--- a/NEWS Wed Jul 13 17:20:21 2016 +0100
+++ b/NEWS Wed Jul 13 20:48:18 2016 +0200
@@ -79,13 +79,19 @@
* Refined folding mode "isabelle" based on Isar syntax: 'next' and 'qed'
are treated as delimiters for fold structure.
-* Improved support for indentation according to Isabelle outer syntax.
-Action "indent-lines" (shortcut C+i) indents the current line according
-to command keywords and some command substructure. Action
+* Syntactic indentation according to Isabelle outer syntax. Action
+"indent-lines" (shortcut C+i) indents the current line according to
+command keywords and some command substructure. Action
"isabelle.newline" (shortcut ENTER) indents the old and the new line
according to command keywords only; see also option
"jedit_indent_newline".
+* Semantic indentation for unstructured proof scripts ('apply' etc.) via
+number of subgoals. This requires information of ongoing document
+processing and may thus lag behind, when the user is editing too
+quickly; see also option "jedit_script_indent" and
+"jedit_script_indent_limit".
+
* Action "isabelle.select-entity" (shortcut CS+ENTER) selects all
occurences of the formal entity at the caret position. This facilitates
systematic renaming.
--- a/src/HOL/Library/AList.thy Wed Jul 13 17:20:21 2016 +0100
+++ b/src/HOL/Library/AList.thy Wed Jul 13 20:48:18 2016 +0200
@@ -73,8 +73,7 @@
@{term "update k' v' (update k v []) = [(k, v), (k', v')]"}.\<close>
lemma update_swap:
- "k \<noteq> k' \<Longrightarrow>
- map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))"
+ "k \<noteq> k' \<Longrightarrow> map_of (update k v (update k' v' al)) = map_of (update k' v' (update k v al))"
by (simp add: update_conv' fun_eq_iff)
lemma update_Some_unfold:
@@ -85,8 +84,8 @@
lemma image_update [simp]: "x \<notin> A \<Longrightarrow> map_of (update x y al) ` A = map_of al ` A"
by (simp add: update_conv')
-qualified definition updates
- :: "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+qualified definition updates ::
+ "'key list \<Rightarrow> 'val list \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where "updates ks vs = fold (case_prod update) (zip ks vs)"
lemma updates_simps [simp]:
@@ -216,8 +215,8 @@
subsection \<open>\<open>update_with_aux\<close> and \<open>delete_aux\<close>\<close>
-qualified primrec update_with_aux
- :: "'val \<Rightarrow> 'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
+qualified primrec update_with_aux ::
+ "'val \<Rightarrow> 'key \<Rightarrow> ('val \<Rightarrow> 'val) \<Rightarrow> ('key \<times> 'val) list \<Rightarrow> ('key \<times> 'val) list"
where
"update_with_aux v k f [] = [(k, f v)]"
| "update_with_aux v k f (p # ps) =
@@ -257,7 +256,7 @@
lemma set_delete_aux: "distinct (map fst xs) \<Longrightarrow> set (delete_aux k xs) = set xs - {k} \<times> UNIV"
apply (induct xs)
- apply simp_all
+ apply simp_all
apply clarsimp
apply (fastforce intro: rev_image_eqI)
done
@@ -291,7 +290,7 @@
lemma map_of_delete_aux':
"distinct (map fst xs) \<Longrightarrow> map_of (delete_aux k xs) = (map_of xs)(k := None)"
apply (induct xs)
- apply (fastforce simp add: map_of_eq_None_iff fun_upd_twist)
+ apply (fastforce simp add: map_of_eq_None_iff fun_upd_twist)
apply (auto intro!: ext)
apply (simp add: map_of_eq_None_iff)
done
@@ -318,9 +317,9 @@
proof
show "map_of (restrict A al) k = ((map_of al)|` A) k" for k
apply (induct al)
- apply simp
+ apply simp
apply (cases "k \<in> A")
- apply auto
+ apply auto
done
qed
--- a/src/HOL/Library/AList_Mapping.thy Wed Jul 13 17:20:21 2016 +0100
+++ b/src/HOL/Library/AList_Mapping.thy Wed Jul 13 20:48:18 2016 +0200
@@ -51,8 +51,11 @@
proof -
have *: "(a, b) \<in> set xs \<Longrightarrow> a \<in> fst ` set xs" for a b xs
by (auto simp add: image_def intro!: bexI)
- show ?thesis apply transfer
- by (auto intro!: map_of_eqI) (auto dest!: map_of_eq_dom intro: *)
+ show ?thesis
+ apply transfer
+ apply (auto intro!: map_of_eqI)
+ apply (auto dest!: map_of_eq_dom intro: *)
+ done
qed
lemma map_values_Mapping [code]:
@@ -72,8 +75,8 @@
apply (rule sym)
subgoal for f xs ys x
apply (cases "map_of xs x"; cases "map_of ys x"; simp)
- apply (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
- dest: map_of_SomeD split: option.splits)+
+ apply (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
+ dest: map_of_SomeD split: option.splits)+
done
done
@@ -86,8 +89,8 @@
apply (rule sym)
subgoal for f xs ys x
apply (cases "map_of xs x"; cases "map_of ys x"; simp)
- apply (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
- dest: map_of_SomeD split: option.splits)+
+ apply (force simp: map_of_eq_None_iff combine_options_def option.the_def o_def image_iff
+ dest: map_of_SomeD split: option.splits)+
done
done
@@ -106,7 +109,7 @@
apply transfer
apply (rule ext)
apply (subst map_of_filter_distinct)
- apply (simp_all add: map_of_clearjunk split: option.split)
+ apply (simp_all add: map_of_clearjunk split: option.split)
done
lemma [code nbe]: "HOL.equal (x :: ('a, 'b) mapping) x \<longleftrightarrow> True"
--- a/src/HOL/Library/BigO.thy Wed Jul 13 17:20:21 2016 +0100
+++ b/src/HOL/Library/BigO.thy Wed Jul 13 20:48:18 2016 +0200
@@ -5,33 +5,35 @@
section \<open>Big O notation\<close>
theory BigO
- imports Complex_Main Function_Algebras Set_Algebras
+ imports
+ Complex_Main
+ Function_Algebras
+ Set_Algebras
begin
text \<open>
-This library is designed to support asymptotic ``big O'' calculations,
-i.e.~reasoning with expressions of the form $f = O(g)$ and $f = g +
-O(h)$. An earlier version of this library is described in detail in
-@{cite "Avigad-Donnelly"}.
+ This library is designed to support asymptotic ``big O'' calculations,
+ i.e.~reasoning with expressions of the form \<open>f = O(g)\<close> and \<open>f = g + O(h)\<close>.
+ An earlier version of this library is described in detail in @{cite
+ "Avigad-Donnelly"}.
+
+ The main changes in this version are as follows:
-The main changes in this version are as follows:
-\begin{itemize}
-\item We have eliminated the \<open>O\<close> operator on sets. (Most uses of this seem
- to be inessential.)
-\item We no longer use \<open>+\<close> as output syntax for \<open>+o\<close>
-\item Lemmas involving \<open>sumr\<close> have been replaced by more general lemmas
- involving `\<open>setsum\<close>.
-\item The library has been expanded, with e.g.~support for expressions of
- the form \<open>f < g + O(h)\<close>.
-\end{itemize}
+ \<^item> We have eliminated the \<open>O\<close> operator on sets. (Most uses of this seem
+ to be inessential.)
+ \<^item> We no longer use \<open>+\<close> as output syntax for \<open>+o\<close>
+ \<^item> Lemmas involving \<open>sumr\<close> have been replaced by more general lemmas
+ involving `\<open>setsum\<close>.
+ \<^item> The library has been expanded, with e.g.~support for expressions of
+ the form \<open>f < g + O(h)\<close>.
-Note also since the Big O library includes rules that demonstrate set
-inclusion, to use the automated reasoners effectively with the library
-one should redeclare the theorem \<open>subsetI\<close> as an intro rule,
-rather than as an \<open>intro!\<close> rule, for example, using
-\<^theory_text>\<open>declare subsetI [del, intro]\<close>.
+ Note also since the Big O library includes rules that demonstrate set
+ inclusion, to use the automated reasoners effectively with the library one
+ should redeclare the theorem \<open>subsetI\<close> as an intro rule, rather than as an
+ \<open>intro!\<close> rule, for example, using \<^theory_text>\<open>declare subsetI [del, intro]\<close>.
\<close>
+
subsection \<open>Definitions\<close>
definition bigo :: "('a \<Rightarrow> 'b::linordered_idom) \<Rightarrow> ('a \<Rightarrow> 'b) set" ("(1O'(_'))")
@@ -42,16 +44,16 @@
(\<exists>c. 0 < c \<and> (\<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>))"
apply auto
apply (case_tac "c = 0")
- apply simp
- apply (rule_tac x = "1" in exI)
- apply simp
+ apply simp
+ apply (rule_tac x = "1" in exI)
+ apply simp
apply (rule_tac x = "\<bar>c\<bar>" in exI)
apply auto
apply (subgoal_tac "c * \<bar>f x\<bar> \<le> \<bar>c\<bar> * \<bar>f x\<bar>")
- apply (erule_tac x = x in allE)
- apply force
+ apply (erule_tac x = x in allE)
+ apply force
apply (rule mult_right_mono)
- apply (rule abs_ge_self)
+ apply (rule abs_ge_self)
apply (rule abs_ge_zero)
done
@@ -62,19 +64,19 @@
apply (auto simp add: bigo_alt_def)
apply (rule_tac x = "ca * c" in exI)
apply (rule conjI)
- apply simp
+ apply simp
apply (rule allI)
apply (drule_tac x = "xa" in spec)+
apply (subgoal_tac "ca * \<bar>f xa\<bar> \<le> ca * (c * \<bar>g xa\<bar>)")
- apply (erule order_trans)
- apply (simp add: ac_simps)
+ apply (erule order_trans)
+ apply (simp add: ac_simps)
apply (rule mult_left_mono, assumption)
apply (rule order_less_imp_le, assumption)
done
lemma bigo_refl [intro]: "f \<in> O(f)"
- apply(auto simp add: bigo_def)
- apply(rule_tac x = 1 in exI)
+ apply (auto simp add: bigo_def)
+ apply (rule_tac x = 1 in exI)
apply simp
done
@@ -93,15 +95,15 @@
apply auto
apply (simp add: ring_distribs func_plus)
apply (rule order_trans)
- apply (rule abs_triangle_ineq)
+ apply (rule abs_triangle_ineq)
apply (rule add_mono)
- apply force
+ apply force
apply force
done
lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
apply (rule equalityI)
- apply (rule bigo_plus_self_subset)
+ apply (rule bigo_plus_self_subset)
apply (rule set_zero_plus2)
apply (rule bigo_zero)
done
@@ -112,73 +114,73 @@
apply (subst bigo_pos_const [symmetric])+
apply (rule_tac x = "\<lambda>n. if \<bar>g n\<bar> \<le> \<bar>f n\<bar> then x n else 0" in exI)
apply (rule conjI)
- apply (rule_tac x = "c + c" in exI)
- apply (clarsimp)
- apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>f xa\<bar>")
- apply (erule_tac x = xa in allE)
- apply (erule order_trans)
- apply (simp)
- apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
- apply (erule order_trans)
- apply (simp add: ring_distribs)
- apply (rule mult_left_mono)
