--- a/NEWS Fri Aug 05 17:36:38 2016 +0200
+++ b/NEWS Fri Aug 05 18:14:34 2016 +0200
@@ -63,6 +63,10 @@
* Cartouche abbreviations work both for " and ` to accomodate typical
situations where old ASCII notation may be updated.
+* Isabelle/ML and Standard ML files are presented in Sidekick with the
+tree structure of section headings: this special comment format is
+described in "implementation" chapter 0, e.g. (*** section ***).
+
* IDE support for the Isabelle/Pure bootstrap process, with the
following independent stages:
@@ -85,6 +89,9 @@
are treated as delimiters for fold structure; 'begin' and 'end'
structure of theory specifications is treated as well.
+* Sidekick parser "isabelle-context" shows nesting of context blocks
+according to 'begin' and 'end' structure.
+
* 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
--- a/src/Doc/Implementation/ML.thy Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Doc/Implementation/ML.thy Fri Aug 05 18:14:34 2016 +0200
@@ -70,11 +70,14 @@
prose description of the purpose of the module. The latter can range from a
single line to several paragraphs of explanations.
- The rest of the file is divided into sections, subsections, subsubsections,
- paragraphs etc.\ using a simple layout via ML comments as follows.
+ The rest of the file is divided into chapters, sections, subsections,
+ subsubsections, paragraphs etc.\ using a simple layout via ML comments as
+ follows.
@{verbatim [display]
-\<open> (*** section ***)
+\<open> (**** chapter ****)
+
+ (*** section ***)
(** subsection **)
--- a/src/HOL/Complete_Partial_Order.thy Fri Aug 05 17:36:38 2016 +0200
+++ b/src/HOL/Complete_Partial_Order.thy Fri Aug 05 18:14:34 2016 +0200
@@ -1,12 +1,12 @@
-(* Title: HOL/Complete_Partial_Order.thy
- Author: Brian Huffman, Portland State University
- Author: Alexander Krauss, TU Muenchen
+(* Title: HOL/Complete_Partial_Order.thy
+ Author: Brian Huffman, Portland State University
+ Author: Alexander Krauss, TU Muenchen
*)
section \<open>Chain-complete partial orders and their fixpoints\<close>
theory Complete_Partial_Order
-imports Product_Type
+ imports Product_Type
begin
subsection \<open>Monotone functions\<close>
@@ -14,131 +14,139 @@
text \<open>Dictionary-passing version of @{const Orderings.mono}.\<close>
definition monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
-where "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
+ where "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))"
-lemma monotoneI[intro?]: "(\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y))
- \<Longrightarrow> monotone orda ordb f"
-unfolding monotone_def by iprover
+lemma monotoneI[intro?]: "(\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)) \<Longrightarrow> monotone orda ordb f"
+ unfolding monotone_def by iprover
lemma monotoneD[dest?]: "monotone orda ordb f \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)"
-unfolding monotone_def by iprover
+ unfolding monotone_def by iprover
subsection \<open>Chains\<close>
-text \<open>A chain is a totally-ordered set. Chains are parameterized over
+text \<open>
+ A chain is a totally-ordered set. Chains are parameterized over
the order for maximal flexibility, since type classes are not enough.
\<close>
-definition
- chain :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
-where
- "chain ord S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"
+definition chain :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool"
+ where "chain ord S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)"
lemma chainI:
assumes "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> ord x y \<or> ord y x"
shows "chain ord S"
-using assms unfolding chain_def by fast
+ using assms unfolding chain_def by fast
lemma chainD:
assumes "chain ord S" and "x \<in> S" and "y \<in> S"
shows "ord x y \<or> ord y x"
-using assms unfolding chain_def by fast
+ using assms unfolding chain_def by fast
lemma chainE:
assumes "chain ord S" and "x \<in> S" and "y \<in> S"
obtains "ord x y" | "ord y x"
-using assms unfolding chain_def by fast
+ using assms unfolding chain_def by fast
lemma chain_empty: "chain ord {}"
-by(simp add: chain_def)
+ by (simp add: chain_def)
lemma chain_equality: "chain op = A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x = y)"
-by(auto simp add: chain_def)
+ by (auto simp add: chain_def)
+
+lemma chain_subset: "chain ord A \<Longrightarrow> B \<subseteq> A \<Longrightarrow> chain ord B"
+ by (rule chainI) (blast dest: chainD)
-lemma chain_subset:
- "\<lbrakk> chain ord A; B \<subseteq> A \<rbrakk>
- \<Longrightarrow> chain ord B"
-by(rule chainI)(blast dest: chainD)
+lemma chain_imageI:
+ assumes chain: "chain le_a Y"
+ and mono: "\<And>x y. x \<in> Y \<Longrightarrow> y \<in> Y \<Longrightarrow> le_a x y \<Longrightarrow> le_b (f x) (f y)"
+ shows "chain le_b (f ` Y)"
+ by (blast intro: chainI dest: chainD[OF chain] mono)
-lemma chain_imageI:
- assumes chain: "chain le_a Y"
- and mono: "\<And>x y. \<lbrakk> x \<in> Y; y \<in> Y; le_a x y \<rbrakk> \<Longrightarrow> le_b (f x) (f y)"
- shows "chain le_b (f ` Y)"
-by(blast intro: chainI dest: chainD[OF chain] mono)
subsection \<open>Chain-complete partial orders\<close>
text \<open>
- A ccpo has a least upper bound for any chain. In particular, the
- empty set is a chain, so every ccpo must have a bottom element.
+ A \<open>ccpo\<close> has a least upper bound for any chain. In particular, the
+ empty set is a chain, so every \<open>ccpo\<close> must have a bottom element.
\<close>
class ccpo = order + Sup +
- assumes ccpo_Sup_upper: "\<lbrakk>chain (op \<le>) A; x \<in> A\<rbrakk> \<Longrightarrow> x \<le> Sup A"
- assumes ccpo_Sup_least: "\<lbrakk>chain (op \<le>) A; \<And>x. x \<in> A \<Longrightarrow> x \<le> z\<rbrakk> \<Longrightarrow> Sup A \<le> z"
+ assumes ccpo_Sup_upper: "chain (op \<le>) A \<Longrightarrow> x \<in> A \<Longrightarrow> x \<le> Sup A"
+ assumes ccpo_Sup_least: "chain (op \<le>) A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup A \<le> z"
begin
lemma chain_singleton: "Complete_Partial_Order.chain op \<le> {x}"
-by(rule chainI) simp
+ by (rule chainI) simp
lemma ccpo_Sup_singleton [simp]: "\<Squnion>{x} = x"
-by(rule antisym)(auto intro: ccpo_Sup_least ccpo_Sup_upper simp add: chain_singleton)
+ by (rule antisym)(auto intro: ccpo_Sup_least ccpo_Sup_upper simp add: chain_singleton)
+
subsection \<open>Transfinite iteration of a function\<close>
-context notes [[inductive_internals]] begin
+context notes [[inductive_internals]]
+begin
inductive_set iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set"
-for f :: "'a \<Rightarrow> 'a"
-where
- step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"
-| Sup: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> Sup M \<in> iterates f"
+ for f :: "'a \<Rightarrow> 'a"
+ where
+ step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f"
+ | Sup: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> Sup M \<in> iterates f"
end
-lemma iterates_le_f:
- "x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x"
-by (induct x rule: iterates.induct)
- (force dest: monotoneD intro!: ccpo_Sup_upper ccpo_Sup_least)+
+lemma iterates_le_f: "x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x"
+ by (induct x rule: iterates.induct)
+ (force dest: monotoneD intro!: ccpo_Sup_upper ccpo_Sup_least)+
lemma chain_iterates:
assumes f: "monotone (op \<le>) (op \<le>) f"
shows "chain (op \<le>) (iterates f)" (is "chain _ ?C")
proof (rule chainI)
- fix x y assume "x \<in> ?C" "y \<in> ?C"
+ fix x y
+ assume "x \<in> ?C" "y \<in> ?C"
then show "x \<le> y \<or> y \<le> x"
proof (induct x arbitrary: y rule: iterates.induct)
- fix x y assume y: "y \<in> ?C"
- and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x"
+ fix x y
+ assume y: "y \<in> ?C"
+ and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x"
from y show "f x \<le> y \<or> y \<le> f x"
proof (induct y rule: iterates.induct)
- case (step y) with IH f show ?case by (auto dest: monotoneD)
+ case (step y)
+ with IH f show ?case by (auto dest: monotoneD)
next
case (Sup M)
then have chM: "chain (op \<le>) M"
and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto
show "f x \<le> Sup M \<or> Sup M \<le> f x"
proof (cases "\<exists>z\<in>M. f x \<le> z")
- case True then have "f x \<le> Sup M"
+ case True
+ then have "f x \<le> Sup M"
apply rule
apply (erule order_trans)
- by (rule ccpo_Sup_upper[OF chM])
- thus ?thesis ..
+ apply (rule ccpo_Sup_upper[OF chM])
+ apply assumption
+ done
+ then show ?thesis ..
next
- case False with IH'
- show ?thesis by (auto intro: ccpo_Sup_least[OF chM])
+ case False
+ with IH' show ?thesis
+ by (auto intro: ccpo_Sup_least[OF chM])
qed
qed
next
case (Sup M y)
show ?case
proof (cases "\<exists>x\<in>M. y \<le> x")
- case True then have "y \<le> Sup M"
+ case True
+ then have "y \<le> Sup M"
apply rule
apply (erule order_trans)
- by (rule ccpo_Sup_upper[OF Sup(1)])
- thus ?thesis ..
+ apply (rule ccpo_Sup_upper[OF Sup(1)])
+ apply assumption
+ done
+ then show ?thesis ..
next
case False with Sup
show ?thesis by (auto intro: ccpo_Sup_least)
@@ -147,19 +155,19 @@
qed
lemma bot_in_iterates: "Sup {} \<in> iterates f"
-by(auto intro: iterates.Sup simp add: chain_empty)
+ by (auto intro: iterates.Sup simp add: chain_empty)
+
subsection \<open>Fixpoint combinator\<close>
-definition
- fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
-where
- "fixp f = Sup (iterates f)"
+definition fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
+ where "fixp f = Sup (iterates f)"
lemma iterates_fixp:
- assumes f: "monotone (op \<le>) (op \<le>) f" shows "fixp f \<in> iterates f"
-unfolding fixp_def
-by (simp add: iterates.Sup chain_iterates f)
+ assumes f: "monotone (op \<le>) (op \<le>) f"
+ shows "fixp f \<in> iterates f"
+ unfolding fixp_def
+ by (simp add: iterates.Sup chain_iterates f)
lemma fixp_unfold:
assumes f: "monotone (op \<le>) (op \<le>) f"
@@ -169,35 +177,45 @@
by (intro iterates_le_f iterates_fixp f)
have "f (fixp f) \<le> Sup (iterates f)"
by (intro ccpo_Sup_upper chain_iterates f iterates.step iterates_fixp)
- thus "f (fixp f) \<le> fixp f"
- unfolding fixp_def .
+ then show "f (fixp f) \<le> fixp f"
+ by (simp only: fixp_def)
qed
lemma fixp_lowerbound:
- assumes f: "monotone (op \<le>) (op \<le>) f" and z: "f z \<le> z" shows "fixp f \<le> z"
-unfolding fixp_def
+ assumes f: "monotone (op \<le>) (op \<le>) f"
+ and z: "f z \<le> z"
+ shows "fixp f \<le> z"
+ unfolding fixp_def
proof (rule ccpo_Sup_least[OF chain_iterates[OF f]])
- fix x assume "x \<in> iterates f"
- thus "x \<le> z"
+ fix x
+ assume "x \<in> iterates f"
+ then show "x \<le> z"
proof (induct x rule: iterates.induct)
- fix x assume "x \<le> z" with f have "f x \<le> f z" by (rule monotoneD)
- also note z finally show "f x \<le> z" .
- qed (auto intro: ccpo_Sup_least)
+ case (step x)
+ from f \<open>x \<le> z\<close> have "f x \<le> f z" by (rule monotoneD)
+ also note z
+ finally show "f x \<le> z" .
