220,7 @@
by simp
qed
-lemma%unimportant det_identical_columns:
+lemma det_identical_columns:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes jk: "j \<noteq> k"
and r: "column j A = column k A"
@@ -300,24 +300,24 @@
finally show "det A = 0" by simp
qed
-lemma%unimportant det_identical_rows:
+lemma det_identical_rows:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes ij: "i \<noteq> j" and r: "row i A = row j A"
shows "det A = 0"
by (metis column_transpose det_identical_columns det_transpose ij r)
-lemma%unimportant det_zero_row:
+lemma det_zero_row:
fixes A :: "'a::{idom, ring_char_0}^'n^'n" and F :: "'b::{field}^'m^'m"
shows "row i A = 0 \<Longrightarrow> det A = 0" and "row j F = 0 \<Longrightarrow> det F = 0"
by (force simp: row_def det_def vec_eq_iff sign_nz intro!: sum.neutral)+
-lemma%unimportant det_zero_column:
+lemma det_zero_column:
fixes A :: "'a::{idom, ring_char_0}^'n^'n" and F :: "'b::{field}^'m^'m"
shows "column i A = 0 \<Longrightarrow> det A = 0" and "column j F = 0 \<Longrightarrow> det F = 0"
unfolding atomize_conj atomize_imp
by (metis det_transpose det_zero_row row_transpose)
-lemma%unimportant det_row_add:
+lemma det_row_add:
fixes a b c :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
@@ -358,7 +358,7 @@
by (simp add: field_simps)
qed auto
-lemma%unimportant det_row_mul:
+lemma det_row_mul:
fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
@@ -395,7 +395,7 @@
by (simp add: field_simps)
qed auto
-lemma%unimportant det_row_0:
+lemma det_row_0:
fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
using det_row_mul[of k 0 "\<lambda>i. 1" b]
@@ -403,11 +403,11 @@
apply (simp only: vector_smult_lzero)
done
-lemma%important det_row_operation:
+lemma det_row_operation:
fixes A :: "'a::{comm_ring_1}^'n^'n"
assumes ij: "i \<noteq> j"
shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
-proof%unimportant -
+proof -
let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
have th: "row i ?Z = row j ?Z" by (vector row_def)
have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
@@ -417,7 +417,7 @@
by simp
qed
-lemma%unimportant det_row_span:
+lemma det_row_span:
fixes A :: "'a::{field}^'n^'n"
assumes x: "x \<in> vec.span {row j A |j. j \<noteq> i}"
shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
@@ -449,10 +449,10 @@
unfolding scalar_mult_eq_scaleR .
qed
-lemma%unimportant matrix_id [simp]: "det (matrix id) = 1"
+lemma matrix_id [simp]: "det (matrix id) = 1"
by (simp add: matrix_id_mat_1)
-lemma%important det_matrix_scaleR [simp]: "det (matrix (((*\<^sub>R) r)) :: real^'n^'n) = r ^ CARD('n::finite)"
+lemma det_matrix_scaleR [simp]: "det (matrix (((*\<^sub>R) r)) :: real^'n^'n) = r ^ CARD('n::finite)"
apply (subst det_diagonal)
apply (auto simp: matrix_def mat_def)
apply (simp add: cart_eq_inner_axis inner_axis_axis)
@@ -463,7 +463,7 @@
exact duplicates by considering the rows/columns as a set.
\<close>
-lemma%unimportant det_dependent_rows:
+lemma det_dependent_rows:
fixes A:: "'a::{field}^'n^'n"
assumes d: "vec.dependent (rows A)"
shows "det A = 0"
@@ -491,23 +491,23 @@
qed
qed
-lemma%unimportant det_dependent_columns:
+lemma det_dependent_columns:
assumes d: "vec.dependent (columns (A::real^'n^'n))"
shows "det A = 0"
by (metis d det_dependent_rows rows_transpose det_transpose)
text \<open>Multilinearity and the multiplication formula\<close>
-lemma%unimportant Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
+lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
by auto
-lemma%unimportant det_linear_row_sum:
+lemma det_linear_row_sum:
assumes fS: "finite S"
shows "det ((\<chi> i. if i = k then sum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
sum (\<lambda>j. det ((\<chi> i. if i = k then a i j else c i)::'a^'n^'n)) S"
using fS by (induct rule: finite_induct; simp add: det_row_0 det_row_add cong: if_cong)
-lemma%unimportant finite_bounded_functions:
+lemma finite_bounded_functions:
assumes fS: "finite S"
shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
proof (induct k)
@@ -532,14 +532,14 @@
qed
-lemma%important det_linear_rows_sum_lemma:
+lemma det_linear_rows_sum_lemma:
assumes fS: "finite S"
and fT: "finite T"
shows "det ((\<chi> i. if i \<in> T then sum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
sum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
{f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
using fT
-proof%unimportant (induct T arbitrary: a c set: finite)
+proof (induct T arbitrary: a c set: finite)
case empty
have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)"
by vector
@@ -577,7 +577,7 @@
(auto intro!: cong[OF refl[of det]] simp: vec_eq_iff)
qed
-lemma%unimportant det_linear_rows_sum:
+lemma det_linear_rows_sum:
fixes S :: "'n::finite set"
assumes fS: "finite S"
shows "det (\<chi> i. sum (a i) S) =
@@ -589,12 +589,12 @@
show ?thesis by simp
qed
-lemma%unimportant matrix_mul_sum_alt:
+lemma matrix_mul_sum_alt:
fixes A B :: "'a::comm_ring_1^'n^'n"
shows "A ** B = (\<chi> i. sum (\<lambda>k. A$i$k *s B $ k) (UNIV :: 'n set))"
by (vector matrix_matrix_mult_def sum_component)
-lemma%unimportant det_rows_mul:
+lemma det_rows_mul:
"det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
prod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
proof (simp add: det_def sum_distrib_left cong add: prod.cong, rule sum.cong)
@@ -612,10 +612,10 @@
by (simp add: field_simps)
qed rule
-lemma%important det_mul:
+proposition det_mul:
fixes A B :: "'a::comm_ring_1^'n^'n"
shows "det (A ** B) = det A * det B"
-proof%unimportant -
+proof -
let ?U = "UNIV :: 'n set"
let ?F = "{f. (\<forall>i \<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
let ?PU = "{p. p permutes ?U}"
@@ -714,10 +714,10 @@
subsection%important \<open>Relation to invertibility\<close>
-lemma%important invertible_det_nz:
+proposition invertible_det_nz:
fixes A::"'a::{field}^'n^'n"
shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
-proof%unimportant (cases "invertible A")
+proof (cases "invertible A")
case True
then obtain B :: "'a^'n^'n" where B: "A ** B = mat 1"
unfolding invertible_right_inverse by blast
@@ -748,7 +748,7 @@
qed
-lemma%unimportant det_nz_iff_inj_gen:
+lemma det_nz_iff_inj_gen:
fixes f :: "'a::field^'n \<Rightarrow> 'a::field^'n"
assumes "Vector_Spaces.linear (*s) (*s) f"
shows "det (matrix f) \<noteq> 0 \<longleftrightarrow> inj f"
@@ -763,14 +763,14 @@
by (metis assms invertible_det_nz invertible_left_inverse matrix_compose_gen matrix_id_mat_1)
qed
-lemma%unimportant det_nz_iff_inj:
+lemma det_nz_iff_inj:
fixes f :: "real^'n \<Rightarrow> real^'n"
assumes "linear f"
shows "det (matrix f) \<noteq> 0 \<longleftrightarrow> inj f"
using det_nz_iff_inj_gen[of f] assms
unfolding linear_matrix_vector_mul_eq .
-lemma%unimportant det_eq_0_rank:
+lemma det_eq_0_rank:
fixes A :: "real^'n^'n"
shows "det A = 0 \<longleftrightarrow> rank A < CARD('n)"
using invertible_det_nz [of A]
@@ -778,11 +778,11 @@
subsubsection%important \<open>Invertibility of matrices and corresponding linear functions\<close>
-lemma%important matrix_left_invertible_gen:
+proposition matrix_left_invertible_gen:
fixes f :: "'a::field^'m \<Rightarrow> 'a::field^'n"
assumes "Vector_Spaces.linear (*s) (*s) f"
shows "((\<exists>B. B ** matrix f = mat 1) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> g \<circ> f = id))"
-proof%unimportant safe
+proof safe
fix B
assume 1: "B ** matrix f = mat 1"
show "\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> g \<circ> f = id"
@@ -801,12 +801,12 @@
then show "\<exists>B. B ** matrix f = mat 1" ..
qed
-lemma%unimportant matrix_left_invertible:
+lemma matrix_left_invertible:
"linear f \<Longrightarrow> ((\<exists>B. B ** matrix f = mat 1) \<longleftrightarrow> (\<exists>g. linear g \<and> g \<circ> f = id))" for f::"real^'m \<Rightarrow> real^'n"
using matrix_left_invertible_gen[of f]
by (auto simp: linear_matrix_vector_mul_eq)
-lemma%unimportant matrix_right_invertible_gen:
+lemma matrix_right_invertible_gen:
fixes f :: "'a::field^'m \<Rightarrow> 'a^'n"
assumes "Vector_Spaces.linear (*s) (*s) f"
shows "((\<exists>B. matrix f ** B = mat 1) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> f \<circ> g = id))"
@@ -829,12 +829,12 @@
then show "\<exists>B. matrix f ** B = mat 1" ..
qed
-lemma%unimportant matrix_right_invertible:
+lemma matrix_right_invertible:
"linear f \<Longrightarrow> ((\<exists>B. matrix f ** B = mat 1) \<longleftrightarrow> (\<exists>g. linear g \<and> f \<circ> g = id))" for f::"real^'m \<Rightarrow> real^'n"
using matrix_right_invertible_gen[of f]
by (auto simp: linear_matrix_vector_mul_eq)
-lemma%unimportant matrix_invertible_gen:
+lemma matrix_invertible_gen:
fixes f :: "'a::field^'m \<Rightarrow> 'a::field^'n"
assumes "Vector_Spaces.linear (*s) (*s) f"
shows "invertible (matrix f) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
@@ -847,13 +847,13 @@
by (metis assms invertible_def matrix_compose_gen matrix_id_mat_1)
qed
-lemma%unimportant matrix_invertible:
+lemma matrix_invertible:
"linear f \<Longrightarrow> invertible (matrix f) \<longleftrightarrow> (\<exists>g. linear g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
for f::"real^'m \<Rightarrow> real^'n"
using matrix_invertible_gen[of f]
by (auto simp: linear_matrix_vector_mul_eq)
-lemma%unimportant invertible_eq_bij:
+lemma invertible_eq_bij:
fixes m :: "'a::field^'m^'n"
shows "invertible m \<longleftrightarrow> bij ((*v) m)"
using matrix_invertible_gen[OF matrix_vector_mul_linear_gen, of m, simplified matrix_of_matrix_vector_mul]
@@ -863,13 +863,13 @@
subsection%important \<open>Cramer's rule\<close>
-lemma%important cramer_lemma_transpose:
+proposition cramer_lemma_transpose:
fixes A:: "'a::{field}^'n^'n"
and x :: "'a::{field}^'n"
shows "det ((\<chi> i. if i = k then sum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
else row i A)::'a::{field}^'n^'n) = x$k * det A"
(is "?lhs = ?rhs")
-proof%unimportant -
+proof -
let ?U = "UNIV :: 'n set"
let ?Uk = "?U - {k}"
have U: "?U = insert k ?Uk"
@@ -899,10 +899,10 @@
done
qed
-lemma%important cramer_lemma:
+proposition cramer_lemma:
fixes A :: "'a::{field}^'n^'n"
shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: 'a::{field}^'n^'n) = x$k * det A"
-proof%unimportant -
+proof -
let ?U = "UNIV :: 'n set"
have *: "\<And>c. sum (\<lambda>i. c i *s row i (transpose A)) ?U = sum (\<lambda>i. c i *s column i A) ?U"
by (auto intro: sum.cong)
@@ -916,11 +916,11 @@
done
qed
-lemma%important cramer:
+theorem cramer:
fixes A ::"'a::{field}^'n^'n"
assumes d0: "det A \<noteq> 0"
shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j) / det A)"
-proof%unimportant -
+proof -
from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
unfolding invertible_det_nz[symmetric] invertible_def
by blast
@@ -948,7 +948,7 @@
definition%important "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow>
transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"
-lemma%unimportant orthogonal_transformation:
+lemma orthogonal_transformation:
"orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v. norm (f v) = norm v)"
unfolding orthogonal_transformation_def
apply auto
@@ -957,70 +957,70 @@
apply (simp add: dot_norm linear_add[symmetric])
done
-lemma%unimportant orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
+lemma orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
by (simp add: linear_iff orthogonal_transformation_def)
-lemma%unimportant orthogonal_orthogonal_transformation:
+lemma orthogonal_orthogonal_transformation:
"orthogonal_transformation f \<Longrightarrow> orthogonal (f x) (f y) \<longleftrightarrow> orthogonal x y"
by (simp add: orthogonal_def orthogonal_transformation_def)
-lemma%unimportant orthogonal_transformation_compose:
+lemma orthogonal_transformation_compose:
"\<lbrakk>orthogonal_transformation f; orthogonal_transformation g\<rbrakk> \<Longrightarrow> orthogonal_transformation(f \<circ> g)"
by (auto simp: orthogonal_transformation_def linear_compose)
-lemma%unimportant orthogonal_transformation_neg:
+lemma orthogonal_transformation_neg:
"orthogonal_transformation(\<lambda>x. -(f x)) \<longleftrightarrow> orthogonal_transformation f"
by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)
-lemma%unimportant orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"
+lemma orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"
by (simp add: linear_iff orthogonal_transformation_def)
-lemma%unimportant orthogonal_transformation_linear:
+lemma orthogonal_transformation_linear:
"orthogonal_transformation f \<Longrightarrow> linear f"
by (simp add: orthogonal_transformation_def)
-lemma%unimportant orthogonal_transformation_inj:
+lemma orthogonal_transformation_inj:
"orthogonal_transformation f \<Longrightarrow> inj f"
unfolding orthogonal_transformation_def inj_on_def
by (metis vector_eq)
-lemma%unimportant orthogonal_transformation_surj:
+lemma orthogonal_transformation_surj:
"orthogonal_transformation f \<Longrightarrow> surj f"
for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
by (simp add: linear_injective_imp_surjective orthogonal_transformation_inj orthogonal_transformation_linear)
-lemma%unimportant orthogonal_transformation_bij:
+lemma orthogonal_transformation_bij:
"orthogonal_transformation f \<Longrightarrow> bij f"
for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
by (simp add: bij_def orthogonal_transformation_inj orthogonal_transformation_surj)
-lemma%unimportant orthogonal_transformation_inv:
+lemma orthogonal_transformation_inv:
"orthogonal_transformation f \<Longrightarrow> orthogonal_transformation (inv f)"
for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
by (metis (no_types, hide_lams) bijection.inv_right bijection_def inj_linear_imp_inv_linear orthogonal_transformation orthogonal_transformation_bij orthogonal_transformation_inj)
-lemma%unimportant orthogonal_transformation_norm:
+lemma orthogonal_transformation_norm:
"orthogonal_transformation f \<Longrightarrow> norm (f x) = norm x"
by (metis orthogonal_transformation)
-lemma%unimportant orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
+lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
by (metis matrix_left_right_inverse orthogonal_matrix_def)
-lemma%unimportant orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
+lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
by (simp add: orthogonal_matrix_def)
-lemma%unimportant orthogonal_matrix_mul:
+lemma orthogonal_matrix_mul:
fixes A :: "real ^'n^'n"
assumes "orthogonal_matrix A" "orthogonal_matrix B"
shows "orthogonal_matrix(A ** B)"
using assms
by (simp add: orthogonal_matrix matrix_transpose_mul matrix_left_right_inverse matrix_mul_assoc)
-lemma%important orthogonal_transformation_matrix:
+proposition orthogonal_transformation_matrix:
fixes f:: "real^'n \<Rightarrow> real^'n"
shows "orthogonal_transformation f \<longleftrightarrow> linear f \<and> orthogonal_matrix(matrix f)"
(is "?lhs \<longleftrightarrow> ?rhs")
-proof%unimportant -
+proof -
let ?mf = "matrix f"
let ?ot = "orthogonal_transformation f"
let ?U = "UNIV :: 'n set"
@@ -1063,11 +1063,11 @@
by (auto simp: linear_def scalar_mult_eq_scaleR)
qed
-lemma%important det_orthogonal_matrix:
+theorem det_orthogonal_matrix:
fixes Q:: "'a::linordered_idom^'n^'n"
assumes oQ: "orthogonal_matrix Q"
shows "det Q = 1 \<or> det Q = - 1"
-proof%unimportant -
+proof -
have "Q ** transpose Q = mat 1"
by (metis oQ orthogonal_matrix_def)
then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
@@ -1078,20 +1078,20 @@
by (simp add: square_eq_1_iff)
qed
-lemma%important orthogonal_transformation_det [simp]:
+lemma orthogonal_transformation_det [simp]:
fixes f :: "real^'n \<Rightarrow> real^'n"
shows "orthogonal_transformation f \<Longrightarrow> \<bar>det (matrix f)\<bar> = 1"
- using%unimportant det_orthogonal_matrix orthogonal_transformation_matrix by fastforce
+ using det_orthogonal_matrix orthogonal_transformation_matrix by fastforce
subsection%important \<open>Linearity of scaling, and hence isometry, that preserves origin\<close>
-lemma%important scaling_linear:
+lemma scaling_linear:
fixes f :: "'a::real_inner \<Rightarrow> 'a::real_inner"
assumes f0: "f 0 = 0"
and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
shows "linear f"
-proof%unimportant -
+proof -
{
fix v w
have "norm (f x) = c * norm x" for x
@@ -1105,13 +1105,13 @@
by (simp add: inner_add field_simps)
qed
-lemma%unimportant isometry_linear:
+lemma isometry_linear:
"f (0::'a::real_inner) = (0::'a) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y \<Longrightarrow> linear f"
by (rule scaling_linear[where c=1]) simp_all
text \<open>Hence another formulation of orthogonal transformation\<close>
-lemma%important orthogonal_transformation_isometry:
+lemma orthogonal_transformation_isometry:
"orthogonal_transformation f \<longleftrightarrow> f(0::'a::real_inner) = (0::'a) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
unfolding orthogonal_transformation
apply (auto simp: linear_0 isometry_linear)
@@ -1119,7 +1119,7 @@
by (metis dist_0_norm)
-lemma%unimportant image_orthogonal_transformation_ball:
+lemma image_orthogonal_transformation_ball:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
assumes "orthogonal_transformation f"
shows "f ` ball x r = ball (f x) r"
@@ -1135,7 +1135,7 @@
by (auto simp: orthogonal_transformation_isometry)
qed
-lemma%unimportant image_orthogonal_transformation_cball:
+lemma image_orthogonal_transformation_cball:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
assumes "orthogonal_transformation f"
shows "f ` cball x r = cball (f x) r"
@@ -1153,29 +1153,29 @@
subsection%important \<open> We can find an orthogonal matrix taking any unit vector to any other\<close>
-lemma%unimportant orthogonal_matrix_transpose [simp]:
+lemma orthogonal_matrix_transpose [simp]:
"orthogonal_matrix(transpose A) \<longleftrightarrow> orthogonal_matrix A"
by (auto simp: orthogonal_matrix_def)
-lemma%unimportant orthogonal_matrix_orthonormal_columns:
+lemma orthogonal_matrix_orthonormal_columns:
fixes A :: "real^'n^'n"
shows "orthogonal_matrix A \<longleftrightarrow>
(\<forall>i. norm(column i A) = 1) \<and>
(\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (column i A) (column j A))"
by (auto simp: orthogonal_matrix matrix_mult_transpose_dot_column vec_eq_iff mat_def norm_eq_1 orthogonal_def)
-lemma%unimportant orthogonal_matrix_orthonormal_rows:
+lemma orthogonal_matrix_orthonormal_rows:
fixes A :: "real^'n^'n"
shows "orthogonal_matrix A \<longleftrightarrow>
(\<forall>i. norm(row i A) = 1) \<and>
(\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (row i A) (row j A))"
using orthogonal_matrix_orthonormal_columns [of "transpose A"] by simp
-lemma%important orthogonal_matrix_exists_basis:
+theorem orthogonal_matrix_exists_basis:
fixes a :: "real^'n"
assumes "norm a = 1"
obtains A where "orthogonal_matrix A" "A *v (axis k 1) = a"
-proof%unimportant -
+proof -
obtain S where "a \<in> S" "pairwise orthogonal S" and noS: "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"
and "independent S" "card S = CARD('n)" "span S = UNIV"
using vector_in_orthonormal_basis assms by force
@@ -1198,11 +1198,11 @@
qed
qed
-lemma%unimportant orthogonal_transformation_exists_1:
+lemma orthogonal_transformation_exists_1:
fixes a b :: "real^'n"
assumes "norm a = 1" "norm b = 1"
obtains f where "orthogonal_transformation f" "f a = b"
-proof%unimportant -
+proof -
obtain k::'n where True
by simp
obtain A B where AB: "orthogonal_matrix A" "orthogonal_matrix B" and eq: "A *v (axis k 1) = a" "B *v (axis k 1) = b"
@@ -1220,11 +1220,11 @@
qed
qed
-lemma%important orthogonal_transformation_exists:
+theorem orthogonal_transformation_exists:
fixes a b :: "real^'n"
assumes "norm a = norm b"
obtains f where "orthogonal_transformation f" "f a = b"
-proof%unimportant (cases "a = 0 \<or> b = 0")
+proof (cases "a = 0 \<or> b = 0")
case True
with assms show ?thesis
using that by force
@@ -1248,12 +1248,12 @@
subsection%important \<open>Can extend an isometry from unit sphere\<close>
-lemma%important isometry_sphere_extend:
+lemma isometry_sphere_extend:
fixes f:: "'a::real_inner \<Rightarrow> 'a"
assumes f1: "\<And>x. norm x = 1 \<Longrightarrow> norm (f x) = 1"
and fd1: "\<And>x y. \<lbrakk>norm x = 1; norm y = 1\<rbrakk> \<Longrightarrow> dist (f x) (f y) = dist x y"
shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
-proof%unimportant -
+proof -
{
fix x y x' y' u v u' v' :: "'a"
assume H: "x = norm x *\<^sub>R u" "y = norm y *\<^sub>R v"
@@ -1291,26 +1291,26 @@
definition%important "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
definition%important "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
-lemma%important orthogonal_rotation_or_rotoinversion:
+lemma orthogonal_rotation_or_rotoinversion:
fixes Q :: "'a::linordered_idom^'n^'n"
shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
- by%unimportant (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
+ by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
text \<open>Explicit formulas for low dimensions\<close>
-lemma%unimportant prod_neutral_const: "prod f {(1::nat)..1} = f 1"
+lemma prod_neutral_const: "prod f {(1::nat)..1} = f 1"
by simp
-lemma%unimportant prod_2: "prod f {(1::nat)..2} = f 1 * f 2"
+lemma prod_2: "prod f {(1::nat)..2} = f 1 * f 2"
by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)
-lemma%unimportant prod_3: "prod f {(1::nat)..3} = f 1 * f 2 * f 3"
+lemma prod_3: "prod f {(1::nat)..3} = f 1 * f 2 * f 3"
by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)
-lemma%unimportant det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
+lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
by (simp add: det_def sign_id)
-lemma%unimportant det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
+lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
proof -
have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
show ?thesis
@@ -1320,7 +1320,7 @@
by (simp add: sign_swap_id sign_id swap_id_eq)
qed
-lemma%unimportant det_3:
+lemma det_3:
"det (A::'a::comm_ring_1^3^3) =
A$1$1 * A$2$2 * A$3$3 +
A$1$2 * A$2$3 * A$3$1 +
@@ -1344,7 +1344,7 @@
text\<open> Slightly stronger results giving rotation, but only in two or more dimensions\<close>
-lemma%unimportant rotation_matrix_exists_basis:
+lemma rotation_matrix_exists_basis:
fixes a :: "real^'n"
assumes 2: "2 \<le> CARD('n)" and "norm a = 1"
obtains A where "rotation_matrix A" "A *v (axis k 1) = a"
@@ -1383,7 +1383,7 @@
qed
qed
-lemma%unimportant rotation_exists_1:
+lemma rotation_exists_1:
fixes a :: "real^'n"
assumes "2 \<le> CARD('n)" "norm a = 1" "norm b = 1"
obtains f where "orthogonal_transformation f" "det(matrix f) = 1" "f a = b"
@@ -1406,7 +1406,7 @@
qed
qed
-lemma%unimportant rotation_exists:
+lemma rotation_exists:
fixes a :: "real^'n"
assumes 2: "2 \<le> CARD('n)" and eq: "norm a = norm b"
obtains f where "orthogonal_transformation f" "det(matrix f) = 1" "f a = b"
@@ -1428,7 +1428,7 @@
with f show thesis ..