- apply (simp add: abs_triangle_ineq)
- apply (simp add: order_less_le)
+ apply (rule_tac x = "c + c" in exI)
+ apply (clarsimp)
+ apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>f xa\<bar>")
+ apply (erule_tac x = xa in allE)
+ apply (erule order_trans)
+ apply (simp)
+ apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
+ apply (erule order_trans)
+ apply (simp add: ring_distribs)
+ apply (rule mult_left_mono)
+ apply (simp add: abs_triangle_ineq)
+ apply (simp add: order_less_le)
apply (rule_tac x = "\<lambda>n. if \<bar>f n\<bar> < \<bar>g n\<bar> then x n else 0" in exI)
apply (rule conjI)
- apply (rule_tac x = "c + c" in exI)
- apply auto
+ apply (rule_tac x = "c + c" in exI)
+ apply auto
apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> (c + c) * \<bar>g xa\<bar>")
- apply (erule_tac x = xa in allE)
- apply (erule order_trans)
- apply simp
+ apply (erule_tac x = xa in allE)
+ apply (erule order_trans)
+ apply simp
apply (subgoal_tac "c * \<bar>f xa + g xa\<bar> \<le> c * (\<bar>f xa\<bar> + \<bar>g xa\<bar>)")
- apply (erule order_trans)
- apply (simp add: ring_distribs)
+ apply (erule order_trans)
+ apply (simp add: ring_distribs)
apply (rule mult_left_mono)
- apply (rule abs_triangle_ineq)
+ apply (rule abs_triangle_ineq)
apply (simp add: order_less_le)
done
lemma bigo_plus_subset2 [intro]: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
apply (subgoal_tac "A + B \<subseteq> O(f) + O(f)")
- apply (erule order_trans)
- apply simp
+ apply (erule order_trans)
+ apply simp
apply (auto del: subsetI simp del: bigo_plus_idemp)
done
lemma bigo_plus_eq: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. 0 \<le> g x \<Longrightarrow> O(f + g) = O(f) + O(g)"
apply (rule equalityI)
- apply (rule bigo_plus_subset)
+ apply (rule bigo_plus_subset)
apply (simp add: bigo_alt_def set_plus_def func_plus)
apply clarify
apply (rule_tac x = "max c ca" in exI)
apply (rule conjI)
- apply (subgoal_tac "c \<le> max c ca")
- apply (erule order_less_le_trans)
- apply assumption
- apply (rule max.cobounded1)
+ apply (subgoal_tac "c \<le> max c ca")
+ apply (erule order_less_le_trans)
+ apply assumption
+ apply (rule max.cobounded1)
apply clarify
apply (drule_tac x = "xa" in spec)+
apply (subgoal_tac "0 \<le> f xa + g xa")
- apply (simp add: ring_distribs)
- apply (subgoal_tac "\<bar>a xa + b xa\<bar> \<le> \<bar>a xa\<bar> + \<bar>b xa\<bar>")
- apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> \<le> max c ca * f xa + max c ca * g xa")
- apply force
- apply (rule add_mono)
- apply (subgoal_tac "c * f xa \<le> max c ca * f xa")
- apply force
- apply (rule mult_right_mono)
- apply (rule max.cobounded1)
- apply assumption
- apply (subgoal_tac "ca * g xa \<le> max c ca * g xa")
- apply force
- apply (rule mult_right_mono)
- apply (rule max.cobounded2)
- apply assumption
- apply (rule abs_triangle_ineq)
+ apply (simp add: ring_distribs)
+ apply (subgoal_tac "\<bar>a xa + b xa\<bar> \<le> \<bar>a xa\<bar> + \<bar>b xa\<bar>")
+ apply (subgoal_tac "\<bar>a xa\<bar> + \<bar>b xa\<bar> \<le> max c ca * f xa + max c ca * g xa")
+ apply force
+ apply (rule add_mono)
+ apply (subgoal_tac "c * f xa \<le> max c ca * f xa")
+ apply force
+ apply (rule mult_right_mono)
+ apply (rule max.cobounded1)
+ apply assumption
+ apply (subgoal_tac "ca * g xa \<le> max c ca * g xa")
+ apply force
+ apply (rule mult_right_mono)
+ apply (rule max.cobounded2)
+ apply assumption
+ apply (rule abs_triangle_ineq)
apply (rule add_nonneg_nonneg)
- apply assumption+
+ apply assumption+
done
lemma bigo_bounded_alt: "\<forall>x. 0 \<le> f x \<Longrightarrow> \<forall>x. f x \<le> c * g x \<Longrightarrow> f \<in> O(g)"
@@ -197,7 +199,7 @@
lemma bigo_bounded2: "\<forall>x. lb x \<le> f x \<Longrightarrow> \<forall>x. f x \<le> lb x + g x \<Longrightarrow> f \<in> lb +o O(g)"
apply (rule set_minus_imp_plus)
apply (rule bigo_bounded)
- apply (auto simp add: fun_Compl_def func_plus)
+ apply (auto simp add: fun_Compl_def func_plus)
apply (drule_tac x = x in spec)+
apply force
done
@@ -218,8 +220,8 @@
lemma bigo_abs3: "O(f) = O(\<lambda>x. \<bar>f x\<bar>)"
apply (rule equalityI)
- apply (rule bigo_elt_subset)
- apply (rule bigo_abs2)
+ apply (rule bigo_elt_subset)
+ apply (rule bigo_abs2)
apply (rule bigo_elt_subset)
apply (rule bigo_abs)
done
@@ -229,13 +231,13 @@
apply (rule set_minus_imp_plus)
apply (subst fun_diff_def)
proof -
- assume a: "f - g \<in> O(h)"
+ assume *: "f - g \<in> O(h)"
have "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) =o O(\<lambda>x. \<bar>\<bar>f x\<bar> - \<bar>g x\<bar>\<bar>)"
by (rule bigo_abs2)
also have "\<dots> \<subseteq> O(\<lambda>x. \<bar>f x - g x\<bar>)"
apply (rule bigo_elt_subset)
apply (rule bigo_bounded)
- apply force
+ apply force
apply (rule allI)
apply (rule abs_triangle_ineq3)
done
@@ -244,23 +246,23 @@
apply (subst fun_diff_def)
apply (rule bigo_abs)
done
- also from a have "\<dots> \<subseteq> O(h)"
+ also from * have "\<dots> \<subseteq> O(h)"
by (rule bigo_elt_subset)
finally show "(\<lambda>x. \<bar>f x\<bar> - \<bar>g x\<bar>) \<in> O(h)".
qed
lemma bigo_abs5: "f =o O(g) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) =o O(g)"
- by (unfold bigo_def, auto)
+ by (auto simp: bigo_def)
-lemma bigo_elt_subset2 [intro]: "f \<in> g +o O(h) \<Longrightarrow> O(f) \<subseteq> O(g) + O(h)"
+lemma bigo_elt_subset2 [intro]:
+ assumes *: "f \<in> g +o O(h)"
+ shows "O(f) \<subseteq> O(g) + O(h)"
proof -
- assume "f \<in> g +o O(h)"
- also have "\<dots> \<subseteq> O(g) + O(h)"
+ note *
+ also have "g +o O(h) \<subseteq> O(g) + O(h)"
by (auto del: subsetI)
also have "\<dots> = O(\<lambda>x. \<bar>g x\<bar>) + O(\<lambda>x. \<bar>h x\<bar>)"
- apply (subst bigo_abs3 [symmetric])+
- apply (rule refl)
- done
+ by (subst bigo_abs3 [symmetric])+ (rule refl)
also have "\<dots> = O((\<lambda>x. \<bar>g x\<bar>) + (\<lambda>x. \<bar>h x\<bar>))"
by (rule bigo_plus_eq [symmetric]) auto
finally have "f \<in> \<dots>" .
@@ -280,11 +282,11 @@
apply (rule allI)
apply (erule_tac x = x in allE)+
apply (subgoal_tac "c * ca * \<bar>f x * g x\<bar> = (c * \<bar>f x\<bar>) * (ca * \<bar>g x\<bar>)")
- apply (erule ssubst)
- apply (subst abs_mult)
- apply (rule mult_mono)
- apply assumption+
- apply auto
+ apply (erule ssubst)
+ apply (subst abs_mult)
+ apply (rule mult_mono)
+ apply assumption+
+ apply auto
apply (simp add: ac_simps abs_mult)
done
@@ -294,14 +296,14 @@
apply auto
apply (drule_tac x = x in spec)
apply (subgoal_tac "\<bar>f x\<bar> * \<bar>b x\<bar> \<le> \<bar>f x\<bar> * (c * \<bar>g x\<bar>)")
- apply (force simp add: ac_simps)
+ apply (force simp add: ac_simps)
apply (rule mult_left_mono, assumption)
apply (rule abs_ge_zero)
done
lemma bigo_mult3: "f \<in> O(h) \<Longrightarrow> g \<in> O(j) \<Longrightarrow> f * g \<in> O(h * j)"
apply (rule subsetD)
- apply (rule bigo_mult)
+ apply (rule bigo_mult)
apply (erule set_times_intro, assumption)
done
@@ -309,7 +311,7 @@
apply (drule set_plus_imp_minus)
apply (rule set_minus_imp_plus)
apply (drule bigo_mult3 [where g = g and j = g])
- apply (auto simp add: algebra_simps)
+ apply (auto simp add: algebra_simps)
done
lemma bigo_mult5:
@@ -339,28 +341,25 @@
finally show "h \<in> f *o O(g)" .
qed
-lemma bigo_mult6:
- fixes f :: "'a \<Rightarrow> 'b::linordered_field"
- shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
+lemma bigo_mult6: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = f *o O(g)"
+ for f :: "'a \<Rightarrow> 'b::linordered_field"
apply (rule equalityI)
- apply (erule bigo_mult5)
+ apply (erule bigo_mult5)
apply (rule bigo_mult2)
done
-lemma bigo_mult7:
- fixes f :: "'a \<Rightarrow> 'b::linordered_field"
- shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
+lemma bigo_mult7: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) \<subseteq> O(f) * O(g)"
+ for f :: "'a \<Rightarrow> 'b::linordered_field"
apply (subst bigo_mult6)
- apply assumption
+ apply assumption
apply (rule set_times_mono3)
apply (rule bigo_refl)
done
-lemma bigo_mult8:
- fixes f :: "'a \<Rightarrow> 'b::linordered_field"
- shows "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
+lemma bigo_mult8: "\<forall>x. f x \<noteq> 0 \<Longrightarrow> O(f * g) = O(f) * O(g)"
+ for f :: "'a \<Rightarrow> 'b::linordered_field"
apply (rule equalityI)
- apply (erule bigo_mult7)
+ apply (erule bigo_mult7)
apply (rule bigo_mult)
done
@@ -377,65 +376,63 @@
lemma bigo_minus3: "O(- f) = O(f)"
by (auto simp add: bigo_def fun_Compl_def)
-lemma bigo_plus_absorb_lemma1: "f \<in> O(g) \<Longrightarrow> f +o O(g) \<subseteq> O(g)"
+lemma bigo_plus_absorb_lemma1:
+ assumes *: "f \<in> O(g)"
+ shows "f +o O(g) \<subseteq> O(g)"
proof -
- assume a: "f \<in> O(g)"
- show "f +o O(g) \<subseteq> O(g)"
+ have "f \<in> O(f)" by auto
+ then have "f +o O(g) \<subseteq> O(f) + O(g)"
+ by (auto del: subsetI)
+ also have "\<dots> \<subseteq> O(g) + O(g)"
proof -
- have "f \<in> O(f)" by auto
- then have "f +o O(g) \<subseteq> O(f) + O(g)"
+ from * have "O(f) \<subseteq> O(g)"
by (auto del: subsetI)
- also have "\<dots> \<subseteq> O(g) + O(g)"
- proof -
- from a have "O(f) \<subseteq> O(g)" by (auto del: subsetI)
- then show ?thesis by (auto del: subsetI)
- qed
- also have "\<dots> \<subseteq> O(g)" by simp
- finally show ?thesis .
+ then show ?thesis
+ by (auto del: subsetI)
qed
+ also have "\<dots> \<subseteq> O(g)" by simp
+ finally show ?thesis .
qed
-lemma bigo_plus_absorb_lemma2: "f \<in> O(g) \<Longrightarrow> O(g) \<subseteq> f +o O(g)"
+lemma bigo_plus_absorb_lemma2:
+ assumes *: "f \<in> O(g)"
+ shows "O(g) \<subseteq> f +o O(g)"
proof -
- assume a: "f \<in> O(g)"
- show "O(g) \<subseteq> f +o O(g)"
- proof -
- from a have "- f \<in> O(g)"
- by auto
- then have "- f +o O(g) \<subseteq> O(g)"
- by (elim bigo_plus_absorb_lemma1)
- then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)"
- by auto
- also have "f +o (- f +o O(g)) = O(g)"
- by (simp add: set_plus_rearranges)
- finally show ?thesis .