+ next
+ case (Sup M)
+ then show ?case
+ by (auto intro: ccpo_Sup_least)
+ qed
qed
end
+
subsection \<open>Fixpoint induction\<close>
setup \<open>Sign.map_naming (Name_Space.mandatory_path "ccpo")\<close>
definition admissible :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
-where "admissible lub ord P = (\<forall>A. chain ord A \<longrightarrow> (A \<noteq> {}) \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
+ where "admissible lub ord P \<longleftrightarrow> (\<forall>A. chain ord A \<longrightarrow> A \<noteq> {} \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
lemma admissibleI:
assumes "\<And>A. chain ord A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (lub A)"
shows "ccpo.admissible lub ord P"
-using assms unfolding ccpo.admissible_def by fast
+ using assms unfolding ccpo.admissible_def by fast
lemma admissibleD:
assumes "ccpo.admissible lub ord P"
@@ -205,7 +223,7 @@
assumes "A \<noteq> {}"
assumes "\<And>x. x \<in> A \<Longrightarrow> P x"
shows "P (lub A)"
-using assms by (auto simp: ccpo.admissible_def)
+ using assms by (auto simp: ccpo.admissible_def)
setup \<open>Sign.map_naming Name_Space.parent_path\<close>
@@ -215,44 +233,54 @@
assumes bot: "P (Sup {})"
assumes step: "\<And>x. P x \<Longrightarrow> P (f x)"
shows "P (fixp f)"
-unfolding fixp_def using adm chain_iterates[OF mono]
+ unfolding fixp_def
+ using adm chain_iterates[OF mono]
proof (rule ccpo.admissibleD)
- show "iterates f \<noteq> {}" using bot_in_iterates by auto
- fix x assume "x \<in> iterates f"
- thus "P x"
- by (induct rule: iterates.induct)
- (case_tac "M = {}", auto intro: step bot ccpo.admissibleD adm)
+ show "iterates f \<noteq> {}"
+ using bot_in_iterates by auto
+next
+ fix x
+ assume "x \<in> iterates f"
+ then show "P x"
+ proof (induct rule: iterates.induct)
+ case prems: (step x)
+ from this(2) show ?case by (rule step)
+ next
+ case (Sup M)
+ then show ?case by (cases "M = {}") (auto intro: step bot ccpo.admissibleD adm)
+ qed
qed
lemma admissible_True: "ccpo.admissible lub ord (\<lambda>x. True)"
-unfolding ccpo.admissible_def by simp
+ unfolding ccpo.admissible_def by simp
(*lemma admissible_False: "\<not> ccpo.admissible lub ord (\<lambda>x. False)"
unfolding ccpo.admissible_def chain_def by simp
*)
lemma admissible_const: "ccpo.admissible lub ord (\<lambda>x. t)"
-by(auto intro: ccpo.admissibleI)
+ by (auto intro: ccpo.admissibleI)
lemma admissible_conj:
assumes "ccpo.admissible lub ord (\<lambda>x. P x)"
assumes "ccpo.admissible lub ord (\<lambda>x. Q x)"
shows "ccpo.admissible lub ord (\<lambda>x. P x \<and> Q x)"
-using assms unfolding ccpo.admissible_def by simp
+ using assms unfolding ccpo.admissible_def by simp
lemma admissible_all:
assumes "\<And>y. ccpo.admissible lub ord (\<lambda>x. P x y)"
shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y. P x y)"
-using assms unfolding ccpo.admissible_def by fast
+ using assms unfolding ccpo.admissible_def by fast
lemma admissible_ball:
assumes "\<And>y. y \<in> A \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. P x y)"
shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y\<in>A. P x y)"
-using assms unfolding ccpo.admissible_def by fast
+ using assms unfolding ccpo.admissible_def by fast
lemma chain_compr: "chain ord A \<Longrightarrow> chain ord {x \<in> A. P x}"
-unfolding chain_def by fast
+ unfolding chain_def by fast
-context ccpo begin
+context ccpo
+begin
lemma admissible_disj_lemma:
assumes A: "chain (op \<le>)A"
@@ -280,17 +308,24 @@
assumes Q: "ccpo.admissible Sup (op \<le>) (\<lambda>x. Q x)"
shows "ccpo.admissible Sup (op \<le>) (\<lambda>x. P x \<or> Q x)"
proof (rule ccpo.admissibleI)
- fix A :: "'a set" assume A: "chain (op \<le>) A"
- assume "A \<noteq> {}"
- and "\<forall>x\<in>A. P x \<or> Q x"
- hence "(\<exists>x\<in>A. P x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<exists>x\<in>A. Q x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)"
- using chainD[OF A] by blast
- hence "(\<exists>x. x \<in> A \<and> P x) \<and> Sup A = Sup {x \<in> A. P x} \<or> (\<exists>x. x \<in> A \<and> Q x) \<and> Sup A = Sup {x \<in> A. Q x}"
+ fix A :: "'a set"
+ assume A: "chain (op \<le>) A"
+ assume "A \<noteq> {}" and "\<forall>x\<in>A. P x \<or> Q x"
+ then have "(\<exists>x\<in>A. P x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<exists>x\<in>A. Q x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)"
+ using chainD[OF A] by blast (* slow *)
+ then have "(\<exists>x. x \<in> A \<and> P x) \<and> Sup A = Sup {x \<in> A. P x} \<or> (\<exists>x. x \<in> A \<and> Q x) \<and> Sup A = Sup {x \<in> A. Q x}"
using admissible_disj_lemma [OF A] by blast
- thus "P (Sup A) \<or> Q (Sup A)"
- apply (rule disjE, simp_all)
- apply (rule disjI1, rule ccpo.admissibleD [OF P chain_compr [OF A]], simp, simp)
- apply (rule disjI2, rule ccpo.admissibleD [OF Q chain_compr [OF A]], simp, simp)
+ then show "P (Sup A) \<or> Q (Sup A)"
+ apply (rule disjE)
+ apply simp_all
+ apply (rule disjI1)
+ apply (rule ccpo.admissibleD [OF P chain_compr [OF A]])
+ apply simp
+ apply simp
+ apply (rule disjI2)
+ apply (rule ccpo.admissibleD [OF Q chain_compr [OF A]])
+ apply simp
+ apply simp
done
qed
@@ -300,7 +335,8 @@
by standard (fast intro: Sup_upper Sup_least)+
lemma lfp_eq_fixp:
- assumes f: "mono f" shows "lfp f = fixp f"
+ assumes f: "mono f"
+ shows "lfp f = fixp f"
proof (rule antisym)
from f have f': "monotone (op \<le>) (op \<le>) f"
unfolding mono_def monotone_def .
--- a/src/HOL/Finite_Set.thy Fri Aug 05 17:36:38 2016 +0200
+++ b/src/HOL/Finite_Set.thy Fri Aug 05 18:14:34 2016 +0200
@@ -1,24 +1,26 @@
(* Title: HOL/Finite_Set.thy
- Author: Tobias Nipkow, Lawrence C Paulson and Markus Wenzel
- with contributions by Jeremy Avigad and Andrei Popescu
+ Author: Tobias Nipkow
+ Author: Lawrence C Paulson
+ Author: Markus Wenzel
+ Author: Jeremy Avigad
+ Author: Andrei Popescu
*)
section \<open>Finite sets\<close>
theory Finite_Set
-imports Product_Type Sum_Type Fields
+ imports Product_Type Sum_Type Fields
begin
subsection \<open>Predicate for finite sets\<close>
-context
- notes [[inductive_internals]]
+context notes [[inductive_internals]]
begin
inductive finite :: "'a set \<Rightarrow> bool"
-where
- emptyI [simp, intro!]: "finite {}"
-| insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
+ where
+ emptyI [simp, intro!]: "finite {}"
+ | insertI [simp, intro!]: "finite A \<Longrightarrow> finite (insert a A)"
end
@@ -145,7 +147,7 @@
shows "\<exists>f n::nat. f ` A = {i. i < n} \<and> inj_on f A"
proof -
from finite_imp_nat_seg_image_inj_on [OF \<open>finite A\<close>]
- obtain f and n::nat where bij: "bij_betw f {i. i<n} A"
+ obtain f and n :: nat where bij: "bij_betw f {i. i<n} A"
by (auto simp: bij_betw_def)
let ?f = "the_inv_into {i. i<n} f"
have "inj_on ?f A \<and> ?f ` A = {i. i<n}"
@@ -317,7 +319,7 @@
next
case insert
then show ?case
- by (clarsimp simp del: image_insert simp add: image_insert [symmetric]) blast
+ by (clarsimp simp del: image_insert simp add: image_insert [symmetric]) blast (* slow *)
qed
lemma finite_vimage_IntI: "finite F \<Longrightarrow> inj_on h A \<Longrightarrow> finite (h -` F \<inter> A)"
@@ -328,7 +330,8 @@
done
lemma finite_finite_vimage_IntI:
- assumes "finite F" and "\<And>y. y \<in> F \<Longrightarrow> finite ((h -` {y}) \<inter> A)"
+ assumes "finite F"
+ and "\<And>y. y \<in> F \<Longrightarrow> finite ((h -` {y}) \<inter> A)"
shows "finite (h -` F \<inter> A)"
proof -
have *: "h -` F \<inter> A = (\<Union> y\<in>F. (h -` {y}) \<inter> A)"
@@ -464,7 +467,7 @@
proof
assume "finite (Pow A)"
then have "finite ((\<lambda>x. {x}) ` A)"
- by (blast intro: finite_subset)
+ by (blast intro: finite_subset) (* somewhat slow *)
then show "finite A"
by (rule finite_imageD [unfolded inj_on_def]) simp
next
@@ -492,7 +495,7 @@
from finite_subset[OF this] assms have 1: "finite (?F ` ?S)"
by simp
have 2: "inj_on ?F ?S"
- by (fastforce simp add: inj_on_def set_eq_iff fun_eq_iff)
+ by (fastforce simp add: inj_on_def set_eq_iff fun_eq_iff) (* somewhat slow *)
show ?thesis
by (rule finite_imageD [OF 1 2])
qed
@@ -524,7 +527,7 @@
lemma finite_subset_induct [consumes 2, case_names empty insert]:
assumes "finite F" and "F \<subseteq> A"
- assumes empty: "P {}"
+ and empty: "P {}"
and insert: "\<And>a F. finite F \<Longrightarrow> a \<in> A \<Longrightarrow> a \<notin> F \<Longrightarrow> P F \<Longrightarrow> P (insert a F)"
shows "P F"
using \<open>finite F\<close> \<open>F \<subseteq> A\<close>
@@ -545,7 +548,7 @@
lemma finite_empty_induct:
assumes "finite A"
- assumes "P A"
+ and "P A"
and remove: "\<And>a A. finite A \<Longrightarrow> a \<in> A \<Longrightarrow> P A \<Longrightarrow> P (A - {a})"
shows "P {}"
proof -
@@ -604,8 +607,8 @@
lemma finite_subset_induct' [consumes 2, case_names empty insert]:
assumes "finite F" and "F \<subseteq> A"
- and empty: "P {}"
- and insert: "\<And>a F. \<lbrakk>finite F; a \<in> A; F \<subseteq> A; a \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert a F)"
+ and empty: "P {}"
+ and insert: "\<And>a F. \<lbrakk>finite F; a \<in> A; F \<subseteq> A; a \<notin> F; P F \<rbrakk> \<Longrightarrow> P (insert a F)"
shows "P F"
proof -
from \<open>finite F\<close>
@@ -632,7 +635,8 @@
subsection \<open>Class \<open>finite\<close>\<close>
-class finite = assumes finite_UNIV: "finite (UNIV :: 'a set)"
+class finite =
+ assumes finite_UNIV: "finite (UNIV :: 'a set)"
begin
lemma finite [simp]: "finite (A :: 'a set)"
@@ -700,9 +704,9 @@
inductive fold_graph :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a set \<Rightarrow> 'b \<Rightarrow> bool"
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: 'b
-where
- emptyI [intro]: "fold_graph f z {} z"
-| insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
+ where
+ emptyI [intro]: "fold_graph f z {} z"
+ | insertI [intro]: "x \<notin> A \<Longrightarrow> fold_graph f z A y \<Longrightarrow> fold_graph f z (insert x A) (f x y)"
inductive_cases empty_fold_graphE [elim!]: "fold_graph f z {} x"
@@ -1069,7 +1073,7 @@
interpret comp_fun_idem Set.remove
by (fact comp_fun_idem_remove)
from \<open>finite A\<close> have "fold Set.remove B A = B - A"
- by (induct A arbitrary: B) auto
+ by (induct A arbitrary: B) auto (* slow *)
then show ?thesis ..
qed
@@ -1124,7 +1128,7 @@
qed
lemma comp_fun_commute_Pow_fold: "comp_fun_commute (\<lambda>x A. A \<union> Set.insert x ` A)"
- by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast
+ by (clarsimp simp: fun_eq_iff comp_fun_commute_def) blast (* somewhat slow *)
lemma Pow_fold:
assumes "finite A"
@@ -1222,13 +1226,12 @@
subsubsection \<open>The natural case\<close>
locale folding =
- fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"
- fixes z :: "'b"
+ fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" and z :: "'b"
assumes comp_fun_commute: "f y \<circ> f x = f x \<circ> f y"
begin
interpretation fold?: comp_fun_commute f
- by standard (insert comp_fun_commute, simp add: fun_eq_iff)
+ by standard (use comp_fun_commute in \<open>simp add: fun_eq_iff\<close>)
definition F :: "'a set \<Rightarrow> 'b"
where eq_fold: "F A = fold f z A"
@@ -1332,14 +1335,14 @@
lemma card_Suc_Diff1: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> Suc (card (A - {x})) = card A"
apply (rule insert_Diff [THEN subst, where t = A])
- apply assumption
+ apply assumption
apply (simp del: insert_Diff_single)
done
lemma card_insert_le_m1: "n > 0 \<Longrightarrow> card y \<le> n - 1 \<Longrightarrow> card (insert x y) \<le> n"
apply (cases "finite y")
- apply (cases "x \<in> y")
- apply (auto simp: insert_absorb)
+ apply (cases "x \<in> y")
+ apply (auto simp: insert_absorb)
done
lemma card_Diff_singleton: "finite A \<Longrightarrow> x \<in> A \<Longrightarrow> card (A - {x}) = card A - 1"
@@ -1397,7 +1400,7 @@
lemma card_seteq: "finite B \<Longrightarrow> (\<And>A. A \<subseteq> B \<Longrightarrow> card B \<le> card A \<Longrightarrow> A = B)"
apply (induct rule: finite_induct)
- apply simp
+ apply simp
apply clarify
apply (subgoal_tac "finite A \<and> A - {x} \<subseteq> F")
prefer 2 apply (blast intro: finite_subset, atomize)
@@ -1430,7 +1433,7 @@
lemma card_Un_le: "card (A \<union> B) \<le> card A + card B"
apply (cases "finite A")
apply (cases "finite B")
- using le_iff_add card_Un_Int apply blast
+ apply (use le_iff_add card_Un_Int in blast)
apply simp
apply simp
done
@@ -1539,7 +1542,7 @@
lemma insert_partition:
"x \<notin> F \<Longrightarrow> \<forall>c1 \<in> insert x F. \<forall>c2 \<in> insert x F. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {} \<Longrightarrow> x \<inter> \<Union>F = {}"
- by auto
+ by auto (* somewhat slow *)
lemma finite_psubset_induct [consumes 1, case_names psubset]:
assumes finite: "finite A"
@@ -1597,13 +1600,13 @@
and remove: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
shows "P B"
proof (cases "finite B")
- assume "\<not>finite B"
+ case False
then show ?thesis by (rule infinite)
next
+ case True
define A where "A = B"
- assume "finite B"
- then have "finite A" "A \<subseteq> B"
- by (simp_all add: A_def)
+ with True have "finite A" "A \<subseteq> B"
+ by simp_all
then show "P A"
proof (induct "card A" arbitrary: A)
case 0
@@ -1623,9 +1626,9 @@
lemma finite_remove_induct [consumes 1, case_names empty remove]:
fixes P :: "'a set \<Rightarrow> bool"
- assumes finite: "finite B"
- and empty: "P {}"
- and rm: "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
+ assumes "finite B"
+ and "P {}"
+ and "\<And>A. finite A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> A \<subseteq> B \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> P (A - {x})) \<Longrightarrow> P A"
defines "B' \<equiv> B"
shows "P B'"
by (induct B' rule: remove_induct) (simp_all add: assms)
@@ -1636,10 +1639,14 @@
"finite C \<Longrightarrow> finite (\<Union>C) \<Longrightarrow> (\<forall>c\<in>C. card c = k) \<Longrightarrow>
(\<forall>c1 \<in> C. \<forall>c2 \<in> C. c1 \<noteq> c2 \<longrightarrow> c1 \<inter> c2 = {}) \<Longrightarrow>
k * card C = card (\<Union>C)"
- apply (induct rule: finite_induct)
- apply simp
- apply (simp add: card_Un_disjoint insert_partition finite_subset [of _ "\<Union>(insert x F)"])
- done
+proof (induct rule: finite_induct)
+ case empty
+ then show ?case by simp
+next
+ case (insert x F)
+ then show ?case
+ by (simp add: card_Un_disjoint insert_partition finite_subset [of _ "\<Union>(insert _ _)"])
+qed
lemma card_eq_UNIV_imp_eq_UNIV:
assumes fin: "finite (UNIV :: 'a set)"
@@ -1679,7 +1686,7 @@
show "b \<notin> A - {b}"
by blast
show "card (A - {b}) = k" and "k = 0 \<longrightarrow> A - {b} = {}"
- using assms b fin by(fastforce dest:mk_disjoint_insert)+
+ using assms b fin by (fastforce dest: mk_disjoint_insert)+
qed
qed
@@ -1688,7 +1695,7 @@
(\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
apply (auto elim!: card_eq_SucD)
apply (subst card.insert)
- apply (auto simp add: intro:ccontr)
+ apply (auto simp add: intro:ccontr)
done
lemma card_1_singletonE:
@@ -1761,7 +1768,7 @@
qed
lemma bij_betw_same_card: "bij_betw f A B \<Longrightarrow> card A = card B"
- by(auto simp: card_image bij_betw_def)
+ by (auto simp: card_image bij_betw_def)
lemma endo_inj_surj: "finite A \<Longrightarrow> f ` A \<subseteq> A \<Longrightarrow> inj_on f A \<Longrightarrow> f ` A = A"
by (simp add: card_seteq card_image)
@@ -1852,12 +1859,12 @@
from finite_Pow_iff[THEN iffD2, OF \<open>finite B\<close>] have "finite (?F ` A)"
by (blast intro: rev_finite_subset)
from pigeonhole_infinite [where f = ?F, OF assms(1) this]
- obtain a0 where "a0 \<in> A" and 1: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
+ obtain a0 where "a0 \<in> A" and infinite: "\<not> finite {a\<in>A. ?F a = ?F a0}" ..