qed
-lemma%unimportant rotation_rightward_line:
+lemma rotation_rightward_line:
fixes a :: "real^'n"
obtains f where "orthogonal_transformation f" "2 \<le> CARD('n) \<Longrightarrow> det(matrix f) = 1"
"f(norm a *\<^sub>R axis k 1) = a"
--- a/src/HOL/Analysis/Extended_Real_Limits.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Extended_Real_Limits.thy Wed Jan 16 21:27:33 2019 +0000
@@ -16,23 +16,23 @@
"HOL-Library.Indicator_Function"
begin
-lemma%important compact_UNIV:
+lemma compact_UNIV:
"compact (UNIV :: 'a::{complete_linorder,linorder_topology,second_countable_topology} set)"
- using%unimportant compact_complete_linorder
+ using compact_complete_linorder
by (auto simp: seq_compact_eq_compact[symmetric] seq_compact_def)
-lemma%important compact_eq_closed:
+lemma compact_eq_closed:
fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
shows "compact S \<longleftrightarrow> closed S"
using%unimportant closed_Int_compact[of S, OF _ compact_UNIV] compact_imp_closed
by auto
-lemma%important closed_contains_Sup_cl:
+lemma closed_contains_Sup_cl:
fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
assumes "closed S"
and "S \<noteq> {}"
shows "Sup S \<in> S"
-proof%unimportant -
+proof -
from compact_eq_closed[of S] compact_attains_sup[of S] assms
obtain s where S: "s \<in> S" "\<forall>t\<in>S. t \<le> s"
by auto
@@ -42,12 +42,12 @@
by simp
qed
-lemma%important closed_contains_Inf_cl:
+lemma closed_contains_Inf_cl:
fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
assumes "closed S"
and "S \<noteq> {}"
shows "Inf S \<in> S"
-proof%unimportant -
+proof -
from compact_eq_closed[of S] compact_attains_inf[of S] assms
obtain s where S: "s \<in> S" "\<forall>t\<in>S. s \<le> t"
by auto
@@ -57,7 +57,7 @@
by simp
qed
-instance enat :: second_countable_topology
+instance%unimportant enat :: second_countable_topology
proof
show "\<exists>B::enat set set. countable B \<and> open = generate_topology B"
proof (intro exI conjI)
@@ -66,8 +66,8 @@
qed (simp add: open_enat_def)
qed
-instance%important ereal :: second_countable_topology
-proof%unimportant (standard, intro exI conjI)
+instance%unimportant ereal :: second_countable_topology
+proof (standard, intro exI conjI)
let ?B = "(\<Union>r\<in>\<rat>. {{..< r}, {r <..}} :: ereal set set)"
show "countable ?B"
by (auto intro: countable_rat)
@@ -98,8 +98,8 @@
text \<open>This is a copy from \<open>ereal :: second_countable_topology\<close>. Maybe find a common super class of
topological spaces where the rational numbers are densely embedded ?\<close>
-instance%important ennreal :: second_countable_topology
-proof%unimportant (standard, intro exI conjI)
+instance ennreal :: second_countable_topology
+proof (standard, intro exI conjI)
let ?B = "(\<Union>r\<in>\<rat>. {{..< r}, {r <..}} :: ennreal set set)"
show "countable ?B"
by (auto intro: countable_rat)
@@ -128,13 +128,13 @@
qed
qed
-lemma%important ereal_open_closed_aux:
+lemma ereal_open_closed_aux:
fixes S :: "ereal set"
assumes "open S"
and "closed S"
and S: "(-\<infinity>) \<notin> S"
shows "S = {}"
-proof%unimportant (rule ccontr)
+proof (rule ccontr)
assume "\<not> ?thesis"
then have *: "Inf S \<in> S"
@@ -172,10 +172,10 @@
by auto
qed
-lemma%important ereal_open_closed:
+lemma ereal_open_closed:
fixes S :: "ereal set"
shows "open S \<and> closed S \<longleftrightarrow> S = {} \<or> S = UNIV"
-proof%unimportant -
+proof -
{
assume lhs: "open S \<and> closed S"
{
@@ -196,10 +196,10 @@
by auto
qed
-lemma%important ereal_open_atLeast:
+lemma ereal_open_atLeast:
fixes x :: ereal
shows "open {x..} \<longleftrightarrow> x = -\<infinity>"
-proof%unimportant
+proof
assume "x = -\<infinity>"
then have "{x..} = UNIV"
by auto
@@ -215,12 +215,12 @@
by (simp add: bot_ereal_def atLeast_eq_UNIV_iff)
qed
-lemma%important mono_closed_real:
+lemma mono_closed_real:
fixes S :: "real set"
assumes mono: "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
and "closed S"
shows "S = {} \<or> S = UNIV \<or> (\<exists>a. S = {a..})"
-proof%unimportant -
+proof -
{
assume "S \<noteq> {}"
{ assume ex: "\<exists>B. \<forall>x\<in>S. B \<le> x"
@@ -261,12 +261,12 @@
by blast
qed
-lemma%important mono_closed_ereal:
+lemma mono_closed_ereal:
fixes S :: "real set"
assumes mono: "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
and "closed S"
shows "\<exists>a. S = {x. a \<le> ereal x}"
-proof%unimportant -
+proof -
{
assume "S = {}"
then have ?thesis
@@ -296,11 +296,11 @@
using mono_closed_real[of S] assms by auto
qed
-lemma%important Liminf_within:
+lemma Liminf_within:
fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
shows "Liminf (at x within S) f = (SUP e\<in>{0<..}. INF y\<in>(S \<inter> ball x e - {x}). f y)"
unfolding Liminf_def eventually_at
-proof%unimportant (rule SUP_eq, simp_all add: Ball_def Bex_def, safe)
+proof (rule SUP_eq, simp_all add: Ball_def Bex_def, safe)
fix P d
assume "0 < d" and "\<forall>y. y \<in> S \<longrightarrow> y \<noteq> x \<and> dist y x < d \<longrightarrow> P y"
then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
@@ -316,11 +316,11 @@
(auto intro!: INF_mono exI[of _ d] simp: dist_commute)
qed
-lemma%important Limsup_within:
+lemma Limsup_within:
fixes f :: "'a::metric_space \<Rightarrow> 'b::complete_lattice"
shows "Limsup (at x within S) f = (INF e\<in>{0<..}. SUP y\<in>(S \<inter> ball x e - {x}). f y)"
unfolding Limsup_def eventually_at
-proof%unimportant (rule INF_eq, simp_all add: Ball_def Bex_def, safe)
+proof (rule INF_eq, simp_all add: Ball_def Bex_def, safe)
fix P d
assume "0 < d" and "\<forall>y. y \<in> S \<longrightarrow> y \<noteq> x \<and> dist y x < d \<longrightarrow> P y"
then have "S \<inter> ball x d - {x} \<subseteq> {x. P x}"
@@ -357,7 +357,7 @@
done
-subsection%important \<open>Extended-Real.thy\<close> (*FIX title *)
+subsection%important \<open>Extended-Real.thy\<close> (*FIX ME change title *)
lemma sum_constant_ereal:
fixes a::ereal
@@ -377,10 +377,10 @@
ultimately show ?thesis by (metis (mono_tags, lifting) eventually_mono)
qed
-lemma%important ereal_Inf_cmult:
+lemma ereal_Inf_cmult:
assumes "c>(0::real)"
shows "Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}"
-proof%unimportant -
+proof -
have "(\<lambda>x::ereal. c * x) (Inf {x::ereal. P x}) = Inf ((\<lambda>x::ereal. c * x)`{x::ereal. P x})"
apply (rule mono_bij_Inf)
apply (simp add: assms ereal_mult_left_mono less_imp_le mono_def)
@@ -399,12 +399,12 @@
It is much more convenient in many situations, see for instance the proof of
\<open>tendsto_sum_ereal\<close> below.\<close>
-lemma%important tendsto_add_ereal_PInf:
+lemma tendsto_add_ereal_PInf:
fixes y :: ereal
assumes y: "y \<noteq> -\<infinity>"
assumes f: "(f \<longlongrightarrow> \<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> \<infinity>) F"
-proof%unimportant -
+proof -
have "\<exists>C. eventually (\<lambda>x. g x > ereal C) F"
proof (cases y)
case (real r)
@@ -437,12 +437,12 @@
that \<open>- (x + y)\<close> is in general different from \<open>(- x) + (- y)\<close> in ereal creates difficulties,
so it is more efficient to copy the previous proof.\<close>
-lemma%important tendsto_add_ereal_MInf:
+lemma tendsto_add_ereal_MInf:
fixes y :: ereal
assumes y: "y \<noteq> \<infinity>"
assumes f: "(f \<longlongrightarrow> -\<infinity>) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> -\<infinity>) F"
-proof%unimportant -
+proof -
have "\<exists>C. eventually (\<lambda>x. g x < ereal C) F"
proof (cases y)
case (real r)
@@ -470,12 +470,12 @@
then show ?thesis by (simp add: tendsto_MInfty)
qed
-lemma%important tendsto_add_ereal_general1:
+lemma tendsto_add_ereal_general1:
fixes x y :: ereal
assumes y: "\<bar>y\<bar> \<noteq> \<infinity>"
assumes f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
-proof%unimportant (cases x)
+proof (cases x)
case (real r)
have a: "\<bar>x\<bar> \<noteq> \<infinity>" by (simp add: real)
show ?thesis by (rule tendsto_add_ereal[OF a, OF y, OF f, OF g])
@@ -488,12 +488,12 @@
by (metis abs_ereal.simps(3) ereal_MInfty_eq_plus)
qed
-lemma%important tendsto_add_ereal_general2:
+lemma tendsto_add_ereal_general2:
fixes x y :: ereal
assumes x: "\<bar>x\<bar> \<noteq> \<infinity>"
and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
-proof%unimportant -
+proof -
have "((\<lambda>x. g x + f x) \<longlongrightarrow> x + y) F"
using tendsto_add_ereal_general1[OF x, OF g, OF f] add.commute[of "y", of "x"] by simp
moreover have "\<And>x. g x + f x = f x + g x" using add.commute by auto
@@ -503,12 +503,12 @@
text \<open>The next lemma says that the addition is continuous on \<open>ereal\<close>, except at
the pairs \<open>(-\<infinity>, \<infinity>)\<close> and \<open>(\<infinity>, -\<infinity>)\<close>.\<close>
-lemma%important tendsto_add_ereal_general [tendsto_intros]:
+lemma tendsto_add_ereal_general [tendsto_intros]:
fixes x y :: ereal
assumes "\<not>((x=\<infinity> \<and> y=-\<infinity>) \<or> (x=-\<infinity> \<and> y=\<infinity>))"
and f: "(f \<longlongrightarrow> x) F" and g: "(g \<longlongrightarrow> y) F"
shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
-proof%unimportant (cases x)
+proof (cases x)
case (real r)
show ?thesis
apply (rule tendsto_add_ereal_general2) using real assms by auto
@@ -528,10 +528,10 @@
ereal times ereal, except at \<open>(\<infinity>, 0)\<close> and \<open>(-\<infinity>, 0)\<close> and \<open>(0, \<infinity>)\<close> and \<open>(0, -\<infinity>)\<close>,
starting with specific situations.\<close>
-lemma%important tendsto_mult_real_ereal:
+lemma tendsto_mult_real_ereal:
assumes "(u \<longlongrightarrow> ereal l) F" "(v \<longlongrightarrow> ereal m) F"
shows "((\<lambda>n. u n * v n) \<longlongrightarrow> ereal l * ereal m) F"
-proof%unimportant -
+proof -
have ureal: "eventually (\<lambda>n. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)])
then have "((\<lambda>n. ereal(real_of_ereal(u n))) \<longlongrightarrow> ereal l) F" using assms by auto
then have limu: "((\<lambda>n. real_of_ereal(u n)) \<longlongrightarrow> l) F" by auto
@@ -551,11 +551,11 @@
then show ?thesis using * filterlim_cong by fastforce
qed
-lemma%important tendsto_mult_ereal_PInf:
+lemma tendsto_mult_ereal_PInf:
fixes f g::"_ \<Rightarrow> ereal"
assumes "(f \<longlongrightarrow> l) F" "l>0" "(g \<longlongrightarrow> \<infinity>) F"
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> \<infinity>) F"
-proof%unimportant -
+proof -
obtain a::real where "0 < ereal a" "a < l" using assms(2) using ereal_dense2 by blast
have *: "eventually (\<lambda>x. f x > a) F" using \<open>a < l\<close> assms(1) by (simp add: order_tendsto_iff)
{
@@ -579,11 +579,11 @@
then show ?thesis by (auto simp add: tendsto_PInfty)
qed
-lemma%important tendsto_mult_ereal_pos:
+lemma tendsto_mult_ereal_pos:
fixes f g::"_ \<Rightarrow> ereal"
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "l>0" "m>0"
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
-proof%unimportant (cases)
+proof (cases)
assume *: "l = \<infinity> \<or> m = \<infinity>"
then show ?thesis
proof (cases)
@@ -618,11 +618,11 @@
shows "sgn(l) * sgn(l) = 1"
apply (cases l) using assms by (auto simp add: one_ereal_def sgn_if)
-lemma%important tendsto_mult_ereal [tendsto_intros]:
+lemma tendsto_mult_ereal [tendsto_intros]:
fixes f g::"_ \<Rightarrow> ereal"
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "\<not>((l=0 \<and> abs(m) = \<infinity>) \<or> (m=0 \<and> abs(l) = \<infinity>))"
shows "((\<lambda>x. f x * g x) \<longlongrightarrow> l * m) F"
-proof%unimportant (cases)
+proof (cases)
assume "l=0 \<or> m=0"
then have "abs(l) \<noteq> \<infinity>" "abs(m) \<noteq> \<infinity>" using assms(3) by auto
then obtain lr mr where "l = ereal lr" "m = ereal mr" by auto
@@ -661,11 +661,11 @@
subsubsection%important \<open>Continuity of division\<close>
-lemma%important tendsto_inverse_ereal_PInf:
+lemma tendsto_inverse_ereal_PInf:
fixes u::"_ \<Rightarrow> ereal"
assumes "(u \<longlongrightarrow> \<infinity>) F"
shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 0) F"
-proof%unimportant -
+proof -
{
fix e::real assume "e>0"
have "1/e < \<infinity>" by auto
@@ -702,11 +702,11 @@
shows "(u \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
using tendsto_inverse unfolding inverse_eq_divide .
-lemma%important tendsto_inverse_ereal [tendsto_intros]:
+lemma tendsto_inverse_ereal [tendsto_intros]:
fixes u::"_ \<Rightarrow> ereal"
assumes "(u \<longlongrightarrow> l) F" "l \<noteq> 0"
shows "((\<lambda>x. 1/ u x) \<longlongrightarrow> 1/l) F"
-proof%unimportant (cases l)
+proof (cases l)
case (real r)
then have "r \<noteq> 0" using assms(2) by auto
then have "1/l = ereal(1/r)" using real by (simp add: one_ereal_def)
@@ -747,11 +747,11 @@
then show ?thesis unfolding v_def using Lim_transform_eventually[OF *] \<open> 1/l = 0 \<close> by auto
qed
-lemma%important tendsto_divide_ereal [tendsto_intros]:
+lemma tendsto_divide_ereal [tendsto_intros]:
fixes f g::"_ \<Rightarrow> ereal"
assumes "(f \<longlongrightarrow> l) F" "(g \<longlongrightarrow> m) F" "m \<noteq> 0" "\<not>(abs(l) = \<infinity> \<and> abs(m) = \<infinity>)"
shows "((\<lambda>x. f x / g x) \<longlongrightarrow> l / m) F"
-proof%unimportant -
+proof -
define h where "h = (\<lambda>x. 1/ g x)"
have *: "(h \<longlongrightarrow> 1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto
have "((\<lambda>x. f x * h x) \<longlongrightarrow> l * (1/m)) F"
@@ -766,11 +766,11 @@
text \<open>The assumptions of @{thm tendsto_diff_ereal} are too strong, we weaken them here.\<close>
-lemma%important tendsto_diff_ereal_general [tendsto_intros]:
+lemma tendsto_diff_ereal_general [tendsto_intros]:
fixes u v::"'a \<Rightarrow> ereal"
assumes "(u \<longlongrightarrow> l) F" "(v \<longlongrightarrow> m) F" "\<not>((l = \<infinity> \<and> m = \<infinity>) \<or> (l = -\<infinity> \<and> m = -\<infinity>))"
shows "((\<lambda>n. u n - v n) \<longlongrightarrow> l - m) F"
-proof%unimportant -
+proof -
have "((\<lambda>n. u n + (-v n)) \<longlongrightarrow> l + (-m)) F"
apply (intro tendsto_intros assms) using assms by (auto simp add: ereal_uminus_eq_reorder)
then show ?thesis by (simp add: minus_ereal_def)
@@ -780,11 +780,11 @@
"(\<lambda> n::nat. real n) \<longlonglongrightarrow> \<infinity>"
by (simp add: filterlim_real_sequentially tendsto_PInfty_eq_at_top)
-lemma%important tendsto_at_top_pseudo_inverse [tendsto_intros]:
+lemma tendsto_at_top_pseudo_inverse [tendsto_intros]:
fixes u::"nat \<Rightarrow> nat"
assumes "LIM n sequentially. u n :> at_top"
shows "LIM n sequentially. Inf {N. u N \<ge> n} :> at_top"
-proof%unimportant -
+proof -
{
fix C::nat
define M where "M = Max {u n| n. n \<le> C}+1"
@@ -811,11 +811,11 @@
then show ?thesis using filterlim_at_top by auto
qed
-lemma%important pseudo_inverse_finite_set:
+lemma pseudo_inverse_finite_set:
fixes u::"nat \<Rightarrow> nat"
assumes "LIM n sequentially. u n :> at_top"
shows "finite {N. u N \<le> n}"
-proof%unimportant -
+proof -
fix n
have "eventually (\<lambda>N. u N \<ge> n+1) sequentially" using assms
by (simp add: filterlim_at_top)
@@ -860,11 +860,11 @@
then show ?thesis by auto
qed
-lemma%important ereal_truncation_real_top [tendsto_intros]:
+lemma ereal_truncation_real_top [tendsto_intros]:
fixes x::ereal
assumes "x \<noteq> - \<infinity>"
shows "(\<lambda>n::nat. real_of_ereal(min x n)) \<longlonglongrightarrow> x"
-proof%unimportant (cases x)
+proof (cases x)
case (real r)
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
then have "min x n = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
@@ -878,10 +878,10 @@
then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
qed (simp add: assms)
-lemma%important ereal_truncation_bottom [tendsto_intros]:
+lemma ereal_truncation_bottom [tendsto_intros]:
fixes x::ereal
shows "(\<lambda>n::nat. max x (- real n)) \<longlonglongrightarrow> x"
-proof%unimportant (cases x)
+proof (cases x)
case (real r)
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
@@ -899,11 +899,11 @@
then show ?thesis by auto
qed
-lemma%important ereal_truncation_real_bottom [tendsto_intros]:
+lemma ereal_truncation_real_bottom [tendsto_intros]:
fixes x::ereal
assumes "x \<noteq> \<infinity>"
shows "(\<lambda>n::nat. real_of_ereal(max x (- real n))) \<longlonglongrightarrow> x"
-proof%unimportant (cases x)
+proof (cases x)
case (real r)
then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
then have "max x (-real n) = x" if "n \<ge> K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
@@ -931,9 +931,9 @@
qed(simp)
-lemma%important continuous_ereal_abs:
+lemma continuous_ereal_abs:
"continuous_on (UNIV::ereal set) abs"
-proof%unimportant -
+proof -
have "continuous_on ({..0} \<union> {(0::ereal)..}) abs"
apply (rule continuous_on_closed_Un, auto)
apply (rule iffD1[OF continuous_on_cong, of "{..0}" _ "\<lambda>x. -x"])
@@ -970,11 +970,11 @@
then show ?thesis by auto
qed
-lemma%important tendsto_mult_ennreal [tendsto_intros]:
+lemma tendsto_mult_ennreal [tendsto_intros]:
fixes l m::ennreal
assumes "(u \<longlongrightarrow> l) F" "(v \<longlongrightarrow> m) F" "\<not>((l = 0 \<and> m = \<infinity>) \<or> (l = \<infinity> \<and> m = 0))"
shows "((\<lambda>n. u n * v n) \<longlongrightarrow> l * m) F"
-proof%unimportant -
+proof -
have "((\<lambda>n. e2ennreal(enn2ereal (u n) * enn2ereal (v n))) \<longlongrightarrow> e2ennreal(enn2ereal l * enn2ereal m)) F"
apply (intro tendsto_intros) using assms apply auto
using enn2ereal_inject zero_ennreal.rep_eq by fastforce+
@@ -987,9 +987,9 @@
qed
-subsection%important \<open>monoset\<close>
+subsection%important \<open>monoset\<close> (*FIX ME title *)
-definition%important (in order) mono_set:
+definition (in order) mono_set:
"mono_set S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto
@@ -997,11 +997,11 @@
lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto
lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto
-lemma%important (in complete_linorder) mono_set_iff:
+lemma (in complete_linorder) mono_set_iff:
fixes S :: "'a set"
defines "a \<equiv> Inf S"
shows "mono_set S \<longleftrightarrow> S = {a <..} \<or> S = {a..}" (is "_ = ?c")
-proof%unimportant
+proof
assume "mono_set S"
then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S"
by (auto simp: mono_set)
@@ -1043,12 +1043,12 @@
by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_ereal_atLeast
ereal_open_closed mono_empty mono_set_iff open_ereal_greaterThan)
-lemma%important ereal_Liminf_Sup_monoset:
+lemma ereal_Liminf_Sup_monoset:
fixes f :: "'a \<Rightarrow> ereal"
shows "Liminf net f =
Sup {l. \<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
(is "_ = Sup ?A")
-proof%unimportant (safe intro!: Liminf_eqI complete_lattice_class.Sup_upper complete_lattice_class.Sup_least)
+proof (safe intro!: Liminf_eqI complete_lattice_class.Sup_upper complete_lattice_class.Sup_least)
fix P
assume P: "eventually P net"
fix S
@@ -1082,12 +1082,12 @@
qed
qed
-lemma%important ereal_Limsup_Inf_monoset:
+lemma ereal_Limsup_Inf_monoset:
fixes f :: "'a \<Rightarrow> ereal"
shows "Limsup net f =
Inf {l. \<forall>S. open S \<longrightarrow> mono_set (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
(is "_ = Inf ?A")
-proof%unimportant (safe intro!: Limsup_eqI complete_lattice_class.Inf_lower complete_lattice_class.Inf_greatest)
+proof (safe intro!: Limsup_eqI complete_lattice_class.Inf_lower complete_lattice_class.Inf_greatest)
fix P
assume P: "eventually P net"
fix S
@@ -1121,11 +1121,11 @@
qed
qed
-lemma%important liminf_bounded_open:
+lemma liminf_bounded_open:
fixes x :: "nat \<Rightarrow> ereal"
shows "x0 \<le> liminf x \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> x0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. x n \<in> S))"
(is "_ \<longleftrightarrow> ?P x0")
-proof%unimportant
+proof
assume "?P x0"
then show "x0 \<le> liminf x"
unfolding ereal_Liminf_Sup_monoset eventually_sequentially
@@ -1159,11 +1159,11 @@
by auto
qed
-lemma%important limsup_finite_then_bounded:
+lemma limsup_finite_then_bounded:
fixes u::"nat \<Rightarrow> real"
assumes "limsup u < \<infinity>"
shows "\<exists>C. \<forall>n. u n \<le> C"
-proof%unimportant -
+proof -
obtain C where C: "limsup u < C" "C < \<infinity>" using assms ereal_dense2 by blast
then have "C = ereal(real_of_ereal C)" using ereal_real by force
have "eventually (\<lambda>n. u n < C) sequentially" using C(1) unfolding Limsup_def
@@ -1273,11 +1273,11 @@
then show ?case using Suc.IH by simp
qed (auto)
-lemma%important Limsup_obtain:
+lemma Limsup_obtain:
fixes u::"_ \<Rightarrow> 'a :: complete_linorder"
assumes "Limsup F u > c"
shows "\<exists>i. u i > c"
-proof%unimportant -
+proof -
have "(INF P\<in>{P. eventually P F}. SUP x\<in>{x. P x}. u x) > c" using assms by (simp add: Limsup_def)
then show ?thesis by (metis eventually_True mem_Collect_eq less_INF_D less_SUP_iff)
qed
@@ -1285,10 +1285,10 @@
text \<open>The next lemma is extremely useful, as it often makes it possible to reduce statements
about limsups to statements about limits.\<close>
-lemma%important limsup_subseq_lim:
+lemma limsup_subseq_lim:
fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (u o r) \<longlonglongrightarrow> limsup u"
-proof%unimportant (cases)
+proof (cases)
assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<le> u p"
then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<le> u (r n)) \<and> r n < r (Suc n)"
by (intro dependent_nat_choice) (auto simp: conj_commute)
@@ -1378,10 +1378,10 @@
then show ?thesis using \<open>strict_mono r\<close> by auto
qed
-lemma%important liminf_subseq_lim:
+lemma liminf_subseq_lim:
fixes u::"nat \<Rightarrow> 'a :: {complete_linorder, linorder_topology}"
shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (u o r) \<longlonglongrightarrow> liminf u"
-proof%unimportant (cases)
+proof (cases)
assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. u m \<ge> u p"