- qed
+ from * have "- f \<in> O(g)"
+ by auto
+ then have "- f +o O(g) \<subseteq> O(g)"
+ by (elim bigo_plus_absorb_lemma1)
+ then have "f +o (- f +o O(g)) \<subseteq> f +o O(g)"
+ by auto
+ also have "f +o (- f +o O(g)) = O(g)"
+ by (simp add: set_plus_rearranges)
+ finally show ?thesis .
qed
lemma bigo_plus_absorb [simp]: "f \<in> O(g) \<Longrightarrow> f +o O(g) = O(g)"
apply (rule equalityI)
- apply (erule bigo_plus_absorb_lemma1)
+ apply (erule bigo_plus_absorb_lemma1)
apply (erule bigo_plus_absorb_lemma2)
done
lemma bigo_plus_absorb2 [intro]: "f \<in> O(g) \<Longrightarrow> A \<subseteq> O(g) \<Longrightarrow> f +o A \<subseteq> O(g)"
apply (subgoal_tac "f +o A \<subseteq> f +o O(g)")
- apply force+
+ apply force+
done
lemma bigo_add_commute_imp: "f \<in> g +o O(h) \<Longrightarrow> g \<in> f +o O(h)"
apply (subst set_minus_plus [symmetric])
apply (subgoal_tac "g - f = - (f - g)")
- apply (erule ssubst)
- apply (rule bigo_minus)
- apply (subst set_minus_plus)
- apply assumption
+ apply (erule ssubst)
+ apply (rule bigo_minus)
+ apply (subst set_minus_plus)
+ apply assumption
apply (simp add: ac_simps)
done
lemma bigo_add_commute: "f \<in> g +o O(h) \<longleftrightarrow> g \<in> f +o O(h)"
apply (rule iffI)
- apply (erule bigo_add_commute_imp)+
+ apply (erule bigo_add_commute_imp)+
done
lemma bigo_const1: "(\<lambda>x. c) \<in> O(\<lambda>x. 1)"
@@ -446,27 +443,24 @@
apply (rule bigo_const1)
done
-lemma bigo_const3:
- fixes c :: "'a::linordered_field"
- shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
+lemma bigo_const3: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. 1) \<in> O(\<lambda>x. c)"
+ for c :: "'a::linordered_field"
apply (simp add: bigo_def)
apply (rule_tac x = "\<bar>inverse c\<bar>" in exI)
apply (simp add: abs_mult [symmetric])
done
-lemma bigo_const4:
- fixes c :: "'a::linordered_field"
- shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
+lemma bigo_const4: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. 1) \<subseteq> O(\<lambda>x. c)"
+ for c :: "'a::linordered_field"
apply (rule bigo_elt_subset)
apply (rule bigo_const3)
apply assumption
done
-lemma bigo_const [simp]:
- fixes c :: "'a::linordered_field"
- shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
+lemma bigo_const [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c) = O(\<lambda>x. 1)"
+ for c :: "'a::linordered_field"
apply (rule equalityI)
- apply (rule bigo_const2)
+ apply (rule bigo_const2)
apply (rule bigo_const4)
apply assumption
done
@@ -482,37 +476,33 @@
apply (rule bigo_const_mult1)
done
-lemma bigo_const_mult3:
- fixes c :: "'a::linordered_field"
- shows "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
+lemma bigo_const_mult3: "c \<noteq> 0 \<Longrightarrow> f \<in> O(\<lambda>x. c * f x)"
+ for c :: "'a::linordered_field"
apply (simp add: bigo_def)
apply (rule_tac x = "\<bar>inverse c\<bar>" in exI)
apply (simp add: abs_mult mult.assoc [symmetric])
done
-lemma bigo_const_mult4:
- fixes c :: "'a::linordered_field"
- shows "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
+lemma bigo_const_mult4: "c \<noteq> 0 \<Longrightarrow> O(f) \<subseteq> O(\<lambda>x. c * f x)"
+ for c :: "'a::linordered_field"
apply (rule bigo_elt_subset)
apply (rule bigo_const_mult3)
apply assumption
done
-lemma bigo_const_mult [simp]:
- fixes c :: "'a::linordered_field"
- shows "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
+lemma bigo_const_mult [simp]: "c \<noteq> 0 \<Longrightarrow> O(\<lambda>x. c * f x) = O(f)"
+ for c :: "'a::linordered_field"
apply (rule equalityI)
- apply (rule bigo_const_mult2)
+ apply (rule bigo_const_mult2)
apply (erule bigo_const_mult4)
done
-lemma bigo_const_mult5 [simp]:
- fixes c :: "'a::linordered_field"
- shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
+lemma bigo_const_mult5 [simp]: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) *o O(f) = O(f)"
+ for c :: "'a::linordered_field"
apply (auto del: subsetI)
- apply (rule order_trans)
- apply (rule bigo_mult2)
- apply (simp add: func_times)
+ apply (rule order_trans)
+ apply (rule bigo_mult2)
+ apply (simp add: func_times)
apply (auto intro!: simp add: bigo_def elt_set_times_def func_times)
apply (rule_tac x = "\<lambda>y. inverse c * x y" in exI)
apply (simp add: mult.assoc [symmetric] abs_mult)
@@ -525,18 +515,19 @@
apply (rule_tac x = "ca * \<bar>c\<bar>" in exI)
apply (rule allI)
apply (subgoal_tac "ca * \<bar>c\<bar> * \<bar>f x\<bar> = \<bar>c\<bar> * (ca * \<bar>f x\<bar>)")
- apply (erule ssubst)
- apply (subst abs_mult)
- apply (rule mult_left_mono)
- apply (erule spec)
- apply simp
- apply(simp add: ac_simps)
+ apply (erule ssubst)
+ apply (subst abs_mult)
+ apply (rule mult_left_mono)
+ apply (erule spec)
+ apply simp
+ apply (simp add: ac_simps)
done
-lemma bigo_const_mult7 [intro]: "f =o O(g) \<Longrightarrow> (\<lambda>x. c * f x) =o O(g)"
+lemma bigo_const_mult7 [intro]:
+ assumes *: "f =o O(g)"
+ shows "(\<lambda>x. c * f x) =o O(g)"
proof -
- assume "f =o O(g)"
- then have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
+ from * have "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
by auto
also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
by (simp add: func_times)
@@ -546,10 +537,9 @@
qed
lemma bigo_compose1: "f =o O(g) \<Longrightarrow> (\<lambda>x. f (k x)) =o O(\<lambda>x. g (k x))"
- unfolding bigo_def by auto
+ by (auto simp: bigo_def)
-lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow>
- (\<lambda>x. f (k x)) =o (\<lambda>x. g (k x)) +o O(\<lambda>x. h(k x))"
+lemma bigo_compose2: "f =o g +o O(h) \<Longrightarrow> (\<lambda>x. f (k x)) =o (\<lambda>x. g (k x)) +o O(\<lambda>x. h(k x))"
apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
apply (drule bigo_compose1)
apply (simp add: fun_diff_def)
@@ -564,21 +554,21 @@
apply (auto simp add: bigo_def)
apply (rule_tac x = "\<bar>c\<bar>" in exI)
apply (subst abs_of_nonneg) back back
- apply (rule setsum_nonneg)
- apply force
+ apply (rule setsum_nonneg)
+ apply force
apply (subst setsum_right_distrib)
apply (rule allI)
apply (rule order_trans)
- apply (rule setsum_abs)
+ apply (rule setsum_abs)
apply (rule setsum_mono)
apply (rule order_trans)
- apply (drule spec)+
- apply (drule bspec)+
- apply assumption+
- apply (drule bspec)
- apply assumption+
+ apply (drule spec)+
+ apply (drule bspec)+
+ apply assumption+
+ apply (drule bspec)
+ apply assumption+
apply (rule mult_right_mono)
- apply (rule abs_ge_self)
+ apply (rule abs_ge_self)
apply force
done
@@ -586,7 +576,7 @@
\<exists>c. \<forall>x y. \<bar>f x y\<bar> \<le> c * h x y \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. f x y) =o O(\<lambda>x. \<Sum>y \<in> A x. h x y)"
apply (rule bigo_setsum_main)
- apply force
+ apply force
apply clarsimp
apply (rule_tac x = c in exI)
apply force
@@ -600,8 +590,8 @@
lemma bigo_setsum3: "f =o O(h) \<Longrightarrow>
(\<lambda>x. \<Sum>y \<in> A x. l x y * f (k x y)) =o O(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)"
apply (rule bigo_setsum1)
- apply (rule allI)+
- apply (rule abs_ge_zero)
+ apply (rule allI)+
+ apply (rule abs_ge_zero)
apply (unfold bigo_def)
apply auto
apply (rule_tac x = c in exI)
@@ -609,7 +599,7 @@
apply (subst abs_mult)+
apply (subst mult.left_commute)
apply (rule mult_left_mono)
- apply (erule spec)
+ apply (erule spec)
apply (rule abs_ge_zero)
done
@@ -632,13 +622,13 @@
O(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y))"
apply (subgoal_tac "(\<lambda>x. \<Sum>y \<in> A x. l x y * h (k x y)) =
(\<lambda>x. \<Sum>y \<in> A x. \<bar>l x y * h (k x y)\<bar>)")
- apply (erule ssubst)
- apply (erule bigo_setsum3)
+ apply (erule ssubst)
+ apply (erule bigo_setsum3)
apply (rule ext)
apply (rule setsum.cong)
- apply (rule refl)
+ apply (rule refl)
apply (subst abs_of_nonneg)
- apply auto
+ apply auto
done
lemma bigo_setsum6: "f =o g +o O(h) \<Longrightarrow> \<forall>x y. 0 \<le> l x y \<Longrightarrow>
@@ -651,9 +641,9 @@
apply (subst setsum_subtractf [symmetric])
apply (subst right_diff_distrib [symmetric])
apply (rule bigo_setsum5)
- apply (subst fun_diff_def [symmetric])
- apply (drule set_plus_imp_minus)
- apply auto
+ apply (subst fun_diff_def [symmetric])
+ apply (drule set_plus_imp_minus)
+ apply auto
done
@@ -662,25 +652,24 @@
lemma bigo_useful_intro: "A \<subseteq> O(f) \<Longrightarrow> B \<subseteq> O(f) \<Longrightarrow> A + B \<subseteq> O(f)"
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_mono2)
- apply assumption+
+ apply assumption+
done
lemma bigo_useful_add: "f =o O(h) \<Longrightarrow> g =o O(h) \<Longrightarrow> f + g =o O(h)"
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_intro)
- apply assumption+
+ apply assumption+
done
-lemma bigo_useful_const_mult:
- fixes c :: "'a::linordered_field"
- shows "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
+lemma bigo_useful_const_mult: "c \<noteq> 0 \<Longrightarrow> (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
+ for c :: "'a::linordered_field"
apply (rule subsetD)
- apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
- apply assumption
- apply (rule bigo_const_mult6)
+ apply (subgoal_tac "(\<lambda>x. 1 / c) *o O(h) \<subseteq> O(h)")
+ apply assumption
+ apply (rule bigo_const_mult6)
apply (subgoal_tac "f = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
- apply (erule ssubst)
- apply (erule set_times_intro2)
+ apply (erule ssubst)
+ apply (erule set_times_intro2)
apply (simp add: func_times)
done
@@ -690,10 +679,10 @@
apply (rule_tac x = c in exI)
apply auto
apply (case_tac "x = 0")
- apply simp
+ apply simp
apply (subgoal_tac "x = Suc (x - 1)")
- apply (erule ssubst) back
- apply (erule spec)
+ apply (erule ssubst) back
+ apply (erule spec)
apply simp
done
@@ -702,10 +691,10 @@
f 0 = g 0 \<Longrightarrow> f =o g +o O(h)"
apply (rule set_minus_imp_plus)
apply (rule bigo_fix)
- apply (subst fun_diff_def)
- apply (subst fun_diff_def [symmetric])
- apply (rule set_plus_imp_minus)
- apply simp
+ apply (subst fun_diff_def)
+ apply (subst fun_diff_def [symmetric])
+ apply (rule set_plus_imp_minus)
+ apply simp
apply (simp add: fun_diff_def)
done
@@ -721,7 +710,7 @@
apply (rule_tac x = c in exI)
apply (rule allI)
apply (rule order_trans)
- apply (erule spec)+
+ apply (erule spec)+
done
lemma bigo_lesseq2: "f =o O(h) \<Longrightarrow> \<forall>x. \<bar>g x\<bar> \<le> f x \<Longrightarrow> g =o O(h)"
@@ -729,7 +718,7 @@
apply (rule allI)
apply (drule_tac x = x in spec)
apply (rule order_trans)
- apply assumption
+ apply assumption
apply (rule abs_ge_self)
done
@@ -737,7 +726,7 @@
apply (erule bigo_lesseq2)
apply (rule allI)
apply (subst abs_of_nonneg)
- apply (erule spec)+
+ apply (erule spec)+
done
lemma bigo_lesseq4: "f =o O(h) \<Longrightarrow>
@@ -745,75 +734,72 @@
apply (erule bigo_lesseq1)
apply (rule allI)
apply (subst abs_of_nonneg)
- apply (erule spec)+
+ apply (erule spec)+
done
lemma bigo_lesso1: "\<forall>x. f x \<le> g x \<Longrightarrow> f <o g =o O(h)"
apply (unfold lesso_def)
apply (subgoal_tac "(\<lambda>x. max (f x - g x) 0) = 0")
- apply (erule ssubst)
- apply (rule bigo_zero)
+ apply (erule ssubst)