obtain b0 where "b0 \<in> B" and "R a0 b0"
using \<open>a0 \<in> A\<close> assms(3) by blast
- have "finite {a\<in>A. ?F a = ?F a0}" if "finite{a:A. R a b0}"
+ have "finite {a\<in>A. ?F a = ?F a0}" if "finite {a\<in>A. R a b0}"
using \<open>b0 \<in> B\<close> \<open>R a0 b0\<close> that by (blast intro: rev_finite_subset)
- with 1 \<open>b0 : B\<close> show ?thesis
+ with infinite \<open>b0 \<in> B\<close> show ?thesis
by blast
qed
@@ -1896,7 +1903,7 @@
then show ?case
apply simp
apply (subst card_Un_disjoint)
- apply (auto simp add: disjoint_eq_subset_Compl)
+ apply (auto simp add: disjoint_eq_subset_Compl)
done
qed
qed
@@ -1914,10 +1921,12 @@
by (simp add: eq_card_imp_inj_on)
qed
-lemma finite_UNIV_surj_inj: "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f" for f :: "'a \<Rightarrow> 'a"
+lemma finite_UNIV_surj_inj: "finite(UNIV:: 'a set) \<Longrightarrow> surj f \<Longrightarrow> inj f"
+ for f :: "'a \<Rightarrow> 'a"
by (blast intro: finite_surj_inj subset_UNIV)
-lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f" for f :: "'a \<Rightarrow> 'a"
+lemma finite_UNIV_inj_surj: "finite(UNIV:: 'a set) \<Longrightarrow> inj f \<Longrightarrow> surj f"
+ for f :: "'a \<Rightarrow> 'a"
by (fastforce simp:surj_def dest!: endo_inj_surj)
corollary infinite_UNIV_nat [iff]: "\<not> finite (UNIV :: nat set)"
@@ -2013,7 +2022,7 @@
using psubset.hyps by blast
show False
apply (rule psubset.IH [where B = "A - {x}"])
- using \<open>x \<in> A\<close> apply blast
+ apply (use \<open>x \<in> A\<close> in blast)
apply (simp add: \<open>X (A - {x})\<close>)
done
qed
--- a/src/HOL/Hilbert_Choice.thy Fri Aug 05 17:36:38 2016 +0200
+++ b/src/HOL/Hilbert_Choice.thy Fri Aug 05 18:14:34 2016 +0200
@@ -6,23 +6,23 @@
section \<open>Hilbert's Epsilon-Operator and the Axiom of Choice\<close>
theory Hilbert_Choice
-imports Wellfounded
-keywords "specification" :: thy_goal
+ imports Wellfounded
+ keywords "specification" :: thy_goal
begin
subsection \<open>Hilbert's epsilon\<close>
-axiomatization Eps :: "('a => bool) => 'a" where
- someI: "P x ==> P (Eps P)"
+axiomatization Eps :: "('a \<Rightarrow> bool) \<Rightarrow> 'a"
+ where someI: "P x \<Longrightarrow> P (Eps P)"
syntax (epsilon)
- "_Eps" :: "[pttrn, bool] => 'a" ("(3\<some>_./ _)" [0, 10] 10)
+ "_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a" ("(3\<some>_./ _)" [0, 10] 10)
syntax (input)
- "_Eps" :: "[pttrn, bool] => 'a" ("(3@ _./ _)" [0, 10] 10)
+ "_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a" ("(3@ _./ _)" [0, 10] 10)
syntax
- "_Eps" :: "[pttrn, bool] => 'a" ("(3SOME _./ _)" [0, 10] 10)
+ "_Eps" :: "pttrn \<Rightarrow> bool \<Rightarrow> 'a" ("(3SOME _./ _)" [0, 10] 10)
translations
- "SOME x. P" == "CONST Eps (%x. P)"
+ "SOME x. P" \<rightleftharpoons> "CONST Eps (\<lambda>x. P)"
print_translation \<open>
[(@{const_syntax Eps}, fn _ => fn [Abs abs] =>
@@ -30,90 +30,92 @@
in Syntax.const @{syntax_const "_Eps"} $ x $ t end)]
\<close> \<comment> \<open>to avoid eta-contraction of body\<close>
-definition inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
-"inv_into A f == %x. SOME y. y : A & f y = x"
+definition inv_into :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
+ where "inv_into A f \<equiv> \<lambda>x. SOME y. y \<in> A \<and> f y = x"
-abbreviation inv :: "('a => 'b) => ('b => 'a)" where
-"inv == inv_into UNIV"
+abbreviation inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)"
+ where "inv \<equiv> inv_into UNIV"
subsection \<open>Hilbert's Epsilon-operator\<close>
-text\<open>Easier to apply than \<open>someI\<close> if the witness comes from an
-existential formula\<close>
-lemma someI_ex [elim?]: "\<exists>x. P x ==> P (SOME x. P x)"
-apply (erule exE)
-apply (erule someI)
-done
+text \<open>
+ Easier to apply than \<open>someI\<close> if the witness comes from an
+ existential formula.
+\<close>
+lemma someI_ex [elim?]: "\<exists>x. P x \<Longrightarrow> P (SOME x. P x)"
+ apply (erule exE)
+ apply (erule someI)
+ done
-text\<open>Easier to apply than \<open>someI\<close> because the conclusion has only one
-occurrence of @{term P}.\<close>
-lemma someI2: "[| P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
+text \<open>
+ Easier to apply than \<open>someI\<close> because the conclusion has only one
+ occurrence of @{term P}.
+\<close>
+lemma someI2: "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. P x)"
by (blast intro: someI)
-text\<open>Easier to apply than \<open>someI2\<close> if the witness comes from an
-existential formula\<close>
-
-lemma someI2_ex: "[| \<exists>a. P a; !!x. P x ==> Q x |] ==> Q (SOME x. P x)"
- by (blast intro: someI2)
-
-lemma someI2_bex: "[| \<exists>a\<in>A. P a; !!x. x \<in> A \<and> P x ==> Q x |] ==> Q (SOME x. x \<in> A \<and> P x)"
+text \<open>
+ Easier to apply than \<open>someI2\<close> if the witness comes from an
+ existential formula.
+\<close>
+lemma someI2_ex: "\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. P x)"
by (blast intro: someI2)
-lemma some_equality [intro]:
- "[| P a; !!x. P x ==> x=a |] ==> (SOME x. P x) = a"
-by (blast intro: someI2)
+lemma someI2_bex: "\<exists>a\<in>A. P a \<Longrightarrow> (\<And>x. x \<in> A \<and> P x \<Longrightarrow> Q x) \<Longrightarrow> Q (SOME x. x \<in> A \<and> P x)"
+ by (blast intro: someI2)
+
+lemma some_equality [intro]: "P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> x = a) \<Longrightarrow> (SOME x. P x) = a"
+ by (blast intro: someI2)
-lemma some1_equality: "[| EX!x. P x; P a |] ==> (SOME x. P x) = a"
-by blast
+lemma some1_equality: "EX!x. P x \<Longrightarrow> P a \<Longrightarrow> (SOME x. P x) = a"
+ by blast
-lemma some_eq_ex: "P (SOME x. P x) = (\<exists>x. P x)"
-by (blast intro: someI)
+lemma some_eq_ex: "P (SOME x. P x) \<longleftrightarrow> (\<exists>x. P x)"
+ by (blast intro: someI)
lemma some_in_eq: "(SOME x. x \<in> A) \<in> A \<longleftrightarrow> A \<noteq> {}"
unfolding ex_in_conv[symmetric] by (rule some_eq_ex)
-lemma some_eq_trivial [simp]: "(SOME y. y=x) = x"
-apply (rule some_equality)
-apply (rule refl, assumption)
-done
+lemma some_eq_trivial [simp]: "(SOME y. y = x) = x"
+ by (rule some_equality) (rule refl)
-lemma some_sym_eq_trivial [simp]: "(SOME y. x=y) = x"
-apply (rule some_equality)
-apply (rule refl)
-apply (erule sym)
-done
+lemma some_sym_eq_trivial [simp]: "(SOME y. x = y) = x"
+ apply (rule some_equality)
+ apply (rule refl)
+ apply (erule sym)
+ done
-subsection\<open>Axiom of Choice, Proved Using the Description Operator\<close>
+subsection \<open>Axiom of Choice, Proved Using the Description Operator\<close>
-lemma choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>f. \<forall>x. Q x (f x)"
-by (fast elim: someI)
+lemma choice: "\<forall>x. \<exists>y. Q x y \<Longrightarrow> \<exists>f. \<forall>x. Q x (f x)"
+ by (fast elim: someI)
-lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y ==> \<exists>f. \<forall>x\<in>S. Q x (f x)"
-by (fast elim: someI)
+lemma bchoice: "\<forall>x\<in>S. \<exists>y. Q x y \<Longrightarrow> \<exists>f. \<forall>x\<in>S. Q x (f x)"
+ by (fast elim: someI)
lemma choice_iff: "(\<forall>x. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x. Q x (f x))"
-by (fast elim: someI)
+ by (fast elim: someI)
lemma choice_iff': "(\<forall>x. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x. P x \<longrightarrow> Q x (f x))"
-by (fast elim: someI)
+ by (fast elim: someI)
lemma bchoice_iff: "(\<forall>x\<in>S. \<exists>y. Q x y) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. Q x (f x))"
-by (fast elim: someI)
+ by (fast elim: someI)
lemma bchoice_iff': "(\<forall>x\<in>S. P x \<longrightarrow> (\<exists>y. Q x y)) \<longleftrightarrow> (\<exists>f. \<forall>x\<in>S. P x \<longrightarrow> Q x (f x))"
-by (fast elim: someI)
+ by (fast elim: someI)
lemma dependent_nat_choice:
- assumes 1: "\<exists>x. P 0 x" and
- 2: "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y"
+ assumes 1: "\<exists>x. P 0 x"
+ and 2: "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y"
shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))"
proof (intro exI allI conjI)
fix n
define f where "f = rec_nat (SOME x. P 0 x) (\<lambda>n x. SOME y. P (Suc n) y \<and> Q n x y)"
- have "P 0 (f 0)" "\<And>n. P n (f n) \<Longrightarrow> P (Suc n) (f (Suc n)) \<and> Q n (f n) (f (Suc n))"
- using someI_ex[OF 1] someI_ex[OF 2] by (simp_all add: f_def)
+ then have "P 0 (f 0)" "\<And>n. P n (f n) \<Longrightarrow> P (Suc n) (f (Suc n)) \<and> Q n (f n) (f (Suc n))"
+ using someI_ex[OF 1] someI_ex[OF 2] by simp_all
then show "P n (f n)" "Q n (f n) (f (Suc n))"
by (induct n) auto
qed
@@ -121,181 +123,172 @@
subsection \<open>Function Inverse\<close>
-lemma inv_def: "inv f = (%y. SOME x. f x = y)"
-by(simp add: inv_into_def)
+lemma inv_def: "inv f = (\<lambda>y. SOME x. f x = y)"
+ by (simp add: inv_into_def)
-lemma inv_into_into: "x : f ` A ==> inv_into A f x : A"
-apply (simp add: inv_into_def)
-apply (fast intro: someI2)
-done
+lemma inv_into_into: "x \<in> f ` A \<Longrightarrow> inv_into A f x \<in> A"
+ by (simp add: inv_into_def) (fast intro: someI2)
-lemma inv_identity [simp]:
- "inv (\<lambda>a. a) = (\<lambda>a. a)"
+lemma inv_identity [simp]: "inv (\<lambda>a. a) = (\<lambda>a. a)"
by (simp add: inv_def)
-lemma inv_id [simp]:
- "inv id = id"
+lemma inv_id [simp]: "inv id = id"
by (simp add: id_def)
-lemma inv_into_f_f [simp]:
- "[| inj_on f A; x : A |] ==> inv_into A f (f x) = x"
-apply (simp add: inv_into_def inj_on_def)
-apply (blast intro: someI2)
-done
+lemma inv_into_f_f [simp]: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> inv_into A f (f x) = x"
+ by (simp add: inv_into_def inj_on_def) (blast intro: someI2)
-lemma inv_f_f: "inj f ==> inv f (f x) = x"
-by simp
+lemma inv_f_f: "inj f \<Longrightarrow> inv f (f x) = x"
+ by simp
-lemma f_inv_into_f: "y : f`A ==> f (inv_into A f y) = y"
-apply (simp add: inv_into_def)
-apply (fast intro: someI2)
-done
+lemma f_inv_into_f: "y : f`A \<Longrightarrow> f (inv_into A f y) = y"
+ by (simp add: inv_into_def) (fast intro: someI2)
-lemma inv_into_f_eq: "[| inj_on f A; x : A; f x = y |] ==> inv_into A f y = x"
-apply (erule subst)
-apply (fast intro: inv_into_f_f)
-done
+lemma inv_into_f_eq: "inj_on f A \<Longrightarrow> x \<in> A \<Longrightarrow> f x = y \<Longrightarrow> inv_into A f y = x"
+ by (erule subst) (fast intro: inv_into_f_f)
-lemma inv_f_eq: "[| inj f; f x = y |] ==> inv f y = x"
-by (simp add:inv_into_f_eq)
+lemma inv_f_eq: "inj f \<Longrightarrow> f x = y \<Longrightarrow> inv f y = x"
+ by (simp add:inv_into_f_eq)
-lemma inj_imp_inv_eq: "[| inj f; ALL x. f(g x) = x |] ==> inv f = g"