then have "\<exists>r. \<forall>n. (\<forall>m\<ge>r n. u m \<ge> u (r n)) \<and> r n < r (Suc n)"
by (intro dependent_nat_choice) (auto simp: conj_commute)
@@ -1476,10 +1476,10 @@
subsequences thanks to \<open>limsup_subseq_lim\<close>. The statement for limits follows for instance from
\<open>tendsto_add_ereal_general\<close>.\<close>
-lemma%important ereal_limsup_add_mono:
+lemma ereal_limsup_add_mono:
fixes u v::"nat \<Rightarrow> ereal"
shows "limsup (\<lambda>n. u n + v n) \<le> limsup u + limsup v"
-proof%unimportant (cases)
+proof (cases)
assume "(limsup u = \<infinity>) \<or> (limsup v = \<infinity>)"
then have "limsup u + limsup v = \<infinity>" by simp
then show ?thesis by auto
@@ -1522,11 +1522,11 @@
This explains why there are more assumptions in the next lemma dealing with liminfs that in the
previous one about limsups.\<close>
-lemma%important ereal_liminf_add_mono:
+lemma ereal_liminf_add_mono:
fixes u v::"nat \<Rightarrow> ereal"
assumes "\<not>((liminf u = \<infinity> \<and> liminf v = -\<infinity>) \<or> (liminf u = -\<infinity> \<and> liminf v = \<infinity>))"
shows "liminf (\<lambda>n. u n + v n) \<ge> liminf u + liminf v"
-proof%unimportant (cases)
+proof (cases)
assume "(liminf u = -\<infinity>) \<or> (liminf v = -\<infinity>)"
then have *: "liminf u + liminf v = -\<infinity>" using assms by auto
show ?thesis by (simp add: *)
@@ -1565,11 +1565,11 @@
then show ?thesis unfolding w_def by simp
qed
-lemma%important ereal_limsup_lim_add:
+lemma ereal_limsup_lim_add:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
shows "limsup (\<lambda>n. u n + v n) = a + limsup v"
-proof%unimportant -
+proof -
have "limsup u = a" using assms(1) using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
then have "limsup (\<lambda>n. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
@@ -1596,11 +1596,11 @@
then show ?thesis using up by simp
qed
-lemma%important ereal_limsup_lim_mult:
+lemma ereal_limsup_lim_mult:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
shows "limsup (\<lambda>n. u n * v n) = a * limsup v"
-proof%unimportant -
+proof -
define w where "w = (\<lambda>n. u n * v n)"
obtain r where r: "strict_mono r" "(v o r) \<longlonglongrightarrow> limsup v" using limsup_subseq_lim by auto
have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
@@ -1626,11 +1626,11 @@
then show ?thesis using I unfolding w_def by auto
qed
-lemma%important ereal_liminf_lim_mult:
+lemma ereal_liminf_lim_mult:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "a>0" "a \<noteq> \<infinity>"
shows "liminf (\<lambda>n. u n * v n) = a * liminf v"
-proof%unimportant -
+proof -
define w where "w = (\<lambda>n. u n * v n)"
obtain r where r: "strict_mono r" "(v o r) \<longlonglongrightarrow> liminf v" using liminf_subseq_lim by auto
have "(u o r) \<longlonglongrightarrow> a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
@@ -1656,11 +1656,11 @@
then show ?thesis using I unfolding w_def by auto
qed
-lemma%important ereal_liminf_lim_add:
+lemma ereal_liminf_lim_add:
fixes u v::"nat \<Rightarrow> ereal"
assumes "u \<longlonglongrightarrow> a" "abs(a) \<noteq> \<infinity>"
shows "liminf (\<lambda>n. u n + v n) = a + liminf v"
-proof%unimportant -
+proof -
have "liminf u = a" using assms(1) tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
then have *: "abs(liminf u) \<noteq> \<infinity>" using assms(2) by auto
have "(\<lambda>n. -u n) \<longlonglongrightarrow> -a" using assms(1) by auto
@@ -1689,10 +1689,10 @@
then show ?thesis using up by simp
qed
-lemma%important ereal_liminf_limsup_add:
+lemma ereal_liminf_limsup_add:
fixes u v::"nat \<Rightarrow> ereal"
shows "liminf (\<lambda>n. u n + v n) \<le> liminf u + limsup v"
-proof%unimportant (cases)
+proof (cases)
assume "limsup v = \<infinity> \<or> liminf u = \<infinity>"
then show ?thesis by auto
next
@@ -1741,12 +1741,12 @@
done
-lemma%important liminf_minus_ennreal:
+lemma liminf_minus_ennreal:
fixes u v::"nat \<Rightarrow> ennreal"
shows "(\<And>n. v n \<le> u n) \<Longrightarrow> liminf (\<lambda>n. u n - v n) \<le> limsup u - limsup v"
unfolding liminf_SUP_INF limsup_INF_SUP
including ennreal.lifting
-proof%unimportant (transfer, clarsimp)
+proof (transfer, clarsimp)
fix v u :: "nat \<Rightarrow> ereal" assume *: "\<forall>x. 0 \<le> v x" "\<forall>x. 0 \<le> u x" "\<And>n. v n \<le> u n"
moreover have "0 \<le> limsup u - limsup v"
using * by (intro ereal_diff_positive Limsup_mono always_eventually) simp
@@ -1759,7 +1759,7 @@
by (auto simp: * ereal_diff_positive max.absorb2 liminf_SUP_INF[symmetric] limsup_INF_SUP[symmetric] ereal_liminf_limsup_minus)
qed
-subsection%unimportant "Relate extended reals and the indicator function"
+subsection%important "Relate extended reals and the indicator function"
lemma ereal_indicator_le_0: "(indicator S x::ereal) \<le> 0 \<longleftrightarrow> x \<notin> S"
by (auto split: split_indicator simp: one_ereal_def)
--- a/src/HOL/Analysis/Fashoda_Theorem.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Fashoda_Theorem.thy Wed Jan 16 21:27:33 2019 +0000
@@ -30,14 +30,14 @@
apply (rule, rule continuous_interval_bij)
done
-lemma%important in_interval_interval_bij:
+lemma in_interval_interval_bij:
fixes a b u v x :: "'a::euclidean_space"
assumes "x \<in> cbox a b"
and "cbox u v \<noteq> {}"
shows "interval_bij (a, b) (u, v) x \<in> cbox u v"
apply (simp only: interval_bij_def split_conv mem_box inner_sum_left_Basis cong: ball_cong)
apply safe
-proof%unimportant -
+proof -
fix i :: 'a
assume i: "i \<in> Basis"
have "cbox a b \<noteq> {}"
@@ -89,7 +89,7 @@
shows "\<bar>x$1\<bar> \<le> 1" and "\<bar>x$2\<bar> \<le> 1"
using assms unfolding infnorm_eq_1_2 by auto
-lemma%important fashoda_unit:
+proposition fashoda_unit:
fixes f g :: "real \<Rightarrow> real^2"
assumes "f ` {-1 .. 1} \<subseteq> cbox (-1) 1"
and "g ` {-1 .. 1} \<subseteq> cbox (-1) 1"
@@ -99,7 +99,7 @@
and "f 1$1 = 1" "g (- 1) $2 = -1"
and "g 1 $2 = 1"
shows "\<exists>s\<in>{-1 .. 1}. \<exists>t\<in>{-1 .. 1}. f s = g t"
-proof%unimportant (rule ccontr)
+proof (rule ccontr)
assume "\<not> ?thesis"
note as = this[unfolded bex_simps,rule_format]
define sqprojection
@@ -256,7 +256,7 @@
qed
qed
-lemma%important fashoda_unit_path:
+proposition fashoda_unit_path:
fixes f g :: "real \<Rightarrow> real^2"
assumes "path f"
and "path g"
@@ -267,7 +267,7 @@
and "(pathstart g)$2 = -1"
and "(pathfinish g)$2 = 1"
obtains z where "z \<in> path_image f" and "z \<in> path_image g"
-proof%unimportant -
+proof -
note assms=assms[unfolded path_def pathstart_def pathfinish_def path_image_def]
define iscale where [abs_def]: "iscale z = inverse 2 *\<^sub>R (z + 1)" for z :: real
have isc: "iscale ` {- 1..1} \<subseteq> {0..1}"
@@ -312,7 +312,7 @@
done
qed
-lemma%important fashoda:
+theorem fashoda:
fixes b :: "real^2"
assumes "path f"
and "path g"
@@ -323,7 +323,7 @@
and "(pathstart g)$2 = a$2"
and "(pathfinish g)$2 = b$2"
obtains z where "z \<in> path_image f" and "z \<in> path_image g"
-proof%unimportant -
+proof -
fix P Q S
presume "P \<or> Q \<or> S" "P \<Longrightarrow> thesis" and "Q \<Longrightarrow> thesis" and "S \<Longrightarrow> thesis"
then show thesis
@@ -632,7 +632,7 @@
subsection%important \<open>Useful Fashoda corollary pointed out to me by Tom Hales\<close>(*FIXME change title? *)
-lemma%important fashoda_interlace:
+corollary fashoda_interlace:
fixes a :: "real^2"
assumes "path f"
and "path g"
@@ -646,7 +646,7 @@
and "(pathstart g)$1 < (pathfinish f)$1"
and "(pathfinish f)$1 < (pathfinish g)$1"
obtains z where "z \<in> path_image f" and "z \<in> path_image g"
-proof%unimportant -
+proof -
have "cbox a b \<noteq> {}"
using path_image_nonempty[of f] using assms(3) by auto
note ab=this[unfolded interval_eq_empty_cart not_ex forall_2 not_less]
--- a/src/HOL/Analysis/Finite_Cartesian_Product.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Finite_Cartesian_Product.thy Wed Jan 16 21:27:33 2019 +0000
@@ -13,7 +13,7 @@
"HOL-Library.FuncSet"
begin
-subsection%unimportant \<open>Finite Cartesian products, with indexing and lambdas\<close>
+subsection%important \<open>Finite Cartesian products, with indexing and lambdas\<close>
typedef ('a, 'b) vec = "UNIV :: ('b::finite \<Rightarrow> 'a) set"
morphisms vec_nth vec_lambda ..
@@ -104,10 +104,10 @@
by (auto elim!: countableE)
qed
-lemma%important infinite_UNIV_vec:
+lemma infinite_UNIV_vec:
assumes "infinite (UNIV :: 'a set)"
shows "infinite (UNIV :: ('a^'b) set)"
-proof%unimportant (subst bij_betw_finite)
+proof (subst bij_betw_finite)
show "bij_betw vec_nth UNIV (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))"
by (intro bij_betwI[of _ _ _ vec_lambda]) (auto simp: vec_eq_iff)
have "infinite (PiE (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" (is "infinite ?A")
@@ -125,9 +125,9 @@
finally show "infinite (Pi (UNIV :: 'b set) (\<lambda>_. UNIV :: 'a set))" .
qed
-lemma%important CARD_vec [simp]:
+lemma CARD_vec [simp]:
"CARD('a^'b) = CARD('a) ^ CARD('b)"
-proof%unimportant (cases "finite (UNIV :: 'a set)")
+proof (cases "finite (UNIV :: 'a set)")
case True
show ?thesis
proof (subst bij_betw_same_card)
@@ -143,7 +143,7 @@
qed
qed (simp_all add: infinite_UNIV_vec)
-lemma%unimportant countable_vector:
+lemma countable_vector:
fixes B:: "'n::finite \<Rightarrow> 'a set"
assumes "\<And>i. countable (B i)"
shows "countable {V. \<forall>i::'n::finite. V $ i \<in> B i}"
@@ -165,7 +165,7 @@
by (simp add: image_comp o_def vec_nth_inverse)
qed
-subsection%unimportant \<open>Group operations and class instances\<close>
+subsection%important \<open>Group operations and class instances\<close>
instantiation vec :: (zero, finite) zero
begin
@@ -230,7 +230,7 @@
by standard (simp_all add: vec_eq_iff)
-subsection%unimportant\<open>Basic componentwise operations on vectors\<close>
+subsection%important\<open>Basic componentwise operations on vectors\<close>
instantiation vec :: (times, finite) times
begin
@@ -295,12 +295,12 @@
text\<open>Also the scalar-vector multiplication.\<close>
-definition vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
+definition%important vector_scalar_mult:: "'a::times \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ 'n" (infixl "*s" 70)
where "c *s x = (\<chi> i. c * (x$i))"
text \<open>scalar product\<close>
-definition scalar_product :: "'a :: semiring_1 ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" where
+definition%important scalar_product :: "'a :: semiring_1 ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" where
"scalar_product v w = (\<Sum> i \<in> UNIV. v $ i * w $ i)"
@@ -311,7 +311,7 @@
definition%important "scaleR \<equiv> (\<lambda> r x. (\<chi> i. scaleR r (x$i)))"
-lemma%important vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
+lemma vector_scaleR_component [simp]: "(scaleR r x)$i = scaleR r (x$i)"
unfolding scaleR_vec_def by simp
instance%unimportant
@@ -330,7 +330,7 @@
(\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x$i \<in> A i) \<and>
(\<forall>y. (\<forall>i. y$i \<in> A i) \<longrightarrow> y \<in> S))"
-instance proof%unimportant
+instance proof
show "open (UNIV :: ('a ^ 'b) set)"
unfolding open_vec_def by auto
next
@@ -358,30 +358,30 @@
end
-lemma%unimportant open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
+lemma open_vector_box: "\<forall>i. open (S i) \<Longrightarrow> open {x. \<forall>i. x $ i \<in> S i}"
unfolding open_vec_def by auto
-lemma%unimportant open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
+lemma open_vimage_vec_nth: "open S \<Longrightarrow> open ((\<lambda>x. x $ i) -` S)"
unfolding open_vec_def
apply clarify
apply (rule_tac x="\<lambda>k. if k = i then S else UNIV" in exI, simp)
done
-lemma%unimportant closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
+lemma closed_vimage_vec_nth: "closed S \<Longrightarrow> closed ((\<lambda>x. x $ i) -` S)"
unfolding closed_open vimage_Compl [symmetric]
by (rule open_vimage_vec_nth)
-lemma%unimportant closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
+lemma closed_vector_box: "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
proof -
have "{x. \<forall>i. x $ i \<in> S i} = (\<Inter>i. (\<lambda>x. x $ i) -` S i)" by auto
thus "\<forall>i. closed (S i) \<Longrightarrow> closed {x. \<forall>i. x $ i \<in> S i}"
by (simp add: closed_INT closed_vimage_vec_nth)
qed
-lemma%important tendsto_vec_nth [tendsto_intros]:
+lemma tendsto_vec_nth [tendsto_intros]:
assumes "((\<lambda>x. f x) \<longlongrightarrow> a) net"
shows "((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
-proof%unimportant (rule topological_tendstoI)
+proof (rule topological_tendstoI)
fix S assume "open S" "a $ i \<in> S"
then have "open ((\<lambda>y. y $ i) -` S)" "a \<in> ((\<lambda>y. y $ i) -` S)"
by (simp_all add: open_vimage_vec_nth)
@@ -391,13 +391,13 @@
by simp
qed
-lemma%unimportant isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
+lemma isCont_vec_nth [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x $ i) a"
unfolding isCont_def by (rule tendsto_vec_nth)
-lemma%important vec_tendstoI:
+lemma vec_tendstoI:
assumes "\<And>i. ((\<lambda>x. f x $ i) \<longlongrightarrow> a $ i) net"
shows "((\<lambda>x. f x) \<longlongrightarrow> a) net"
-proof%unimportant (rule topological_tendstoI)
+proof (rule topological_tendstoI)
fix S assume "open S" and "a \<in> S"
then obtain A where A: "\<And>i. open (A i)" "\<And>i. a $ i \<in> A i"
and S: "\<And>y. \<forall>i. y $ i \<in> A i \<Longrightarrow> y \<in> S"
@@ -410,13 +410,13 @@
by (rule eventually_mono, simp add: S)
qed
-lemma%unimportant tendsto_vec_lambda [tendsto_intros]:
+lemma tendsto_vec_lambda [tendsto_intros]:
assumes "\<And>i. ((\<lambda>x. f x i) \<longlongrightarrow> a i) net"
shows "((\<lambda>x. \<chi> i. f x i) \<longlongrightarrow> (\<chi> i. a i)) net"
using assms by (simp add: vec_tendstoI)
-lemma%important open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
-proof%unimportant (rule openI)
+lemma open_image_vec_nth: assumes "open S" shows "open ((\<lambda>x. x $ i) ` S)"
+proof (rule openI)
fix a assume "a \<in> (\<lambda>x. x $ i) ` S"
then obtain z where "a = z $ i" and "z \<in> S" ..
then obtain A where A: "\<forall>i. open (A i) \<and> z $ i \<in> A i"
@@ -470,10 +470,10 @@
instantiation%unimportant vec :: (metric_space, finite) metric_space
begin
-lemma%important dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
+lemma dist_vec_nth_le: "dist (x $ i) (y $ i) \<le> dist x y"
unfolding dist_vec_def by (rule member_le_L2_set) simp_all
-instance proof%unimportant
+instance proof
fix x y :: "'a ^ 'b"
show "dist x y = 0 \<longleftrightarrow> x = y"
unfolding dist_vec_def
@@ -532,15 +532,15 @@
end
-lemma%important Cauchy_vec_nth:
+lemma Cauchy_vec_nth:
"Cauchy (\<lambda>n. X n) \<Longrightarrow> Cauchy (\<lambda>n. X n $ i)"
unfolding Cauchy_def by (fast intro: le_less_trans [OF dist_vec_nth_le])
-lemma%important vec_CauchyI:
+lemma vec_CauchyI:
fixes X :: "nat \<Rightarrow> 'a::metric_space ^ 'n"
assumes X: "\<And>i. Cauchy (\<lambda>n. X n $ i)"
shows "Cauchy (\<lambda>n. X n)"
-proof%unimportant (rule metric_CauchyI)
+proof (rule metric_CauchyI)
fix r :: real assume "0 < r"
hence "0 < r / of_nat CARD('n)" (is "0 < ?s") by simp
define N where "N i = (LEAST N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m $ i) (X n $ i) < ?s)" for i
@@ -591,7 +591,7 @@
definition%important "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
-instance proof%unimportant
+instance proof
fix a :: real and x y :: "'a ^ 'b"
show "norm x = 0 \<longleftrightarrow> x = 0"
unfolding norm_vec_def
@@ -613,32 +613,32 @@
end
-lemma%unimportant norm_nth_le: "norm (x $ i) \<le> norm x"
+lemma norm_nth_le: "norm (x $ i) \<le> norm x"
unfolding norm_vec_def
by (rule member_le_L2_set) simp_all
-lemma%important norm_le_componentwise_cart:
+lemma norm_le_componentwise_cart:
fixes x :: "'a::real_normed_vector^'n"
assumes "\<And>i. norm(x$i) \<le> norm(y$i)"
shows "norm x \<le> norm y"
- unfolding%unimportant norm_vec_def
- by%unimportant (rule L2_set_mono) (auto simp: assms)
+ unfolding norm_vec_def
+ by (rule L2_set_mono) (auto simp: assms)
-lemma%unimportant component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
+lemma component_le_norm_cart: "\<bar>x$i\<bar> \<le> norm x"
apply (simp add: norm_vec_def)
apply (rule member_le_L2_set, simp_all)
done
-lemma%unimportant norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
+lemma norm_bound_component_le_cart: "norm x \<le> e ==> \<bar>x$i\<bar> \<le> e"
by (metis component_le_norm_cart order_trans)
-lemma%unimportant norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
+lemma norm_bound_component_lt_cart: "norm x < e ==> \<bar>x$i\<bar> < e"
by (metis component_le_norm_cart le_less_trans)
-lemma%unimportant norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
+lemma norm_le_l1_cart: "norm x \<le> sum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
by (simp add: norm_vec_def L2_set_le_sum)
-lemma%unimportant bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x $ i)"
+lemma bounded_linear_vec_nth: "bounded_linear (\<lambda>x. x $ i)"
apply standard
apply (rule vector_add_component)
apply (rule vector_scaleR_component)
@@ -655,7 +655,7 @@
definition%important "inner x y = sum (\<lambda>i. inner (x$i) (y$i)) UNIV"
-instance proof%unimportant
+instance proof
fix r :: real and x y z :: "'a ^ 'b"
show "inner x y = inner y x"
unfolding inner_vec_def
@@ -686,13 +686,13 @@
definition%important "axis k x = (\<chi> i. if i = k then x else 0)"
-lemma%unimportant axis_nth [simp]: "axis i x $ i = x"
+lemma axis_nth [simp]: "axis i x $ i = x"
unfolding axis_def by simp
-lemma%unimportant axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
+lemma axis_eq_axis: "axis i x = axis j y \<longleftrightarrow> x = y \<and> i = j \<or> x = 0 \<and> y = 0"
unfolding axis_def vec_eq_iff by auto
-lemma%unimportant inner_axis_axis:
+lemma inner_axis_axis:
"inner (axis i x) (axis j y) = (if i = j then inner x y else 0)"
unfolding inner_vec_def
apply (cases "i = j")
@@ -702,10 +702,10 @@
apply (rule sum.neutral, simp add: axis_def)
done
-lemma%unimportant inner_axis: "inner x (axis i y) = inner (x $ i) y"
+lemma inner_axis: "inner x (axis i y) = inner (x $ i) y"
by (simp add: inner_vec_def axis_def sum.remove [where x=i])
-lemma%unimportant inner_axis': "inner(axis i y) x = inner y (x $ i)"
+lemma inner_axis': "inner(axis i y) x = inner y (x $ i)"
by (simp add: inner_axis inner_commute)
instantiation%unimportant vec :: (euclidean_space, finite) euclidean_space
@@ -713,7 +713,7 @@
definition%important "Basis = (\<Union>i. \<Union>u\<in>Basis. {axis i u})"
-instance proof%unimportant
+instance proof
show "(Basis :: ('a ^ 'b) set) \<noteq> {}"
unfolding Basis_vec_def by simp
next
@@ -732,8 +732,8 @@
by (simp add: inner_axis euclidean_all_zero_iff vec_eq_iff)
qed
-lemma%important DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
-proof%unimportant -
+proposition DIM_cart [simp]: "DIM('a^'b) = CARD('b) * DIM('a)"
+proof -
have "card (\<Union>i::'b. \<Union>u::'a\<in>Basis. {axis i u}) = (\<Sum>i::'b\<in>UNIV. card (\<Union>u::'a\<in>Basis. {axis i u}))"
by (rule card_UN_disjoint) (auto simp: axis_eq_axis)
also have "... = CARD('b) * DIM('a)"
@@ -744,30 +744,30 @@
end
-lemma%unimportant norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
+lemma norm_axis_1 [simp]: "norm (axis m (1::real)) = 1"
by (simp add: inner_axis' norm_eq_1)
-lemma%important sum_norm_allsubsets_bound_cart:
+lemma sum_norm_allsubsets_bound_cart:
fixes f:: "'a \<Rightarrow> real ^'n"
assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (sum f Q) \<le> e"
shows "sum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) * e"
- using%unimportant sum_norm_allsubsets_bound[OF assms]
- by%unimportant simp
+ using sum_norm_allsubsets_bound[OF assms]
+ by simp
-lemma%unimportant cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
+lemma cart_eq_inner_axis: "a $ i = inner a (axis i 1)"
by (simp add: inner_axis)
-lemma%unimportant axis_eq_0_iff [simp]:
+lemma axis_eq_0_iff [simp]:
shows "axis m x = 0 \<longleftrightarrow> x = 0"
by (simp add: axis_def vec_eq_iff)
-lemma%unimportant axis_in_Basis_iff [simp]: "axis i a \<in> Basis \<longleftrightarrow> a \<in> Basis"
+lemma axis_in_Basis_iff [simp]: "axis i a \<in> Basis \<longleftrightarrow> a \<in> Basis"
by (auto simp: Basis_vec_def axis_eq_axis)
text\<open>Mapping each basis element to the corresponding finite index\<close>
definition axis_index :: "('a::comm_ring_1)^'n \<Rightarrow> 'n" where "axis_index v \<equiv> SOME i. v = axis i 1"
-lemma%unimportant axis_inverse:
+lemma axis_inverse:
fixes v :: "real^'n"
assumes "v \<in> Basis"
shows "\<exists>i. v = axis i 1"
@@ -778,20 +778,20 @@
by (force simp add: vec_eq_iff)
qed
-lemma%unimportant axis_index:
+lemma axis_index:
fixes v :: "real^'n"
assumes "v \<in> Basis"
shows "v = axis (axis_index v) 1"
by (metis (mono_tags) assms axis_inverse axis_index_def someI_ex)
-lemma%unimportant axis_index_axis [simp]:
+lemma axis_index_axis [simp]:
fixes UU :: "real^'n"
shows "(axis_index (axis u 1 :: real^'n)) = (u::'n)"
by (simp add: axis_eq_axis axis_index_def)
subsection%unimportant \<open>A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space\<close>
-lemma%unimportant sum_cong_aux:
+lemma sum_cong_aux:
"(\<And>x. x \<in> A \<Longrightarrow> f x = g x) \<Longrightarrow> sum f A = sum g A"
by (auto intro: sum.cong)
@@ -823,22 +823,22 @@
end
\<close> "lift trivial vector statements to real arith statements"
-lemma%unimportant vec_0[simp]: "vec 0 = 0" by vector
-lemma%unimportant vec_1[simp]: "vec 1 = 1" by vector
+lemma vec_0[simp]: "vec 0 = 0" by vector
+lemma vec_1[simp]: "vec 1 = 1" by vector
-lemma%unimportant vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
+lemma vec_inj[simp]: "vec x = vec y \<longleftrightarrow> x = y" by vector