+ apply (rule bigo_zero)
apply (unfold func_zero)
apply (rule ext)
apply (simp split: split_max)
done
-lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow>
- \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. k x \<le> f x \<Longrightarrow> k <o g =o O(h)"
+lemma bigo_lesso2: "f =o g +o O(h) \<Longrightarrow> \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. k x \<le> f x \<Longrightarrow> k <o g =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq4)
- apply (erule set_plus_imp_minus)
- apply (rule allI)
- apply (rule max.cobounded2)
+ apply (erule set_plus_imp_minus)
+ apply (rule allI)
+ apply (rule max.cobounded2)
apply (rule allI)
apply (subst fun_diff_def)
apply (case_tac "0 \<le> k x - g x")
- apply simp
- apply (subst abs_of_nonneg)
- apply (drule_tac x = x in spec) back
- apply (simp add: algebra_simps)
- apply (subst diff_conv_add_uminus)+
- apply (rule add_right_mono)
- apply (erule spec)
+ apply simp
+ apply (subst abs_of_nonneg)
+ apply (drule_tac x = x in spec) back
+ apply (simp add: algebra_simps)
+ apply (subst diff_conv_add_uminus)+
+ apply (rule add_right_mono)
+ apply (erule spec)
apply (rule order_trans)
- prefer 2
- apply (rule abs_ge_zero)
+ prefer 2
+ apply (rule abs_ge_zero)
apply (simp add: algebra_simps)
done
-lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
- \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. g x \<le> k x \<Longrightarrow> f <o k =o O(h)"
+lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow> \<forall>x. 0 \<le> k x \<Longrightarrow> \<forall>x. g x \<le> k x \<Longrightarrow> f <o k =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq4)
- apply (erule set_plus_imp_minus)
- apply (rule allI)
- apply (rule max.cobounded2)
+ apply (erule set_plus_imp_minus)
+ apply (rule allI)
+ apply (rule max.cobounded2)
apply (rule allI)
apply (subst fun_diff_def)
apply (case_tac "0 \<le> f x - k x")
- apply simp
- apply (subst abs_of_nonneg)
- apply (drule_tac x = x in spec) back
- apply (simp add: algebra_simps)
- apply (subst diff_conv_add_uminus)+
- apply (rule add_left_mono)
- apply (rule le_imp_neg_le)
- apply (erule spec)
+ apply simp
+ apply (subst abs_of_nonneg)
+ apply (drule_tac x = x in spec) back
+ apply (simp add: algebra_simps)
+ apply (subst diff_conv_add_uminus)+
+ apply (rule add_left_mono)
+ apply (rule le_imp_neg_le)
+ apply (erule spec)
apply (rule order_trans)
- prefer 2
- apply (rule abs_ge_zero)
+ prefer 2
+ apply (rule abs_ge_zero)
apply (simp add: algebra_simps)
done
-lemma bigo_lesso4:
- fixes k :: "'a \<Rightarrow> 'b::linordered_field"
- shows "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
+lemma bigo_lesso4: "f <o g =o O(k) \<Longrightarrow> g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
+ for k :: "'a \<Rightarrow> 'b::linordered_field"
apply (unfold lesso_def)
apply (drule set_plus_imp_minus)
apply (drule bigo_abs5) back
apply (simp add: fun_diff_def)
apply (drule bigo_useful_add)
- apply assumption
+ apply assumption
apply (erule bigo_lesseq2) back
apply (rule allI)
apply (auto simp add: func_plus fun_diff_def algebra_simps split: split_max abs_split)
@@ -826,7 +812,7 @@
apply (rule allI)
apply (drule_tac x = x in spec)
apply (subgoal_tac "\<bar>max (f x - g x) 0\<bar> = max (f x - g x) 0")
- apply (clarsimp simp add: algebra_simps)
+ apply (clarsimp simp add: algebra_simps)
apply (rule abs_of_nonneg)
apply (rule max.cobounded2)
done
@@ -834,39 +820,41 @@
lemma lesso_add: "f <o g =o O(h) \<Longrightarrow> k <o l =o O(h) \<Longrightarrow> (f + k) <o (g + l) =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq3)
- apply (erule bigo_useful_add)
- apply assumption
- apply (force split: split_max)
+ apply (erule bigo_useful_add)
+ apply assumption
+ apply (force split: split_max)
apply (auto split: split_max simp add: func_plus)
done
-lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> (0::real)"
+lemma bigo_LIMSEQ1: "f =o O(g) \<Longrightarrow> g \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> 0"
+ for f g :: "nat \<Rightarrow> real"
apply (simp add: LIMSEQ_iff bigo_alt_def)
apply clarify
apply (drule_tac x = "r / c" in spec)
apply (drule mp)
- apply simp
+ apply simp
apply clarify
apply (rule_tac x = no in exI)
apply (rule allI)
apply (drule_tac x = n in spec)+
apply (rule impI)
apply (drule mp)
- apply assumption
+ apply assumption
apply (rule order_le_less_trans)
- apply assumption
+ apply assumption
apply (rule order_less_le_trans)
- apply (subgoal_tac "c * \<bar>g n\<bar> < c * (r / c)")
- apply assumption
- apply (erule mult_strict_left_mono)
- apply assumption
+ apply (subgoal_tac "c * \<bar>g n\<bar> < c * (r / c)")
+ apply assumption
+ apply (erule mult_strict_left_mono)
+ apply assumption
apply simp
done
-lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> g \<longlonglongrightarrow> (a::real)"
+lemma bigo_LIMSEQ2: "f =o g +o O(h) \<Longrightarrow> h \<longlonglongrightarrow> 0 \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> g \<longlonglongrightarrow> a"
+ for f g h :: "nat \<Rightarrow> real"
apply (drule set_plus_imp_minus)
apply (drule bigo_LIMSEQ1)
- apply assumption
+ apply assumption
apply (simp only: fun_diff_def)
apply (erule Lim_transform2)
apply assumption
--- a/src/HOL/Library/Convex.thy Wed Jul 13 17:20:21 2016 +0100
+++ b/src/HOL/Library/Convex.thy Wed Jul 13 20:48:18 2016 +0200
@@ -6,7 +6,7 @@
section \<open>Convexity in real vector spaces\<close>
theory Convex
-imports Product_Vector
+ imports Product_Vector
begin
subsection \<open>Convexity\<close>
@@ -24,24 +24,27 @@
shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
using assms unfolding convex_def by fast
-lemma convex_alt:
- "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
+lemma convex_alt: "convex s \<longleftrightarrow> (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. 0 \<le> u \<and> u \<le> 1 \<longrightarrow> ((1 - u) *\<^sub>R x + u *\<^sub>R y) \<in> s)"
(is "_ \<longleftrightarrow> ?alt")
proof
- assume alt[rule_format]: ?alt
- {
- fix x y and u v :: real
- assume mem: "x \<in> s" "y \<in> s"
- assume "0 \<le> u" "0 \<le> v"
- moreover
- assume "u + v = 1"
- then have "u = 1 - v" by auto
- ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
- using alt[OF mem] by auto
- }
- then show "convex s"
- unfolding convex_def by auto
-qed (auto simp: convex_def)
+ show "convex s" if alt: ?alt
+ proof -
+ {
+ fix x y and u v :: real
+ assume mem: "x \<in> s" "y \<in> s"
+ assume "0 \<le> u" "0 \<le> v"
+ moreover
+ assume "u + v = 1"
+ then have "u = 1 - v" by auto
+ ultimately have "u *\<^sub>R x + v *\<^sub>R y \<in> s"
+ using alt [rule_format, OF mem] by auto
+ }
+ then show ?thesis
+ unfolding convex_def by auto
+ qed
+ show ?alt if "convex s"
+ using that by (auto simp: convex_def)
+qed
lemma convexD_alt:
assumes "convex s" "a \<in> s" "b \<in> s" "0 \<le> u" "u \<le> 1"
@@ -53,7 +56,7 @@
shows "((u/(u+v)) *\<^sub>R x + (v/(u+v)) *\<^sub>R y) \<in> S"
apply (rule convexD)
using assms
- apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
+ apply (simp_all add: zero_le_divide_iff add_divide_distrib [symmetric])
done
lemma convex_empty[intro,simp]: "convex {}"
@@ -270,12 +273,12 @@
case False
then show ?thesis
using *[rule_format, of "{x, y}" "\<lambda> z. if z = x then 1 - \<mu> else \<mu>"] **
- by auto
+ by auto
next
case True
then show ?thesis
using *[rule_format, of "{x, y}" "\<lambda> z. 1"] **
- by (auto simp: field_simps real_vector.scale_left_diff_distrib)
+ by (auto simp: field_simps real_vector.scale_left_diff_distrib)
qed
qed
qed
@@ -293,8 +296,8 @@
have if_distrib_arg: "\<And>P f g x. (if P then f else g) x = (if P then f x else g x)"
by simp
show "(\<Sum>x\<in>t. u x *\<^sub>R x) \<in> s"
- using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
- by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
+ using sum[THEN spec[where x="\<lambda>x. if x\<in>t then u x else 0"]] as *
+ by (auto simp: assms setsum.If_cases if_distrib if_distrib_arg)
qed (erule_tac x=s in allE, erule_tac x=u in allE, auto)
@@ -306,39 +309,45 @@
lemma convex_onI [intro?]:
assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- shows "convex_on A f"
+ f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ shows "convex_on A f"
unfolding convex_on_def
proof clarify
- fix x y u v assume A: "x \<in> A" "y \<in> A" "(u::real) \<ge> 0" "v \<ge> 0" "u + v = 1"
- from A(5) have [simp]: "v = 1 - u" by (simp add: algebra_simps)
- from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" using assms[of u y x]
+ fix x y
+ fix u v :: real
+ assume A: "x \<in> A" "y \<in> A" "u \<ge> 0" "v \<ge> 0" "u + v = 1"
+ from A(5) have [simp]: "v = 1 - u"
+ by (simp add: algebra_simps)
+ from A(1-4) show "f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
+ using assms[of u y x]
by (cases "u = 0 \<or> u = 1") (auto simp: algebra_simps)
qed
lemma convex_on_linorderI [intro?]:
fixes A :: "('a::{linorder,real_vector}) set"
assumes "\<And>t x y. t > 0 \<Longrightarrow> t < 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x < y \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- shows "convex_on A f"
+ f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ shows "convex_on A f"
proof
- fix t x y assume A: "x \<in> A" "y \<in> A" "(t::real) > 0" "t < 1"
- with assms[of t x y] assms[of "1 - t" y x]
+ fix x y
+ fix t :: real
+ assume A: "x \<in> A" "y \<in> A" "t > 0" "t < 1"
+ with assms [of t x y] assms [of "1 - t" y x]
show "f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
by (cases x y rule: linorder_cases) (auto simp: algebra_simps)
qed
lemma convex_onD:
assumes "convex_on A f"
- shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- using assms unfolding convex_on_def by auto
+ shows "\<And>t x y. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow> x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow>
+ f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ using assms by (auto simp: convex_on_def)
lemma convex_onD_Icc:
assumes "convex_on {x..y} f" "x \<le> (y :: _ :: {real_vector,preorder})"
- shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
- f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
- using assms(2) by (intro convex_onD[OF assms(1)]) simp_all
+ shows "\<And>t. t \<ge> 0 \<Longrightarrow> t \<le> 1 \<Longrightarrow>
+ f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y"
+ using assms(2) by (intro convex_onD [OF assms(1)]) simp_all
lemma convex_on_subset: "convex_on t f \<Longrightarrow> s \<subseteq> t \<Longrightarrow> convex_on s f"
unfolding convex_on_def by auto
@@ -370,7 +379,8 @@
and "convex_on s f"
shows "convex_on s (\<lambda>x. c * f x)"
proof -
- have *: "\<And>u c fx v fy :: real. u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
+ have *: "u * (c * fx) + v * (c * fy) = c * (u * fx + v * fy)"
+ for u c fx v fy :: real
by (simp add: field_simps)
show ?thesis using assms(2) and mult_left_mono [OF _ assms(1)]
unfolding convex_on_def and * by auto
@@ -517,20 +527,24 @@
assume *: "y > 0" "x > 0" "\<mu> \<ge> 0" "\<mu> \<le> 1"
{
assume "\<mu> = 0"