+lemma inj_imp_inv_eq: "inj f \<Longrightarrow> \<forall>x. f (g x) = x \<Longrightarrow> inv f = g"
by (blast intro: inv_into_f_eq)
-text\<open>But is it useful?\<close>
+text \<open>But is it useful?\<close>
lemma inj_transfer:
- assumes injf: "inj f" and minor: "!!y. y \<in> range(f) ==> P(inv f y)"
+ assumes inj: "inj f"
+ and minor: "\<And>y. y \<in> range f \<Longrightarrow> P (inv f y)"
shows "P x"
proof -
have "f x \<in> range f" by auto
- hence "P(inv f (f x))" by (rule minor)
- thus "P x" by (simp add: inv_into_f_f [OF injf])
+ then have "P(inv f (f x))" by (rule minor)
+ then show "P x" by (simp add: inv_into_f_f [OF inj])
qed
-lemma inj_iff: "(inj f) = (inv f o f = id)"
-apply (simp add: o_def fun_eq_iff)
-apply (blast intro: inj_on_inverseI inv_into_f_f)
-done
+lemma inj_iff: "inj f \<longleftrightarrow> inv f \<circ> f = id"
+ by (simp add: o_def fun_eq_iff) (blast intro: inj_on_inverseI inv_into_f_f)
-lemma inv_o_cancel[simp]: "inj f ==> inv f o f = id"
-by (simp add: inj_iff)
+lemma inv_o_cancel[simp]: "inj f \<Longrightarrow> inv f \<circ> f = id"
+ by (simp add: inj_iff)
+
+lemma o_inv_o_cancel[simp]: "inj f \<Longrightarrow> g \<circ> inv f \<circ> f = g"
+ by (simp add: comp_assoc)
-lemma o_inv_o_cancel[simp]: "inj f ==> g o inv f o f = g"
-by (simp add: comp_assoc)
+lemma inv_into_image_cancel[simp]: "inj_on f A \<Longrightarrow> S \<subseteq> A \<Longrightarrow> inv_into A f ` f ` S = S"
+ by (fastforce simp: image_def)
-lemma inv_into_image_cancel[simp]:
- "inj_on f A ==> S <= A ==> inv_into A f ` f ` S = S"
-by(fastforce simp: image_def)
+lemma inj_imp_surj_inv: "inj f \<Longrightarrow> surj (inv f)"
+ by (blast intro!: surjI inv_into_f_f)
-lemma inj_imp_surj_inv: "inj f ==> surj (inv f)"
-by (blast intro!: surjI inv_into_f_f)
-
-lemma surj_f_inv_f: "surj f ==> f(inv f y) = y"
-by (simp add: f_inv_into_f)
+lemma surj_f_inv_f: "surj f \<Longrightarrow> f (inv f y) = y"
+ by (simp add: f_inv_into_f)
lemma inv_into_injective:
assumes eq: "inv_into A f x = inv_into A f y"
- and x: "x: f`A"
- and y: "y: f`A"
- shows "x=y"
+ and x: "x \<in> f`A"
+ and y: "y \<in> f`A"
+ shows "x = y"
proof -
- have "f (inv_into A f x) = f (inv_into A f y)" using eq by simp
- thus ?thesis by (simp add: f_inv_into_f x y)
+ from eq have "f (inv_into A f x) = f (inv_into A f y)"
+ by simp
+ with x y show ?thesis
+ by (simp add: f_inv_into_f)
qed
-lemma inj_on_inv_into: "B <= f`A ==> inj_on (inv_into A f) B"
-by (blast intro: inj_onI dest: inv_into_injective injD)
+lemma inj_on_inv_into: "B \<subseteq> f`A \<Longrightarrow> inj_on (inv_into A f) B"
+ by (blast intro: inj_onI dest: inv_into_injective injD)
-lemma bij_betw_inv_into: "bij_betw f A B ==> bij_betw (inv_into A f) B A"
-by (auto simp add: bij_betw_def inj_on_inv_into)
+lemma bij_betw_inv_into: "bij_betw f A B \<Longrightarrow> bij_betw (inv_into A f) B A"
+ by (auto simp add: bij_betw_def inj_on_inv_into)
-lemma surj_imp_inj_inv: "surj f ==> inj (inv f)"
-by (simp add: inj_on_inv_into)
+lemma surj_imp_inj_inv: "surj f \<Longrightarrow> inj (inv f)"
+ by (simp add: inj_on_inv_into)
-lemma surj_iff: "(surj f) = (f o inv f = id)"
-by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
+lemma surj_iff: "surj f \<longleftrightarrow> f \<circ> inv f = id"
+ by (auto intro!: surjI simp: surj_f_inv_f fun_eq_iff[where 'b='a])
lemma surj_iff_all: "surj f \<longleftrightarrow> (\<forall>x. f (inv f x) = x)"
- unfolding surj_iff by (simp add: o_def fun_eq_iff)
+ by (simp add: o_def surj_iff fun_eq_iff)
-lemma surj_imp_inv_eq: "[| surj f; \<forall>x. g(f x) = x |] ==> inv f = g"
-apply (rule ext)
-apply (drule_tac x = "inv f x" in spec)
-apply (simp add: surj_f_inv_f)
-done
+lemma surj_imp_inv_eq: "surj f \<Longrightarrow> \<forall>x. g (f x) = x \<Longrightarrow> inv f = g"
+ apply (rule ext)
+ apply (drule_tac x = "inv f x" in spec)
+ apply (simp add: surj_f_inv_f)
+ done
-lemma bij_imp_bij_inv: "bij f ==> bij (inv f)"
-by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
+lemma bij_imp_bij_inv: "bij f \<Longrightarrow> bij (inv f)"
+ by (simp add: bij_def inj_imp_surj_inv surj_imp_inj_inv)
-lemma inv_equality: "[| !!x. g (f x) = x; !!y. f (g y) = y |] ==> inv f = g"
-apply (rule ext)
-apply (auto simp add: inv_into_def)
-done
+lemma inv_equality: "(\<And>x. g (f x) = x) \<Longrightarrow> (\<And>y. f (g y) = y) \<Longrightarrow> inv f = g"
+ by (rule ext) (auto simp add: inv_into_def)
+
+lemma inv_inv_eq: "bij f \<Longrightarrow> inv (inv f) = f"
+ by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f)
-lemma inv_inv_eq: "bij f ==> inv (inv f) = f"
-apply (rule inv_equality)
-apply (auto simp add: bij_def surj_f_inv_f)
-done
-
-(** bij(inv f) implies little about f. Consider f::bool=>bool such that
- f(True)=f(False)=True. Then it's consistent with axiom someI that
- inv f could be any function at all, including the identity function.
- If inv f=id then inv f is a bijection, but inj f, surj(f) and
- inv(inv f)=f all fail.
-**)
+text \<open>
+ \<open>bij (inv f)\<close> implies little about \<open>f\<close>. Consider \<open>f :: bool \<Rightarrow> bool\<close> such
+ that \<open>f True = f False = True\<close>. Then it ia consistent with axiom \<open>someI\<close>
+ that \<open>inv f\<close> could be any function at all, including the identity function.
+ If \<open>inv f = id\<close> then \<open>inv f\<close> is a bijection, but \<open>inj f\<close>, \<open>surj f\<close> and \<open>inv
+ (inv f) = f\<close> all fail.
+\<close>
lemma inv_into_comp:
- "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
- inv_into A (f o g) x = (inv_into A g o inv_into (g ` A) f) x"
-apply (rule inv_into_f_eq)
- apply (fast intro: comp_inj_on)
- apply (simp add: inv_into_into)
-apply (simp add: f_inv_into_f inv_into_into)
-done
+ "inj_on f (g ` A) \<Longrightarrow> inj_on g A \<Longrightarrow> x \<in> f ` g ` A \<Longrightarrow>
+ inv_into A (f \<circ> g) x = (inv_into A g \<circ> inv_into (g ` A) f) x"
+ apply (rule inv_into_f_eq)
+ apply (fast intro: comp_inj_on)
+ apply (simp add: inv_into_into)
+ apply (simp add: f_inv_into_f inv_into_into)
+ done
-lemma o_inv_distrib: "[| bij f; bij g |] ==> inv (f o g) = inv g o inv f"
-apply (rule inv_equality)
-apply (auto simp add: bij_def surj_f_inv_f)
-done
+lemma o_inv_distrib: "bij f \<Longrightarrow> bij g \<Longrightarrow> inv (f \<circ> g) = inv g \<circ> inv f"
+ by (rule inv_equality) (auto simp add: bij_def surj_f_inv_f)
-lemma image_surj_f_inv_f: "surj f ==> f ` (inv f ` A) = A"
+lemma image_surj_f_inv_f: "surj f \<Longrightarrow> f ` (inv f ` A) = A"
by (simp add: surj_f_inv_f image_comp comp_def)
-lemma image_inv_f_f: "inj f ==> inv f ` (f ` A) = A"
+lemma image_inv_f_f: "inj f \<Longrightarrow> inv f ` (f ` A) = A"
by simp
-lemma inv_image_comp: "inj f ==> inv f ` (f ` X) = X"
+lemma inv_image_comp: "inj f \<Longrightarrow> inv f ` (f ` X) = X"
by (fact image_inv_f_f)
-lemma bij_image_Collect_eq: "bij f ==> f ` Collect P = {y. P (inv f y)}"
-apply auto
-apply (force simp add: bij_is_inj)
-apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
-done
+lemma bij_image_Collect_eq: "bij f \<Longrightarrow> f ` Collect P = {y. P (inv f y)}"
+ apply auto
+ apply (force simp add: bij_is_inj)
+ apply (blast intro: bij_is_surj [THEN surj_f_inv_f, symmetric])
+ done
-lemma bij_vimage_eq_inv_image: "bij f ==> f -` A = inv f ` A"
-apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
-apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
-done
+lemma bij_vimage_eq_inv_image: "bij f \<Longrightarrow> f -` A = inv f ` A"
+ apply (auto simp add: bij_is_surj [THEN surj_f_inv_f])
+ apply (blast intro: bij_is_inj [THEN inv_into_f_f, symmetric])
+ done
lemma finite_fun_UNIVD1:
assumes fin: "finite (UNIV :: ('a \<Rightarrow> 'b) set)"
- and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
+ and card: "card (UNIV :: 'b set) \<noteq> Suc 0"
shows "finite (UNIV :: 'a set)"
proof -
- from fin have finb: "finite (UNIV :: 'b set)" by (rule finite_fun_UNIVD2)
+ from fin have finb: "finite (UNIV :: 'b set)"
+ by (rule finite_fun_UNIVD2)
with card have "card (UNIV :: 'b set) \<ge> Suc (Suc 0)"
by (cases "card (UNIV :: 'b set)") (auto simp add: card_eq_0_iff)
- then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)" by auto
- then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)" by (auto simp add: card_Suc_eq)
- from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))" by (rule finite_imageI)
+ then obtain n where "card (UNIV :: 'b set) = Suc (Suc n)" "n = card (UNIV :: 'b set) - Suc (Suc 0)"
+ by auto
+ then obtain b1 b2 where b1b2: "(b1 :: 'b) \<noteq> (b2 :: 'b)"
+ by (auto simp add: card_Suc_eq)
+ from fin have "finite (range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1))"
+ by (rule finite_imageI)
moreover have "UNIV = range (\<lambda>f :: 'a \<Rightarrow> 'b. inv f b1)"
proof (rule UNIV_eq_I)
fix x :: 'a
- from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1" by (simp add: inv_into_def)
- thus "x \<in> range (\<lambda>f::'a \<Rightarrow> 'b. inv f b1)" by blast
+ from b1b2 have "x = inv (\<lambda>y. if y = x then b1 else b2) b1"
+ by (simp add: inv_into_def)
+ then show "x \<in> range (\<lambda>f::'a \<Rightarrow> 'b. inv f b1)"
+ by blast
qed
- ultimately show "finite (UNIV :: 'a set)" by simp
+ ultimately show "finite (UNIV :: 'a set)"
+ by simp
qed
text \<open>
@@ -318,18 +311,18 @@
define Sseq where "Sseq = rec_nat S (\<lambda>n T. T - {SOME e. e \<in> T})"
define pick where "pick n = (SOME e. e \<in> Sseq n)" for n
have *: "Sseq n \<subseteq> S" "\<not> finite (Sseq n)" for n
- by (induct n) (auto simp add: Sseq_def inf)
+ by (induct n) (auto simp: Sseq_def inf)
then have **: "\<And>n. pick n \<in> Sseq n"
unfolding pick_def by (subst (asm) finite.simps) (auto simp add: ex_in_conv intro: someI_ex)
with * have "range pick \<subseteq> S" by auto
- moreover
- {
- fix n m
+ moreover have "pick n \<noteq> pick (n + Suc m)" for m n
+ proof -
have "pick n \<notin> Sseq (n + Suc m)"
by (induct m) (auto simp add: Sseq_def pick_def)
- with ** have "pick n \<noteq> pick (n + Suc m)" by auto
- }
- then have "inj pick" by (intro linorder_injI) (auto simp add: less_iff_Suc_add)
+ with ** show ?thesis by auto
+ qed
+ then have "inj pick"
+ by (intro linorder_injI) (auto simp add: less_iff_Suc_add)
ultimately show ?thesis by blast
qed
@@ -338,31 +331,35 @@
using finite_imageD finite_subset infinite_UNIV_char_0 infinite_countable_subset by auto
lemma image_inv_into_cancel:
- assumes SURJ: "f`A=A'" and SUB: "B' \<le> A'"
+ assumes surj: "f`A = A'"
+ and sub: "B' \<subseteq> A'"