-lemma%unimportant vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
+lemma vec_in_image_vec: "vec x \<in> (vec ` S) \<longleftrightarrow> x \<in> S" by auto
-lemma%unimportant vec_add: "vec(x + y) = vec x + vec y" by vector
-lemma%unimportant vec_sub: "vec(x - y) = vec x - vec y" by vector
-lemma%unimportant vec_cmul: "vec(c * x) = c *s vec x " by vector
-lemma%unimportant vec_neg: "vec(- x) = - vec x " by vector
+lemma vec_add: "vec(x + y) = vec x + vec y" by vector
+lemma vec_sub: "vec(x - y) = vec x - vec y" by vector
+lemma vec_cmul: "vec(c * x) = c *s vec x " by vector
+lemma vec_neg: "vec(- x) = - vec x " by vector
-lemma%unimportant vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
+lemma vec_scaleR: "vec(c * x) = c *\<^sub>R vec x"
by vector
-lemma%unimportant vec_sum:
+lemma vec_sum:
assumes "finite S"
shows "vec(sum f S) = sum (vec \<circ> f) S"
using assms
@@ -852,24 +852,24 @@
text\<open>Obvious "component-pushing".\<close>
-lemma%unimportant vec_component [simp]: "vec x $ i = x"
+lemma vec_component [simp]: "vec x $ i = x"
by vector
-lemma%unimportant vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
+lemma vector_mult_component [simp]: "(x * y)$i = x$i * y$i"
by vector
-lemma%unimportant vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
+lemma vector_smult_component [simp]: "(c *s y)$i = c * (y$i)"
by vector
-lemma%unimportant cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
+lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
-lemmas%unimportant vector_component =
+lemmas vector_component =
vec_component vector_add_component vector_mult_component
vector_smult_component vector_minus_component vector_uminus_component
vector_scaleR_component cond_component
-subsection%unimportant \<open>Some frequently useful arithmetic lemmas over vectors\<close>
+subsection%important \<open>Some frequently useful arithmetic lemmas over vectors\<close>
instance vec :: (semigroup_mult, finite) semigroup_mult
by standard (vector mult.assoc)
@@ -1011,7 +1011,7 @@
definition%important map_matrix::"('a \<Rightarrow> 'b) \<Rightarrow> (('a, 'i::finite)vec, 'j::finite) vec \<Rightarrow> (('b, 'i)vec, 'j) vec" where
"map_matrix f x = (\<chi> i j. f (x $ i $ j))"
-lemma%unimportant nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
+lemma nth_map_matrix[simp]: "map_matrix f x $ i $ j = f (x $ i $ j)"
by (simp add: map_matrix_def)
definition%important matrix_matrix_mult :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"
@@ -1034,17 +1034,17 @@
definition%unimportant "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
definition%unimportant "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
-lemma%unimportant times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0"
+lemma times0_left [simp]: "(0::'a::semiring_1^'n^'m) ** (A::'a ^'p^'n) = 0"
by (simp add: matrix_matrix_mult_def zero_vec_def)
-lemma%unimportant times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0"
+lemma times0_right [simp]: "(A::'a::semiring_1^'n^'m) ** (0::'a ^'p^'n) = 0"
by (simp add: matrix_matrix_mult_def zero_vec_def)
-lemma%unimportant mat_0[simp]: "mat 0 = 0" by (vector mat_def)
-lemma%unimportant matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
+lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
+lemma matrix_add_ldistrib: "(A ** (B + C)) = (A ** B) + (A ** C)"
by (vector matrix_matrix_mult_def sum.distrib[symmetric] field_simps)
-lemma%unimportant matrix_mul_lid [simp]:
+lemma matrix_mul_lid [simp]:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "mat 1 ** A = A"
apply (simp add: matrix_matrix_mult_def mat_def)
@@ -1053,7 +1053,7 @@
mult_1_left mult_zero_left if_True UNIV_I)
done
-lemma%unimportant matrix_mul_rid [simp]:
+lemma matrix_mul_rid [simp]:
fixes A :: "'a::semiring_1 ^ 'm ^ 'n"
shows "A ** mat 1 = A"
apply (simp add: matrix_matrix_mult_def mat_def)
@@ -1062,39 +1062,39 @@
mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
done
-lemma%unimportant matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
+proposition matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
apply (vector matrix_matrix_mult_def sum_distrib_left sum_distrib_right mult.assoc)
apply (subst sum.swap)
apply simp
done
-lemma%unimportant matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
+proposition matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
apply (vector matrix_matrix_mult_def matrix_vector_mult_def
sum_distrib_left sum_distrib_right mult.assoc)
apply (subst sum.swap)
apply simp
done
-lemma%unimportant scalar_matrix_assoc:
+lemma scalar_matrix_assoc:
fixes A :: "('a::real_algebra_1)^'m^'n"
shows "k *\<^sub>R (A ** B) = (k *\<^sub>R A) ** B"
by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff scaleR_sum_right)
-lemma%unimportant matrix_scalar_ac:
+lemma matrix_scalar_ac:
fixes A :: "('a::real_algebra_1)^'m^'n"
shows "A ** (k *\<^sub>R B) = k *\<^sub>R A ** B"
by (simp add: matrix_matrix_mult_def sum_distrib_left mult_ac vec_eq_iff)
-lemma%unimportant matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
+lemma matrix_vector_mul_lid [simp]: "mat 1 *v x = (x::'a::semiring_1 ^ 'n)"
apply (vector matrix_vector_mult_def mat_def)
apply (simp add: if_distrib if_distribR sum.delta' cong del: if_weak_cong)
done
-lemma%unimportant matrix_transpose_mul:
+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%unimportant matrix_eq:
+lemma matrix_eq:
fixes A B :: "'a::semiring_1 ^ 'n ^ 'm"
shows "A = B \<longleftrightarrow> (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
apply auto
@@ -1107,49 +1107,49 @@
sum.delta[OF finite] cong del: if_weak_cong)
done
-lemma%unimportant matrix_vector_mul_component: "(A *v x)$k = inner (A$k) x"
+lemma matrix_vector_mul_component: "(A *v x)$k = inner (A$k) x"
by (simp add: matrix_vector_mult_def inner_vec_def)
-lemma%unimportant dot_lmul_matrix: "inner ((x::real ^_) v* A) y = inner x (A *v y)"
+lemma dot_lmul_matrix: "inner ((x::real ^_) v* A) y = inner x (A *v y)"
apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def sum_distrib_right sum_distrib_left ac_simps)
apply (subst sum.swap)
apply simp
done
-lemma%unimportant transpose_mat [simp]: "transpose (mat n) = mat n"
+lemma transpose_mat [simp]: "transpose (mat n) = mat n"
by (vector transpose_def mat_def)
-lemma%unimportant transpose_transpose [simp]: "transpose(transpose A) = A"
+lemma transpose_transpose [simp]: "transpose(transpose A) = A"
by (vector transpose_def)
-lemma%unimportant row_transpose [simp]: "row i (transpose A) = column i A"
+lemma row_transpose [simp]: "row i (transpose A) = column i A"
by (simp add: row_def column_def transpose_def vec_eq_iff)
-lemma%unimportant column_transpose [simp]: "column i (transpose A) = row i A"
+lemma column_transpose [simp]: "column i (transpose A) = row i A"
by (simp add: row_def column_def transpose_def vec_eq_iff)
-lemma%unimportant rows_transpose [simp]: "rows(transpose A) = columns A"
+lemma rows_transpose [simp]: "rows(transpose A) = columns A"
by (auto simp add: rows_def columns_def intro: set_eqI)
-lemma%unimportant columns_transpose [simp]: "columns(transpose A) = rows A"
+lemma columns_transpose [simp]: "columns(transpose A) = rows A"
by (metis transpose_transpose rows_transpose)
-lemma%unimportant transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
+lemma transpose_scalar: "transpose (k *\<^sub>R A) = k *\<^sub>R transpose A"
unfolding transpose_def
by (simp add: vec_eq_iff)
-lemma%unimportant transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
+lemma transpose_iff [iff]: "transpose A = transpose B \<longleftrightarrow> A = B"
by (metis transpose_transpose)
-lemma%unimportant matrix_mult_sum:
+lemma matrix_mult_sum:
"(A::'a::comm_semiring_1^'n^'m) *v x = sum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
by (simp add: matrix_vector_mult_def vec_eq_iff column_def mult.commute)
-lemma%unimportant vector_componentwise:
+lemma vector_componentwise:
"(x::'a::ring_1^'n) = (\<chi> j. \<Sum>i\<in>UNIV. (x$i) * (axis i 1 :: 'a^'n) $ j)"
by (simp add: axis_def if_distrib sum.If_cases vec_eq_iff)
-lemma%unimportant basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
+lemma basis_expansion: "sum (\<lambda>i. (x$i) *s axis i 1) UNIV = (x::('a::ring_1) ^'n)"
by (auto simp add: axis_def vec_eq_iff if_distrib sum.If_cases cong del: if_weak_cong)
@@ -1158,51 +1158,51 @@
definition%important matrix :: "('a::{plus,times, one, zero}^'m \<Rightarrow> 'a ^ 'n) \<Rightarrow> 'a^'m^'n"
where "matrix f = (\<chi> i j. (f(axis j 1))$i)"
-lemma%unimportant matrix_id_mat_1: "matrix id = mat 1"
+lemma matrix_id_mat_1: "matrix id = mat 1"
by (simp add: mat_def matrix_def axis_def)
-lemma%unimportant matrix_scaleR: "(matrix ((*\<^sub>R) r)) = mat r"
+lemma matrix_scaleR: "(matrix ((*\<^sub>R) r)) = mat r"
by (simp add: mat_def matrix_def axis_def if_distrib cong: if_cong)
-lemma%unimportant matrix_vector_mul_linear[intro, simp]: "linear (\<lambda>x. A *v (x::'a::real_algebra_1 ^ _))"
+lemma matrix_vector_mul_linear[intro, simp]: "linear (\<lambda>x. A *v (x::'a::real_algebra_1 ^ _))"
by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff field_simps sum_distrib_left
sum.distrib scaleR_right.sum)
-lemma%unimportant vector_matrix_left_distrib [algebra_simps]:
+lemma vector_matrix_left_distrib [algebra_simps]:
shows "(x + y) v* A = x v* A + y v* A"
unfolding vector_matrix_mult_def
by (simp add: algebra_simps sum.distrib vec_eq_iff)
-lemma%unimportant matrix_vector_right_distrib [algebra_simps]:
+lemma matrix_vector_right_distrib [algebra_simps]:
"A *v (x + y) = A *v x + A *v y"
by (vector matrix_vector_mult_def sum.distrib distrib_left)
-lemma%unimportant matrix_vector_mult_diff_distrib [algebra_simps]:
+lemma matrix_vector_mult_diff_distrib [algebra_simps]:
fixes A :: "'a::ring_1^'n^'m"
shows "A *v (x - y) = A *v x - A *v y"
by (vector matrix_vector_mult_def sum_subtractf right_diff_distrib)
-lemma%unimportant matrix_vector_mult_scaleR[algebra_simps]:
+lemma matrix_vector_mult_scaleR[algebra_simps]:
fixes A :: "real^'n^'m"
shows "A *v (c *\<^sub>R x) = c *\<^sub>R (A *v x)"
using linear_iff matrix_vector_mul_linear by blast
-lemma%unimportant matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
+lemma matrix_vector_mult_0_right [simp]: "A *v 0 = 0"
by (simp add: matrix_vector_mult_def vec_eq_iff)
-lemma%unimportant matrix_vector_mult_0 [simp]: "0 *v w = 0"
+lemma matrix_vector_mult_0 [simp]: "0 *v w = 0"
by (simp add: matrix_vector_mult_def vec_eq_iff)
-lemma%unimportant matrix_vector_mult_add_rdistrib [algebra_simps]:
+lemma matrix_vector_mult_add_rdistrib [algebra_simps]:
"(A + B) *v x = (A *v x) + (B *v x)"
by (vector matrix_vector_mult_def sum.distrib distrib_right)
-lemma%unimportant matrix_vector_mult_diff_rdistrib [algebra_simps]:
+lemma matrix_vector_mult_diff_rdistrib [algebra_simps]:
fixes A :: "'a :: ring_1^'n^'m"
shows "(A - B) *v x = (A *v x) - (B *v x)"
by (vector matrix_vector_mult_def sum_subtractf left_diff_distrib)
-lemma%unimportant matrix_vector_column:
+lemma matrix_vector_column:
"(A::'a::comm_semiring_1^'n^_) *v x = sum (\<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)
@@ -1215,17 +1215,17 @@
"matrix_inv(A:: 'a::semiring_1^'n^'m) =
(SOME A'::'a^'m^'n. A ** A' = mat 1 \<and> A' ** A = mat 1)"
-lemma%unimportant inj_matrix_vector_mult:
+lemma inj_matrix_vector_mult:
fixes A::"'a::field^'n^'m"
assumes "invertible A"
shows "inj ((*v) A)"
by (metis assms inj_on_inverseI invertible_def matrix_vector_mul_assoc matrix_vector_mul_lid)
-lemma%important scalar_invertible:
+lemma scalar_invertible:
fixes A :: "('a::real_algebra_1)^'m^'n"
assumes "k \<noteq> 0" and "invertible A"
shows "invertible (k *\<^sub>R A)"
-proof%unimportant -
+proof -
obtain A' where "A ** A' = mat 1" and "A' ** A = mat 1"
using assms unfolding invertible_def by auto
with \<open>k \<noteq> 0\<close>
@@ -1235,50 +1235,50 @@
unfolding invertible_def by auto
qed
-lemma%unimportant scalar_invertible_iff:
+lemma scalar_invertible_iff:
fixes A :: "('a::real_algebra_1)^'m^'n"
assumes "k \<noteq> 0" and "invertible A"
shows "invertible (k *\<^sub>R A) \<longleftrightarrow> k \<noteq> 0 \<and> invertible A"
by (simp add: assms scalar_invertible)
-lemma%unimportant vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
+lemma vector_transpose_matrix [simp]: "x v* transpose A = A *v x"
unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
by simp
-lemma%unimportant transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
+lemma transpose_matrix_vector [simp]: "transpose A *v x = x v* A"
unfolding transpose_def vector_matrix_mult_def matrix_vector_mult_def
by simp
-lemma%unimportant vector_scalar_commute:
+lemma vector_scalar_commute:
fixes A :: "'a::{field}^'m^'n"
shows "A *v (c *s x) = c *s (A *v x)"
by (simp add: vector_scalar_mult_def matrix_vector_mult_def mult_ac sum_distrib_left)
-lemma%unimportant scalar_vector_matrix_assoc:
+lemma scalar_vector_matrix_assoc:
fixes k :: "'a::{field}" and x :: "'a::{field}^'n" and A :: "'a^'m^'n"
shows "(k *s x) v* A = k *s (x v* A)"
by (metis transpose_matrix_vector vector_scalar_commute)
-lemma%unimportant vector_matrix_mult_0 [simp]: "0 v* A = 0"
+lemma vector_matrix_mult_0 [simp]: "0 v* A = 0"
unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
-lemma%unimportant vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
+lemma vector_matrix_mult_0_right [simp]: "x v* 0 = 0"
unfolding vector_matrix_mult_def by (simp add: zero_vec_def)
-lemma%unimportant vector_matrix_mul_rid [simp]:
+lemma vector_matrix_mul_rid [simp]:
fixes v :: "('a::semiring_1)^'n"
shows "v v* mat 1 = v"
by (metis matrix_vector_mul_lid transpose_mat vector_transpose_matrix)
-lemma%unimportant scaleR_vector_matrix_assoc:
+lemma scaleR_vector_matrix_assoc:
fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
shows "(k *\<^sub>R x) v* A = k *\<^sub>R (x v* A)"
by (metis matrix_vector_mult_scaleR transpose_matrix_vector)
-lemma%important vector_scaleR_matrix_ac:
+lemma vector_scaleR_matrix_ac:
fixes k :: real and x :: "real^'n" and A :: "real^'m^'n"
shows "x v* (k *\<^sub>R A) = k *\<^sub>R (x v* A)"
-proof%unimportant -
+proof -
have "x v* (k *\<^sub>R A) = (k *\<^sub>R x) v* A"
unfolding vector_matrix_mult_def
by (simp add: algebra_simps)
--- a/src/HOL/Analysis/Finite_Product_Measure.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Finite_Product_Measure.thy Wed Jan 16 21:27:33 2019 +0000
@@ -8,14 +8,14 @@
imports Binary_Product_Measure
begin
-lemma%unimportant PiE_choice: "(\<exists>f\<in>Pi\<^sub>E I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)"
+lemma PiE_choice: "(\<exists>f\<in>Pi\<^sub>E I F. \<forall>i\<in>I. P i (f i)) \<longleftrightarrow> (\<forall>i\<in>I. \<exists>x\<in>F i. P i x)"
by (auto simp: Bex_def PiE_iff Ball_def dest!: choice_iff'[THEN iffD1])
(force intro: exI[of _ "restrict f I" for f])
-lemma%unimportant case_prod_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
+lemma case_prod_const: "(\<lambda>(i, j). c) = (\<lambda>_. c)"
by auto
-subsubsection%unimportant \<open>More about Function restricted by \<^const>\<open>extensional\<close>\<close>
+subsubsection \<open>More about Function restricted by \<^const>\<open>extensional\<close>\<close>
definition
"merge I J = (\<lambda>(x, y) i. if i \<in> I then x i else if i \<in> J then y i else undefined)"
@@ -116,42 +116,42 @@
definition%important prod_emb where
"prod_emb I M K X = (\<lambda>x. restrict x K) -` X \<inter> (\<Pi>\<^sub>E i\<in>I. space (M i))"
-lemma%important prod_emb_iff:
+lemma prod_emb_iff:
"f \<in> prod_emb I M K X \<longleftrightarrow> f \<in> extensional I \<and> (restrict f K \<in> X) \<and> (\<forall>i\<in>I. f i \<in> space (M i))"
unfolding%unimportant prod_emb_def PiE_def by auto
-lemma%unimportant
+lemma
shows prod_emb_empty[simp]: "prod_emb M L K {} = {}"
and prod_emb_Un[simp]: "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B"
and prod_emb_Int: "prod_emb M L K (A \<inter> B) = prod_emb M L K A \<inter> prod_emb M L K B"
and prod_emb_UN[simp]: "prod_emb M L K (\<Union>i\<in>I. F i) = (\<Union>i\<in>I. prod_emb M L K (F i))"
and prod_emb_INT[simp]: "I \<noteq> {} \<Longrightarrow> prod_emb M L K (\<Inter>i\<in>I. F i) = (\<Inter>i\<in>I. prod_emb M L K (F i))"
and prod_emb_Diff[simp]: "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
- by%unimportant (auto simp: prod_emb_def)
+ by (auto simp: prod_emb_def)
-lemma%unimportant prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
+lemma prod_emb_PiE: "J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow>
prod_emb I M J (\<Pi>\<^sub>E i\<in>J. E i) = (\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i))"
by (force simp: prod_emb_def PiE_iff if_split_mem2)
-lemma%unimportant prod_emb_PiE_same_index[simp]:
+lemma prod_emb_PiE_same_index[simp]:
"(\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> space (M i)) \<Longrightarrow> prod_emb I M I (Pi\<^sub>E I E) = Pi\<^sub>E I E"
by (auto simp: prod_emb_def PiE_iff)
-lemma%unimportant prod_emb_trans[simp]:
+lemma prod_emb_trans[simp]:
"J \<subseteq> K \<Longrightarrow> K \<subseteq> L \<Longrightarrow> prod_emb L M K (prod_emb K M J X) = prod_emb L M J X"
by (auto simp add: Int_absorb1 prod_emb_def PiE_def)
-lemma%unimportant prod_emb_Pi:
+lemma prod_emb_Pi:
assumes "X \<in> (\<Pi> j\<in>J. sets (M j))" "J \<subseteq> K"
shows "prod_emb K M J (Pi\<^sub>E J X) = (\<Pi>\<^sub>E i\<in>K. if i \<in> J then X i else space (M i))"
using assms sets.space_closed
by (auto simp: prod_emb_def PiE_iff split: if_split_asm) blast+
-lemma%unimportant prod_emb_id:
+lemma prod_emb_id:
"B \<subseteq> (\<Pi>\<^sub>E i\<in>L. space (M i)) \<Longrightarrow> prod_emb L M L B = B"
by (auto simp: prod_emb_def subset_eq extensional_restrict)
-lemma%unimportant prod_emb_mono:
+lemma prod_emb_mono:
"F \<subseteq> G \<Longrightarrow> prod_emb A M B F \<subseteq> prod_emb A M B G"
by (auto simp: prod_emb_def)
@@ -173,20 +173,20 @@
translations
"\<Pi>\<^sub>M x\<in>I. M" == "CONST PiM I (%x. M)"
-lemma%important extend_measure_cong:
+lemma extend_measure_cong:
assumes "\<Omega> = \<Omega>'" "I = I'" "G = G'" "\<And>i. i \<in> I' \<Longrightarrow> \<mu> i = \<mu>' i"
shows "extend_measure \<Omega> I G \<mu> = extend_measure \<Omega>' I' G' \<mu>'"
- unfolding%unimportant extend_measure_def by (auto simp add: assms)
+ unfolding extend_measure_def by (auto simp add: assms)
-lemma%unimportant Pi_cong_sets:
+lemma Pi_cong_sets:
"\<lbrakk>I = J; \<And>x. x \<in> I \<Longrightarrow> M x = N x\<rbrakk> \<Longrightarrow> Pi I M = Pi J N"
unfolding Pi_def by auto
-lemma%important PiM_cong:
+lemma PiM_cong:
assumes "I = J" "\<And>x. x \<in> I \<Longrightarrow> M x = N x"
shows "PiM I M = PiM J N"
unfolding PiM_def
-proof%unimportant (rule extend_measure_cong, goal_cases)
+proof (rule extend_measure_cong, goal_cases)
case 1
show ?case using assms
by (subst assms(1), intro PiE_cong[of J "\<lambda>i. space (M i)" "\<lambda>i. space (N i)"]) simp_all
@@ -206,14 +206,14 @@
qed
-lemma%unimportant prod_algebra_sets_into_space:
+lemma prod_algebra_sets_into_space:
"prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
by (auto simp: prod_emb_def prod_algebra_def)
-lemma%important prod_algebra_eq_finite:
+lemma prod_algebra_eq_finite:
assumes I: "finite I"
shows "prod_algebra I M = {(\<Pi>\<^sub>E i\<in>I. X i) |X. X \<in> (\<Pi> j\<in>I. sets (M j))}" (is "?L = ?R")
-proof%unimportant (intro iffI set_eqI)
+proof (intro iffI set_eqI)
fix A assume "A \<in> ?L"
then obtain J E where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
and A: "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
@@ -232,32 +232,32 @@
by (auto simp: prod_algebra_def)
qed
-lemma%unimportant prod_algebraI:
+lemma prod_algebraI:
"finite J \<Longrightarrow> (J \<noteq> {} \<or> I = {}) \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i))
\<Longrightarrow> prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> prod_algebra I M"
by (auto simp: prod_algebra_def)
-lemma%unimportant prod_algebraI_finite:
+lemma prod_algebraI_finite:
"finite I \<Longrightarrow> (\<forall>i\<in>I. E i \<in> sets (M i)) \<Longrightarrow> (Pi\<^sub>E I E) \<in> prod_algebra I M"
using prod_algebraI[of I I E M] prod_emb_PiE_same_index[of I E M, OF sets.sets_into_space] by simp
-lemma%unimportant Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
+lemma Int_stable_PiE: "Int_stable {Pi\<^sub>E J E | E. \<forall>i\<in>I. E i \<in> sets (M i)}"
proof (safe intro!: Int_stableI)
fix E F assume "\<forall>i\<in>I. E i \<in> sets (M i)" "\<forall>i\<in>I. F i \<in> sets (M i)"
then show "\<exists>G. Pi\<^sub>E J E \<inter> Pi\<^sub>E J F = Pi\<^sub>E J G \<and> (\<forall>i\<in>I. G i \<in> sets (M i))"
by (auto intro!: exI[of _ "\<lambda>i. E i \<inter> F i"] simp: PiE_Int)
qed
-lemma%unimportant prod_algebraE:
+lemma prod_algebraE:
assumes A: "A \<in> prod_algebra I M"
obtains J E where "A = prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j)"
"finite J" "J \<noteq> {} \<or> I = {}" "J \<subseteq> I" "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
using A by (auto simp: prod_algebra_def)
-lemma%important prod_algebraE_all:
+lemma prod_algebraE_all:
assumes A: "A \<in> prod_algebra I M"
obtains E where "A = Pi\<^sub>E I E" "E \<in> (\<Pi> i\<in>I. sets (M i))"
-proof%unimportant -
+proof -
from A obtain E J where A: "A = prod_emb I M J (Pi\<^sub>E J E)"
and J: "J \<subseteq> I" and E: "E \<in> (\<Pi> i\<in>J. sets (M i))"
by (auto simp: prod_algebra_def)
@@ -270,7 +270,7 @@
ultimately show ?thesis using that by auto
qed
-lemma%unimportant Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
+lemma Int_stable_prod_algebra: "Int_stable (prod_algebra I M)"
proof (unfold Int_stable_def, safe)
fix A assume "A \<in> prod_algebra I M"
from prod_algebraE[OF this] guess J E . note A = this
@@ -291,11 +291,11 @@
finally show "A \<inter> B \<in> prod_algebra I M" .