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y" by simp
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using * by simp
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y = y"
+ by simp
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+ using * by simp
}
moreover
{
assume "\<mu> = 1"
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using * by simp
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+ using * by simp
}
moreover
{
assume "\<mu> \<noteq> 1" "\<mu> \<noteq> 0"
- then have "\<mu> > 0" "(1 - \<mu>) > 0" using * by auto
- then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0" using *
- by (auto simp: add_pos_pos)
+ then have "\<mu> > 0" "(1 - \<mu>) > 0"
+ using * by auto
+ then have "\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y > 0"
+ using * by (auto simp: add_pos_pos)
}
ultimately show "(1 - \<mu>) *\<^sub>R y + \<mu> *\<^sub>R x > 0"
by fastforce
@@ -550,11 +564,14 @@
using assms
proof (induct s arbitrary: a rule: finite_ne_induct)
case (singleton i)
- then have ai: "a i = 1" by auto
- then show ?case by auto
+ then have ai: "a i = 1"
+ by auto
+ then show ?case
+ by auto
next
case (insert i s)
- then have "convex_on C f" by simp
+ then have "convex_on C f"
+ by simp
from this[unfolded convex_on_def, rule_format]
have conv: "\<And>x y \<mu>. x \<in> C \<Longrightarrow> y \<in> C \<Longrightarrow> 0 \<le> \<mu> \<Longrightarrow> \<mu> \<le> 1 \<Longrightarrow>
f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
@@ -593,8 +610,7 @@
unfolding setsum_divide_distrib by simp
have "convex C" using insert by auto
then have asum: "(\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<in> C"
- using insert convex_setsum[OF \<open>finite s\<close>
- \<open>convex C\<close> a1 a_nonneg] by auto
+ using insert convex_setsum [OF \<open>finite s\<close> \<open>convex C\<close> a1 a_nonneg] by auto
have asum_le: "f (\<Sum> j \<in> s. ?a j *\<^sub>R y j) \<le> (\<Sum> j \<in> s. ?a j * f (y j))"
using a_nonneg a1 insert by blast
have "f (\<Sum> j \<in> insert i s. a j *\<^sub>R y j) = f ((\<Sum> j \<in> s. a j *\<^sub>R y j) + a i *\<^sub>R y i)"
@@ -611,10 +627,12 @@
using conv[of "y i" "(\<Sum> j \<in> s. ?a j *\<^sub>R y j)" "a i", OF yai(1) asum yai(2) ai1]
by (auto simp: add.commute)
also have "\<dots> \<le> (1 - a i) * (\<Sum> j \<in> s. ?a j * f (y j)) + a i * f (y i)"
- using add_right_mono[OF mult_left_mono[of _ _ "1 - a i",
- OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"] by simp
+ using add_right_mono [OF mult_left_mono [of _ _ "1 - a i",
+ OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
+ by simp
also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
- unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"] using i0 by auto
+ unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
+ using i0 by auto
also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
using i0 by auto
also have "\<dots> = (\<Sum> j \<in> insert i s. a j * f (y j))"
@@ -635,9 +653,9 @@
fix \<mu> :: real
assume *: "convex_on C f" "x \<in> C" "y \<in> C" "0 \<le> \<mu>" "\<mu> \<le> 1"
from this[unfolded convex_on_def, rule_format]
- have "\<And>u v. 0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y"
+ have "0 \<le> u \<Longrightarrow> 0 \<le> v \<Longrightarrow> u + v = 1 \<Longrightarrow> f (u *\<^sub>R x + v *\<^sub>R y) \<le> u * f x + v * f y" for u v
by auto
- from this[of "\<mu>" "1 - \<mu>", simplified] *
+ from this [of "\<mu>" "1 - \<mu>", simplified] *
show "f (\<mu> *\<^sub>R x + (1 - \<mu>) *\<^sub>R y) \<le> \<mu> * f x + (1 - \<mu>) * f y"
by auto
next
@@ -701,8 +719,8 @@
using * unfolding convex_alt by fastforce
have geq: "\<mu> * (f x - f ?x) + (1 - \<mu>) * (f y - f ?x) \<ge>
\<mu> * f' ?x * (x - ?x) + (1 - \<mu>) * f' ?x * (y - ?x)"
- using add_mono[OF mult_left_mono[OF leq[OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
- mult_left_mono[OF leq[OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
+ using add_mono [OF mult_left_mono [OF leq [OF xpos *(2)] \<open>\<mu> \<ge> 0\<close>]
+ mult_left_mono [OF leq [OF xpos *(3)] \<open>1 - \<mu> \<ge> 0\<close>]]
by auto
then have "\<mu> * f x + (1 - \<mu>) * f y - f ?x \<ge> 0"
by (auto simp: field_simps)
@@ -728,14 +746,14 @@
have "?\<mu> * x + (1 - ?\<mu>) * y = (y - z) * x / (y - x) + (1 - (y - z) / (y - x)) * y"
by (auto simp: field_simps)
also have "\<dots> = ((y - z) * x + (y - x - (y - z)) * y) / (y - x)"
- using assms unfolding add_divide_distrib by (auto simp: field_simps)
+ using assms by (simp only: add_divide_distrib) (auto simp: field_simps)
also have "\<dots> = z"
using assms by (auto simp: field_simps)
finally show ?thesis
using comb by auto
qed
- show "z \<in> C" using z less assms
- unfolding atLeastAtMost_iff le_less by auto
+ show "z \<in> C"
+ using z less assms by (auto simp: le_less)
qed
lemma f''_imp_f':
@@ -744,20 +762,21 @@
and f': "\<And>x. x \<in> C \<Longrightarrow> DERIV f x :> (f' x)"
and f'': "\<And>x. x \<in> C \<Longrightarrow> DERIV f' x :> (f'' x)"
and pos: "\<And>x. x \<in> C \<Longrightarrow> f'' x \<ge> 0"
- and "x \<in> C" "y \<in> C"
+ and x: "x \<in> C"
+ and y: "y \<in> C"
shows "f' x * (y - x) \<le> f y - f x"
using assms
proof -
- {
- fix x y :: real
- assume *: "x \<in> C" "y \<in> C" "y > x"
- then have ge: "y - x > 0" "y - x \<ge> 0"
+ have less_imp: "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
+ if *: "x \<in> C" "y \<in> C" "y > x" for x y :: real
+ proof -
+ from * have ge: "y - x > 0" "y - x \<ge> 0"
by auto
from * have le: "x - y < 0" "x - y \<le> 0"
by auto
then obtain z1 where z1: "z1 > x" "z1 < y" "f y - f x = (y - x) * f' z1"
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>],
- THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
+ THEN f', THEN MVT2[OF \<open>x < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]]
by auto
then have "z1 \<in> C"
using atMostAtLeast_subset_convex \<open>convex C\<close> \<open>x \<in> C\<close> \<open>y \<in> C\<close> \<open>x < y\<close>
@@ -766,11 +785,11 @@
by (simp add: field_simps)
obtain z2 where z2: "z2 > x" "z2 < z1" "f' z1 - f' x = (z1 - x) * f'' z2"
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>x \<in> C\<close> \<open>z1 \<in> C\<close> \<open>x < z1\<close>],
- THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
+ THEN f'', THEN MVT2[OF \<open>x < z1\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
by auto
obtain z3 where z3: "z3 > z1" "z3 < y" "f' y - f' z1 = (y - z1) * f'' z3"
using subsetD[OF atMostAtLeast_subset_convex[OF \<open>convex C\<close> \<open>z1 \<in> C\<close> \<open>y \<in> C\<close> \<open>z1 < y\<close>],
- THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
+ THEN f'', THEN MVT2[OF \<open>z1 < y\<close>, rule_format, unfolded atLeastAtMost_iff[symmetric]]] z1
by auto
have "f' y - (f x - f y) / (x - y) = f' y - f' z1"
using * z1' by auto
@@ -818,22 +837,18 @@
by (simp add: algebra_simps)
then have "f y - f x - f' x * (y - x) \<ge> 0"
using ge by auto
- then have "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
+ then show "f y - f x \<ge> f' x * (y - x)" "f' y * (x - y) \<le> f x - f y"
using res by auto
- } note less_imp = this
- {
- fix x y :: real
- assume "x \<in> C" "y \<in> C" "x \<noteq> y"
- then have"f y - f x \<ge> f' x * (y - x)"
- unfolding neq_iff using less_imp by auto
- }
- moreover
- {
- fix x y :: real
- assume "x \<in> C" "y \<in> C" "x = y"
- then have "f y - f x \<ge> f' x * (y - x)" by auto
- }
- ultimately show ?thesis using assms by blast
+ qed
+ show ?thesis
+ proof (cases "x = y")
+ case True
+ with x y show ?thesis by auto
+ next
+ case False
+ with less_imp x y show ?thesis
+ by (auto simp: neq_iff)
+ qed
qed
lemma f''_ge0_imp_convex:
@@ -855,10 +870,10 @@
using DERIV_log by auto
then have f': "\<And>z. z > 0 \<Longrightarrow> DERIV (\<lambda> z. - log b z) z :> - 1 / (ln b * z)"
by (auto simp: DERIV_minus)
- have "\<And>z :: real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
+ have "\<And>z::real. z > 0 \<Longrightarrow> DERIV inverse z :> - (inverse z ^ Suc (Suc 0))"
using less_imp_neq[THEN not_sym, THEN DERIV_inverse] by auto
from this[THEN DERIV_cmult, of _ "- 1 / ln b"]
- have "\<And>z :: real. z > 0 \<Longrightarrow>
+ have "\<And>z::real. z > 0 \<Longrightarrow>
DERIV (\<lambda> z. (- 1 / ln b) * inverse z) z :> (- 1 / ln b) * (- (inverse z ^ Suc (Suc 0)))"
by auto
then have f''0: "\<And>z::real. z > 0 \<Longrightarrow>
@@ -866,9 +881,9 @@
unfolding inverse_eq_divide by (auto simp: mult.assoc)
have f''_ge0: "\<And>z::real. z > 0 \<Longrightarrow> 1 / (ln b * z * z) \<ge> 0"
using \<open>b > 1\<close> by (auto intro!: less_imp_le)
- from f''_ge0_imp_convex[OF pos_is_convex,
- unfolded greaterThan_iff, OF f' f''0 f''_ge0]
- show ?thesis by auto
+ from f''_ge0_imp_convex[OF pos_is_convex, unfolded greaterThan_iff, OF f' f''0 f''_ge0]
+ show ?thesis
+ by auto
qed
@@ -876,45 +891,59 @@
lemma convex_on_realI:
assumes "connected A"
- assumes "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
- assumes "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
- shows "convex_on A f"
+ and "\<And>x. x \<in> A \<Longrightarrow> (f has_real_derivative f' x) (at x)"
+ and "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f' x \<le> f' y"
+ shows "convex_on A f"
proof (rule convex_on_linorderI)
fix t x y :: real
- assume t: "t > 0" "t < 1" and xy: "x \<in> A" "y \<in> A" "x < y"
+ assume t: "t > 0" "t < 1"
+ assume xy: "x \<in> A" "y \<in> A" "x < y"
define z where "z = (1 - t) * x + t * y"
- with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A" using connected_contains_Icc by blast
+ with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A"
+ using connected_contains_Icc by blast
- from xy t have xz: "z > x" by (simp add: z_def algebra_simps)
- have "y - z = (1 - t) * (y - x)" by (simp add: z_def algebra_simps)
- also from xy t have "... > 0" by (intro mult_pos_pos) simp_all
- finally have yz: "z < y" by simp
+ from xy t have xz: "z > x"
+ by (simp add: z_def algebra_simps)
+ have "y - z = (1 - t) * (y - x)"
+ by (simp add: z_def algebra_simps)
+ also from xy t have "\<dots> > 0"
+ by (intro mult_pos_pos) simp_all
+ finally have yz: "z < y"
+ by simp
from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
by (intro MVT2) (auto intro!: assms(2))
- then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)" by auto
+ then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)"
+ by auto
from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
by (intro MVT2) (auto intro!: assms(2))
- then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)" by auto
+ then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)"
+ by auto
from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
- also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A" by auto
- with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>" by (intro assms(3)) auto
+ also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A"
+ by auto
+ with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>"
+ by (intro assms(3)) auto
also from \<xi>(3) have "f' \<xi> = (f z - f x) / (z - x)" .