shows "f `((inv_into A f)`B') = B'"
using assms
-proof (auto simp add: f_inv_into_f)
- let ?f' = "(inv_into A f)"
- fix a' assume *: "a' \<in> B'"
- then have "a' \<in> A'" using SUB by auto
- then have "a' = f (?f' a')"
- using SURJ by (auto simp add: f_inv_into_f)
- then show "a' \<in> f ` (?f' ` B')" using * by blast
+proof (auto simp: f_inv_into_f)
+ let ?f' = "inv_into A f"
+ fix a'
+ assume *: "a' \<in> B'"
+ with sub have "a' \<in> A'" by auto
+ with surj have "a' = f (?f' a')"
+ by (auto simp: f_inv_into_f)
+ with * show "a' \<in> f ` (?f' ` B')" by blast
qed
lemma inv_into_inv_into_eq:
- assumes "bij_betw f A A'" "a \<in> A"
+ assumes "bij_betw f A A'"
+ and a: "a \<in> A"
shows "inv_into A' (inv_into A f) a = f a"
proof -
- let ?f' = "inv_into A f" let ?f'' = "inv_into A' ?f'"
- have 1: "bij_betw ?f' A' A" using assms
- by (auto simp add: bij_betw_inv_into)
- obtain a' where 2: "a' \<in> A'" and 3: "?f' a' = a"
- using 1 \<open>a \<in> A\<close> unfolding bij_betw_def by force
- hence "?f'' a = a'"
- using \<open>a \<in> A\<close> 1 3 by (auto simp add: f_inv_into_f bij_betw_def)
- moreover have "f a = a'" using assms 2 3
- by (auto simp add: bij_betw_def)
+ let ?f' = "inv_into A f"
+ let ?f'' = "inv_into A' ?f'"
+ from assms have *: "bij_betw ?f' A' A"
+ by (auto simp: bij_betw_inv_into)
+ with a obtain a' where a': "a' \<in> A'" "?f' a' = a"
+ unfolding bij_betw_def by force
+ with a * have "?f'' a = a'"
+ by (auto simp: f_inv_into_f bij_betw_def)
+ moreover from assms a' have "f a = a'"
+ by (auto simp: bij_betw_def)
ultimately show "?f'' a = f a" by simp
qed
@@ -370,72 +367,82 @@
assumes "A \<noteq> {}"
shows "(\<exists>f. inj_on f A \<and> f ` A \<le> A') \<longleftrightarrow> (\<exists>g. g ` A' = A)"
proof safe
- fix f assume INJ: "inj_on f A" and INCL: "f ` A \<le> A'"
- let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'" let ?csi = "\<lambda>a. a \<in> A"
+ fix f
+ assume inj: "inj_on f A" and incl: "f ` A \<subseteq> A'"
+ let ?phi = "\<lambda>a' a. a \<in> A \<and> f a = a'"
+ let ?csi = "\<lambda>a. a \<in> A"
let ?g = "\<lambda>a'. if a' \<in> f ` A then (SOME a. ?phi a' a) else (SOME a. ?csi a)"
have "?g ` A' = A"
proof
- show "?g ` A' \<le> A"
+ show "?g ` A' \<subseteq> A"
proof clarify
- fix a' assume *: "a' \<in> A'"
+ fix a'
+ assume *: "a' \<in> A'"
show "?g a' \<in> A"
- proof cases
- assume Case1: "a' \<in> f ` A"
+ proof (cases "a' \<in> f ` A")
+ case True
then obtain a where "?phi a' a" by blast
- hence "?phi a' (SOME a. ?phi a' a)" using someI[of "?phi a'" a] by blast
- with Case1 show ?thesis by auto
+ then have "?phi a' (SOME a. ?phi a' a)"
+ using someI[of "?phi a'" a] by blast
+ with True show ?thesis by auto
next
- assume Case2: "a' \<notin> f ` A"
- hence "?csi (SOME a. ?csi a)" using assms someI_ex[of ?csi] by blast
- with Case2 show ?thesis by auto
+ case False
+ with assms have "?csi (SOME a. ?csi a)"
+ using someI_ex[of ?csi] by blast
+ with False show ?thesis by auto
qed
qed
next
- show "A \<le> ?g ` A'"
- proof-
- {fix a assume *: "a \<in> A"
- let ?b = "SOME aa. ?phi (f a) aa"
- have "?phi (f a) a" using * by auto
- hence 1: "?phi (f a) ?b" using someI[of "?phi(f a)" a] by blast
- hence "?g(f a) = ?b" using * by auto
- moreover have "a = ?b" using 1 INJ * by (auto simp add: inj_on_def)
- ultimately have "?g(f a) = a" by simp
- with INCL * have "?g(f a) = a \<and> f a \<in> A'" by auto
- }
- thus ?thesis by force
+ show "A \<subseteq> ?g ` A'"
+ proof -
+ have "?g (f a) = a \<and> f a \<in> A'" if a: "a \<in> A" for a
+ proof -
+ let ?b = "SOME aa. ?phi (f a) aa"
+ from a have "?phi (f a) a" by auto
+ then have *: "?phi (f a) ?b"
+ using someI[of "?phi(f a)" a] by blast
+ then have "?g (f a) = ?b" using a by auto
+ moreover from inj * a have "a = ?b"
+ by (auto simp add: inj_on_def)
+ ultimately have "?g(f a) = a" by simp
+ with incl a show ?thesis by auto
+ qed
+ then show ?thesis by force
qed
qed
- thus "\<exists>g. g ` A' = A" by blast
+ then show "\<exists>g. g ` A' = A" by blast
next
- fix g let ?f = "inv_into A' g"
+ fix g
+ let ?f = "inv_into A' g"
have "inj_on ?f (g ` A')"
- by (auto simp add: inj_on_inv_into)
- moreover
- {fix a' assume *: "a' \<in> A'"
- let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
- have "?phi a'" using * by auto
- hence "?phi(SOME b'. ?phi b')" using someI[of ?phi] by blast
- hence "?f(g a') \<in> A'" unfolding inv_into_def by auto
- }
- ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'" by auto
+ by (auto simp: inj_on_inv_into)
+ moreover have "?f (g a') \<in> A'" if a': "a' \<in> A'" for a'
+ proof -
+ let ?phi = "\<lambda> b'. b' \<in> A' \<and> g b' = g a'"
+ from a' have "?phi a'" by auto
+ then have "?phi (SOME b'. ?phi b')"
+ using someI[of ?phi] by blast
+ then show ?thesis by (auto simp: inv_into_def)
+ qed
+ ultimately show "\<exists>f. inj_on f (g ` A') \<and> f ` g ` A' \<subseteq> A'"
+ by auto
qed
lemma Ex_inj_on_UNION_Sigma:
"\<exists>f. (inj_on f (\<Union>i \<in> I. A i) \<and> f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i))"
proof
- let ?phi = "\<lambda> a i. i \<in> I \<and> a \<in> A i"
- let ?sm = "\<lambda> a. SOME i. ?phi a i"
+ let ?phi = "\<lambda>a i. i \<in> I \<and> a \<in> A i"
+ let ?sm = "\<lambda>a. SOME i. ?phi a i"
let ?f = "\<lambda>a. (?sm a, a)"
- have "inj_on ?f (\<Union>i \<in> I. A i)" unfolding inj_on_def by auto
+ have "inj_on ?f (\<Union>i \<in> I. A i)"
+ by (auto simp: inj_on_def)
moreover
- { { fix i a assume "i \<in> I" and "a \<in> A i"
- hence "?sm a \<in> I \<and> a \<in> A(?sm a)" using someI[of "?phi a" i] by auto
- }
- hence "?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)" by auto
- }
- ultimately
- show "inj_on ?f (\<Union>i \<in> I. A i) \<and> ?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)"
- by auto
+ have "?sm a \<in> I \<and> a \<in> A(?sm a)" if "i \<in> I" and "a \<in> A i" for i a
+ using that someI[of "?phi a" i] by auto
+ then have "?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)"
+ by auto
+ ultimately show "inj_on ?f (\<Union>i \<in> I. A i) \<and> ?f ` (\<Union>i \<in> I. A i) \<le> (SIGMA i : I. A i)"
+ by auto
qed
lemma inv_unique_comp:
@@ -448,190 +455,199 @@
subsection \<open>The Cantor-Bernstein Theorem\<close>
lemma Cantor_Bernstein_aux:
- shows "\<exists>A' h. A' \<le> A \<and>
- (\<forall>a \<in> A'. a \<notin> g`(B - f ` A')) \<and>
- (\<forall>a \<in> A'. h a = f a) \<and>
- (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a))"
-proof-
- obtain H where H_def: "H = (\<lambda> A'. A - (g`(B - (f ` A'))))" by blast
- have 0: "mono H" unfolding mono_def H_def by blast
- then obtain A' where 1: "H A' = A'" using lfp_unfold by blast
- hence 2: "A' = A - (g`(B - (f ` A')))" unfolding H_def by simp
- hence 3: "A' \<le> A" by blast
- have 4: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')"
- using 2 by blast
- have 5: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
- using 2 by blast
- (* *)
- obtain h where h_def:
- "h = (\<lambda> a. if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" by blast
- hence "\<forall>a \<in> A'. h a = f a" by auto
- moreover
- have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
+ "\<exists>A' h. A' \<subseteq> A \<and>
+ (\<forall>a \<in> A'. a \<notin> g ` (B - f ` A')) \<and>
+ (\<forall>a \<in> A'. h a = f a) \<and>
+ (\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g (h a))"
+proof -
+ define H where "H A' = A - (g ` (B - (f ` A')))" for A'
+ have "mono H" unfolding mono_def H_def by blast
+ from lfp_unfold [OF this] obtain A' where "H A' = A'" by blast
+ then have "A' = A - (g ` (B - (f ` A')))" by (simp add: H_def)
+ then have 1: "A' \<subseteq> A"
+ and 2: "\<forall>a \<in> A'. a \<notin> g ` (B - f ` A')"
+ and 3: "\<forall>a \<in> A - A'. \<exists>b \<in> B - (f ` A'). a = g b"
+ by blast+
+ define h where "h a = (if a \<in> A' then f a else (SOME b. b \<in> B - (f ` A') \<and> a = g b))" for a
+ then have 4: "\<forall>a \<in> A'. h a = f a" by simp
+ have "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g (h a)"
proof
- fix a assume *: "a \<in> A - A'"
- let ?phi = "\<lambda> b. b \<in> B - (f ` A') \<and> a = g b"
- have "h a = (SOME b. ?phi b)" using h_def * by auto
- moreover have "\<exists>b. ?phi b" using 5 * by auto
- ultimately show "?phi (h a)" using someI_ex[of ?phi] by auto
+ fix a
+ let ?phi = "\<lambda>b. b \<in> B - (f ` A') \<and> a = g b"
+ assume *: "a \<in> A - A'"
+ from * have "h a = (SOME b. ?phi b)" by (auto simp: h_def)
+ moreover from 3 * have "\<exists>b. ?phi b" by auto
+ ultimately show "?phi (h a)"
+ using someI_ex[of ?phi] by auto
qed
- ultimately show ?thesis using 3 4 by blast
+ with 1 2 4 show ?thesis by blast
qed
theorem Cantor_Bernstein:
- assumes INJ1: "inj_on f A" and SUB1: "f ` A \<le> B" and
- INJ2: "inj_on g B" and SUB2: "g ` B \<le> A"
+ assumes inj1: "inj_on f A" and sub1: "f ` A \<subseteq> B"
+ and inj2: "inj_on g B" and sub2: "g ` B \<subseteq> A"
shows "\<exists>h. bij_betw h A B"
proof-
- obtain A' and h where 0: "A' \<le> A" and
- 1: "\<forall>a \<in> A'. a \<notin> g`(B - f ` A')" and
- 2: "\<forall>a \<in> A'. h a = f a" and
- 3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g(h a)"
- using Cantor_Bernstein_aux[of A g B f] by blast
+ obtain A' and h where "A' \<subseteq> A"
+ and 1: "\<forall>a \<in> A'. a \<notin> g ` (B - f ` A')"
+ and 2: "\<forall>a \<in> A'. h a = f a"
+ and 3: "\<forall>a \<in> A - A'. h a \<in> B - (f ` A') \<and> a = g (h a)"
+ using Cantor_Bernstein_aux [of A g B f] by blast
have "inj_on h A"
proof (intro inj_onI)
fix a1 a2
assume 4: "a1 \<in> A" and 5: "a2 \<in> A" and 6: "h a1 = h a2"
show "a1 = a2"
- proof(cases "a1 \<in> A'")
- assume Case1: "a1 \<in> A'"
+ proof (cases "a1 \<in> A'")
+ case True
show ?thesis
- proof(cases "a2 \<in> A'")
- assume Case11: "a2 \<in> A'"
- hence "f a1 = f a2" using Case1 2 6 by auto
- thus ?thesis using INJ1 Case1 Case11 0
- unfolding inj_on_def by blast
+ proof (cases "a2 \<in> A'")
+ case True': True
+ with True 2 6 have "f a1 = f a2" by auto
+ with inj1 \<open>A' \<subseteq> A\<close> True True' show ?thesis
+ unfolding inj_on_def by blast
next
- assume Case12: "a2 \<notin> A'"
- hence False using 3 5 2 6 Case1 by force
- thus ?thesis by simp
+ case False
+ with 2 3 5 6 True have False by force
+ then show ?thesis ..
qed
next
- assume Case2: "a1 \<notin> A'"
+ case False
show ?thesis
- proof(cases "a2 \<in> A'")
- assume Case21: "a2 \<in> A'"
- hence False using 3 4 2 6 Case2 by auto
- thus ?thesis by simp
+ proof (cases "a2 \<in> A'")
+ case True
+ with 2 3 4 6 False have False by auto
+ then show ?thesis ..