qed
-lemma%unimportant prod_algebra_mono:
+proposition prod_algebra_mono:
assumes space: "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"
assumes sets: "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)"
shows "prod_algebra I E \<subseteq> prod_algebra I F"
-proof%unimportant
+proof
fix A assume "A \<in> prod_algebra I E"
then obtain J G where J: "J \<noteq> {} \<or> I = {}" "finite J" "J \<subseteq> I"
and A: "A = prod_emb I E J (\<Pi>\<^sub>E i\<in>J. G i)"
@@ -315,18 +315,17 @@
apply auto
done
qed
-
-lemma%unimportant prod_algebra_cong:
+proposition prod_algebra_cong:
assumes "I = J" and sets: "(\<And>i. i \<in> I \<Longrightarrow> sets (M i) = sets (N i))"
shows "prod_algebra I M = prod_algebra J N"
-proof%unimportant -
+proof -
have space: "\<And>i. i \<in> I \<Longrightarrow> space (M i) = space (N i)"
using sets_eq_imp_space_eq[OF sets] by auto
with sets show ?thesis unfolding \<open>I = J\<close>
by (intro antisym prod_algebra_mono) auto
qed
-lemma%unimportant space_in_prod_algebra:
+lemma space_in_prod_algebra:
"(\<Pi>\<^sub>E i\<in>I. space (M i)) \<in> prod_algebra I M"
proof cases
assume "I = {}" then show ?thesis
@@ -341,26 +340,26 @@
finally show ?thesis .
qed
-lemma%unimportant space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
+lemma space_PiM: "space (\<Pi>\<^sub>M i\<in>I. M i) = (\<Pi>\<^sub>E i\<in>I. space (M i))"
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro space_extend_measure) simp
-lemma%unimportant prod_emb_subset_PiM[simp]: "prod_emb I M K X \<subseteq> space (PiM I M)"
+lemma prod_emb_subset_PiM[simp]: "prod_emb I M K X \<subseteq> space (PiM I M)"
by (auto simp: prod_emb_def space_PiM)
-lemma%unimportant space_PiM_empty_iff[simp]: "space (PiM I M) = {} \<longleftrightarrow> (\<exists>i\<in>I. space (M i) = {})"
+lemma space_PiM_empty_iff[simp]: "space (PiM I M) = {} \<longleftrightarrow> (\<exists>i\<in>I. space (M i) = {})"
by (auto simp: space_PiM PiE_eq_empty_iff)
-lemma%unimportant undefined_in_PiM_empty[simp]: "(\<lambda>x. undefined) \<in> space (PiM {} M)"
+lemma undefined_in_PiM_empty[simp]: "(\<lambda>x. undefined) \<in> space (PiM {} M)"
by (auto simp: space_PiM)
-lemma%unimportant sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
+lemma sets_PiM: "sets (\<Pi>\<^sub>M i\<in>I. M i) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) (prod_algebra I M)"
using prod_algebra_sets_into_space unfolding PiM_def prod_algebra_def by (intro sets_extend_measure) simp
-lemma%important sets_PiM_single: "sets (PiM I M) =
+proposition sets_PiM_single: "sets (PiM I M) =
sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f\<in>\<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} | i A. i \<in> I \<and> A \<in> sets (M i)}"
(is "_ = sigma_sets ?\<Omega> ?R")
unfolding sets_PiM
-proof%unimportant (rule sigma_sets_eqI)
+proof (rule sigma_sets_eqI)
interpret R: sigma_algebra ?\<Omega> "sigma_sets ?\<Omega> ?R" by (rule sigma_algebra_sigma_sets) auto
fix A assume "A \<in> prod_algebra I M"
from prod_algebraE[OF this] guess J X . note X = this
@@ -388,7 +387,7 @@
finally show "A \<in> sigma_sets ?\<Omega> (prod_algebra I M)" .
qed
-lemma%unimportant sets_PiM_eq_proj:
+lemma sets_PiM_eq_proj:
"I \<noteq> {} \<Longrightarrow> sets (PiM I M) = sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))"
apply (simp add: sets_PiM_single)
apply (subst sets_Sup_eq[where X="\<Pi>\<^sub>E i\<in>I. space (M i)"])
@@ -400,18 +399,18 @@
apply (auto intro!: arg_cong2[where f=sigma_sets])
done
-lemma%unimportant (*FIX ME needs name *)
+lemma (*FIX ME needs name *)
shows space_PiM_empty: "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
and sets_PiM_empty: "sets (Pi\<^sub>M {} M) = { {}, {\<lambda>k. undefined} }"
by (simp_all add: space_PiM sets_PiM_single image_constant sigma_sets_empty_eq)
-lemma%important sets_PiM_sigma:
+proposition sets_PiM_sigma:
assumes \<Omega>_cover: "\<And>i. i \<in> I \<Longrightarrow> \<exists>S\<subseteq>E i. countable S \<and> \<Omega> i = \<Union>S"
assumes E: "\<And>i. i \<in> I \<Longrightarrow> E i \<subseteq> Pow (\<Omega> i)"
assumes J: "\<And>j. j \<in> J \<Longrightarrow> finite j" "\<Union>J = I"
defines "P \<equiv> {{f\<in>(\<Pi>\<^sub>E i\<in>I. \<Omega> i). \<forall>i\<in>j. f i \<in> A i} | A j. j \<in> J \<and> A \<in> Pi j E}"
shows "sets (\<Pi>\<^sub>M i\<in>I. sigma (\<Omega> i) (E i)) = sets (sigma (\<Pi>\<^sub>E i\<in>I. \<Omega> i) P)"
-proof%unimportant cases
+proof cases
assume "I = {}"
with \<open>\<Union>J = I\<close> have "P = {{\<lambda>_. undefined}} \<or> P = {}"
by (auto simp: P_def)
@@ -495,21 +494,21 @@
finally show "?thesis" .
qed
-lemma%unimportant sets_PiM_in_sets:
+lemma sets_PiM_in_sets:
assumes space: "space N = (\<Pi>\<^sub>E i\<in>I. space (M i))"
assumes sets: "\<And>i A. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {x\<in>space N. x i \<in> A} \<in> sets N"
shows "sets (\<Pi>\<^sub>M i \<in> I. M i) \<subseteq> sets N"
unfolding sets_PiM_single space[symmetric]
by (intro sets.sigma_sets_subset subsetI) (auto intro: sets)
-lemma%unimportant sets_PiM_cong[measurable_cong]:
+lemma sets_PiM_cong[measurable_cong]:
assumes "I = J" "\<And>i. i \<in> J \<Longrightarrow> sets (M i) = sets (N i)" shows "sets (PiM I M) = sets (PiM J N)"
using assms sets_eq_imp_space_eq[OF assms(2)] by (simp add: sets_PiM_single cong: PiE_cong conj_cong)
-lemma%important sets_PiM_I:
+lemma sets_PiM_I:
assumes "finite J" "J \<subseteq> I" "\<forall>i\<in>J. E i \<in> sets (M i)"
shows "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)"
-proof%unimportant cases
+proof cases
assume "J = {}"
then have "prod_emb I M J (\<Pi>\<^sub>E j\<in>J. E j) = (\<Pi>\<^sub>E j\<in>I. space (M j))"
by (auto simp: prod_emb_def)
@@ -520,7 +519,7 @@
by (force simp add: sets_PiM prod_algebra_def)
qed
-lemma%unimportant measurable_PiM:
+proposition measurable_PiM:
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
f -` prod_emb I M J (Pi\<^sub>E J X) \<inter> space N \<in> sets N"
@@ -532,13 +531,13 @@
with sets[of J X] show "f -` A \<inter> space N \<in> sets N" by auto
qed
-lemma%important measurable_PiM_Collect:
+lemma measurable_PiM_Collect:
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
assumes sets: "\<And>X J. J \<noteq> {} \<or> I = {} \<Longrightarrow> finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> X i \<in> sets (M i)) \<Longrightarrow>
{\<omega>\<in>space N. \<forall>i\<in>J. f \<omega> i \<in> X i} \<in> sets N"
shows "f \<in> measurable N (PiM I M)"
using sets_PiM prod_algebra_sets_into_space space
-proof%unimportant (rule measurable_sigma_sets)
+proof (rule measurable_sigma_sets)
fix A assume "A \<in> prod_algebra I M"
from prod_algebraE[OF this] guess J X . note X = this
then have "f -` A \<inter> space N = {\<omega> \<in> space N. \<forall>i\<in>J. f \<omega> i \<in> X i}"
@@ -547,7 +546,7 @@
finally show "f -` A \<inter> space N \<in> sets N" .
qed
-lemma%unimportant measurable_PiM_single:
+lemma measurable_PiM_single:
assumes space: "f \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
assumes sets: "\<And>A i. i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> {\<omega> \<in> space N. f \<omega> i \<in> A} \<in> sets N"
shows "f \<in> measurable N (PiM I M)"
@@ -561,11 +560,11 @@
finally show "f -` A \<inter> space N \<in> sets N" .
qed (auto simp: space)
-lemma%important measurable_PiM_single':
+lemma measurable_PiM_single':
assumes f: "\<And>i. i \<in> I \<Longrightarrow> f i \<in> measurable N (M i)"
and "(\<lambda>\<omega> i. f i \<omega>) \<in> space N \<rightarrow> (\<Pi>\<^sub>E i\<in>I. space (M i))"
shows "(\<lambda>\<omega> i. f i \<omega>) \<in> measurable N (Pi\<^sub>M I M)"
-proof%unimportant (rule measurable_PiM_single)
+proof (rule measurable_PiM_single)
fix A i assume A: "i \<in> I" "A \<in> sets (M i)"
then have "{\<omega> \<in> space N. f i \<omega> \<in> A} = f i -` A \<inter> space N"
by auto
@@ -573,12 +572,12 @@
using A f by (auto intro!: measurable_sets)
qed fact
-lemma%unimportant sets_PiM_I_finite[measurable]:
+lemma sets_PiM_I_finite[measurable]:
assumes "finite I" and sets: "(\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i))"
shows "(\<Pi>\<^sub>E j\<in>I. E j) \<in> sets (\<Pi>\<^sub>M i\<in>I. M i)"
using sets_PiM_I[of I I E M] sets.sets_into_space[OF sets] \<open>finite I\<close> sets by auto
-lemma%unimportant measurable_component_singleton[measurable (raw)]:
+lemma measurable_component_singleton[measurable (raw)]:
assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> measurable (Pi\<^sub>M I M) (M i)"
proof (unfold measurable_def, intro CollectI conjI ballI)
fix A assume "A \<in> sets (M i)"
@@ -589,30 +588,30 @@
using \<open>A \<in> sets (M i)\<close> \<open>i \<in> I\<close> by (auto intro!: sets_PiM_I)
qed (insert \<open>i \<in> I\<close>, auto simp: space_PiM)
-lemma%unimportant measurable_component_singleton'[measurable_dest]:
+lemma measurable_component_singleton'[measurable_dest]:
assumes f: "f \<in> measurable N (Pi\<^sub>M I M)"
assumes g: "g \<in> measurable L N"
assumes i: "i \<in> I"
shows "(\<lambda>x. (f (g x)) i) \<in> measurable L (M i)"
using measurable_compose[OF measurable_compose[OF g f] measurable_component_singleton, OF i] .
-lemma%unimportant measurable_PiM_component_rev:
+lemma measurable_PiM_component_rev:
"i \<in> I \<Longrightarrow> f \<in> measurable (M i) N \<Longrightarrow> (\<lambda>x. f (x i)) \<in> measurable (PiM I M) N"
by simp
-lemma%unimportant measurable_case_nat[measurable (raw)]:
+lemma measurable_case_nat[measurable (raw)]:
assumes [measurable (raw)]: "i = 0 \<Longrightarrow> f \<in> measurable M N"
"\<And>j. i = Suc j \<Longrightarrow> (\<lambda>x. g x j) \<in> measurable M N"
shows "(\<lambda>x. case_nat (f x) (g x) i) \<in> measurable M N"
by (cases i) simp_all
-lemma%unimportant measurable_case_nat'[measurable (raw)]:
+lemma measurable_case_nat'[measurable (raw)]:
assumes fg[measurable]: "f \<in> measurable N M" "g \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
shows "(\<lambda>x. case_nat (f x) (g x)) \<in> measurable N (\<Pi>\<^sub>M i\<in>UNIV. M)"
using fg[THEN measurable_space]
by (auto intro!: measurable_PiM_single' simp add: space_PiM PiE_iff split: nat.split)
-lemma%unimportant measurable_add_dim[measurable]:
+lemma measurable_add_dim[measurable]:
"(\<lambda>(f, y). f(i := y)) \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M M i) (Pi\<^sub>M (insert i I) M)"
(is "?f \<in> measurable ?P ?I")
proof (rule measurable_PiM_single)
@@ -626,12 +625,12 @@
finally show "{\<omega> \<in> space ?P. case_prod (\<lambda>f. fun_upd f i) \<omega> j \<in> A} \<in> sets ?P" .
qed (auto simp: space_pair_measure space_PiM PiE_def)
-lemma%important measurable_fun_upd:
+proposition measurable_fun_upd:
assumes I: "I = J \<union> {i}"
assumes f[measurable]: "f \<in> measurable N (PiM J M)"
assumes h[measurable]: "h \<in> measurable N (M i)"
shows "(\<lambda>x. (f x) (i := h x)) \<in> measurable N (PiM I M)"
-proof%unimportant (intro measurable_PiM_single')
+proof (intro measurable_PiM_single')
fix j assume "j \<in> I" then show "(\<lambda>\<omega>. ((f \<omega>)(i := h \<omega>)) j) \<in> measurable N (M j)"
unfolding I by (cases "j = i") auto
next
@@ -640,14 +639,14 @@
by (auto simp: space_PiM PiE_iff extensional_def)
qed
-lemma%unimportant measurable_component_update:
+lemma measurable_component_update:
"x \<in> space (Pi\<^sub>M I M) \<Longrightarrow> i \<notin> I \<Longrightarrow> (\<lambda>v. x(i := v)) \<in> measurable (M i) (Pi\<^sub>M (insert i I) M)"
by simp
-lemma%important measurable_merge[measurable]:
+lemma measurable_merge[measurable]:
"merge I J \<in> measurable (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M)"
(is "?f \<in> measurable ?P ?U")
-proof%unimportant (rule measurable_PiM_single)
+proof (rule measurable_PiM_single)
fix i A assume A: "A \<in> sets (M i)" "i \<in> I \<union> J"
then have "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} =
(if i \<in> I then ((\<lambda>x. x i) \<circ> fst) -` A \<inter> space ?P else ((\<lambda>x. x i) \<circ> snd) -` A \<inter> space ?P)"
@@ -658,7 +657,7 @@
finally show "{\<omega> \<in> space ?P. merge I J \<omega> i \<in> A} \<in> sets ?P" .
qed (auto simp: space_pair_measure space_PiM PiE_iff merge_def extensional_def)
-lemma%unimportant measurable_restrict[measurable (raw)]:
+lemma measurable_restrict[measurable (raw)]:
assumes X: "\<And>i. i \<in> I \<Longrightarrow> X i \<in> measurable N (M i)"
shows "(\<lambda>x. \<lambda>i\<in>I. X i x) \<in> measurable N (Pi\<^sub>M I M)"
proof (rule measurable_PiM_single)
@@ -669,14 +668,14 @@
using A X by (auto intro!: measurable_sets)
qed (insert X, auto simp add: PiE_def dest: measurable_space)
-lemma%unimportant measurable_abs_UNIV:
+lemma measurable_abs_UNIV:
"(\<And>n. (\<lambda>\<omega>. f n \<omega>) \<in> measurable M (N n)) \<Longrightarrow> (\<lambda>\<omega> n. f n \<omega>) \<in> measurable M (PiM UNIV N)"
by (intro measurable_PiM_single) (auto dest: measurable_space)
-lemma%unimportant measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
+lemma measurable_restrict_subset: "J \<subseteq> L \<Longrightarrow> (\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J M)"
by (intro measurable_restrict measurable_component_singleton) auto
-lemma%unimportant measurable_restrict_subset':
+lemma measurable_restrict_subset':
assumes "J \<subseteq> L" "\<And>x. x \<in> J \<Longrightarrow> sets (M x) = sets (N x)"
shows "(\<lambda>f. restrict f J) \<in> measurable (Pi\<^sub>M L M) (Pi\<^sub>M J N)"
proof-
@@ -687,12 +686,12 @@
finally show ?thesis .
qed
-lemma%unimportant measurable_prod_emb[intro, simp]:
+lemma measurable_prod_emb[intro, simp]:
"J \<subseteq> L \<Longrightarrow> X \<in> sets (Pi\<^sub>M J M) \<Longrightarrow> prod_emb L M J X \<in> sets (Pi\<^sub>M L M)"
unfolding prod_emb_def space_PiM[symmetric]
by (auto intro!: measurable_sets measurable_restrict measurable_component_singleton)
-lemma%unimportant merge_in_prod_emb:
+lemma merge_in_prod_emb:
assumes "y \<in> space (PiM I M)" "x \<in> X" and X: "X \<in> sets (Pi\<^sub>M J M)" and "J \<subseteq> I"
shows "merge J I (x, y) \<in> prod_emb I M J X"
using assms sets.sets_into_space[OF X]
@@ -700,7 +699,7 @@
cong: if_cong restrict_cong)
(simp add: extensional_def)
-lemma%unimportant prod_emb_eq_emptyD:
+lemma prod_emb_eq_emptyD:
assumes J: "J \<subseteq> I" and ne: "space (PiM I M) \<noteq> {}" and X: "X \<in> sets (Pi\<^sub>M J M)"
and *: "prod_emb I M J X = {}"
shows "X = {}"
@@ -711,19 +710,19 @@
from merge_in_prod_emb[OF this \<open>x\<in>X\<close> X J] * show "x \<in> {}" by auto
qed
-lemma%unimportant sets_in_Pi_aux:
+lemma sets_in_Pi_aux:
"finite I \<Longrightarrow> (\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
{x\<in>space (PiM I M). x \<in> Pi I F} \<in> sets (PiM I M)"
by (simp add: subset_eq Pi_iff)
-lemma%unimportant sets_in_Pi[measurable (raw)]:
+lemma sets_in_Pi[measurable (raw)]:
"finite I \<Longrightarrow> f \<in> measurable N (PiM I M) \<Longrightarrow>
(\<And>j. j \<in> I \<Longrightarrow> {x\<in>space (M j). x \<in> F j} \<in> sets (M j)) \<Longrightarrow>
Measurable.pred N (\<lambda>x. f x \<in> Pi I F)"
unfolding pred_def
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_Pi_aux]) auto
-lemma%unimportant sets_in_extensional_aux:
+lemma sets_in_extensional_aux:
"{x\<in>space (PiM I M). x \<in> extensional I} \<in> sets (PiM I M)"
proof -
have "{x\<in>space (PiM I M). x \<in> extensional I} = space (PiM I M)"
@@ -731,12 +730,12 @@
then show ?thesis by simp
qed
-lemma%unimportant sets_in_extensional[measurable (raw)]:
+lemma sets_in_extensional[measurable (raw)]:
"f \<in> measurable N (PiM I M) \<Longrightarrow> Measurable.pred N (\<lambda>x. f x \<in> extensional I)"
unfolding pred_def
by (rule measurable_sets_Collect[of f N "PiM I M", OF _ sets_in_extensional_aux]) auto
-lemma%unimportant sets_PiM_I_countable:
+lemma sets_PiM_I_countable:
assumes I: "countable I" and E: "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
proof cases
assume "I \<noteq> {}"
@@ -747,11 +746,11 @@
finally show ?thesis .
qed (simp add: sets_PiM_empty)
-lemma%important sets_PiM_D_countable:
+lemma sets_PiM_D_countable:
assumes A: "A \<in> PiM I M"
shows "\<exists>J\<subseteq>I. \<exists>X\<in>PiM J M. countable J \<and> A = prod_emb I M J X"
using A[unfolded sets_PiM_single]
-proof%unimportant induction
+proof induction
case (Basic A)
then obtain i X where *: "i \<in> I" "X \<in> sets (M i)" and "A = {f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> X}"
by auto
@@ -783,12 +782,12 @@
qed(auto intro: X)
qed
-lemma%important measure_eqI_PiM_finite:
+proposition measure_eqI_PiM_finite:
assumes [simp]: "finite I" "sets P = PiM I M" "sets Q = PiM I M"
assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = Q (Pi\<^sub>E I A)"
assumes A: "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = space (PiM I M)" "\<And>i::nat. P (A i) \<noteq> \<infinity>"
shows "P = Q"
-proof%unimportant (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
+proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
show "range A \<subseteq> prod_algebra I M" "(\<Union>i. A i) = (\<Pi>\<^sub>E i\<in>I. space (M i))" "\<And>i. P (A i) \<noteq> \<infinity>"
unfolding space_PiM[symmetric] by fact+
fix X assume "X \<in> prod_algebra I M"
@@ -799,13 +798,13 @@
unfolding X by (subst (1 2) prod_emb_Pi) (auto simp: eq)
qed (simp_all add: sets_PiM)
-lemma%important measure_eqI_PiM_infinite:
+proposition measure_eqI_PiM_infinite:
assumes [simp]: "sets P = PiM I M" "sets Q = PiM I M"
assumes eq: "\<And>A J. finite J \<Longrightarrow> J \<subseteq> I \<Longrightarrow> (\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow>
P (prod_emb I M J (Pi\<^sub>E J A)) = Q (prod_emb I M J (Pi\<^sub>E J A))"
assumes A: "finite_measure P"
shows "P = Q"
-proof%unimportant (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
+proof (rule measure_eqI_generator_eq[OF Int_stable_prod_algebra prod_algebra_sets_into_space])
interpret finite_measure P by fact
define i where "i = (SOME i. i \<in> I)"
have i: "I \<noteq> {} \<Longrightarrow> i \<in> I"
@@ -839,12 +838,12 @@
fixes I :: "'i set"
assumes finite_index: "finite I"
-lemma%important (in finite_product_sigma_finite) sigma_finite_pairs:
+proposition (in finite_product_sigma_finite) sigma_finite_pairs:
"\<exists>F::'i \<Rightarrow> nat \<Rightarrow> 'a set.
(\<forall>i\<in>I. range (F i) \<subseteq> sets (M i)) \<and>
(\<forall>k. \<forall>i\<in>I. emeasure (M i) (F i k) \<noteq> \<infinity>) \<and> incseq (\<lambda>k. \<Pi>\<^sub>E i\<in>I. F i k) \<and>
(\<Union>k. \<Pi>\<^sub>E i\<in>I. F i k) = space (PiM I M)"
-proof%unimportant -
+proof -
have "\<forall>i::'i. \<exists>F::nat \<Rightarrow> 'a set. range F \<subseteq> sets (M i) \<and> incseq F \<and> (\<Union>i. F i) = space (M i) \<and> (\<forall>k. emeasure (M i) (F k) \<noteq> \<infinity>)"
using M.sigma_finite_incseq by metis
from choice[OF this] guess F :: "'i \<Rightarrow> nat \<Rightarrow> 'a set" ..