finally have "(f y - f z) * (z - x) \<ge> (f z - f x) * (y - z)"
using xz yz by (simp add: field_simps)
- also have "z - x = t * (y - x)" by (simp add: z_def algebra_simps)
- also have "y - z = (1 - t) * (y - x)" by (simp add: z_def algebra_simps)
- finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)" using xy by simp
- thus "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
+ also have "z - x = t * (y - x)"
+ by (simp add: z_def algebra_simps)
+ also have "y - z = (1 - t) * (y - x)"
+ by (simp add: z_def algebra_simps)
+ finally have "(f y - f z) * t \<ge> (f z - f x) * (1 - t)"
+ using xy by simp
+ then show "(1 - t) * f x + t * f y \<ge> f ((1 - t) *\<^sub>R x + t *\<^sub>R y)"
by (simp add: z_def algebra_simps)
qed
lemma convex_on_inverse:
assumes "A \<subseteq> {0<..}"
- shows "convex_on A (inverse :: real \<Rightarrow> real)"
+ shows "convex_on A (inverse :: real \<Rightarrow> real)"
proof (rule convex_on_subset[OF _ assms], intro convex_on_realI[of _ _ "\<lambda>x. -inverse (x^2)"])
- fix u v :: real assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
+ fix u v :: real
+ assume "u \<in> {0<..}" "v \<in> {0<..}" "u \<le> v"
with assms show "-inverse (u^2) \<le> -inverse (v^2)"
by (intro le_imp_neg_le le_imp_inverse_le power_mono) (simp_all)
qed (insert assms, auto intro!: derivative_eq_intros simp: divide_simps power2_eq_square)
@@ -922,40 +951,47 @@
lemma convex_onD_Icc':
assumes "convex_on {x..y} f" "c \<in> {x..y}"
defines "d \<equiv> y - x"
- shows "f c \<le> (f y - f x) / d * (c - x) + f x"
-proof (cases y x rule: linorder_cases)
- assume less: "x < y"
- hence d: "d > 0" by (simp add: d_def)
+ shows "f c \<le> (f y - f x) / d * (c - x) + f x"
+proof (cases x y rule: linorder_cases)
+ case less
+ then have d: "d > 0"
+ by (simp add: d_def)
from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
by (simp_all add: d_def divide_simps)
- have "f c = f (x + (c - x) * 1)" by simp
- also from less have "1 = ((y - x) / d)" by (simp add: d_def)
+ have "f c = f (x + (c - x) * 1)"
+ by simp
+ also from less have "1 = ((y - x) / d)"
+ by (simp add: d_def)
also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
by (simp add: field_simps)
- also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y" using assms less
- by (intro convex_onD_Icc) simp_all
- also from d have "\<dots> = (f y - f x) / d * (c - x) + f x" by (simp add: field_simps)
+ also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y"
+ using assms less by (intro convex_onD_Icc) simp_all
+ also from d have "\<dots> = (f y - f x) / d * (c - x) + f x"
+ by (simp add: field_simps)
finally show ?thesis .
qed (insert assms(2), simp_all)
lemma convex_onD_Icc'':
assumes "convex_on {x..y} f" "c \<in> {x..y}"
defines "d \<equiv> y - x"
- shows "f c \<le> (f x - f y) / d * (y - c) + f y"
-proof (cases y x rule: linorder_cases)
- assume less: "x < y"
- hence d: "d > 0" by (simp add: d_def)
+ shows "f c \<le> (f x - f y) / d * (y - c) + f y"
+proof (cases x y rule: linorder_cases)
+ case less
+ then have d: "d > 0"
+ by (simp add: d_def)
from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
by (simp_all add: d_def divide_simps)
- have "f c = f (y - (y - c) * 1)" by simp
- also from less have "1 = ((y - x) / d)" by (simp add: d_def)
+ have "f c = f (y - (y - c) * 1)"
+ by simp
+ also from less have "1 = ((y - x) / d)"
+ by (simp add: d_def)
also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
by (simp add: field_simps)
- also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y" using assms less
- by (intro convex_onD_Icc) (simp_all add: field_simps)
- also from d have "\<dots> = (f x - f y) / d * (y - c) + f y" by (simp add: field_simps)
+ also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y"
+ using assms less by (intro convex_onD_Icc) (simp_all add: field_simps)
+ also from d have "\<dots> = (f x - f y) / d * (y - c) + f y"
+ by (simp add: field_simps)
finally show ?thesis .
qed (insert assms(2), simp_all)
-
end
--- a/src/HOL/Library/Mapping.thy Wed Jul 13 17:20:21 2016 +0100
+++ b/src/HOL/Library/Mapping.thy Wed Jul 13 20:48:18 2016 +0200
@@ -74,11 +74,11 @@
(\<lambda>k. if k < length xs then Some (xs ! k) else None)
(\<lambda>k. if k < length ys then Some (ys ! k) else None)"
apply induct
- apply auto
+ apply auto
unfolding rel_fun_def
apply clarsimp
apply (case_tac xa)
- apply (auto dest: list_all2_lengthD list_all2_nthD)
+ apply (auto dest: list_all2_lengthD list_all2_nthD)
done
qed
--- a/src/HOL/Library/Set_Algebras.thy Wed Jul 13 17:20:21 2016 +0100
+++ b/src/HOL/Library/Set_Algebras.thy Wed Jul 13 20:48:18 2016 +0200
@@ -1,24 +1,26 @@
(* Title: HOL/Library/Set_Algebras.thy
- Author: Jeremy Avigad and Kevin Donnelly; Florian Haftmann, TUM
+ Author: Jeremy Avigad
+ Author: Kevin Donnelly
+ Author: Florian Haftmann, TUM
*)
section \<open>Algebraic operations on sets\<close>
theory Set_Algebras
-imports Main
+ imports Main
begin
text \<open>
- This library lifts operations like addition and multiplication to
- sets. It was designed to support asymptotic calculations. See the
- comments at the top of theory \<open>BigO\<close>.
+ This library lifts operations like addition and multiplication to sets. It
+ was designed to support asymptotic calculations. See the comments at the top
+ of @{file "BigO.thy"}.
\<close>
instantiation set :: (plus) plus
begin
-definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
- set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
+definition plus_set :: "'a::plus set \<Rightarrow> 'a set \<Rightarrow> 'a set"
+ where set_plus_def: "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
instance ..
@@ -27,8 +29,8 @@
instantiation set :: (times) times
begin
-definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set" where
- set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
+definition times_set :: "'a::times set \<Rightarrow> 'a set \<Rightarrow> 'a set"
+ where set_times_def: "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
instance ..
@@ -37,8 +39,7 @@
instantiation set :: (zero) zero
begin
-definition
- set_zero[simp]: "(0::'a::zero set) = {0}"
+definition set_zero[simp]: "(0::'a::zero set) = {0}"
instance ..
@@ -47,21 +48,20 @@
instantiation set :: (one) one
begin
-definition
- set_one[simp]: "(1::'a::one set) = {1}"
+definition set_one[simp]: "(1::'a::one set) = {1}"
instance ..
end
-definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "+o" 70) where
- "a +o B = {c. \<exists>b\<in>B. c = a + b}"
+definition elt_set_plus :: "'a::plus \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "+o" 70)
+ where "a +o B = {c. \<exists>b\<in>B. c = a + b}"
-definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "*o" 80) where
- "a *o B = {c. \<exists>b\<in>B. c = a * b}"
+definition elt_set_times :: "'a::times \<Rightarrow> 'a set \<Rightarrow> 'a set" (infixl "*o" 80)
+ where "a *o B = {c. \<exists>b\<in>B. c = a * b}"
-abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infix "=o" 50) where
- "x =o A \<equiv> x \<in> A"
+abbreviation (input) elt_set_eq :: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infix "=o" 50)
+ where "x =o A \<equiv> x \<in> A"
instance set :: (semigroup_add) semigroup_add
by standard (force simp add: set_plus_def add.assoc)
@@ -98,19 +98,21 @@
lemma set_plus_intro2 [intro]: "b \<in> C \<Longrightarrow> a + b \<in> a +o C"
by (auto simp add: elt_set_plus_def)
-lemma set_plus_rearrange:
- "((a::'a::comm_monoid_add) +o C) + (b +o D) = (a + b) +o (C + D)"
+lemma set_plus_rearrange: "(a +o C) + (b +o D) = (a + b) +o (C + D)"
+ for a b :: "'a::comm_monoid_add"
apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
apply (rule_tac x = "ba + bb" in exI)
- apply (auto simp add: ac_simps)
+ apply (auto simp add: ac_simps)
apply (rule_tac x = "aa + a" in exI)
apply (auto simp add: ac_simps)
done
-lemma set_plus_rearrange2: "(a::'a::semigroup_add) +o (b +o C) = (a + b) +o C"
+lemma set_plus_rearrange2: "a +o (b +o C) = (a + b) +o C"
+ for a b :: "'a::semigroup_add"
by (auto simp add: elt_set_plus_def add.assoc)
-lemma set_plus_rearrange3: "((a::'a::semigroup_add) +o B) + C = a +o (B + C)"
+lemma set_plus_rearrange3: "(a +o B) + C = a +o (B + C)"
+ for a :: "'a::semigroup_add"
apply (auto simp add: elt_set_plus_def set_plus_def)
apply (blast intro: ac_simps)
apply (rule_tac x = "a + aa" in exI)
@@ -121,7 +123,8 @@
apply (auto simp add: ac_simps)
done
-theorem set_plus_rearrange4: "C + ((a::'a::comm_monoid_add) +o D) = a +o (C + D)"
+theorem set_plus_rearrange4: "C + (a +o D) = a +o (C + D)"
+ for a :: "'a::comm_monoid_add"
apply (auto simp add: elt_set_plus_def set_plus_def ac_simps)
apply (rule_tac x = "aa + ba" in exI)
apply (auto simp add: ac_simps)
@@ -133,13 +136,15 @@
lemma set_plus_mono [intro!]: "C \<subseteq> D \<Longrightarrow> a +o C \<subseteq> a +o D"
by (auto simp add: elt_set_plus_def)
-lemma set_plus_mono2 [intro]: "(C::'a::plus set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F"
+lemma set_plus_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C + E \<subseteq> D + F"
+ for C D E F :: "'a::plus set"
by (auto simp add: set_plus_def)
lemma set_plus_mono3 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> C + D"
by (auto simp add: elt_set_plus_def set_plus_def)
-lemma set_plus_mono4 [intro]: "(a::'a::comm_monoid_add) \<in> C \<Longrightarrow> a +o D \<subseteq> D + C"
+lemma set_plus_mono4 [intro]: "a \<in> C \<Longrightarrow> a +o D \<subseteq> D + C"
+ for a :: "'a::comm_monoid_add"
by (auto simp add: elt_set_plus_def set_plus_def ac_simps)
lemma set_plus_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a +o B \<subseteq> C + D"
@@ -166,33 +171,45 @@
apply auto
done
-lemma set_plus_mono4_b: "(a::'a::comm_monoid_add) : C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C"
+lemma set_plus_mono4_b: "a \<in> C \<Longrightarrow> x \<in> a +o D \<Longrightarrow> x \<in> D + C"
+ for a x :: "'a::comm_monoid_add"
apply (frule set_plus_mono4)
apply auto
done
-lemma set_zero_plus [simp]: "(0::'a::comm_monoid_add) +o C = C"
+lemma set_zero_plus [simp]: "0 +o C = C"
+ for C :: "'a::comm_monoid_add set"
by (auto simp add: elt_set_plus_def)
-lemma set_zero_plus2: "(0::'a::comm_monoid_add) \<in> A \<Longrightarrow> B \<subseteq> A + B"
+lemma set_zero_plus2: "0 \<in> A \<Longrightarrow> B \<subseteq> A + B"
+ for A B :: "'a::comm_monoid_add set"
apply (auto simp add: set_plus_def)
apply (rule_tac x = 0 in bexI)
apply (rule_tac x = x in bexI)
apply (auto simp add: ac_simps)
done
-lemma set_plus_imp_minus: "(a::'a::ab_group_add) : b +o C \<Longrightarrow> (a - b) \<in> C"
+lemma set_plus_imp_minus: "a \<in> b +o C \<Longrightarrow> a - b \<in> C"
+ for a b :: "'a::ab_group_add"
by (auto simp add: elt_set_plus_def ac_simps)
-lemma set_minus_imp_plus: "(a::'a::ab_group_add) - b : C \<Longrightarrow> a \<in> b +o C"
+lemma set_minus_imp_plus: "a - b \<in> C \<Longrightarrow> a \<in> b +o C"
+ for a b :: "'a::ab_group_add"
apply (auto simp add: elt_set_plus_def ac_simps)
apply (subgoal_tac "a = (a + - b) + b")
- apply (rule bexI, assumption)
- apply (auto simp add: ac_simps)
+ apply (rule bexI)
+ apply assumption
+ apply (auto simp add: ac_simps)
done
-lemma set_minus_plus: "(a::'a::ab_group_add) - b \<in> C \<longleftrightarrow> a \<in> b +o C"
- by (rule iffI, rule set_minus_imp_plus, assumption, rule set_plus_imp_minus)
+lemma set_minus_plus: "a - b \<in> C \<longleftrightarrow> a \<in> b +o C"
+ for a b :: "'a::ab_group_add"
+ apply (rule iffI)
+ apply (rule set_minus_imp_plus)