next
- assume Case22: "a2 \<notin> A'"
- hence "a1 = g(h a1) \<and> a2 = g(h a2)" using Case2 4 5 3 by auto
- thus ?thesis using 6 by simp
+ case False': False
+ with False 3 4 5 have "a1 = g (h a1)" "a2 = g (h a2)" by auto
+ with 6 show ?thesis by simp
qed
qed
qed
- (* *)
- moreover
- have "h ` A = B"
+ moreover have "h ` A = B"
proof safe
- fix a assume "a \<in> A"
- thus "h a \<in> B" using SUB1 2 3 by (cases "a \<in> A'") auto
+ fix a
+ assume "a \<in> A"
+ with sub1 2 3 show "h a \<in> B" by (cases "a \<in> A'") auto
next
- fix b assume *: "b \<in> B"
+ fix b
+ assume *: "b \<in> B"
show "b \<in> h ` A"
- proof(cases "b \<in> f ` A'")
- assume Case1: "b \<in> f ` A'"
- then obtain a where "a \<in> A' \<and> b = f a" by blast
- thus ?thesis using 2 0 by force
+ proof (cases "b \<in> f ` A'")
+ case True
+ then obtain a where "a \<in> A'" "b = f a" by blast
+ with \<open>A' \<subseteq> A\<close> 2 show ?thesis by force
next
- assume Case2: "b \<notin> f ` A'"
- hence "g b \<notin> A'" using 1 * by auto
- hence 4: "g b \<in> A - A'" using * SUB2 by auto
- hence "h(g b) \<in> B \<and> g(h(g b)) = g b"
- using 3 by auto
- hence "h(g b) = b" using * INJ2 unfolding inj_on_def by auto
- thus ?thesis using 4 by force
+ case False
+ with 1 * have "g b \<notin> A'" by auto
+ with sub2 * have 4: "g b \<in> A - A'" by auto
+ with 3 have "h (g b) \<in> B" "g (h (g b)) = g b" by auto
+ with inj2 * have "h (g b) = b" by (auto simp: inj_on_def)
+ with 4 show ?thesis by force
qed
qed
- (* *)
- ultimately show ?thesis unfolding bij_betw_def by auto
+ ultimately show ?thesis
+ by (auto simp: bij_betw_def)
qed
+
subsection \<open>Other Consequences of Hilbert's Epsilon\<close>
text \<open>Hilbert's Epsilon and the @{term split} Operator\<close>
-text\<open>Looping simprule\<close>
-lemma split_paired_Eps: "(SOME x. P x) = (SOME (a,b). P(a,b))"
+text \<open>Looping simprule!\<close>
+lemma split_paired_Eps: "(SOME x. P x) = (SOME (a, b). P (a, b))"
by simp
lemma Eps_case_prod: "Eps (case_prod P) = (SOME xy. P (fst xy) (snd xy))"
by (simp add: split_def)
-lemma Eps_case_prod_eq [simp]: "(@(x',y'). x = x' & y = y') = (x,y)"
+lemma Eps_case_prod_eq [simp]: "(SOME (x', y'). x = x' \<and> y = y') = (x, y)"
by blast
-text\<open>A relation is wellfounded iff it has no infinite descending chain\<close>
-lemma wf_iff_no_infinite_down_chain:
- "wf r = (~(\<exists>f. \<forall>i. (f(Suc i),f i) \<in> r))"
-apply (simp only: wf_eq_minimal)
-apply (rule iffI)
- apply (rule notI)
- apply (erule exE)
- apply (erule_tac x = "{w. \<exists>i. w=f i}" in allE, blast)
-apply (erule contrapos_np, simp, clarify)
-apply (subgoal_tac "\<forall>n. rec_nat x (%i y. @z. z:Q & (z,y) :r) n \<in> Q")
- apply (rule_tac x = "rec_nat x (%i y. @z. z:Q & (z,y) :r)" in exI)
- apply (rule allI, simp)
- apply (rule someI2_ex, blast, blast)
-apply (rule allI)
-apply (induct_tac "n", simp_all)
-apply (rule someI2_ex, blast+)
-done
+text \<open>A relation is wellfounded iff it has no infinite descending chain.\<close>
+lemma wf_iff_no_infinite_down_chain: "wf r \<longleftrightarrow> (\<not> (\<exists>f. \<forall>i. (f (Suc i), f i) \<in> r))"
+ apply (simp only: wf_eq_minimal)
+ apply (rule iffI)
+ apply (rule notI)
+ apply (erule exE)
+ apply (erule_tac x = "{w. \<exists>i. w = f i}" in allE)
+ apply blast
+ apply (erule contrapos_np)
+ apply simp
+ apply clarify
+ apply (subgoal_tac "\<forall>n. rec_nat x (\<lambda>i y. SOME z. z \<in> Q \<and> (z, y) \<in> r) n \<in> Q")
+ apply (rule_tac x = "rec_nat x (\<lambda>i y. SOME z. z \<in> Q \<and> (z, y) \<in> r)" in exI)
+ apply (rule allI)
+ apply simp
+ apply (rule someI2_ex)
+ apply blast
+ apply blast
+ apply (rule allI)
+ apply (induct_tac n)
+ apply simp_all
+ apply (rule someI2_ex)
+ apply blast
+ apply blast
+ done
lemma wf_no_infinite_down_chainE:
- assumes "wf r" obtains k where "(f (Suc k), f k) \<notin> r"
-using \<open>wf r\<close> wf_iff_no_infinite_down_chain[of r] by blast
+ assumes "wf r"
+ obtains k where "(f (Suc k), f k) \<notin> r"
+ using assms wf_iff_no_infinite_down_chain[of r] by blast
-text\<open>A dynamically-scoped fact for TFL\<close>
-lemma tfl_some: "\<forall>P x. P x --> P (Eps P)"
+text \<open>A dynamically-scoped fact for TFL\<close>
+lemma tfl_some: "\<forall>P x. P x \<longrightarrow> P (Eps P)"
by (blast intro: someI)
subsection \<open>Least value operator\<close>
-definition
- LeastM :: "['a => 'b::ord, 'a => bool] => 'a" where
- "LeastM m P == SOME x. P x & (\<forall>y. P y --> m x <= m y)"
+definition LeastM :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a"
+ where "LeastM m P \<equiv> (SOME x. P x \<and> (\<forall>y. P y \<longrightarrow> m x \<le> m y))"
syntax
- "_LeastM" :: "[pttrn, 'a => 'b::ord, bool] => 'a" ("LEAST _ WRT _. _" [0, 4, 10] 10)
+ "_LeastM" :: "pttrn \<Rightarrow> ('a \<Rightarrow> 'b::ord) \<Rightarrow> bool \<Rightarrow> 'a" ("LEAST _ WRT _. _" [0, 4, 10] 10)
translations
- "LEAST x WRT m. P" == "CONST LeastM m (%x. P)"
+ "LEAST x WRT m. P" \<rightleftharpoons> "CONST LeastM m (\<lambda>x. P)"
lemma LeastMI2:
- "P x ==> (!!y. P y ==> m x <= m y)
- ==> (!!x. P x ==> \<forall>y. P y --> m x \<le> m y ==> Q x)
- ==> Q (LeastM m P)"
+ "P x \<Longrightarrow>
+ (\<And>y. P y \<Longrightarrow> m x \<le> m y) \<Longrightarrow>
+ (\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> m x \<le> m y \<Longrightarrow> Q x) \<Longrightarrow>
+ Q (LeastM m P)"
apply (simp add: LeastM_def)
- apply (rule someI2_ex, blast, blast)
+ apply (rule someI2_ex)
+ apply blast
+ apply blast
done
lemma LeastM_equality:
- "P k ==> (!!x. P x ==> m k <= m x)
- ==> m (LEAST x WRT m. P x) = (m k::'a::order)"
- apply (rule LeastMI2, assumption, blast)
+ "P k \<Longrightarrow> (\<And>x. P x \<Longrightarrow> m k \<le> m x) \<Longrightarrow> m (LEAST x WRT m. P x) = (m k :: 'a::order)"
+ apply (rule LeastMI2)
+ apply assumption
+ apply blast
apply (blast intro!: order_antisym)
done
lemma wf_linord_ex_has_least:
- "wf r ==> \<forall>x y. ((x,y):r^+) = ((y,x)~:r^*) ==> P k
- ==> \<exists>x. P x & (!y. P y --> (m x,m y):r^*)"
+ "wf r \<Longrightarrow> \<forall>x y. (x, y) \<in> r\<^sup>+ \<longleftrightarrow> (y, x) \<notin> r\<^sup>* \<Longrightarrow> P k \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> (m x, m y) \<in> r\<^sup>*)"
apply (drule wf_trancl [THEN wf_eq_minimal [THEN iffD1]])
- apply (drule_tac x = "m`Collect P" in spec, force)
+ apply (drule_tac x = "m ` Collect P" in spec)
+ apply force
done
-lemma ex_has_least_nat:
- "P k ==> \<exists>x. P x & (\<forall>y. P y --> m x <= (m y::nat))"
+lemma ex_has_least_nat: "P k \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> m x \<le> (m y :: nat))"
apply (simp only: pred_nat_trancl_eq_le [symmetric])
apply (rule wf_pred_nat [THEN wf_linord_ex_has_least])
- apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le, assumption)
+ apply (simp add: less_eq linorder_not_le pred_nat_trancl_eq_le)
+ apply assumption
done
-lemma LeastM_nat_lemma:
- "P k ==> P (LeastM m P) & (\<forall>y. P y --> m (LeastM m P) <= (m y::nat))"
+lemma LeastM_nat_lemma: "P k \<Longrightarrow> P (LeastM m P) \<and> (\<forall>y. P y \<longrightarrow> m (LeastM m P) \<le> (m y :: nat))"
apply (simp add: LeastM_def)
apply (rule someI_ex)
apply (erule ex_has_least_nat)
@@ -639,91 +655,87 @@
lemmas LeastM_natI = LeastM_nat_lemma [THEN conjunct1]
-lemma LeastM_nat_le: "P x ==> m (LeastM m P) <= (m x::nat)"
-by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp], assumption, assumption)
+lemma LeastM_nat_le: "P x \<Longrightarrow> m (LeastM m P) \<le> (m x :: nat)"
+ by (rule LeastM_nat_lemma [THEN conjunct2, THEN spec, THEN mp])
subsection \<open>Greatest value operator\<close>
-definition
- GreatestM :: "['a => 'b::ord, 'a => bool] => 'a" where
- "GreatestM m P == SOME x. P x & (\<forall>y. P y --> m y <= m x)"
+definition GreatestM :: "('a \<Rightarrow> 'b::ord) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a"
+ where "GreatestM m P \<equiv> SOME x. P x \<and> (\<forall>y. P y \<longrightarrow> m y \<le> m x)"
-definition
- Greatest :: "('a::ord => bool) => 'a" (binder "GREATEST " 10) where
- "Greatest == GreatestM (%x. x)"
+definition Greatest :: "('a::ord \<Rightarrow> bool) \<Rightarrow> 'a" (binder "GREATEST " 10)
+ where "Greatest \<equiv> GreatestM (\<lambda>x. x)"
syntax
- "_GreatestM" :: "[pttrn, 'a => 'b::ord, bool] => 'a"
- ("GREATEST _ WRT _. _" [0, 4, 10] 10)
+ "_GreatestM" :: "pttrn \<Rightarrow> ('a \<Rightarrow> 'b::ord) \<Rightarrow> bool \<Rightarrow> 'a" ("GREATEST _ WRT _. _" [0, 4, 10] 10)
translations
- "GREATEST x WRT m. P" == "CONST GreatestM m (%x. P)"
+ "GREATEST x WRT m. P" \<rightleftharpoons> "CONST GreatestM m (\<lambda>x. P)"
lemma GreatestMI2:
- "P x ==> (!!y. P y ==> m y <= m x)
- ==> (!!x. P x ==> \<forall>y. P y --> m y \<le> m x ==> Q x)
- ==> Q (GreatestM m P)"
+ "P x \<Longrightarrow>
+ (\<And>y. P y \<Longrightarrow> m y \<le> m x) \<Longrightarrow>
+ (\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> m y \<le> m x \<Longrightarrow> Q x) \<Longrightarrow>
+ Q (GreatestM m P)"
apply (simp add: GreatestM_def)
- apply (rule someI2_ex, blast, blast)
+ apply (rule someI2_ex)
+ apply blast
+ apply blast
done
lemma GreatestM_equality:
- "P k ==> (!!x. P x ==> m x <= m k)
- ==> m (GREATEST x WRT m. P x) = (m k::'a::order)"
- apply (rule_tac m = m in GreatestMI2, assumption, blast)
+ "P k \<Longrightarrow>
+ (\<And>x. P x \<Longrightarrow> m x \<le> m k) \<Longrightarrow>
+ m (GREATEST x WRT m. P x) = (m k :: 'a::order)"
+ apply (rule GreatestMI2 [where m = m])
+ apply assumption
+ apply blast
apply (blast intro!: order_antisym)
done
-lemma Greatest_equality:
- "P (k::'a::order) ==> (!!x. P x ==> x <= k) ==> (GREATEST x. P x) = k"
+lemma Greatest_equality: "P k \<Longrightarrow> (\<And>x. P x \<Longrightarrow> x \<le> k) \<Longrightarrow> (GREATEST x. P x) = k"
+ for k :: "'a::order"
apply (simp add: Greatest_def)
- apply (erule GreatestM_equality, blast)
+ apply (erule GreatestM_equality)
+ apply blast
done
lemma ex_has_greatest_nat_lemma:
- "P k ==> \<forall>x. P x --> (\<exists>y. P y & ~ ((m y::nat) <= m x))
- ==> \<exists>y. P y & ~ (m y < m k + n)"
- apply (induct n, force)
- apply (force simp add: le_Suc_eq)
- done
+ "P k \<Longrightarrow> \<forall>x. P x \<longrightarrow> (\<exists>y. P y \<and> \<not> m y \<le> (m x :: nat)) \<Longrightarrow> \<exists>y. P y \<and> \<not> m y < m k + n"
+ by (induct n) (force simp: le_Suc_eq)+
lemma ex_has_greatest_nat:
- "P k ==> \<forall>y. P y --> m y < b
- ==> \<exists>x. P x & (\<forall>y. P y --> (m y::nat) <= m x)"
+ "P k \<Longrightarrow> \<forall>y. P y \<longrightarrow> m y < b \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> m y \<le> (m x :: nat))"
apply (rule ccontr)
apply (cut_tac P = P and n = "b - m k" in ex_has_greatest_nat_lemma)
- apply (subgoal_tac [3] "m k <= b", auto)
+ apply (subgoal_tac [3] "m k \<le> b")
+ apply auto
done
lemma GreatestM_nat_lemma:
- "P k ==> \<forall>y. P y --> m y < b
- ==> P (GreatestM m P) & (\<forall>y. P y --> (m y::nat) <= m (GreatestM m P))"
+ "P k \<Longrightarrow> \<forall>y. P y \<longrightarrow> m y < b \<Longrightarrow>
+ P (GreatestM m P) \<and> (\<forall>y. P y \<longrightarrow> (m y :: nat) \<le> m (GreatestM m P))"
apply (simp add: GreatestM_def)
apply (rule someI_ex)
- apply (erule ex_has_greatest_nat, assumption)
+ apply (erule ex_has_greatest_nat)
+ apply assumption
done
lemmas GreatestM_natI = GreatestM_nat_lemma [THEN conjunct1]
-lemma GreatestM_nat_le:
- "P x ==> \<forall>y. P y --> m y < b
- ==> (m x::nat) <= m (GreatestM m P)"
- apply (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