@@ -870,7 +869,7 @@
qed
qed
-lemma%unimportant emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
+lemma emeasure_PiM_empty[simp]: "emeasure (PiM {} M) {\<lambda>_. undefined} = 1"
proof -
let ?\<mu> = "\<lambda>A. if A = {} then 0 else (1::ennreal)"
have "emeasure (Pi\<^sub>M {} M) (prod_emb {} M {} (\<Pi>\<^sub>E i\<in>{}. {})) = 1"
@@ -887,12 +886,12 @@
by simp
qed
-lemma%unimportant PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
+lemma PiM_empty: "PiM {} M = count_space {\<lambda>_. undefined}"
by (rule measure_eqI) (auto simp add: sets_PiM_empty)
-lemma%important (in product_sigma_finite) emeasure_PiM:
+lemma (in product_sigma_finite) emeasure_PiM:
"finite I \<Longrightarrow> (\<And>i. i\<in>I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (PiM I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
-proof%unimportant (induct I arbitrary: A rule: finite_induct)
+proof (induct I arbitrary: A rule: finite_induct)
case (insert i I)
interpret finite_product_sigma_finite M I by standard fact
have "finite (insert i I)" using \<open>finite I\<close> by auto
@@ -947,7 +946,7 @@
by (subst (asm) prod_emb_PiE_same_index) (auto intro!: sets.sets_into_space)
qed simp
-lemma%unimportant (in product_sigma_finite) PiM_eqI:
+lemma (in product_sigma_finite) PiM_eqI:
assumes I[simp]: "finite I" and P: "sets P = PiM I M"
assumes eq: "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
shows "P = PiM I M"
@@ -965,7 +964,7 @@
qed
qed
-lemma%unimportant (in product_sigma_finite) sigma_finite:
+lemma (in product_sigma_finite) sigma_finite:
assumes "finite I"
shows "sigma_finite_measure (PiM I M)"
proof
@@ -986,11 +985,11 @@
sublocale finite_product_sigma_finite \<subseteq> sigma_finite_measure "Pi\<^sub>M I M"
using sigma_finite[OF finite_index] .
-lemma%unimportant (in finite_product_sigma_finite) measure_times:
+lemma (in finite_product_sigma_finite) measure_times:
"(\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
using emeasure_PiM[OF finite_index] by auto
-lemma%unimportant (in product_sigma_finite) nn_integral_empty:
+lemma (in product_sigma_finite) nn_integral_empty:
"0 \<le> f (\<lambda>k. undefined) \<Longrightarrow> integral\<^sup>N (Pi\<^sub>M {} M) f = f (\<lambda>k. undefined)"
by (simp add: PiM_empty nn_integral_count_space_finite max.absorb2)
@@ -998,7 +997,7 @@
assumes IJ[simp]: "I \<inter> J = {}" and fin: "finite I" "finite J"
shows "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J) = Pi\<^sub>M (I \<union> J) M"
(is "?D = ?P")
-proof%unimportant (rule PiM_eqI)
+proof (rule PiM_eqI)
interpret I: finite_product_sigma_finite M I by standard fact
interpret J: finite_product_sigma_finite M J by standard fact
fix A assume A: "\<And>i. i \<in> I \<union> J \<Longrightarrow> A i \<in> sets (M i)"
@@ -1010,12 +1009,12 @@
(auto simp: * J.emeasure_pair_measure_Times I.measure_times J.measure_times prod.union_disjoint)
qed (insert fin, simp_all)
-lemma%important (in product_sigma_finite) product_nn_integral_fold:
+proposition (in product_sigma_finite) product_nn_integral_fold:
assumes IJ: "I \<inter> J = {}" "finite I" "finite J"
and f[measurable]: "f \<in> borel_measurable (Pi\<^sub>M (I \<union> J) M)"
shows "integral\<^sup>N (Pi\<^sub>M (I \<union> J) M) f =
(\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (merge I J (x, y)) \<partial>(Pi\<^sub>M J M)) \<partial>(Pi\<^sub>M I M))"
-proof%unimportant -
+proof -
interpret I: finite_product_sigma_finite M I by standard fact
interpret J: finite_product_sigma_finite M J by standard fact
interpret P: pair_sigma_finite "Pi\<^sub>M I M" "Pi\<^sub>M J M" by standard
@@ -1030,7 +1029,7 @@
done
qed
-lemma%unimportant (in product_sigma_finite) distr_singleton:
+lemma (in product_sigma_finite) distr_singleton:
"distr (Pi\<^sub>M {i} M) (M i) (\<lambda>x. x i) = M i" (is "?D = _")
proof (intro measure_eqI[symmetric])
interpret I: finite_product_sigma_finite M "{i}" by standard simp
@@ -1042,7 +1041,7 @@
by (simp add: emeasure_distr measurable_component_singleton)
qed simp
-lemma%unimportant (in product_sigma_finite) product_nn_integral_singleton:
+lemma (in product_sigma_finite) product_nn_integral_singleton:
assumes f: "f \<in> borel_measurable (M i)"
shows "integral\<^sup>N (Pi\<^sub>M {i} M) (\<lambda>x. f (x i)) = integral\<^sup>N (M i) f"
proof -
@@ -1054,11 +1053,11 @@
done
qed
-lemma%important (in product_sigma_finite) product_nn_integral_insert:
+proposition (in product_sigma_finite) product_nn_integral_insert:
assumes I[simp]: "finite I" "i \<notin> I"
and f: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ x. (\<integral>\<^sup>+ y. f (x(i := y)) \<partial>(M i)) \<partial>(Pi\<^sub>M I M))"
-proof%unimportant -
+proof -
interpret I: finite_product_sigma_finite M I by standard auto
interpret i: finite_product_sigma_finite M "{i}" by standard auto
have IJ: "I \<inter> {i} = {}" and insert: "I \<union> {i} = insert i I"
@@ -1078,7 +1077,7 @@
qed
qed
-lemma%unimportant (in product_sigma_finite) product_nn_integral_insert_rev:
+lemma (in product_sigma_finite) product_nn_integral_insert_rev:
assumes I[simp]: "finite I" "i \<notin> I"
and [measurable]: "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
shows "integral\<^sup>N (Pi\<^sub>M (insert i I) M) f = (\<integral>\<^sup>+ y. (\<integral>\<^sup>+ x. f (x(i := y)) \<partial>(Pi\<^sub>M I M)) \<partial>(M i))"
@@ -1089,7 +1088,7 @@
apply measurable
done
-lemma%unimportant (in product_sigma_finite) product_nn_integral_prod:
+lemma (in product_sigma_finite) product_nn_integral_prod:
assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)"
shows "(\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M) = (\<Prod>i\<in>I. integral\<^sup>N (M i) (f i))"
using assms proof (induction I)
@@ -1112,11 +1111,11 @@
done
qed (simp add: space_PiM)
-lemma%important (in product_sigma_finite) product_nn_integral_pair:
+proposition (in product_sigma_finite) product_nn_integral_pair:
assumes [measurable]: "case_prod f \<in> borel_measurable (M x \<Otimes>\<^sub>M M y)"
assumes xy: "x \<noteq> y"
shows "(\<integral>\<^sup>+\<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>PiM {x, y} M) = (\<integral>\<^sup>+z. f (fst z) (snd z) \<partial>(M x \<Otimes>\<^sub>M M y))"
-proof%unimportant -
+proof -
interpret psm: pair_sigma_finite "M x" "M y"
unfolding pair_sigma_finite_def using sigma_finite_measures by simp_all
have "{x, y} = {y, x}" by auto
@@ -1129,7 +1128,7 @@
finally show ?thesis .
qed
-lemma%unimportant (in product_sigma_finite) distr_component:
+lemma (in product_sigma_finite) distr_component:
"distr (M i) (Pi\<^sub>M {i} M) (\<lambda>x. \<lambda>i\<in>{i}. x) = Pi\<^sub>M {i} M" (is "?D = ?P")
proof (intro PiM_eqI)
fix A assume A: "\<And>ia. ia \<in> {i} \<Longrightarrow> A ia \<in> sets (M ia)"
@@ -1139,7 +1138,7 @@
by (subst emeasure_distr) (auto intro!: sets_PiM_I_finite measurable_restrict)
qed simp_all
-lemma%unimportant (in product_sigma_finite)
+lemma (in product_sigma_finite)
assumes IJ: "I \<inter> J = {}" "finite I" "finite J" and A: "A \<in> sets (Pi\<^sub>M (I \<union> J) M)"
shows emeasure_fold_integral:
"emeasure (Pi\<^sub>M (I \<union> J) M) A = (\<integral>\<^sup>+x. emeasure (Pi\<^sub>M J M) ((\<lambda>y. merge I J (x, y)) -` A \<inter> space (Pi\<^sub>M J M)) \<partial>Pi\<^sub>M I M)" (is ?I)
@@ -1164,11 +1163,11 @@
by (simp add: vimage_comp comp_def space_pair_measure cong: measurable_cong)
qed
-lemma%unimportant sets_Collect_single:
+lemma sets_Collect_single:
"i \<in> I \<Longrightarrow> A \<in> sets (M i) \<Longrightarrow> { x \<in> space (Pi\<^sub>M I M). x i \<in> A } \<in> sets (Pi\<^sub>M I M)"
by simp
-lemma%unimportant pair_measure_eq_distr_PiM:
+lemma pair_measure_eq_distr_PiM:
fixes M1 :: "'a measure" and M2 :: "'a measure"
assumes "sigma_finite_measure M1" "sigma_finite_measure M2"
shows "(M1 \<Otimes>\<^sub>M M2) = distr (Pi\<^sub>M UNIV (case_bool M1 M2)) (M1 \<Otimes>\<^sub>M M2) (\<lambda>x. (x True, x False))"
@@ -1195,7 +1194,7 @@
by (subst emeasure_distr) (auto simp: measurable_pair_iff)
qed simp
-lemma%unimportant infprod_in_sets[intro]:
+lemma infprod_in_sets[intro]:
fixes E :: "nat \<Rightarrow> 'a set" assumes E: "\<And>i. E i \<in> sets (M i)"
shows "Pi UNIV E \<in> sets (\<Pi>\<^sub>M i\<in>UNIV::nat set. M i)"
proof -
--- a/src/HOL/Analysis/Function_Topology.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Function_Topology.thy Wed Jan 16 21:27:33 2019 +0000
@@ -61,6 +61,8 @@
I realized afterwards that this theory has a lot in common with \<^file>\<open>~~/src/HOL/Library/Finite_Map.thy\<close>.
\<close>
+
+
subsection%important \<open>Topology without type classes\<close>
subsubsection%important \<open>The topology generated by some (open) subsets\<close>
@@ -78,14 +80,14 @@
| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology_on S k) \<Longrightarrow> generate_topology_on S (\<Union>K)"
| Basis: "s \<in> S \<Longrightarrow> generate_topology_on S s"
-lemma%unimportant istopology_generate_topology_on:
+lemma istopology_generate_topology_on:
"istopology (generate_topology_on S)"
unfolding istopology_def by (auto intro: generate_topology_on.intros)
text \<open>The basic property of the topology generated by a set \<open>S\<close> is that it is the
smallest topology containing all the elements of \<open>S\<close>:\<close>
-lemma%unimportant generate_topology_on_coarsest:
+lemma generate_topology_on_coarsest:
assumes "istopology T"
"\<And>s. s \<in> S \<Longrightarrow> T s"
"generate_topology_on S s0"
@@ -96,17 +98,17 @@
abbreviation%unimportant topology_generated_by::"('a set set) \<Rightarrow> ('a topology)"
where "topology_generated_by S \<equiv> topology (generate_topology_on S)"
-lemma%unimportant openin_topology_generated_by_iff:
+lemma openin_topology_generated_by_iff:
"openin (topology_generated_by S) s \<longleftrightarrow> generate_topology_on S s"
using topology_inverse'[OF istopology_generate_topology_on[of S]] by simp
-lemma%unimportant openin_topology_generated_by:
+lemma openin_topology_generated_by:
"openin (topology_generated_by S) s \<Longrightarrow> generate_topology_on S s"
using openin_topology_generated_by_iff by auto
-lemma%important topology_generated_by_topspace:
+lemma topology_generated_by_topspace:
"topspace (topology_generated_by S) = (\<Union>S)"
-proof%unimportant
+proof
{
fix s assume "openin (topology_generated_by S) s"
then have "generate_topology_on S s" by (rule openin_topology_generated_by)
@@ -121,9 +123,9 @@
unfolding topspace_def using openin_topology_generated_by_iff by auto
qed
-lemma%important topology_generated_by_Basis:
+lemma topology_generated_by_Basis:
"s \<in> S \<Longrightarrow> openin (topology_generated_by S) s"
-by%unimportant (simp only: openin_topology_generated_by_iff, auto simp: generate_topology_on.Basis)
+ by (simp only: openin_topology_generated_by_iff, auto simp: generate_topology_on.Basis)
lemma generate_topology_on_Inter:
"\<lbrakk>finite \<F>; \<And>K. K \<in> \<F> \<Longrightarrow> generate_topology_on \<S> K; \<F> \<noteq> {}\<rbrakk> \<Longrightarrow> generate_topology_on \<S> (\<Inter>\<F>)"
@@ -266,28 +268,28 @@
where "continuous_on_topo T1 T2 f = ((\<forall> U. openin T2 U \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1)))
\<and> (f`(topspace T1) \<subseteq> (topspace T2)))"
-lemma%important continuous_on_continuous_on_topo:
+lemma continuous_on_continuous_on_topo:
"continuous_on s f \<longleftrightarrow> continuous_on_topo (subtopology euclidean s) euclidean f"
-unfolding%unimportant continuous_on_open_invariant openin_open vimage_def continuous_on_topo_def
+ unfolding continuous_on_open_invariant openin_open vimage_def continuous_on_topo_def
topspace_euclidean_subtopology open_openin topspace_euclidean by fast
-lemma%unimportant continuous_on_topo_UNIV:
+lemma continuous_on_topo_UNIV:
"continuous_on UNIV f \<longleftrightarrow> continuous_on_topo euclidean euclidean f"
using continuous_on_continuous_on_topo[of UNIV f] subtopology_UNIV[of euclidean] by auto
-lemma%important continuous_on_topo_open [intro]:
+lemma continuous_on_topo_open [intro]:
"continuous_on_topo T1 T2 f \<Longrightarrow> openin T2 U \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
-unfolding%unimportant continuous_on_topo_def by auto
+ unfolding continuous_on_topo_def by auto
-lemma%unimportant continuous_on_topo_topspace [intro]:
+lemma continuous_on_topo_topspace [intro]:
"continuous_on_topo T1 T2 f \<Longrightarrow> f`(topspace T1) \<subseteq> (topspace T2)"
unfolding continuous_on_topo_def by auto
-lemma%important continuous_on_generated_topo_iff:
+lemma continuous_on_generated_topo_iff:
"continuous_on_topo T1 (topology_generated_by S) f \<longleftrightarrow>
((\<forall>U. U \<in> S \<longrightarrow> openin T1 (f-`U \<inter> topspace(T1))) \<and> (f`(topspace T1) \<subseteq> (\<Union> S)))"
unfolding continuous_on_topo_def topology_generated_by_topspace
-proof%unimportant (auto simp add: topology_generated_by_Basis)
+proof (auto simp add: topology_generated_by_Basis)
assume H: "\<forall>U. U \<in> S \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)"
fix U assume "openin (topology_generated_by S) U"
then have "generate_topology_on S U" by (rule openin_topology_generated_by)
@@ -309,17 +311,17 @@
qed (auto simp add: H)
qed
-lemma%important continuous_on_generated_topo:
+lemma continuous_on_generated_topo:
assumes "\<And>U. U \<in>S \<Longrightarrow> openin T1 (f-`U \<inter> topspace(T1))"
"f`(topspace T1) \<subseteq> (\<Union> S)"
shows "continuous_on_topo T1 (topology_generated_by S) f"
-using%unimportant assms continuous_on_generated_topo_iff by blast
+ using assms continuous_on_generated_topo_iff by blast
-lemma%important continuous_on_topo_compose:
+proposition continuous_on_topo_compose:
assumes "continuous_on_topo T1 T2 f" "continuous_on_topo T2 T3 g"
shows "continuous_on_topo T1 T3 (g o f)"
-using%unimportant assms unfolding continuous_on_topo_def
-proof%unimportant (auto)
+ using assms unfolding continuous_on_topo_def
+proof (auto)
fix U :: "'c set"
assume H: "openin T3 U"
have "openin T1 (f -` (g -` U \<inter> topspace T2) \<inter> topspace T1)"
@@ -330,7 +332,7 @@
by simp
qed (blast)
-lemma%unimportant continuous_on_topo_preimage_topspace [intro]:
+lemma continuous_on_topo_preimage_topspace [intro]:
assumes "continuous_on_topo T1 T2 f"
shows "f-`(topspace T2) \<inter> topspace T1 = topspace T1"
using assms unfolding continuous_on_topo_def by auto
@@ -349,9 +351,9 @@
definition%important pullback_topology::"('a set) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b topology) \<Rightarrow> ('a topology)"
where "pullback_topology A f T = topology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
-lemma%important istopology_pullback_topology:
+lemma istopology_pullback_topology:
"istopology (\<lambda>S. \<exists>U. openin T U \<and> S = f-`U \<inter> A)"
-unfolding%unimportant istopology_def proof (auto)
+ unfolding istopology_def proof (auto)
fix K assume "\<forall>S\<in>K. \<exists>U. openin T U \<and> S = f -` U \<inter> A"
then have "\<exists>U. \<forall>S\<in>K. openin T (U S) \<and> S = f-`(U S) \<inter> A"
by (rule bchoice)
@@ -362,19 +364,19 @@
then show "\<exists>V. openin T V \<and> \<Union>K = f -` V \<inter> A" by auto
qed
-lemma%unimportant openin_pullback_topology:
+lemma openin_pullback_topology:
"openin (pullback_topology A f T) S \<longleftrightarrow> (\<exists>U. openin T U \<and> S = f-`U \<inter> A)"
unfolding pullback_topology_def topology_inverse'[OF istopology_pullback_topology] by auto
-lemma%unimportant topspace_pullback_topology:
+lemma topspace_pullback_topology:
"topspace (pullback_topology A f T) = f-`(topspace T) \<inter> A"
by (auto simp add: topspace_def openin_pullback_topology)
-lemma%important continuous_on_topo_pullback [intro]:
+proposition continuous_on_topo_pullback [intro]:
assumes "continuous_on_topo T1 T2 g"
shows "continuous_on_topo (pullback_topology A f T1) T2 (g o f)"
unfolding continuous_on_topo_def
-proof%unimportant (auto)
+proof (auto)
fix U::"'b set" assume "openin T2 U"
then have "openin T1 (g-`U \<inter> topspace T1)"
using assms unfolding continuous_on_topo_def by auto
@@ -394,11 +396,11 @@
using assms unfolding continuous_on_topo_def by auto
qed
-lemma%important continuous_on_topo_pullback' [intro]:
+proposition continuous_on_topo_pullback' [intro]:
assumes "continuous_on_topo T1 T2 (f o g)" "topspace T1 \<subseteq> g-`A"
shows "continuous_on_topo T1 (pullback_topology A f T2) g"
unfolding continuous_on_topo_def
-proof%unimportant (auto)
+proof (auto)
fix U assume "openin (pullback_topology A f T2) U"
then have "\<exists>V. openin T2 V \<and> U = f-`V \<inter> A"
unfolding openin_pullback_topology by auto
@@ -547,9 +549,9 @@
done
qed
-lemma%important topspace_product_topology:
+lemma topspace_product_topology:
"topspace (product_topology T I) = (\<Pi>\<^sub>E i\<in>I. topspace(T i))"
-proof%unimportant
+proof
show "topspace (product_topology T I) \<subseteq> (\<Pi>\<^sub>E i\<in>I. topspace (T i))"
unfolding product_topology_def topology_generated_by_topspace
unfolding topspace_def by auto
@@ -559,16 +561,16 @@
unfolding product_topology_def using PiE_def by (auto simp add: topology_generated_by_topspace)
qed
-lemma%unimportant product_topology_basis:
+lemma product_topology_basis:
assumes "\<And>i. openin (T i) (X i)" "finite {i. X i \<noteq> topspace (T i)}"
shows "openin (product_topology T I) (\<Pi>\<^sub>E i\<in>I. X i)"
unfolding product_topology_def
by (rule topology_generated_by_Basis) (use assms in auto)
-lemma%important product_topology_open_contains_basis:
+proposition product_topology_open_contains_basis:
assumes "openin (product_topology T I) U" "x \<in> U"
shows "\<exists>X. x \<in> (\<Pi>\<^sub>E i\<in>I. X i) \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> (\<Pi>\<^sub>E i\<in>I. X i) \<subseteq> U"
-proof%unimportant -
+proof -
have "generate_topology_on {(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}} U"
using assms unfolding product_topology_def by (intro openin_topology_generated_by) auto
then have "\<And>x. x\<in>U \<Longrightarrow> \<exists>X. x \<in> (\<Pi>\<^sub>E i\<in>I. X i) \<and> (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)} \<and> (\<Pi>\<^sub>E i\<in>I. X i) \<subseteq> U"
@@ -717,7 +719,7 @@
text \<open>The basic property of the product topology is the continuity of projections:\<close>
-lemma%unimportant continuous_on_topo_product_coordinates [simp]:
+lemma continuous_on_topo_product_coordinates [simp]:
assumes "i \<in> I"
shows "continuous_on_topo (product_topology T I) (T i) (\<lambda>x. x i)"
proof -
@@ -739,12 +741,12 @@
by (auto simp add: assms topspace_product_topology PiE_iff)
qed
-lemma%important continuous_on_topo_coordinatewise_then_product [intro]:
+lemma continuous_on_topo_coordinatewise_then_product [intro]:
assumes "\<And>i. i \<in> I \<Longrightarrow> continuous_on_topo T1 (T i) (\<lambda>x. f x i)"
"\<And>i x. i \<notin> I \<Longrightarrow> x \<in> topspace T1 \<Longrightarrow> f x i = undefined"
shows "continuous_on_topo T1 (product_topology T I) f"
unfolding product_topology_def
-proof%unimportant (rule continuous_on_generated_topo)
+proof (rule continuous_on_generated_topo)
fix U assume "U \<in> {Pi\<^sub>E I X |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
then obtain X where H: "U = Pi\<^sub>E I X" "\<And>i. openin (T i) (X i)" "finite {i. X i \<noteq> topspace (T i)}"
by blast
@@ -772,7 +774,7 @@
using assms unfolding continuous_on_topo_def by auto
qed
-lemma%unimportant continuous_on_topo_product_then_coordinatewise [intro]:
+lemma continuous_on_topo_product_then_coordinatewise [intro]:
assumes "continuous_on_topo T1 (product_topology T I) f"
shows "\<And>i. i \<in> I \<Longrightarrow> continuous_on_topo T1 (T i) (\<lambda>x. f x i)"
"\<And>i x. i \<notin> I \<Longrightarrow> x \<in> topspace T1 \<Longrightarrow> f x i = undefined"
@@ -794,7 +796,7 @@
using \<open>i \<notin> I\<close> by (auto simp add: PiE_iff extensional_def)
qed
-lemma%unimportant continuous_on_restrict:
+lemma continuous_on_restrict:
assumes "J \<subseteq> I"
shows "continuous_on_topo (product_topology T I) (product_topology T J) (\<lambda>x. restrict x J)"
proof (rule continuous_on_topo_coordinatewise_then_product)
@@ -817,7 +819,7 @@
definition%important open_fun_def:
"open U = openin (product_topology (\<lambda>i. euclidean) UNIV) U"
-instance proof%unimportant
+instance proof
have "topspace (product_topology (\<lambda>(i::'a). euclidean::('b topology)) UNIV) = UNIV"
unfolding topspace_product_topology topspace_euclidean by auto
then show "open (UNIV::('a \<Rightarrow> 'b) set)"
@@ -826,15 +828,15 @@
end
-lemma%unimportant euclidean_product_topology:
+lemma euclidean_product_topology:
"euclidean = product_topology (\<lambda>i. euclidean::('b::topological_space) topology) UNIV"
by (metis open_openin topology_eq open_fun_def)
-lemma%important product_topology_basis':
+proposition product_topology_basis':
fixes x::"'i \<Rightarrow> 'a" and U::"'i \<Rightarrow> ('b::topological_space) set"
assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> open (U i)"
shows "open {f. \<forall>i\<in>I. f (x i) \<in> U i}"
-proof%unimportant -
+proof -
define J where "J = x`I"
define V where "V = (\<lambda>y. if y \<in> J then \<Inter>{U i|i. i\<in>I \<and> x i = y} else UNIV)"
define X where "X = (\<lambda>y. if y \<in> J then V y else UNIV)"
@@ -864,24 +866,24 @@
text \<open>The results proved in the general situation of products of possibly different
spaces have their counterparts in this simpler setting.\<close>
-lemma%unimportant continuous_on_product_coordinates [simp]:
+lemma continuous_on_product_coordinates [simp]:
"continuous_on UNIV (\<lambda>x. x i::('b::topological_space))"
unfolding continuous_on_topo_UNIV euclidean_product_topology
by (rule continuous_on_topo_product_coordinates, simp)
-lemma%unimportant continuous_on_coordinatewise_then_product [intro, continuous_intros]:
+lemma continuous_on_coordinatewise_then_product [intro, continuous_intros]:
assumes "\<And>i. continuous_on UNIV (\<lambda>x. f x i)"
shows "continuous_on UNIV f"
using assms unfolding continuous_on_topo_UNIV euclidean_product_topology
by (rule continuous_on_topo_coordinatewise_then_product, simp)
-lemma%unimportant continuous_on_product_then_coordinatewise:
+lemma continuous_on_product_then_coordinatewise:
assumes "continuous_on UNIV f"
shows "continuous_on UNIV (\<lambda>x. f x i)"
using assms unfolding continuous_on_topo_UNIV euclidean_product_topology
by (rule continuous_on_topo_product_then_coordinatewise(1), simp)
-lemma%unimportant continuous_on_product_coordinatewise_iff:
+lemma continuous_on_product_coordinatewise_iff:
"continuous_on UNIV f \<longleftrightarrow> (\<forall>i. continuous_on UNIV (\<lambda>x. f x i))"
by (auto intro: continuous_on_product_then_coordinatewise)
@@ -891,7 +893,7 @@
of product spaces, but they have more to do with countability and could
be put in the corresponding theory.\<close>
-lemma%unimportant countable_nat_product_event_const:
+lemma countable_nat_product_event_const:
fixes F::"'a set" and a::'a
assumes "a \<in> F" "countable F"
shows "countable {x::(nat \<Rightarrow> 'a). (\<forall>i. x i \<in> F) \<and> finite {i. x i \<noteq> a}}"
@@ -928,7 +930,7 @@
then show ?thesis using countable_subset[OF *] by auto
qed
-lemma%unimportant countable_product_event_const:
+lemma countable_product_event_const:
fixes F::"('a::countable) \<Rightarrow> 'b set" and b::'b
assumes "\<And>i. countable (F i)"
shows "countable {f::('a \<Rightarrow> 'b). (\<forall>i. f i \<in> F i) \<and> (finite {i. f i \<noteq> b})}"
@@ -961,7 +963,7 @@
qed
instance "fun" :: (countable, first_countable_topology) first_countable_topology
-proof%unimportant
+proof
fix x::"'a \<Rightarrow> 'b"
have "\<exists>A::('b \<Rightarrow> nat \<Rightarrow> 'b set). \<forall>x. (\<forall>i. x \<in> A x i \<and> open (A x i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A x i \<subseteq> S))"
apply (rule choice) using first_countable_basis by auto
@@ -1033,11 +1035,11 @@
using \<open>countable K\<close> \<open>\<And>k. k \<in> K \<Longrightarrow> x \<in> k\<close> \<open>\<And>k. k \<in> K \<Longrightarrow> open k\<close> Inc by auto
qed
-lemma%important product_topology_countable_basis:
+proposition product_topology_countable_basis:
shows "\<exists>K::(('a::countable \<Rightarrow> 'b::second_countable_topology) set set).