+ apply assumption
+ apply (rule set_plus_imp_minus)
+ apply assumption
+ done
lemma set_times_intro [intro]: "a \<in> C \<Longrightarrow> b \<in> D \<Longrightarrow> a * b \<in> C * D"
by (auto simp add: set_times_def)
@@ -205,8 +222,8 @@
lemma set_times_intro2 [intro!]: "b \<in> C \<Longrightarrow> a * b \<in> a *o C"
by (auto simp add: elt_set_times_def)
-lemma set_times_rearrange:
- "((a::'a::comm_monoid_mult) *o C) * (b *o D) = (a * b) *o (C * D)"
+lemma set_times_rearrange: "(a *o C) * (b *o D) = (a * b) *o (C * D)"
+ for a b :: "'a::comm_monoid_mult"
apply (auto simp add: elt_set_times_def set_times_def)
apply (rule_tac x = "ba * bb" in exI)
apply (auto simp add: ac_simps)
@@ -214,12 +231,12 @@
apply (auto simp add: ac_simps)
done
-lemma set_times_rearrange2:
- "(a::'a::semigroup_mult) *o (b *o C) = (a * b) *o C"
+lemma set_times_rearrange2: "a *o (b *o C) = (a * b) *o C"
+ for a b :: "'a::semigroup_mult"
by (auto simp add: elt_set_times_def mult.assoc)
-lemma set_times_rearrange3:
- "((a::'a::semigroup_mult) *o B) * C = a *o (B * C)"
+lemma set_times_rearrange3: "(a *o B) * C = a *o (B * C)"
+ for a :: "'a::semigroup_mult"
apply (auto simp add: elt_set_times_def set_times_def)
apply (blast intro: ac_simps)
apply (rule_tac x = "a * aa" in exI)
@@ -230,8 +247,8 @@
apply (auto simp add: ac_simps)
done
-theorem set_times_rearrange4:
- "C * ((a::'a::comm_monoid_mult) *o D) = a *o (C * D)"
+theorem set_times_rearrange4: "C * (a *o D) = a *o (C * D)"
+ for a :: "'a::comm_monoid_mult"
apply (auto simp add: elt_set_times_def set_times_def ac_simps)
apply (rule_tac x = "aa * ba" in exI)
apply (auto simp add: ac_simps)
@@ -243,13 +260,15 @@
lemma set_times_mono [intro]: "C \<subseteq> D \<Longrightarrow> a *o C \<subseteq> a *o D"
by (auto simp add: elt_set_times_def)
-lemma set_times_mono2 [intro]: "(C::'a::times set) \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
+lemma set_times_mono2 [intro]: "C \<subseteq> D \<Longrightarrow> E \<subseteq> F \<Longrightarrow> C * E \<subseteq> D * F"
+ for C D E F :: "'a::times set"
by (auto simp add: set_times_def)
lemma set_times_mono3 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> C * D"
by (auto simp add: elt_set_times_def set_times_def)
-lemma set_times_mono4 [intro]: "(a::'a::comm_monoid_mult) : C \<Longrightarrow> a *o D \<subseteq> D * C"
+lemma set_times_mono4 [intro]: "a \<in> C \<Longrightarrow> a *o D \<subseteq> D * C"
+ for a :: "'a::comm_monoid_mult"
by (auto simp add: elt_set_times_def set_times_def ac_simps)
lemma set_times_mono5: "a \<in> C \<Longrightarrow> B \<subseteq> D \<Longrightarrow> a *o B \<subseteq> C * D"
@@ -276,30 +295,31 @@
apply auto
done
-lemma set_times_mono4_b: "(a::'a::comm_monoid_mult) \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
+lemma set_times_mono4_b: "a \<in> C \<Longrightarrow> x \<in> a *o D \<Longrightarrow> x \<in> D * C"
+ for a x :: "'a::comm_monoid_mult"
apply (frule set_times_mono4)
apply auto
done
-lemma set_one_times [simp]: "(1::'a::comm_monoid_mult) *o C = C"
+lemma set_one_times [simp]: "1 *o C = C"
+ for C :: "'a::comm_monoid_mult set"
by (auto simp add: elt_set_times_def)
-lemma set_times_plus_distrib:
- "(a::'a::semiring) *o (b +o C) = (a * b) +o (a *o C)"
+lemma set_times_plus_distrib: "a *o (b +o C) = (a * b) +o (a *o C)"
+ for a b :: "'a::semiring"
by (auto simp add: elt_set_plus_def elt_set_times_def ring_distribs)
-lemma set_times_plus_distrib2:
- "(a::'a::semiring) *o (B + C) = (a *o B) + (a *o C)"
+lemma set_times_plus_distrib2: "a *o (B + C) = (a *o B) + (a *o C)"
+ for a :: "'a::semiring"
apply (auto simp add: set_plus_def elt_set_times_def ring_distribs)
apply blast
apply (rule_tac x = "b + bb" in exI)
apply (auto simp add: ring_distribs)
done
-lemma set_times_plus_distrib3: "((a::'a::semiring) +o C) * D \<subseteq> a *o D + C * D"
- apply (auto simp add:
- elt_set_plus_def elt_set_times_def set_times_def
- set_plus_def ring_distribs)
+lemma set_times_plus_distrib3: "(a +o C) * D \<subseteq> a *o D + C * D"
+ for a :: "'a::semiring"
+ apply (auto simp: elt_set_plus_def elt_set_times_def set_times_def set_plus_def ring_distribs)
apply auto
done
@@ -307,23 +327,25 @@
set_times_plus_distrib
set_times_plus_distrib2
-lemma set_neg_intro: "(a::'a::ring_1) \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
+lemma set_neg_intro: "a \<in> (- 1) *o C \<Longrightarrow> - a \<in> C"
+ for a :: "'a::ring_1"
by (auto simp add: elt_set_times_def)
-lemma set_neg_intro2: "(a::'a::ring_1) \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
+lemma set_neg_intro2: "a \<in> C \<Longrightarrow> - a \<in> (- 1) *o C"
+ for a :: "'a::ring_1"
by (auto simp add: elt_set_times_def)
lemma set_plus_image: "S + T = (\<lambda>(x, y). x + y) ` (S \<times> T)"
- unfolding set_plus_def by (fastforce simp: image_iff)
+ by (fastforce simp: set_plus_def image_iff)
lemma set_times_image: "S * T = (\<lambda>(x, y). x * y) ` (S \<times> T)"
- unfolding set_times_def by (fastforce simp: image_iff)
+ by (fastforce simp: set_times_def image_iff)
lemma finite_set_plus: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s + t)"
- unfolding set_plus_image by simp
+ by (simp add: set_plus_image)
lemma finite_set_times: "finite s \<Longrightarrow> finite t \<Longrightarrow> finite (s * t)"
- unfolding set_times_image by simp
+ by (simp add: set_times_image)
lemma set_setsum_alt:
assumes fin: "finite I"
--- a/src/Pure/Isar/keyword.scala Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Pure/Isar/keyword.scala Wed Jul 13 20:48:18 2016 +0200
@@ -66,6 +66,8 @@
val theory_body = Set(THY_LOAD, THY_DECL, THY_DECL_BLOCK, THY_GOAL)
+ val prf_script = Set(PRF_SCRIPT)
+
val proof =
Set(QED, QED_SCRIPT, QED_BLOCK, QED_GLOBAL, PRF_GOAL, PRF_BLOCK, NEXT_BLOCK, PRF_OPEN,
PRF_CLOSE, PRF_CHAIN, PRF_DECL, PRF_ASM, PRF_ASM_GOAL, PRF_SCRIPT, PRF_SCRIPT_GOAL,
--- a/src/Pure/Isar/outer_syntax.scala Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Pure/Isar/outer_syntax.scala Wed Jul 13 20:48:18 2016 +0200
@@ -174,9 +174,9 @@
if (keywords.is_command(tok, Keyword.theory_goal)) (2, 1)
else if (keywords.is_command(tok, Keyword.theory)) (1, 0)
else if (keywords.is_command(tok, Keyword.proof_open)) (y + 2, y + 1)
- else if (keywords.is_command(tok, Keyword.PRF_BLOCK == _)) (y + 2, y + 1)
- else if (keywords.is_command(tok, Keyword.QED_BLOCK == _)) (y - 1, y - 2)
- else if (keywords.is_command(tok, Keyword.PRF_CLOSE == _)) (y, y - 1)
+ else if (keywords.is_command(tok, Set(Keyword.PRF_BLOCK))) (y + 2, y + 1)
+ else if (keywords.is_command(tok, Set(Keyword.QED_BLOCK))) (y - 1, y - 2)
+ else if (keywords.is_command(tok, Set(Keyword.PRF_CLOSE))) (y, y - 1)
else if (keywords.is_command(tok, Keyword.proof_close)) (y + 1, y - 1)
else if (keywords.is_command(tok, Keyword.qed_global)) (1, 0)
else (x, y)
--- a/src/Pure/Isar/proof.ML Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Pure/Isar/proof.ML Wed Jul 13 20:48:18 2016 +0200
@@ -42,7 +42,6 @@
val raw_goal: state -> {context: context, facts: thm list, goal: thm}
val goal: state -> {context: context, facts: thm list, goal: thm}
val simple_goal: state -> {context: context, goal: thm}
- val status_markup: state -> Markup.T
val let_bind: (term list * term) list -> state -> state
val let_bind_cmd: (string list * string) list -> state -> state
val write: Syntax.mode -> (term * mixfix) list -> state -> state
@@ -561,11 +560,6 @@
val (ctxt, (_, goal)) = get_goal (refine_insert using state);
in {context = ctxt, goal = goal} end;
-fun status_markup state =
- (case try goal state of
- SOME {goal, ...} => Markup.proof_state (Thm.nprems_of goal)
- | NONE => Markup.empty);
-
fun method_error kind pos state =
Seq.single (Proof_Display.method_error kind pos (raw_goal state));
--- a/src/Pure/Isar/token.scala Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Pure/Isar/token.scala Wed Jul 13 20:48:18 2016 +0200
@@ -263,6 +263,7 @@
def is_begin: Boolean = is_keyword("begin")
def is_end: Boolean = is_command("end")
+ def is_begin_or_command: Boolean = is_begin || is_command
def content: String =
if (kind == Token.Kind.STRING) Scan.Parsers.quoted_content("\"", source)
--- a/src/Pure/PIDE/command.ML Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Pure/PIDE/command.ML Wed Jul 13 20:48:18 2016 +0200
@@ -209,17 +209,28 @@
fun status tr m =
Toplevel.setmp_thread_position tr (fn () => Output.status (Markup.markup_only m)) ();
-fun proof_status tr st =
+fun command_indent tr st =
(case try Toplevel.proof_of st of
- SOME prf => status tr (Proof.status_markup prf)
+ SOME prf =>
+ let val keywords = Thy_Header.get_keywords (Proof.theory_of prf) in
+ if Keyword.command_kind keywords (Toplevel.name_of tr) = SOME Keyword.prf_script then
+ (case try Proof.goal prf of
+ SOME {goal, ...} =>
+ let val n = Thm.nprems_of goal
+ in if n > 1 then report tr (Markup.command_indent (n - 1)) else () end
+ | NONE => ())
+ else ()
+ end
| NONE => ());
+
fun eval_state keywords span tr ({state, ...}: eval_state) =
let
val _ = Thread_Attributes.expose_interrupt ();
val st = reset_state keywords tr state;
+ val _ = command_indent tr st;
val _ = status tr Markup.running;
val (errs1, result) = run keywords true tr st;
val errs2 = (case result of NONE => [] | SOME st' => check_cmts span tr st');
@@ -235,7 +246,6 @@
in {failed = true, command = tr, state = st} end
| SOME st' =>
let
- val _ = proof_status tr st';
val _ = status tr Markup.finished;
in {failed = false, command = tr, state = st'} end)
end;
--- a/src/Pure/PIDE/markup.ML Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Pure/PIDE/markup.ML Wed Jul 13 20:48:18 2016 +0200
@@ -155,8 +155,7 @@
val parse_command_timing_properties:
Properties.T -> ({file: string, offset: int, name: string} * Time.time) option
val timingN: string val timing: {elapsed: Time.time, cpu: Time.time, gc: Time.time} -> T
- val subgoalsN: string
- val proof_stateN: string val proof_state: int -> T
+ val command_indentN: string val command_indent: int -> T
val goalN: string val goal: T
val subgoalN: string val subgoal: string -> T
val taskN: string
@@ -576,10 +575,12 @@
| _ => NONE);
-(* toplevel *)
+(* indentation *)
-val subgoalsN = "subgoals";
-val (proof_stateN, proof_state) = markup_int "proof_state" subgoalsN;
+val (command_indentN, command_indent) = markup_int "command_indent" indentN;
+
+
+(* goals *)
val (goalN, goal) = markup_elem "goal";
val (subgoalN, subgoal) = markup_string "subgoal" nameN;
--- a/src/Pure/PIDE/markup.scala Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Pure/PIDE/markup.scala Wed Jul 13 20:48:18 2016 +0200
@@ -372,10 +372,17 @@
val COMMAND_TIMING = "command_timing"
- /* toplevel */
+ /* command indentation */
- val SUBGOALS = "subgoals"
- val PROOF_STATE = "proof_state"
+ object Command_Indent
+ {
+ val name = "command_indent"
+ def unapply(markup: Markup): Option[Int] =
+ if (markup.name == name) Indent.unapply(markup.properties) else None
+ }
+
+
+ /* goals */
val GOAL = "goal"
val SUBGOAL = "subgoal"
--- a/src/Tools/jEdit/etc/options Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Tools/jEdit/etc/options Wed Jul 13 20:48:18 2016 +0200
@@ -45,6 +45,12 @@
public option jedit_indent_newline : bool = true
-- "indentation of Isabelle keywords on ENTER (action isabelle.newline)"
+public option jedit_indent_script : bool = true
+ -- "indent unstructured proof script ('apply' etc.) via number of subgoals"
+
+public option jedit_indent_script_limit : int = 20
+ -- "maximum indentation of unstructured proof script ('apply' etc.)"