- done
+lemma GreatestM_nat_le: "P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> m y < b \<Longrightarrow> (m x :: nat) \<le> m (GreatestM m P)"
+ by (blast dest: GreatestM_nat_lemma [THEN conjunct2, THEN spec, of P])
-text \<open>\medskip Specialization to \<open>GREATEST\<close>.\<close>
+text \<open>\<^medskip> Specialization to \<open>GREATEST\<close>.\<close>
-lemma GreatestI: "P (k::nat) ==> \<forall>y. P y --> y < b ==> P (GREATEST x. P x)"
- apply (simp add: Greatest_def)
- apply (rule GreatestM_natI, auto)
- done
+lemma GreatestI: "P k \<Longrightarrow> \<forall>y. P y \<longrightarrow> y < b \<Longrightarrow> P (GREATEST x. P x)"
+ for k :: nat
+ unfolding Greatest_def by (rule GreatestM_natI) auto
-lemma Greatest_le:
- "P x ==> \<forall>y. P y --> y < b ==> (x::nat) <= (GREATEST x. P x)"
- apply (simp add: Greatest_def)
- apply (rule GreatestM_nat_le, auto)
- done
+lemma Greatest_le: "P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> y < b \<Longrightarrow> x \<le> (GREATEST x. P x)"
+ for x :: nat
+ unfolding Greatest_def by (rule GreatestM_nat_le) auto
subsection \<open>An aside: bounded accessible part\<close>
@@ -732,7 +744,8 @@
lemma finite_mono_remains_stable_implies_strict_prefix:
fixes f :: "nat \<Rightarrow> 'a::order"
- assumes S: "finite (range f)" "mono f" and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
+ assumes S: "finite (range f)" "mono f"
+ and eq: "\<forall>n. f n = f (Suc n) \<longrightarrow> f (Suc n) = f (Suc (Suc n))"
shows "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m < f n) \<and> (\<forall>n\<ge>N. f N = f n)"
using assms
proof -
@@ -740,15 +753,16 @@
proof (rule ccontr)
assume "\<not> ?thesis"
then have "\<And>n. f n \<noteq> f (Suc n)" by auto
- then have "\<And>n. f n < f (Suc n)"
- using \<open>mono f\<close> by (auto simp: le_less mono_iff_le_Suc)
- with lift_Suc_mono_less_iff[of f]
- have *: "\<And>n m. n < m \<Longrightarrow> f n < f m" by auto
+ with \<open>mono f\<close> have "\<And>n. f n < f (Suc n)"
+ by (auto simp: le_less mono_iff_le_Suc)
+ with lift_Suc_mono_less_iff[of f] have *: "\<And>n m. n < m \<Longrightarrow> f n < f m"
+ by auto
have "inj f"
proof (intro injI)
fix x y
assume "f x = f y"
- then show "x = y" by (cases x y rule: linorder_cases) (auto dest: *)
+ then show "x = y"
+ by (cases x y rule: linorder_cases) (auto dest: *)
qed
with \<open>finite (range f)\<close> have "finite (UNIV::nat set)"
by (rule finite_imageD)
@@ -760,16 +774,22 @@
unfolding N_def using n by (rule LeastI)
show ?thesis
proof (intro exI[of _ N] conjI allI impI)
- fix n assume "N \<le> n"
+ fix n
+ assume "N \<le> n"
then have "\<And>m. N \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m = f N"
proof (induct rule: dec_induct)
- case (step n) then show ?case
- using eq[rule_format, of "n - 1"] N
+ case base
+ then show ?case by simp
+ next
+ case (step n)
+ then show ?case
+ using eq [rule_format, of "n - 1"] N
by (cases n) (auto simp add: le_Suc_eq)
- qed simp
+ qed
from this[of n] \<open>N \<le> n\<close> show "f N = f n" by auto
next
- fix n m :: nat assume "m < n" "n \<le> N"
+ fix n m :: nat
+ assume "m < n" "n \<le> N"
then show "f m < f n"
proof (induct rule: less_Suc_induct)
case (1 i)
@@ -777,37 +797,41 @@
then have "f i \<noteq> f (Suc i)"
unfolding N_def by (rule not_less_Least)
with \<open>mono f\<close> show ?case by (simp add: mono_iff_le_Suc less_le)
- qed auto
+ next
+ case 2
+ then show ?case by simp
+ qed
qed
qed
lemma finite_mono_strict_prefix_implies_finite_fixpoint:
fixes f :: "nat \<Rightarrow> 'a set"
assumes S: "\<And>i. f i \<subseteq> S" "finite S"
- and inj: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)"
+ and ex: "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)"
shows "f (card S) = (\<Union>n. f n)"
proof -
- from inj obtain N where inj: "(\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n)" and eq: "(\<forall>n\<ge>N. f N = f n)" by auto
-
- { fix i have "i \<le> N \<Longrightarrow> i \<le> card (f i)"
- proof (induct i)
- case 0 then show ?case by simp
- next
- case (Suc i)
- with inj[rule_format, of "Suc i" i]
- have "(f i) \<subset> (f (Suc i))" by auto
- moreover have "finite (f (Suc i))" using S by (rule finite_subset)
- ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
- with Suc show ?case using inj by auto
- qed
- }
+ from ex obtain N where inj: "\<And>n m. n \<le> N \<Longrightarrow> m \<le> N \<Longrightarrow> m < n \<Longrightarrow> f m \<subset> f n"
+ and eq: "\<forall>n\<ge>N. f N = f n"
+ by atomize auto
+ have "i \<le> N \<Longrightarrow> i \<le> card (f i)" for i
+ proof (induct i)
+ case 0
+ then show ?case by simp
+ next
+ case (Suc i)
+ with inj [of "Suc i" i] have "(f i) \<subset> (f (Suc i))" by auto
+ moreover have "finite (f (Suc i))" using S by (rule finite_subset)
+ ultimately have "card (f i) < card (f (Suc i))" by (intro psubset_card_mono)
+ with Suc inj show ?case by auto
+ qed
then have "N \<le> card (f N)" by simp
also have "\<dots> \<le> card S" using S by (intro card_mono)
finally have "f (card S) = f N" using eq by auto
- then show ?thesis using eq inj[rule_format, of N]
+ then show ?thesis
+ using eq inj [of N]
apply auto
apply (case_tac "n < N")
- apply (auto simp: not_less)
+ apply (auto simp: not_less)
done
qed
@@ -819,183 +843,174 @@
assumes bij: "bij f"
begin
-lemma bij_inv:
- "bij (inv f)"
+lemma bij_inv: "bij (inv f)"
using bij by (rule bij_imp_bij_inv)
-lemma surj [simp]:
- "surj f"
+lemma surj [simp]: "surj f"
using bij by (rule bij_is_surj)
-lemma inj:
- "inj f"
+lemma inj: "inj f"
using bij by (rule bij_is_inj)
-lemma surj_inv [simp]:
- "surj (inv f)"
+lemma surj_inv [simp]: "surj (inv f)"
using inj by (rule inj_imp_surj_inv)
-lemma inj_inv:
- "inj (inv f)"
+lemma inj_inv: "inj (inv f)"
using surj by (rule surj_imp_inj_inv)
-lemma eqI:
- "f a = f b \<Longrightarrow> a = b"
+lemma eqI: "f a = f b \<Longrightarrow> a = b"
using inj by (rule injD)
-lemma eq_iff [simp]:
- "f a = f b \<longleftrightarrow> a = b"
+lemma eq_iff [simp]: "f a = f b \<longleftrightarrow> a = b"
by (auto intro: eqI)
-lemma eq_invI:
- "inv f a = inv f b \<Longrightarrow> a = b"
+lemma eq_invI: "inv f a = inv f b \<Longrightarrow> a = b"
using inj_inv by (rule injD)
-lemma eq_inv_iff [simp]:
- "inv f a = inv f b \<longleftrightarrow> a = b"
+lemma eq_inv_iff [simp]: "inv f a = inv f b \<longleftrightarrow> a = b"
by (auto intro: eq_invI)
-lemma inv_left [simp]:
- "inv f (f a) = a"
+lemma inv_left [simp]: "inv f (f a) = a"
using inj by (simp add: inv_f_eq)
-lemma inv_comp_left [simp]:
- "inv f \<circ> f = id"
+lemma inv_comp_left [simp]: "inv f \<circ> f = id"
by (simp add: fun_eq_iff)
-lemma inv_right [simp]:
- "f (inv f a) = a"
+lemma inv_right [simp]: "f (inv f a) = a"
using surj by (simp add: surj_f_inv_f)
-lemma inv_comp_right [simp]:
- "f \<circ> inv f = id"
+lemma inv_comp_right [simp]: "f \<circ> inv f = id"
by (simp add: fun_eq_iff)
-lemma inv_left_eq_iff [simp]:
- "inv f a = b \<longleftrightarrow> f b = a"
+lemma inv_left_eq_iff [simp]: "inv f a = b \<longleftrightarrow> f b = a"
by auto
-lemma inv_right_eq_iff [simp]:
- "b = inv f a \<longleftrightarrow> f b = a"
+lemma inv_right_eq_iff [simp]: "b = inv f a \<longleftrightarrow> f b = a"
by auto
end
lemma infinite_imp_bij_betw:
-assumes INF: "\<not> finite A"
-shows "\<exists>h. bij_betw h A (A - {a})"
-proof(cases "a \<in> A")
- assume Case1: "a \<notin> A" hence "A - {a} = A" by blast
- thus ?thesis using bij_betw_id[of A] by auto
+ assumes infinite: "\<not> finite A"
+ shows "\<exists>h. bij_betw h A (A - {a})"
+proof (cases "a \<in> A")
+ case False
+ then have "A - {a} = A" by blast
+ then show ?thesis
+ using bij_betw_id[of A] by auto
next
- assume Case2: "a \<in> A"
- have "\<not> finite (A - {a})" using INF by auto
- with infinite_iff_countable_subset[of "A - {a}"] obtain f::"nat \<Rightarrow> 'a"
- where 1: "inj f" and 2: "f ` UNIV \<le> A - {a}" by blast
- obtain g where g_def: "g = (\<lambda> n. if n = 0 then a else f (Suc n))" by blast
- obtain A' where A'_def: "A' = g ` UNIV" by blast
- have temp: "\<forall>y. f y \<noteq> a" using 2 by blast
- have 3: "inj_on g UNIV \<and> g ` UNIV \<le> A \<and> a \<in> g ` UNIV"
- proof(auto simp add: Case2 g_def, unfold inj_on_def, intro ballI impI,
- case_tac "x = 0", auto simp add: 2)
- fix y assume "a = (if y = 0 then a else f (Suc y))"
- thus "y = 0" using temp by (case_tac "y = 0", auto)
+ case True
+ with infinite have "\<not> finite (A - {a})" by auto
+ with infinite_iff_countable_subset[of "A - {a}"]
+ obtain f :: "nat \<Rightarrow> 'a" where 1: "inj f" and 2: "f ` UNIV \<subseteq> A - {a}" by blast
+ define g where "g n = (if n = 0 then a else f (Suc n))" for n
+ define A' where "A' = g ` UNIV"
+ have *: "\<forall>y. f y \<noteq> a" using 2 by blast
+ have 3: "inj_on g UNIV \<and> g ` UNIV \<subseteq> A \<and> a \<in> g ` UNIV"
+ apply (auto simp add: True g_def [abs_def])
+ apply (unfold inj_on_def)
+ apply (intro ballI impI)
+ apply (case_tac "x = 0")
+ apply (auto simp add: 2)
+ proof -
+ fix y
+ assume "a = (if y = 0 then a else f (Suc y))"
+ then show "y = 0" by (cases "y = 0") (use * in auto)
next
fix x y
assume "f (Suc x) = (if y = 0 then a else f (Suc y))"
- thus "x = y" using 1 temp unfolding inj_on_def by (case_tac "y = 0", auto)
+ with 1 * show "x = y" by (cases "y = 0") (auto simp: inj_on_def)
next
- fix n show "f (Suc n) \<in> A" using 2 by blast
+ fix n
+ from 2 show "f (Suc n) \<in> A" by blast
qed
- hence 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<le> A"
- using inj_on_imp_bij_betw[of g] unfolding A'_def by auto
- hence 5: "bij_betw (inv g) A' UNIV"
- by (auto simp add: bij_betw_inv_into)
- (* *)
- obtain n where "g n = a" using 3 by auto
- hence 6: "bij_betw g (UNIV - {n}) (A' - {a})"
- using 3 4 unfolding A'_def
- by clarify (rule bij_betw_subset, auto simp: image_set_diff)
- (* *)
- obtain v where v_def: "v = (\<lambda> m. if m < n then m else Suc m)" by blast
+ then have 4: "bij_betw g UNIV A' \<and> a \<in> A' \<and> A' \<subseteq> A"
+ using inj_on_imp_bij_betw[of g] by (auto simp: A'_def)
+ then have 5: "bij_betw (inv g) A' UNIV"
+ by (auto simp add: bij_betw_inv_into)
+ from 3 obtain n where n: "g n = a" by auto
+ have 6: "bij_betw g (UNIV - {n}) (A' - {a})"
+ by (rule bij_betw_subset) (use 3 4 n in \<open>auto simp: image_set_diff A'_def\<close>)
+ define v where "v m = (if m < n then m else Suc m)" for m
have 7: "bij_betw v UNIV (UNIV - {n})"
- proof(unfold bij_betw_def inj_on_def, intro conjI, clarify)
- fix m1 m2 assume "v m1 = v m2"
- thus "m1 = m2"
- by(case_tac "m1 < n", case_tac "m2 < n",
- auto simp add: inj_on_def v_def, case_tac "m2 < n", auto)
+ proof (unfold bij_betw_def inj_on_def, intro conjI, clarify)
+ fix m1 m2
+ assume "v m1 = v m2"
+ then show "m1 = m2"
+ apply (cases "m1 < n")
+ apply (cases "m2 < n")
+ apply (auto simp: inj_on_def v_def [abs_def])
+ apply (cases "m2 < n")
+ apply auto
+ done
next
show "v ` UNIV = UNIV - {n}"
- proof(auto simp add: v_def)
- fix m assume *: "m \<noteq> n" and **: "m \<notin> Suc ` {m'. \<not> m' < n}"
- {assume "n \<le> m" with * have 71: "Suc n \<le> m" by auto
- then obtain m' where 72: "m = Suc m'" using Suc_le_D by auto
- with 71 have "n \<le> m'" by auto
- with 72 ** have False by auto
- }
- thus "m < n" by force
+ proof (auto simp: v_def [abs_def])
+ fix m
+ assume "m \<noteq> n"
+ assume *: "m \<notin> Suc ` {m'. \<not> m' < n}"
+ have False if "n \<le> m"
+ proof -
+ from \<open>m \<noteq> n\<close> that have **: "Suc n \<le> m" by auto
+ from Suc_le_D [OF this] obtain m' where m': "m = Suc m'" ..