topological_basis K \<and> countable K \<and>
(\<forall>k\<in>K. \<exists>X. (k = Pi\<^sub>E UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV})"
-proof%unimportant -
+proof -
obtain B::"'b set set" where B: "countable B \<and> topological_basis B"
using ex_countable_basis by auto
then have "B \<noteq> {}" by (meson UNIV_I empty_iff open_UNIV topological_basisE)
@@ -1140,15 +1142,15 @@
definition%important strong_operator_topology::"('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_normed_vector) topology"
where "strong_operator_topology = pullback_topology UNIV blinfun_apply euclidean"
-lemma%unimportant strong_operator_topology_topspace:
+lemma strong_operator_topology_topspace:
"topspace strong_operator_topology = UNIV"
unfolding strong_operator_topology_def topspace_pullback_topology topspace_euclidean by auto
-lemma%important strong_operator_topology_basis:
+lemma strong_operator_topology_basis:
fixes f::"('a::real_normed_vector \<Rightarrow>\<^sub>L'b::real_normed_vector)" and U::"'i \<Rightarrow> 'b set" and x::"'i \<Rightarrow> 'a"
assumes "finite I" "\<And>i. i \<in> I \<Longrightarrow> open (U i)"
shows "openin strong_operator_topology {f. \<forall>i\<in>I. blinfun_apply f (x i) \<in> U i}"
-proof%unimportant -
+proof -
have "open {g::('a\<Rightarrow>'b). \<forall>i\<in>I. g (x i) \<in> U i}"
by (rule product_topology_basis'[OF assms])
moreover have "{f. \<forall>i\<in>I. blinfun_apply f (x i) \<in> U i}
@@ -1158,16 +1160,16 @@
unfolding strong_operator_topology_def open_openin apply (subst openin_pullback_topology) by auto
qed
-lemma%important strong_operator_topology_continuous_evaluation:
+lemma strong_operator_topology_continuous_evaluation:
"continuous_on_topo strong_operator_topology euclidean (\<lambda>f. blinfun_apply f x)"
-proof%unimportant -
+proof -
have "continuous_on_topo strong_operator_topology euclidean ((\<lambda>f. f x) o blinfun_apply)"
unfolding strong_operator_topology_def apply (rule continuous_on_topo_pullback)
using continuous_on_topo_UNIV continuous_on_product_coordinates by fastforce
then show ?thesis unfolding comp_def by simp
qed
-lemma%unimportant continuous_on_strong_operator_topo_iff_coordinatewise:
+lemma continuous_on_strong_operator_topo_iff_coordinatewise:
"continuous_on_topo T strong_operator_topology f
\<longleftrightarrow> (\<forall>x. continuous_on_topo T euclidean (\<lambda>y. blinfun_apply (f y) x))"
proof (auto)
@@ -1190,9 +1192,9 @@
by (auto simp add: \<open>continuous_on_topo T euclidean (blinfun_apply o f)\<close>)
qed
-lemma%important strong_operator_topology_weaker_than_euclidean:
+lemma strong_operator_topology_weaker_than_euclidean:
"continuous_on_topo euclidean strong_operator_topology (\<lambda>f. f)"
-by%unimportant (subst continuous_on_strong_operator_topo_iff_coordinatewise,
+ by (subst continuous_on_strong_operator_topo_iff_coordinatewise,
auto simp add: continuous_on_topo_UNIV[symmetric] linear_continuous_on)
@@ -1226,9 +1228,9 @@
spaces. It would thus be desirable to hide them once the instance is proved, but I do not know how
to do this.\<close>
-lemma%important dist_fun_le_dist_first_terms:
+lemma dist_fun_le_dist_first_terms:
"dist x y \<le> 2 * Max {dist (x (from_nat n)) (y (from_nat n))|n. n \<le> N} + (1/2)^N"
-proof%unimportant -
+proof -
have "(\<Sum>n. (1 / 2) ^ (n+Suc N) * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1)
= (\<Sum>n. (1 / 2) ^ (Suc N) * ((1/2) ^ n * min (dist (x (from_nat (n+Suc N))) (y (from_nat (n+Suc N)))) 1))"
by (rule suminf_cong, simp add: power_add)
@@ -1276,7 +1278,7 @@
finally show ?thesis unfolding M_def by simp
qed
-lemma%unimportant open_fun_contains_ball_aux:
+lemma open_fun_contains_ball_aux:
assumes "open (U::(('a \<Rightarrow> 'b) set))"
"x \<in> U"
shows "\<exists>e>0. {y. dist x y < e} \<subseteq> U"
@@ -1339,7 +1341,7 @@
then show "\<exists>m>0. {y. dist x y < m} \<subseteq> U" using * by blast
qed
-lemma%unimportant ball_fun_contains_open_aux:
+lemma ball_fun_contains_open_aux:
fixes x::"('a \<Rightarrow> 'b)" and e::real
assumes "e>0"
shows "\<exists>U. open U \<and> x \<in> U \<and> U \<subseteq> {y. dist x y < e}"
@@ -1386,7 +1388,7 @@
ultimately show ?thesis by auto
qed
-lemma%unimportant fun_open_ball_aux:
+lemma fun_open_ball_aux:
fixes U::"('a \<Rightarrow> 'b) set"
shows "open U \<longleftrightarrow> (\<forall>x\<in>U. \<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> y \<in> U)"
proof (auto)
@@ -1575,7 +1577,7 @@
-subsection%important \<open>Measurability\<close> (*FIX ME mv *)
+subsection%important \<open>Measurability\<close> (*FIX ME move? *)
text \<open>There are two natural sigma-algebras on a product space: the borel sigma algebra,
generated by open sets in the product, and the product sigma algebra, countably generated by
@@ -1593,11 +1595,11 @@
compare it with the product sigma algebra as explained above.
\<close>
-lemma%unimportant measurable_product_coordinates [measurable (raw)]:
+lemma measurable_product_coordinates [measurable (raw)]:
"(\<lambda>x. x i) \<in> measurable borel borel"
by (rule borel_measurable_continuous_on1[OF continuous_on_product_coordinates])
-lemma%unimportant measurable_product_then_coordinatewise:
+lemma measurable_product_then_coordinatewise:
fixes f::"'a \<Rightarrow> 'b \<Rightarrow> ('c::topological_space)"
assumes [measurable]: "f \<in> borel_measurable M"
shows "(\<lambda>x. f x i) \<in> borel_measurable M"
@@ -1611,10 +1613,10 @@
of the product sigma algebra that is more similar to the one we used above for the product
topology.\<close>
-lemma%important sets_PiM_finite:
+lemma sets_PiM_finite:
"sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i))
{(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}}"
-proof%unimportant
+proof
have "{(\<Pi>\<^sub>E i\<in>I. X i) |X. (\<forall>i. X i \<in> sets (M i)) \<and> finite {i. X i \<noteq> space (M i)}} \<subseteq> sets (Pi\<^sub>M I M)"
proof (auto)
fix X assume H: "\<forall>i. X i \<in> sets (M i)" "finite {i. X i \<noteq> space (M i)}"
@@ -1654,9 +1656,9 @@
done
qed
-lemma%important sets_PiM_subset_borel:
+lemma sets_PiM_subset_borel:
"sets (Pi\<^sub>M UNIV (\<lambda>_. borel)) \<subseteq> sets borel"
-proof%unimportant -
+proof -
have *: "Pi\<^sub>E UNIV X \<in> sets borel" if [measurable]: "\<And>i. X i \<in> sets borel" "finite {i. X i \<noteq> UNIV}" for X::"'a \<Rightarrow> 'b set"
proof -
define I where "I = {i. X i \<noteq> UNIV}"
@@ -1673,9 +1675,9 @@
by (simp add: * sets.sigma_sets_subset')
qed
-lemma%important sets_PiM_equal_borel:
+proposition sets_PiM_equal_borel:
"sets (Pi\<^sub>M UNIV (\<lambda>i::('a::countable). borel::('b::second_countable_topology measure))) = sets borel"
-proof%unimportant
+proof
obtain K::"('a \<Rightarrow> 'b) set set" where K: "topological_basis K" "countable K"
"\<And>k. k \<in> K \<Longrightarrow> \<exists>X. (k = Pi\<^sub>E UNIV X) \<and> (\<forall>i. open (X i)) \<and> finite {i. X i \<noteq> UNIV}"
using product_topology_countable_basis by fast
@@ -1700,11 +1702,11 @@
unfolding borel_def by auto
qed (simp add: sets_PiM_subset_borel)
-lemma%important measurable_coordinatewise_then_product:
+lemma measurable_coordinatewise_then_product:
fixes f::"'a \<Rightarrow> ('b::countable) \<Rightarrow> ('c::second_countable_topology)"
assumes [measurable]: "\<And>i. (\<lambda>x. f x i) \<in> borel_measurable M"
shows "f \<in> borel_measurable M"
-proof%unimportant -
+proof -
have "f \<in> measurable M (Pi\<^sub>M UNIV (\<lambda>_. borel))"
by (rule measurable_PiM_single', auto simp add: assms)
then show ?thesis using sets_PiM_equal_borel measurable_cong_sets by blast
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Wed Jan 16 21:27:33 2019 +0000
@@ -7489,7 +7489,7 @@
fixes f :: "'n::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] .
+ using integral_linear[OF assms(1) bounded_linear_vec_nth,unfolded o_def] .
lemma content_closed_interval:
fixes a :: "'a::ordered_euclidean_space"
--- a/src/HOL/Analysis/Tagged_Division.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Analysis/Tagged_Division.thy Wed Jan 16 21:27:33 2019 +0000
@@ -32,7 +32,7 @@
subsection%unimportant \<open>Sundries\<close>
-(*FIXME restructure this entire section consists of single lemma *)
+
text\<open>A transitive relation is well-founded if all initial segments are finite.\<close>
lemma wf_finite_segments:
@@ -55,7 +55,7 @@
by (simp add: field_simps)
qed
-subsection%unimportant \<open>Some useful lemmas about intervals\<close>
+subsection%important \<open>Some useful lemmas about intervals\<close>
lemma interior_subset_union_intervals:
assumes "i = cbox a b"
@@ -152,22 +152,22 @@
lemmas interval_bounds = interval_upperbound interval_lowerbound
-lemma%important
+lemma
fixes X::"real set"
shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
- by%unimportant (auto simp: interval_upperbound_def interval_lowerbound_def)
+ by (auto simp: interval_upperbound_def interval_lowerbound_def)
-lemma%important interval_bounds'[simp]:
+lemma interval_bounds'[simp]:
assumes "cbox a b \<noteq> {}"
shows "interval_upperbound (cbox a b) = b"
and "interval_lowerbound (cbox a b) = a"
- using%unimportant assms unfolding box_ne_empty by auto
+ using assms unfolding box_ne_empty by auto
-lemma%important interval_upperbound_Times:
+lemma interval_upperbound_Times:
assumes "A \<noteq> {}" and "B \<noteq> {}"
shows "interval_upperbound (A \<times> B) = (interval_upperbound A, interval_upperbound B)"
-proof%unimportant-
+proof-
from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
have "(\<Sum>i\<in>Basis. (SUP x\<in>A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x\<in>A. x \<bullet> i) *\<^sub>R i)"
by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
@@ -178,10 +178,10 @@
by (subst sum_Basis_prod_eq) (auto simp add: sum_prod)
qed
-lemma%important interval_lowerbound_Times:
+lemma interval_lowerbound_Times:
assumes "A \<noteq> {}" and "B \<noteq> {}"
shows "interval_lowerbound (A \<times> B) = (interval_lowerbound A, interval_lowerbound B)"
-proof%unimportant-
+proof-
from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
have "(\<Sum>i\<in>Basis. (INF x\<in>A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x\<in>A. x \<bullet> i) *\<^sub>R i)"
by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
@@ -226,11 +226,11 @@
using equation_minus_iff
by (auto simp: gauge_def surj_def intro!: open_surjective_linear_image linear_uminus)
-lemma%important gauge_Inter:
+lemma gauge_Inter:
assumes "finite S"
and "\<And>u. u\<in>S \<Longrightarrow> gauge (f u)"
shows "gauge (\<lambda>x. \<Inter>{f u x | u. u \<in> S})"
-proof%unimportant -
+proof -
have *: "\<And>x. {f u x |u. u \<in> S} = (\<lambda>u. f u x) ` S"
by auto
show ?thesis
@@ -238,12 +238,12 @@
using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
qed
-lemma%important gauge_existence_lemma:
+lemma gauge_existence_lemma:
"(\<forall>x. \<exists>d :: real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)"
- by%unimportant (metis zero_less_one)
+ by (metis zero_less_one)
-subsection%unimportant \<open>Attempt a systematic general set of "offset" results for components\<close>
-(*FIXME Restructure *)
+subsection%important \<open>Attempt a systematic general set of "offset" results for components\<close>
+(*FIXME Restructure, the subsection consists only of 1 lemma *)
lemma gauge_modify:
assumes "(\<forall>S. open S \<longrightarrow> open {x. f(x) \<in> S})" "gauge d"
shows "gauge (\<lambda>x. {y. f y \<in> d (f x)})"
@@ -605,10 +605,10 @@
qed
qed
-proposition%important partial_division_extend_interval:
+proposition partial_division_extend_interval:
assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
-proof%unimportant (cases "p = {}")
+proof (cases "p = {}")
case True
obtain q where "q division_of (cbox a b)"
by (rule elementary_interval)
@@ -687,11 +687,11 @@
"p division_of S \<Longrightarrow> \<exists>a b. S \<subseteq> cbox a (b::'a::euclidean_space)"
by (meson bounded_subset_cbox_symmetric elementary_bounded)
-lemma%important division_union_intervals_exists:
+proposition division_union_intervals_exists:
fixes a b :: "'a::euclidean_space"
assumes "cbox a b \<noteq> {}"
obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
-proof%unimportant (cases "cbox c d = {}")
+proof (cases "cbox c d = {}")
case True
with assms that show ?thesis by force
next
@@ -739,11 +739,11 @@
shows "\<Union>f division_of \<Union>\<Union>f"
using assms by (auto intro!: division_ofI)
-lemma%important elementary_union_interval:
+lemma elementary_union_interval:
fixes a b :: "'a::euclidean_space"
assumes "p division_of \<Union>p"
obtains q where "q division_of (cbox a b \<union> \<Union>p)"
-proof%unimportant (cases "p = {} \<or> cbox a b = {}")
+proof (cases "p = {} \<or> cbox a b = {}")
case True
obtain p where "p division_of (cbox a b)"
by (rule elementary_interval)
@@ -855,11 +855,11 @@
using that by auto
qed
-lemma%important elementary_union:
+lemma elementary_union:
fixes S T :: "'a::euclidean_space set"
assumes "ps division_of S" "pt division_of T"
obtains p where "p division_of (S \<union> T)"
-proof%unimportant -
+proof -
have *: "S \<union> T = \<Union>ps \<union> \<Union>pt"
using assms unfolding division_of_def by auto
show ?thesis
@@ -868,13 +868,13 @@
by (simp add: * that)
qed
-lemma%important partial_division_extend:
+lemma partial_division_extend:
fixes T :: "'a::euclidean_space set"
assumes "p division_of S"
and "q division_of T"
and "S \<subseteq> T"
obtains r where "p \<subseteq> r" and "r division_of T"
-proof%unimportant -
+proof -
note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
obtain a b where ab: "T \<subseteq> cbox a b"
using elementary_subset_cbox[OF assms(2)] by auto
@@ -931,7 +931,7 @@
qed
-lemma%important division_split:
+lemma division_split:
fixes a :: "'a::euclidean_space"
assumes "p division_of (cbox a b)"
and k: "k\<in>Basis"
@@ -939,7 +939,7 @@
(is "?p1 division_of ?I1")
and "{l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> p \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}} division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
(is "?p2 division_of ?I2")
-proof%unimportant (rule_tac[!] division_ofI)
+proof (rule_tac[!] division_ofI)
note p = division_ofD[OF assms(1)]
show "finite ?p1" "finite ?p2"
using p(1) by auto
@@ -1020,14 +1020,14 @@
lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
unfolding tagged_division_of_def tagged_partial_division_of_def by auto
-lemma%important tagged_division_of:
+lemma tagged_division_of:
"s tagged_division_of i \<longleftrightarrow>
finite s \<and>
(\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and>
(\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow>
interior K1 \<inter> interior K2 = {}) \<and>
(\<Union>{K. \<exists>x. (x,K) \<in> s} = i)"
- unfolding%unimportant tagged_division_of_def tagged_partial_division_of_def by auto
+ unfolding tagged_division_of_def tagged_partial_division_of_def by auto
lemma tagged_division_ofI:
assumes "finite s"
@@ -1052,10 +1052,10 @@
and "(\<Union>{K. \<exists>x. (x,K) \<in> s} = i)"
using assms unfolding tagged_division_of by blast+
-lemma%important division_of_tagged_division:
+lemma division_of_tagged_division:
assumes "s tagged_division_of i"
shows "(snd ` s) division_of i"
-proof%unimportant (rule division_ofI)
+proof (rule division_ofI)
note assm = tagged_division_ofD[OF assms]
show "\<Union>(snd ` s) = i" "finite (snd ` s)"
using assm by auto
@@ -1073,10 +1073,10 @@
using assm(5) k'(2) xk by blast
qed
-lemma%important partial_division_of_tagged_division:
+lemma partial_division_of_tagged_division:
assumes "s tagged_partial_division_of i"
shows "(snd ` s) division_of \<Union>(snd ` s)"
-proof%unimportant (rule division_ofI)
+proof (rule division_ofI)
note assm = tagged_partial_division_ofD[OF assms]
show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
using assm by auto
@@ -1121,12 +1121,12 @@
unfolding box_real[symmetric]
by (rule tagged_division_of_self)
-lemma%important tagged_division_Un:
+lemma tagged_division_Un:
assumes "p1 tagged_division_of s1"
and "p2 tagged_division_of s2"
and "interior s1 \<inter> interior s2 = {}"
shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
-proof%unimportant (rule tagged_division_ofI)
+proof (rule tagged_division_ofI)
note p1 = tagged_division_ofD[OF assms(1)]
note p2 = tagged_division_ofD[OF assms(2)]
show "finite (p1 \<union> p2)"
@@ -1149,12 +1149,12 @@
by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
qed
-lemma%important tagged_division_Union:
+lemma tagged_division_Union:
assumes "finite I"
and tag: "\<And>i. i\<in>I \<Longrightarrow> pfn i tagged_division_of i"
and disj: "\<And>i1 i2. \<lbrakk>i1 \<in> I; i2 \<in> I; i1 \<noteq> i2\<rbrakk> \<Longrightarrow> interior(i1) \<inter> interior(i2) = {}"
shows "\<Union>(pfn ` I) tagged_division_of (\<Union>I)"
-proof%unimportant (rule tagged_division_ofI)
+proof (rule tagged_division_ofI)
note assm = tagged_division_ofD[OF tag]
show "finite (\<Union>(pfn ` I))"
using assms by auto
@@ -1229,13 +1229,13 @@
unfolding box_real[symmetric]
by (rule tagged_division_Un_interval)
-lemma%important tagged_division_split_left_inj:
+lemma tagged_division_split_left_inj:
assumes d: "d tagged_division_of i"
and tags: "(x1, K1) \<in> d" "(x2, K2) \<in> d"
and "K1 \<noteq> K2"
and eq: "K1 \<inter> {x. x \<bullet> k \<le> c} = K2 \<inter> {x. x \<bullet> k \<le> c}"
shows "interior (K1 \<inter> {x. x\<bullet>k \<le> c}) = {}"
-proof%unimportant -
+proof -
have "interior (K1 \<inter> K2) = {} \<or> (x2, K2) = (x1, K1)"
using tags d by (metis (no_types) interior_Int tagged_division_ofD(5))
then show ?thesis
@@ -1255,11 +1255,11 @@
using eq \<open>K1 \<noteq> K2\<close> by (metis (no_types) inf_assoc inf_bot_left inf_left_idem interior_Int old.prod.inject)
qed
-lemma%important (in comm_monoid_set) over_tagged_division_lemma:
+lemma (in comm_monoid_set) over_tagged_division_lemma:
assumes "p tagged_division_of i"
and "\<And>u v. box u v = {} \<Longrightarrow> d (cbox u v) = \<^bold>1"
shows "F (\<lambda>(_, k). d k) p = F d (snd ` p)"
-proof%unimportant -
+proof -
have *: "(\<lambda>(_ ,k). d k) = d \<circ> snd"
by (simp add: fun_eq_iff)
note assm = tagged_division_ofD[OF assms(1)]
@@ -1293,7 +1293,7 @@
\<^typ>\<open>bool\<close>.\<close>
paragraph%important \<open>Using additivity of lifted function to encode definedness.\<close>
-text%important \<open>%whitespace\<close>
+text \<open>%whitespace\<close>
definition%important lift_option :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> 'c option"
where
"lift_option f a' b' = Option.bind a' (\<lambda>a. Option.bind b' (\<lambda>b. Some (f a b)))"
@@ -1307,7 +1307,7 @@
lemma%important comm_monoid_lift_option:
assumes "comm_monoid f z"
shows "comm_monoid (lift_option f) (Some z)"
-proof%unimportant -
+proof -
from assms interpret comm_monoid f z .