+
section "Completion"
--- a/src/Tools/jEdit/src/isabelle.scala Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Tools/jEdit/src/isabelle.scala Wed Jul 13 20:48:18 2016 +0200
@@ -265,6 +265,8 @@
{
Isabelle.buffer_syntax(buffer) match {
case Some(syntax) if buffer.isInstanceOf[Buffer] =>
+ val keywords = syntax.keywords
+
val caret = text_area.getCaretPosition
val line = text_area.getCaretLine
val line_range = JEdit_Lib.line_range(buffer, line)
@@ -282,9 +284,12 @@
val (tokens1, context1) = line_content(line_range.start, caret, context0)
val (tokens2, _) = line_content(caret, line_range.stop, context1)
- if (tokens1.nonEmpty && tokens1.head.is_command) buffer.indentLine(line, true)
+ if (tokens1.nonEmpty &&
+ (tokens1.head.is_begin_or_command || keywords.is_quasi_command(tokens1.head)))
+ buffer.indentLine(line, true)
- if (tokens2.isEmpty || tokens2.head.is_command)
+ if (tokens2.isEmpty ||
+ tokens2.head.is_begin_or_command || keywords.is_quasi_command(tokens2.head))
JEdit_Lib.buffer_edit(buffer) {
text_area.setSelectedText("\n")
if (!buffer.indentLine(line + 1, true)) text_area.goToStartOfWhiteSpace(false)
--- a/src/Tools/jEdit/src/rendering.scala Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Tools/jEdit/src/rendering.scala Wed Jul 13 20:48:18 2016 +0200
@@ -137,6 +137,9 @@
/* markup elements */
+ private val indentation_elements =
+ Markup.Elements(Markup.Command_Indent.name)
+
private val semantic_completion_elements =
Markup.Elements(Markup.COMPLETION, Markup.NO_COMPLETION)
@@ -295,6 +298,16 @@
val markdown_item_color4 = color_value("markdown_item_color4")
+ /* indentation */
+
+ def indentation(range: Text.Range): Int =
+ snapshot.select(range, Rendering.indentation_elements, _ =>
+ {
+ case Text.Info(_, XML.Elem(Markup.Command_Indent(i), _)) => Some(i)
+ case _ => None
+ }).headOption.map(_.info).getOrElse(0)
+
+
/* completion */
def semantic_completion(completed_range: Option[Text.Range], range: Text.Range)
--- a/src/Tools/jEdit/src/text_structure.scala Wed Jul 13 17:20:21 2016 +0100
+++ b/src/Tools/jEdit/src/text_structure.scala Wed Jul 13 20:48:18 2016 +0200
@@ -26,13 +26,13 @@
def iterator(line: Int, lim: Int = limit): Iterator[Text.Info[Token]] =
{
val it = Token_Markup.line_token_iterator(syntax, buffer, line, line + lim)
- if (comments) it.filterNot(_.info.is_space) else it.filter(_.info.is_proper)
+ if (comments) it.filterNot(_.info.is_space) else it.filterNot(_.info.is_improper)
}
def reverse_iterator(line: Int, lim: Int = limit): Iterator[Text.Info[Token]] =
{
val it = Token_Markup.line_token_reverse_iterator(syntax, buffer, line, line - lim)
- if (comments) it.filterNot(_.info.is_space) else it.filter(_.info.is_proper)
+ if (comments) it.filterNot(_.info.is_space) else it.filterNot(_.info.is_improper)
}
}
@@ -52,31 +52,52 @@
val keywords = syntax.keywords
val nav = new Navigator(syntax, buffer.asInstanceOf[Buffer], true)
- def head_token(line: Int): Option[Token] =
- nav.iterator(line, 1).toStream.headOption.map(_.info)
-
- def head_is_quasi_command(line: Int): Boolean =
- head_token(line) match {
- case None => false
- case Some(tok) => keywords.is_quasi_command(tok)
- }
-
- def prev_command: Option[Token] =
- nav.reverse_iterator(prev_line, 1).
- collectFirst({ case Text.Info(_, tok) if tok.is_command => tok })
-
- def prev_span: Iterator[Token] =
- nav.reverse_iterator(prev_line).map(_.info).takeWhile(tok => !tok.is_command)
-
- def prev_line_span: Iterator[Token] =
- nav.reverse_iterator(prev_line, 1).map(_.info).takeWhile(tok => !tok.is_command)
+ val indent_size = buffer.getIndentSize
def line_indent(line: Int): Int =
if (line < 0 || line >= buffer.getLineCount) 0
else buffer.getCurrentIndentForLine(line, null)
- val indent_size = buffer.getIndentSize
+ def line_head(line: Int): Option[Text.Info[Token]] =
+ nav.iterator(line, 1).toStream.headOption
+
+ def head_is_quasi_command(line: Int): Boolean =
+ line_head(line) match {
+ case None => false
+ case Some(Text.Info(_, tok)) => keywords.is_quasi_command(tok)
+ }
+
+ def prev_line_command: Option[Token] =
+ nav.reverse_iterator(prev_line, 1).
+ collectFirst({ case Text.Info(_, tok) if tok.is_begin_or_command => tok })
+
+ def prev_line_span: Iterator[Token] =
+ nav.reverse_iterator(prev_line, 1).map(_.info).takeWhile(tok => !tok.is_begin_or_command)
+
+ def prev_span: Iterator[Token] =
+ nav.reverse_iterator(prev_line).map(_.info).takeWhile(tok => !tok.is_begin_or_command)
+
+
+ val script_indent: Text.Info[Token] => Int =
+ {
+ val opt_rendering: Option[Rendering] =
+ if (PIDE.options.value.bool("jedit_indent_script"))
+ GUI_Thread.now {
+ (for {
+ text_area <- JEdit_Lib.jedit_text_areas(buffer)
+ doc_view <- PIDE.document_view(text_area)
+ } yield doc_view.get_rendering).toStream.headOption
+ }
+ else None
+ val limit = PIDE.options.value.int("jedit_indent_script_limit")
+ (info: Text.Info[Token]) =>
+ opt_rendering match {
+ case Some(rendering) if keywords.is_command(info.info, Keyword.prf_script) =>
+ (rendering.indentation(info.range) min limit) max 0
+ case _ => 0
+ }
+ }
def indent_indent(tok: Token): Int =
if (keywords.is_command(tok, keyword_open)) indent_size
@@ -84,9 +105,24 @@
else 0
def indent_offset(tok: Token): Int =
- if (keywords.is_command(tok, Keyword.proof_enclose) || tok.is_begin) indent_size
+ if (keywords.is_command(tok, Keyword.proof_enclose)) indent_size
else 0
+ def indent_structure: Int =
+ nav.reverse_iterator(current_line - 1).scanLeft((0, false))(
+ { case ((ind, _), Text.Info(range, tok)) =>
+ val ind1 = ind + indent_indent(tok)
+ if (tok.is_begin_or_command && !keywords.is_command(tok, Keyword.prf_script)) {
+ val line = buffer.getLineOfOffset(range.start)
+ line_head(line) match {
+ case Some(info) if info.info == tok =>
+ (ind1 + indent_offset(tok) + line_indent(line), true)
+ case _ => (ind1, false)
+ }
+ }
+ else (ind1, false)
+ }).collectFirst({ case (i, true) => i }).getOrElse(0)
+
def indent_brackets: Int =
(0 /: prev_line_span)(
{ case (i, tok) =>
@@ -98,35 +134,20 @@
if (prev_span.exists(keywords.is_quasi_command(_))) indent_size
else 0
- def indent_structure: Int =
- nav.reverse_iterator(current_line - 1).scanLeft((0, false))(
- { case ((ind, _), Text.Info(range, tok)) =>
- val ind1 = ind + indent_indent(tok)
- if (tok.is_command) {
- val line = buffer.getLineOfOffset(range.start)
- if (head_token(line) == Some(tok))
- (ind1 + indent_offset(tok) + line_indent(line), true)
- else (ind1, false)
- }
- else (ind1, false)
- }).collectFirst({ case (i, true) => i }).getOrElse(0)
-
- def nesting(it: Iterator[Token], open: Token => Boolean, close: Token => Boolean): Int =
- (0 /: it)({ case (d, tok) => if (open(tok)) d + 1 else if (close(tok)) d - 1 else d })
-
- def indent_begin: Int =
- (nesting(nav.iterator(current_line - 1, 1).map(_.info), _.is_begin, _.is_end) max 0) *
- indent_size
-
val indent =
- head_token(current_line) match {
+ line_head(current_line) match {
case None => indent_structure + indent_brackets + indent_extra
- case Some(tok) =>
- if (keywords.is_before_command(tok) ||
- keywords.is_command(tok, Keyword.theory)) indent_begin
- else if (tok.is_command) indent_structure + indent_begin - indent_offset(tok)
+ case Some(info @ Text.Info(range, tok)) =>
+ if (tok.is_begin ||
+ keywords.is_before_command(tok) ||
+ keywords.is_command(tok, Keyword.theory)) 0
+ else if (keywords.is_command(tok, Keyword.proof_enclose))
+ indent_structure + script_indent(info) - indent_offset(tok)
+ else if (keywords.is_command(tok, Keyword.proof))
+ (indent_structure + script_indent(info) - indent_offset(tok)) max indent_size
+ else if (tok.is_command) indent_structure - indent_offset(tok)
else {
- prev_command match {
+ prev_line_command match {
case None =>
val extra =
(keywords.is_quasi_command(tok), head_is_quasi_command(prev_line)) match {
@@ -134,10 +155,10 @@
case (true, false) => - indent_extra
case (false, true) => indent_extra
}
- line_indent(prev_line) - indent_offset(tok) + indent_brackets + extra
+ line_indent(prev_line) + indent_brackets + extra - indent_offset(tok)
case Some(prev_tok) =>
- indent_structure - indent_offset(tok) - indent_offset(prev_tok) +
- indent_brackets - indent_indent(prev_tok) + indent_size
+ indent_structure + indent_brackets + indent_size - indent_offset(tok) -
+ indent_offset(prev_tok) - indent_indent(prev_tok)
}
}
}