show ?thesis
by standard (auto simp: lift_option_def ac_simps split: bind_split)
@@ -1334,16 +1334,16 @@
paragraph \<open>Division points\<close>
-text%important \<open>%whitespace\<close>
+text \<open>%whitespace\<close>
definition%important "division_points (k::('a::euclidean_space) set) d =
{(j,x). j \<in> Basis \<and> (interval_lowerbound k)\<bullet>j < x \<and> x < (interval_upperbound k)\<bullet>j \<and>
(\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
-lemma%important division_points_finite:
+lemma division_points_finite:
fixes i :: "'a::euclidean_space set"
assumes "d division_of i"
shows "finite (division_points i d)"
-proof%unimportant -
+proof -
note assm = division_ofD[OF assms]
let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)\<bullet>j < x \<and> x < (interval_upperbound i)\<bullet>j \<and>
(\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
@@ -1353,7 +1353,7 @@
unfolding * using assm by auto
qed
-lemma%important division_points_subset:
+lemma division_points_subset:
fixes a :: "'a::euclidean_space"
assumes "d division_of (cbox a b)"
and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "a\<bullet>k < c" "c < b\<bullet>k"
@@ -1362,7 +1362,7 @@
division_points (cbox a b) d" (is ?t1)
and "division_points (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l . l \<in> d \<and> \<not>(l \<inter> {x. x\<bullet>k \<ge> c} = {})} \<subseteq>
division_points (cbox a b) d" (is ?t2)
-proof%unimportant -
+proof -
note assm = division_ofD[OF assms(1)]
have *: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
"\<forall>i\<in>Basis. a\<bullet>i \<le> (\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i"
@@ -1423,7 +1423,7 @@
by force
qed
-lemma%important division_points_psubset:
+lemma division_points_psubset:
fixes a :: "'a::euclidean_space"
assumes d: "d division_of (cbox a b)"
and altb: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" "a\<bullet>k < c" "c < b\<bullet>k"
@@ -1434,7 +1434,7 @@
division_points (cbox a b) d" (is "?D1 \<subset> ?D")
and "division_points (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}} \<subset>
division_points (cbox a b) d" (is "?D2 \<subset> ?D")
-proof%unimportant -
+proof -
have ab: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
using altb by (auto intro!:less_imp_le)
obtain u v where l: "l = cbox u v"
@@ -1485,13 +1485,13 @@
using \<open>K1 \<in> \<D>\<close> \<open>K1 \<noteq> K2\<close> by auto
qed
-lemma%important division_split_right_inj:
+lemma division_split_right_inj:
fixes S :: "'a::euclidean_space set"
assumes div: "\<D> division_of S"
and eq: "K1 \<inter> {x::'a. x\<bullet>k \<ge> c} = K2 \<inter> {x. x\<bullet>k \<ge> c}"
and "K1 \<in> \<D>" "K2 \<in> \<D>" "K1 \<noteq> K2"
shows "interior (K1 \<inter> {x. x\<bullet>k \<ge> c}) = {}"
-proof%unimportant -
+proof -
have "interior K2 \<inter> interior {a. a \<bullet> k \<ge> c} = interior K1 \<inter> interior {a. a \<bullet> k \<ge> c}"
by (metis (no_types) eq interior_Int)
moreover have "\<And>A. interior A \<inter> interior K2 = {} \<or> A = K2 \<or> A \<notin> \<D>"
@@ -1515,13 +1515,13 @@
unfolding * ** interval_split[OF assms] by (rule refl)
qed
-lemma%important division_doublesplit:
+lemma division_doublesplit:
fixes a :: "'a::euclidean_space"
assumes "p division_of (cbox a b)"
and k: "k \<in> Basis"
shows "(\<lambda>l. l \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e}) ` {l\<in>p. l \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e} \<noteq> {}}
division_of (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e})"
-proof%unimportant -
+proof -
have *: "\<And>x c. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e"
by auto
have **: "\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'"
@@ -1560,9 +1560,9 @@
using box_empty_imp [of One "-One"] by simp
qed
-lemma%important division:
+lemma division:
"F g d = g (cbox a b)" if "d division_of (cbox a b)"
-proof%unimportant -
+proof -
define C where [abs_def]: "C = card (division_points (cbox a b) d)"
then show ?thesis
using that proof (induction C arbitrary: a b d rule: less_induct)
@@ -1747,10 +1747,10 @@
qed
qed
-lemma%important tagged_division:
+proposition tagged_division:
assumes "d tagged_division_of (cbox a b)"
shows "F (\<lambda>(_, l). g l) d = g (cbox a b)"
-proof%unimportant -
+proof -
have "F (\<lambda>(_, k). g k) d = F g (snd ` d)"
using assms box_empty_imp by (rule over_tagged_division_lemma)
then show ?thesis
@@ -1830,12 +1830,12 @@
subsection%important \<open>Special case of additivity we need for the FTC\<close>
(*fix add explanation here *)
-lemma%important additive_tagged_division_1:
+lemma additive_tagged_division_1:
fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
assumes "a \<le> b"
and "p tagged_division_of {a..b}"
shows "sum (\<lambda>(x,k). f(Sup k) - f(Inf k)) p = f b - f a"
-proof%unimportant -
+proof -
let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
interpret operative_real plus 0 ?f
rewrites "comm_monoid_set.F (+) 0 = sum"
@@ -1881,21 +1881,21 @@
lemma fine_Un: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
unfolding fine_def by blast
-lemma%important fine_Union: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
+lemma fine_Union: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
unfolding fine_def by auto
-lemma%unimportant fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
+lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
unfolding fine_def by blast
subsection%important \<open>Some basic combining lemmas\<close>
-lemma%important tagged_division_Union_exists:
+lemma tagged_division_Union_exists:
assumes "finite I"
and "\<forall>i\<in>I. \<exists>p. p tagged_division_of i \<and> d fine p"
and "\<forall>i1\<in>I. \<forall>i2\<in>I. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
and "\<Union>I = i"
obtains p where "p tagged_division_of i" and "d fine p"
-proof%unimportant -
+proof -
obtain pfn where pfn:
"\<And>x. x \<in> I \<Longrightarrow> pfn x tagged_division_of x"
"\<And>x. x \<in> I \<Longrightarrow> d fine pfn x"
@@ -1913,17 +1913,17 @@
fixes i :: "'n::euclidean_space set"
shows "s division_of i \<Longrightarrow> closed i"
unfolding division_of_def by fastforce
-
+(* FIXME structure here, do not have one lemma in each subsection *)
subsection%important \<open>General bisection principle for intervals; might be useful elsewhere\<close>
-
-lemma%important interval_bisection_step:
+(* FIXME move? *)
+lemma interval_bisection_step:
fixes type :: "'a::euclidean_space"
assumes emp: "P {}"
and Un: "\<And>S T. \<lbrakk>P S; P T; interior(S) \<inter> interior(T) = {}\<rbrakk> \<Longrightarrow> P (S \<union> T)"
and non: "\<not> P (cbox a (b::'a))"
obtains c d where "\<not> P (cbox c d)"
and "\<And>i. i \<in> Basis \<Longrightarrow> a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
-proof%unimportant -
+proof -
have "cbox a b \<noteq> {}"
using emp non by metis
then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
@@ -2029,14 +2029,14 @@
by (metis (no_types, lifting) assms(3) that)
qed
-lemma%important interval_bisection:
+lemma interval_bisection:
fixes type :: "'a::euclidean_space"
assumes "P {}"
and Un: "\<And>S T. \<lbrakk>P S; P T; interior(S) \<inter> interior(T) = {}\<rbrakk> \<Longrightarrow> P (S \<union> T)"
and "\<not> P (cbox a (b::'a))"
obtains x where "x \<in> cbox a b"
and "\<forall>e>0. \<exists>c d. x \<in> cbox c d \<and> cbox c d \<subseteq> ball x e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
-proof%unimportant -
+proof -
have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and>
(\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and>
2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" (is "\<forall>x. ?P x")
@@ -2182,11 +2182,11 @@
subsection%important \<open>Cousin's lemma\<close>
-lemma%important fine_division_exists:
+lemma fine_division_exists: (*FIXME rename(?) *)
fixes a b :: "'a::euclidean_space"
assumes "gauge g"
obtains p where "p tagged_division_of (cbox a b)" "g fine p"
-proof%unimportant (cases "\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p")
+proof (cases "\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p")
case True
then show ?thesis
using that by auto
@@ -2228,14 +2228,14 @@
subsection%important \<open>A technical lemma about "refinement" of division\<close>
-lemma%important tagged_division_finer:
+lemma tagged_division_finer:
fixes p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set"
assumes ptag: "p tagged_division_of (cbox a b)"
and "gauge d"
obtains q where "q tagged_division_of (cbox a b)"
and "d fine q"
and "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q"
-proof%unimportant -
+proof -
have p: "finite p" "p tagged_partial_division_of (cbox a b)"
using ptag unfolding tagged_division_of_def by auto
have "(\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and> (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))"
@@ -2307,7 +2307,7 @@
lemma is an obvious multidimensional generalization of Lemma 3, p65 of Swartz's
"Introduction to Gauge Integrals". \<close>
-proposition%important covering_lemma:
+proposition covering_lemma:
assumes "S \<subseteq> cbox a b" "box a b \<noteq> {}" "gauge g"
obtains \<D> where
"countable \<D>" "\<Union>\<D> \<subseteq> cbox a b"
@@ -2316,7 +2316,7 @@
"\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> g x"
"\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n"
"S \<subseteq> \<Union>\<D>"
-proof%unimportant -
+proof -
have aibi: "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i" and normab: "0 < norm(b - a)"
using \<open>box a b \<noteq> {}\<close> box_eq_empty box_sing by fastforce+
let ?K0 = "\<lambda>(n, f::'a\<Rightarrow>nat).
@@ -2578,11 +2578,11 @@
definition%important division_filter :: "'a::euclidean_space set \<Rightarrow> ('a \<times> 'a set) set filter"
where "division_filter s = (INF g\<in>{g. gauge g}. principal {p. p tagged_division_of s \<and> g fine p})"
-lemma%important eventually_division_filter:
+proposition eventually_division_filter:
"(\<forall>\<^sub>F p in division_filter s. P p) \<longleftrightarrow>
(\<exists>g. gauge g \<and> (\<forall>p. p tagged_division_of s \<and> g fine p \<longrightarrow> P p))"
- unfolding%unimportant division_filter_def
-proof%unimportant (subst eventually_INF_base; clarsimp)
+ unfolding division_filter_def
+proof (subst eventually_INF_base; clarsimp)
fix g1 g2 :: "'a \<Rightarrow> 'a set" show "gauge g1 \<Longrightarrow> gauge g2 \<Longrightarrow> \<exists>x. gauge x \<and>
{p. p tagged_division_of s \<and> x fine p} \<subseteq> {p. p tagged_division_of s \<and> g1 fine p} \<and>
{p. p tagged_division_of s \<and> x fine p} \<subseteq> {p. p tagged_division_of s \<and> g2 fine p}"
@@ -2593,7 +2593,7 @@
unfolding trivial_limit_def eventually_division_filter
by (auto elim: fine_division_exists)
-lemma%important eventually_division_filter_tagged_division:
+lemma eventually_division_filter_tagged_division:
"eventually (\<lambda>p. p tagged_division_of s) (division_filter s)"
unfolding eventually_division_filter by (intro exI[of _ "\<lambda>x. ball x 1"]) auto
--- a/src/HOL/Library/Cardinality.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Library/Cardinality.thy Wed Jan 16 21:27:33 2019 +0000
@@ -153,6 +153,19 @@
lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"
by (simp add: less_Suc_eq_le [symmetric])
+
+class CARD_1 =
+ assumes CARD_1: "CARD ('a) = 1"
+begin
+
+subclass finite
+proof
+ from CARD_1 show "finite (UNIV :: 'a set)"
+ by (auto intro!: card_ge_0_finite)
+qed
+
+end
+
text \<open>Class for cardinality "at least 2"\<close>
class card2 = finite +
--- a/src/HOL/Library/Numeral_Type.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Library/Numeral_Type.thy Wed Jan 16 21:27:33 2019 +0000
@@ -45,6 +45,8 @@
unfolding type_definition.card [OF type_definition_bit1]
by simp
+subsection \<open>@{typ num1}\<close>
+
instance num1 :: finite
proof
show "finite (UNIV::num1 set)"
@@ -52,6 +54,52 @@
using finite by (rule finite_imageI)
qed
+instantiation num1 :: CARD_1
+begin
+
+instance
+proof
+ show "CARD(num1) = 1" by auto
+qed
+
+end
+
+lemma num1_eq_iff: "(x::num1) = (y::num1) \<longleftrightarrow> True"
+ by (induct x, induct y) simp
+
+instantiation num1 :: "{comm_ring,comm_monoid_mult,numeral}"
+begin
+
+instance
+ by standard (simp_all add: num1_eq_iff)
+
+end
+
+lemma num1_eqI:
+ fixes a::num1 shows "a = b"
+by(simp add: num1_eq_iff)
+
+lemma num1_eq1 [simp]:
+ fixes a::num1 shows "a = 1"
+ by (rule num1_eqI)
+
+lemma forall_1[simp]: "(\<forall>i::num1. P i) \<longleftrightarrow> P 1"
+ by (metis (full_types) num1_eq_iff)
+
+lemma ex_1[simp]: "(\<exists>x::num1. P x) \<longleftrightarrow> P 1"
+ by auto (metis (full_types) num1_eq_iff)
+
+instantiation num1 :: linorder begin
+definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b"
+definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b"
+instance
+ by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI)
+end
+
+instance num1 :: wellorder
+ by intro_classes (auto simp: less_eq_num1_def less_num1_def)
+
+
instance bit0 :: (finite) card2
proof
show "finite (UNIV::'a bit0 set)"
@@ -185,17 +233,6 @@
\<^typ>\<open>num1\<close>, since 0 and 1 are not distinct.
\<close>
-instantiation num1 :: "{comm_ring,comm_monoid_mult,numeral}"
-begin
-
-lemma num1_eq_iff: "(x::num1) = (y::num1) \<longleftrightarrow> True"
- by (induct x, induct y) simp
-
-instance
- by standard (simp_all add: num1_eq_iff)
-
-end
-
instantiation
bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus}"
begin
@@ -350,8 +387,7 @@
definition "enum_class.enum_ex P = P (1 :: num1)"
instance
by intro_classes
- (auto simp add: enum_num1_def enum_all_num1_def enum_ex_num1_def num1_eq_iff Ball_def,
- (metis (full_types) num1_eq_iff)+)
+ (auto simp add: enum_num1_def enum_all_num1_def enum_ex_num1_def num1_eq_iff Ball_def)
end
instantiation num0 and num1 :: card_UNIV begin
--- a/src/Pure/General/path.ML Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Pure/General/path.ML Wed Jan 16 21:27:33 2019 +0000
@@ -14,6 +14,7 @@
val root: T
val named_root: string -> T
val parent: T
+ val make: string list -> T
val basic: string -> T
val variable: string -> T
val has_parent: T -> bool
@@ -22,7 +23,6 @@
val starts_basic: T -> bool
val append: T -> T -> T
val appends: T list -> T
- val make: string list -> T
val implode: T -> string
val explode: string -> T
val decode: T XML.Decode.T
@@ -88,6 +88,7 @@
val current = Path [];
val root = Path [Root ""];
fun named_root s = Path [root_elem s];
+val make = Path o rev o map basic_elem;
fun basic s = Path [basic_elem s];
fun variable s = Path [variable_elem s];
val parent = Path [Parent];
@@ -117,7 +118,6 @@
fun append (Path xs) (Path ys) = Path (fold_rev apply ys xs);
fun appends paths = Library.foldl (uncurry append) (current, paths);
-val make = appends o map basic;
fun norm elems = fold_rev apply elems [];
--- a/src/Pure/General/path.scala Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Pure/General/path.scala Wed Jan 16 21:27:33 2019 +0000
@@ -73,6 +73,7 @@
val current: Path = new Path(Nil)
val root: Path = new Path(List(Root("")))
def named_root(s: String): Path = new Path(List(root_elem(s)))
+ def make(elems: List[String]): Path = new Path(elems.reverse.map(basic_elem(_)))
def basic(s: String): Path = new Path(List(basic_elem(s)))
def variable(s: String): Path = new Path(List(variable_elem(s)))
val parent: Path = new Path(List(Parent))
--- a/src/Pure/Pure.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Pure/Pure.thy Wed Jan 16 21:27:33 2019 +0000
@@ -22,6 +22,7 @@
"hide_class" "hide_type" "hide_const" "hide_fact" :: thy_decl
and "external_file" "bibtex_file" :: thy_load
and "generate_file" :: thy_decl
+ and "export_generated_files" :: diag
and "ML_file" "ML_file_debug" "ML_file_no_debug" :: thy_load % "ML"
and "SML_file" "SML_file_debug" "SML_file_no_debug" :: thy_load % "ML"
and "SML_import" "SML_export" "ML_export" :: thy_decl % "ML"
@@ -125,6 +126,26 @@
(Parse.position Parse.path -- (\<^keyword>\<open>=\<close> |-- Parse.input Parse.embedded)
>> Generated_Files.generate_file_cmd);
+ val _ =
+ Outer_Syntax.command \<^command_keyword>\<open>export_generated_files\<close>
+ "export generated files from this or other theories"
+ (Scan.repeat Parse.name_position >> (fn names =>
+ Toplevel.keep (fn st =>
+ let
+ val ctxt = Toplevel.context_of st;
+ val thy = Toplevel.theory_of st;
+ val other_thys =
+ if null names then [thy]
+ else map (Theory.check {long = false} ctxt) names;
+ val paths = Generated_Files.export_files thy other_thys;
+ in
+ if not (null paths) then
+ (writeln (Export.message thy "");
+ writeln (cat_lines (paths |> map (fn (path, pos) =>
+ Markup.markup (Markup.entityN, Position.def_properties_of pos)
+ (quote (Path.implode path))))))
+ else ()
+ end)));
in end\<close>
--- a/src/Pure/Thy/export.scala Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Pure/Thy/export.scala Wed Jan 16 21:27:33 2019 +0000
@@ -259,6 +259,7 @@
session_name: String,
export_dir: Path,
progress: Progress = No_Progress,
+ export_prune: Int = 0,
export_list: Boolean = false,
export_patterns: List[String] = Nil,
export_prefix: String = "")
@@ -287,7 +288,13 @@
name <- group.map(_._2).sorted
entry <- read_entry(db, session_name, theory_name, name)
} {
- val path = export_dir + Path.basic(theory_name) + Path.explode(name)
+ val elems = theory_name :: space_explode('/', name)
+ val path =
+ if (elems.length < export_prune + 1) {
+ error("Cannot prune path by " + export_prune + " element(s): " + Path.make(elems))
+ }
+ else export_dir + Path.make(elems.drop(export_prune))
+
progress.echo(export_prefix + "export " + path)
Isabelle_System.mkdirs(path.dir)
Bytes.write(path, entry.uncompressed(cache = store.xz_cache))
@@ -311,6 +318,7 @@
var export_list = false
var no_build = false
var options = Options.init()
+ var export_prune = 0
var system_mode = false
var export_patterns: List[String] = Nil
@@ -323,6 +331,7 @@
-l list exports
-n no build of session
-o OPTION override Isabelle system OPTION (via NAME=VAL or NAME)
+ -p NUM prune path of exported files by NUM elements
-s system build mode for session image
-x PATTERN extract files matching pattern (e.g. "*:**" for all)
@@ -338,6 +347,7 @@
"l" -> (_ => export_list = true),
"n" -> (_ => no_build = true),
"o:" -> (arg => options = options + arg),
+ "p:" -> (arg => export_prune = Value.Int.parse(arg)),
"s" -> (_ => system_mode = true),
"x:" -> (arg => export_patterns ::= arg))
@@ -366,7 +376,7 @@
/* export files */
val store = Sessions.store(options, system_mode)
- export_files(store, session_name, export_dir, progress = progress,
+ export_files(store, session_name, export_dir, progress = progress, export_prune = export_prune,
export_list = export_list, export_patterns = export_patterns)
})
}
--- a/src/Pure/Thy/sessions.ML Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Pure/Thy/sessions.ML Wed Jan 16 21:27:33 2019 +0000
@@ -35,9 +35,13 @@
Parse.!!! (Scan.optional in_path ("document", Position.none)
-- Scan.repeat1 (Parse.position Parse.path));
+val prune =
+ Scan.optional (Parse.$$$ "[" |-- Parse.!!! (Parse.nat --| Parse.$$$ "]")) 0;
+
val export_files =
Parse.$$$ "export_files" |--
- Parse.!!! (Scan.optional in_path ("export", Position.none) -- Scan.repeat1 Parse.embedded);
+ Parse.!!!
+ (Scan.optional in_path ("export", Position.none) -- prune -- Scan.repeat1 Parse.embedded);
in
@@ -95,7 +99,7 @@
in () end);
val _ =
- export_files |> List.app (fn (export_dir, _) =>
+ export_files |> List.app (fn ((export_dir, _), _) =>
ignore (Resources.check_path ctxt session_dir export_dir));
in () end));
--- a/src/Pure/Thy/sessions.scala Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Pure/Thy/sessions.scala Wed Jan 16 21:27:33 2019 +0000
@@ -515,7 +515,7 @@
theories: List[(Options, List[(String, Position.T)])],
global_theories: List[String],
document_files: List[(Path, Path)],
- export_files: List[(Path, List[String])],
+ export_files: List[(Path, Int, List[String])],
meta_digest: SHA1.Digest)
{
def deps: List[String] = parent.toList ::: imports
@@ -568,7 +568,7 @@
entry.document_files.map({ case (s1, s2) => (Path.explode(s1), Path.explode(s2)) })
val export_files =
- entry.export_files.map({ case (dir, pats) => (Path.explode(dir), pats) })
+ entry.export_files.map({ case (dir, prune, pats) => (Path.explode(dir), prune, pats) })
val meta_digest =
SHA1.digest((name, chapter, entry.parent, entry.options, entry.imports,
@@ -831,7 +831,7 @@
imports: List[String],
theories: List[(List[Options.Spec], List[((String, Position.T), Boolean)])],
document_files: List[(String, String)],
- export_files: List[(String, List[String])]) extends Entry
+ export_files: List[(String, Int, List[String])]) extends Entry
{
def theories_no_position: List[(List[Options.Spec], List[(String, Boolean)])] =
theories.map({ case (a, b) => (a, b.map({ case ((c, _), d) => (c, d) })) })
@@ -870,9 +870,11 @@
$$$(DOCUMENT_FILES) ~!
((in_path | success("document")) ~ rep1(path)) ^^ { case _ ~ (x ~ y) => y.map((x, _)) }
+ val prune = $$$("[") ~! (nat ~ $$$("]")) ^^ { case _ ~ (x ~ _) => x } | success(0)
+
val export_files =
- $$$(EXPORT_FILES) ~! ((in_path | success("export")) ~ rep1(embedded)) ^^
- { case _ ~ (x ~ y) => (x, y) }
+ $$$(EXPORT_FILES) ~! ((in_path | success("export")) ~ prune ~ rep1(embedded)) ^^
+ { case _ ~ (x ~ y ~ z) => (x, y, z) }
command(SESSION) ~!
(position(session_name) ~
--- a/src/Pure/Tools/build.scala Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Pure/Tools/build.scala Wed Jan 16 21:27:33 2019 +0000
@@ -335,9 +335,10 @@
Isabelle_System.rm_tree(export_tmp_dir)
if (result1.ok) {
- for ((dir, pats) <- info.export_files) {
+ for ((dir, prune, pats) <- info.export_files) {
Export.export_files(store, name, info.dir + dir,
progress = if (verbose) progress else No_Progress,
+ export_prune = prune,
export_patterns = pats,
export_prefix = name + ": ")
}
--- a/src/Pure/Tools/generated_files.ML Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Pure/Tools/generated_files.ML Wed Jan 16 21:27:33 2019 +0000
@@ -7,9 +7,9 @@
signature GENERATED_FILES =
sig
val add_files: (Path.T * Position.T) * string -> theory -> theory
- val get_files: theory -> (Path.T * string) list
- val write_files: theory -> Path.T -> Path.T list
- val export_files: theory -> Path.T list
+ val get_files: theory -> {path: Path.T, pos: Position.T, text: string} list
+ val write_files: theory -> Path.T -> (Path.T * Position.T) list
+ val export_files: theory -> theory list -> (Path.T * Position.T) list
type file_type =
{name: string, ext: string, make_comment: string -> string, make_string: string -> string}
val file_type:
@@ -67,18 +67,21 @@
| NONE => (path, (pos, text)) :: files))
end;
-val get_files = map (apsnd #2) o rev o #1 o Data.get;
+val get_files =
+ map (fn (path, (pos, text)) => {path = path, pos = pos, text = text}) o rev o #1 o Data.get;
fun write_files thy dir =
- get_files thy |> map (fn (path, text) =>
+ get_files thy |> map (fn {path, pos, text} =>
let
val path' = Path.expand (Path.append dir path);
val _ = Isabelle_System.mkdirs (Path.dir path');
val _ = File.generate path' text;
- in path end);
+ in (path, pos) end);
-fun export_files thy =
- get_files thy |> map (fn (path, text) => (Export.export thy (Path.implode path) [text]; path));
+fun export_files thy other_thys =
+ other_thys |> maps (fn other_thy =>
+ get_files other_thy |> map (fn {path, pos, text} =>
+ (Export.export thy (Path.implode path) [text]; (path, pos))));
(* file types *)
--- a/src/Tools/Haskell/Haskell.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Tools/Haskell/Haskell.thy Wed Jan 16 21:27:33 2019 +0000
@@ -1924,6 +1924,6 @@
return ()
\<close>
-ML_command \<open>Generated_Files.export_files \<^theory>\<close>
+export_generated_files
end
--- a/src/Tools/Haskell/Test.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/Tools/Haskell/Test.thy Wed Jan 16 21:27:33 2019 +0000
@@ -14,7 +14,7 @@
val files = Generated_Files.write_files \<^theory>\<open>Haskell\<close> src_dir;
val modules = files
- |> map (Path.implode #> unsuffix ".hs" #> space_explode "/" #> space_implode ".");
+ |> map (#1 #> Path.implode #> unsuffix ".hs" #> space_explode "/" #> space_implode ".");
val _ =
GHC.new_project tmp_dir
{name = "isabelle",