+ with ** have "n \<le> m'" by auto
+ with m' * show ?thesis by auto
+ qed
+ then show "m < n" by force
qed
qed
- (* *)
- obtain h' where h'_def: "h' = g o v o (inv g)" by blast
- hence 8: "bij_betw h' A' (A' - {a})" using 5 7 6
- by (auto simp add: bij_betw_trans)
- (* *)
- obtain h where h_def: "h = (\<lambda> b. if b \<in> A' then h' b else b)" by blast
- have "\<forall>b \<in> A'. h b = h' b" unfolding h_def by auto
- hence "bij_betw h A' (A' - {a})" using 8 bij_betw_cong[of A' h] by auto
+ define h' where "h' = g \<circ> v \<circ> (inv g)"
+ with 5 6 7 have 8: "bij_betw h' A' (A' - {a})"
+ by (auto simp add: bij_betw_trans)
+ define h where "h b = (if b \<in> A' then h' b else b)" for b
+ then have "\<forall>b \<in> A'. h b = h' b" by simp
+ with 8 have "bij_betw h A' (A' - {a})"
+ using bij_betw_cong[of A' h] by auto
moreover
- {have "\<forall>b \<in> A - A'. h b = b" unfolding h_def by auto
- hence "bij_betw h (A - A') (A - A')"
- using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
- }
+ have "\<forall>b \<in> A - A'. h b = b" by (auto simp: h_def)
+ then have "bij_betw h (A - A') (A - A')"
+ using bij_betw_cong[of "A - A'" h id] bij_betw_id[of "A - A'"] by auto
moreover
- have "(A' Int (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
- ((A' - {a}) Int (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
- using 4 by blast
+ from 4 have "(A' \<inter> (A - A') = {} \<and> A' \<union> (A - A') = A) \<and>
+ ((A' - {a}) \<inter> (A - A') = {} \<and> (A' - {a}) \<union> (A - A') = A - {a})"
+ by blast
ultimately have "bij_betw h A (A - {a})"
- using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
- thus ?thesis by blast
+ using bij_betw_combine[of h A' "A' - {a}" "A - A'" "A - A'"] by simp
+ then show ?thesis by blast
qed
lemma infinite_imp_bij_betw2:
-assumes INF: "\<not> finite A"
-shows "\<exists>h. bij_betw h A (A \<union> {a})"
-proof(cases "a \<in> A")
- assume Case1: "a \<in> A" hence "A \<union> {a} = A" by blast
- thus ?thesis using bij_betw_id[of A] by auto
+ assumes "\<not> finite A"
+ shows "\<exists>h. bij_betw h A (A \<union> {a})"
+proof (cases "a \<in> A")
+ case True
+ then have "A \<union> {a} = A" by blast
+ then show ?thesis using bij_betw_id[of A] by auto
next
+ case False
let ?A' = "A \<union> {a}"
- assume Case2: "a \<notin> A" hence "A = ?A' - {a}" by blast
- moreover have "\<not> finite ?A'" using INF by auto
+ from False have "A = ?A' - {a}" by blast
+ moreover from assms have "\<not> finite ?A'" by auto
ultimately obtain f where "bij_betw f ?A' A"
- using infinite_imp_bij_betw[of ?A' a] by auto
- hence "bij_betw(inv_into ?A' f) A ?A'" using bij_betw_inv_into by blast
- thus ?thesis by auto
+ using infinite_imp_bij_betw[of ?A' a] by auto
+ then have "bij_betw (inv_into ?A' f) A ?A'" by (rule bij_betw_inv_into)
+ then show ?thesis by auto
qed
-lemma bij_betw_inv_into_left:
-assumes BIJ: "bij_betw f A A'" and IN: "a \<in> A"
-shows "(inv_into A f) (f a) = a"
-using assms unfolding bij_betw_def
-by clarify (rule inv_into_f_f)
+lemma bij_betw_inv_into_left: "bij_betw f A A' \<Longrightarrow> a \<in> A \<Longrightarrow> inv_into A f (f a) = a"
+ unfolding bij_betw_def by clarify (rule inv_into_f_f)
-lemma bij_betw_inv_into_right:
-assumes "bij_betw f A A'" "a' \<in> A'"
-shows "f(inv_into A f a') = a'"
-using assms unfolding bij_betw_def using f_inv_into_f by force
+lemma bij_betw_inv_into_right: "bij_betw f A A' \<Longrightarrow> a' \<in> A' \<Longrightarrow> f (inv_into A f a') = a'"
+ unfolding bij_betw_def using f_inv_into_f by force
lemma bij_betw_inv_into_subset:
-assumes BIJ: "bij_betw f A A'" and
- SUB: "B \<le> A" and IM: "f ` B = B'"
-shows "bij_betw (inv_into A f) B' B"
-using assms unfolding bij_betw_def
-by (auto intro: inj_on_inv_into)
+ "bij_betw f A A' \<Longrightarrow> B \<subseteq> A \<Longrightarrow> f ` B = B' \<Longrightarrow> bij_betw (inv_into A f) B' B"
+ by (auto simp: bij_betw_def intro: inj_on_inv_into)
subsection \<open>Specification package -- Hilbertized version\<close>
-lemma exE_some: "[| Ex P ; c == Eps P |] ==> P c"
+lemma exE_some: "Ex P \<Longrightarrow> c \<equiv> Eps P \<Longrightarrow> P c"
by (simp only: someI_ex)
ML_file "Tools/choice_specification.ML"
--- a/src/HOL/Relation.thy Fri Aug 05 17:36:38 2016 +0200
+++ b/src/HOL/Relation.thy Fri Aug 05 18:14:34 2016 +0200
@@ -1,11 +1,12 @@
(* Title: HOL/Relation.thy
- Author: Lawrence C Paulson, Cambridge University Computer Laboratory; Stefan Berghofer, TU Muenchen
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Author: Stefan Berghofer, TU Muenchen
*)
section \<open>Relations -- as sets of pairs, and binary predicates\<close>
theory Relation
-imports Finite_Set
+ imports Finite_Set
begin
text \<open>A preliminary: classical rules for reasoning on predicates\<close>
@@ -400,16 +401,17 @@
by (auto intro: transpI)
lemma trans_empty [simp]: "trans {}"
-by (blast intro: transI)
+ by (blast intro: transI)
lemma transp_empty [simp]: "transp (\<lambda>x y. False)"
-using trans_empty[to_pred] by(simp add: bot_fun_def)
+ using trans_empty[to_pred] by (simp add: bot_fun_def)
lemma trans_singleton [simp]: "trans {(a, a)}"
-by (blast intro: transI)
+ by (blast intro: transI)
lemma transp_singleton [simp]: "transp (\<lambda>x y. x = a \<and> y = a)"
-by(simp add: transp_def)
+ by (simp add: transp_def)
+
subsubsection \<open>Totality\<close>
@@ -418,7 +420,7 @@
lemma total_onI [intro?]:
"(\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r) \<Longrightarrow> total_on A r"
-unfolding total_on_def by blast
+ unfolding total_on_def by blast
abbreviation "total \<equiv> total_on UNIV"
@@ -426,7 +428,8 @@
by (simp add: total_on_def)
lemma total_on_singleton [simp]: "total_on {x} {(x, x)}"
-unfolding total_on_def by blast
+ unfolding total_on_def by blast
+
subsubsection \<open>Single valued relations\<close>
@@ -496,7 +499,7 @@
subsubsection \<open>Diagonal: identity over a set\<close>
-definition Id_on :: "'a set \<Rightarrow> 'a rel"
+definition Id_on :: "'a set \<Rightarrow> 'a rel"
where "Id_on A = (\<Union>x\<in>A. {(x, x)})"
lemma Id_on_empty [simp]: "Id_on {} = {}"
@@ -633,7 +636,7 @@
lemma relcompp_apply: "(R OO S) a c \<longleftrightarrow> (\<exists>b. R a b \<and> S b c)"
unfolding relcomp_unfold [to_pred] ..
-lemma eq_OO: "op= OO R = R"
+lemma eq_OO: "op = OO R = R"
by blast
lemma OO_eq: "R OO op = = R"
@@ -728,10 +731,10 @@
lemma conversep_inject[simp]: "r\<inverse>\<inverse> = s \<inverse>\<inverse> \<longleftrightarrow> r = s"
by (fact converse_inject[to_pred])
-lemma converse_subset_swap: "r \<subseteq> s \<inverse> = (r \<inverse> \<subseteq> s)"
+lemma converse_subset_swap: "r \<subseteq> s \<inverse> \<longleftrightarrow> r \<inverse> \<subseteq> s"
by auto
-lemma conversep_le_swap: "r \<le> s \<inverse>\<inverse> = (r \<inverse>\<inverse> \<le> s)"
+lemma conversep_le_swap: "r \<le> s \<inverse>\<inverse> \<longleftrightarrow> r \<inverse>\<inverse> \<le> s"
by (fact converse_subset_swap[to_pred])
lemma converse_Id [simp]: "Id\<inverse> = Id"
@@ -800,7 +803,7 @@
where "Field r = Domain r \<union> Range r"
lemma FieldI1: "(i, j) \<in> R \<Longrightarrow> i \<in> Field R"
-unfolding Field_def by blast
+ unfolding Field_def by blast
lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
unfolding Field_def by auto
@@ -932,7 +935,7 @@
by blast
lemma Field_square [simp]: "Field (x \<times> x) = x"
-unfolding Field_def by blast
+ unfolding Field_def by blast
subsubsection \<open>Image of a set under a relation\<close>
@@ -972,7 +975,7 @@
by blast
lemma Image_Int_eq: "single_valued (converse R) \<Longrightarrow> R `` (A \<inter> B) = R `` A \<inter> R `` B"
- by (simp add: single_valued_def, blast)
+ by (auto simp: single_valued_def)
lemma Image_Un: "R `` (A \<union> B) = R `` A \<union> R `` B"
by blast
@@ -1079,7 +1082,7 @@
interpret comp_fun_idem Set.insert
by (fact comp_fun_idem_insert)
show ?thesis
- by standard (auto simp add: fun_eq_iff comp_fun_commute split: prod.split)
+ by standard (auto simp: fun_eq_iff comp_fun_commute split: prod.split)
qed
lemma Image_fold:
--- a/src/HOL/Transitive_Closure.thy Fri Aug 05 17:36:38 2016 +0200
+++ b/src/HOL/Transitive_Closure.thy Fri Aug 05 18:14:34 2016 +0200
@@ -6,7 +6,7 @@
section \<open>Reflexive and Transitive closure of a relation\<close>
theory Transitive_Closure
-imports Relation
+ imports Relation
begin
ML_file "~~/src/Provers/trancl.ML"
@@ -16,25 +16,24 @@
\<open>trancl\<close> is transitive closure,
\<open>reflcl\<close> is reflexive closure.
- These postfix operators have \emph{maximum priority}, forcing their
+ These postfix operators have \<^emph>\<open>maximum priority\<close>, forcing their
operands to be atomic.
\<close>
-context
- notes [[inductive_internals]]
+context notes [[inductive_internals]]
begin
inductive_set rtrancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_\<^sup>*)" [1000] 999)
for r :: "('a \<times> 'a) set"
-where
- rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) \<in> r\<^sup>*"
-| rtrancl_into_rtrancl [Pure.intro]: "(a, b) \<in> r\<^sup>* \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>*"
+ where
+ rtrancl_refl [intro!, Pure.intro!, simp]: "(a, a) \<in> r\<^sup>*"
+ | rtrancl_into_rtrancl [Pure.intro]: "(a, b) \<in> r\<^sup>* \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>*"
inductive_set trancl :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" ("(_\<^sup>+)" [1000] 999)
for r :: "('a \<times> 'a) set"
-where
- r_into_trancl [intro, Pure.intro]: "(a, b) \<in> r \<Longrightarrow> (a, b) \<in> r\<^sup>+"
-| trancl_into_trancl [Pure.intro]: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
+ where
+ r_into_trancl [intro, Pure.intro]: "(a, b) \<in> r \<Longrightarrow> (a, b) \<in> r\<^sup>+"
+ | trancl_into_trancl [Pure.intro]: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
notation
rtranclp ("(_\<^sup>*\<^sup>*)" [1000] 1000) and
@@ -215,7 +214,9 @@
lemma rtrancl_r_diff_Id: "(r - Id)\<^sup>* = r\<^sup>*"
apply (rule sym)
- apply (rule rtrancl_subset, blast, clarify)
+ apply (rule rtrancl_subset)
+ apply blast
+ apply clarify
apply (rename_tac a b)
apply (case_tac "a = b")
apply blast
@@ -258,10 +259,10 @@
lemmas converse_rtrancl_induct = converse_rtranclp_induct [to_set]
lemmas converse_rtranclp_induct2 =
- converse_rtranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule, consumes 1, case_names refl step]
+ converse_rtranclp_induct [of _ "(ax, ay)" "(bx, by)", split_rule, consumes 1, case_names refl step]
lemmas converse_rtrancl_induct2 =
- converse_rtrancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
+ converse_rtrancl_induct [of "(ax, ay)" "(bx, by)", split_format (complete),
consumes 1, case_names refl step]
lemma converse_rtranclpE [consumes 1, case_names base step]:
@@ -283,24 +284,30 @@
lemma r_comp_rtrancl_eq: "r O r\<^sup>* = r\<^sup>* O r"
by (blast elim: rtranclE converse_rtranclE
- intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
+ intro: rtrancl_into_rtrancl converse_rtrancl_into_rtrancl)
lemma rtrancl_unfold: "r\<^sup>* = Id \<union> r\<^sup>* O r"
by (auto intro: rtrancl_into_rtrancl elim: rtranclE)
lemma rtrancl_Un_separatorE:
"(a, b) \<in> (P \<union> Q)\<^sup>* \<Longrightarrow> \<forall>x y. (a, x) \<in> P\<^sup>* \<longrightarrow> (x, y) \<in> Q \<longrightarrow> x = y \<Longrightarrow> (a, b) \<in> P\<^sup>*"
- apply (induct rule:rtrancl.induct)
- apply blast
- apply (blast intro:rtrancl_trans)
- done
+proof (induct rule: rtrancl.induct)
+ case rtrancl_refl
+ then show ?case by blast
+next
+ case rtrancl_into_rtrancl
+ then show ?case by (blast intro: rtrancl_trans)
+qed
lemma rtrancl_Un_separator_converseE:
"(a, b) \<in> (P \<union> Q)\<^sup>* \<Longrightarrow> \<forall>x y. (x, b) \<in> P\<^sup>* \<longrightarrow> (y, x) \<in> Q \<longrightarrow> y = x \<Longrightarrow> (a, b) \<in> P\<^sup>*"
- apply (induct rule:converse_rtrancl_induct)
- apply blast
- apply (blast intro:rtrancl_trans)
- done
+proof (induct rule: converse_rtrancl_induct)
+ case base
+ then show ?case by blast
+next
+ case step
+ then show ?case by (blast intro: rtrancl_trans)
+qed
lemma Image_closed_trancl:
assumes "r `` X \<subseteq> X"
@@ -368,10 +375,10 @@
lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]
lemmas tranclp_induct2 =
- tranclp_induct [of _ "(ax,ay)" "(bx,by)", split_rule, consumes 1, case_names base step]
+ tranclp_induct [of _ "(ax, ay)" "(bx, by)", split_rule, consumes 1, case_names base step]
lemmas trancl_induct2 =
- trancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
+ trancl_induct [of "(ax, ay)" "(bx, by)", split_format (complete),
consumes 1, case_names base step]
lemma tranclp_trans_induct:
@@ -394,7 +401,7 @@
apply (rule subsetI)
apply auto
apply (erule trancl_induct)
- apply auto
+ apply auto
done
lemma trancl_unfold: "r\<^sup>+ = r \<union> r\<^sup>+ O r"
@@ -449,7 +456,7 @@
lemma tranclp_converseI: "(r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y \<Longrightarrow> (r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y"
apply (drule conversepD)
apply (erule tranclp_induct)
- apply (iprover intro: conversepI tranclp_trans)+
+ apply (iprover intro: conversepI tranclp_trans)+
done
lemmas trancl_converseI = tranclp_converseI [to_set]
@@ -457,7 +464,7 @@
lemma tranclp_converseD: "(r\<inverse>\<inverse>)\<^sup>+\<^sup>+ x y \<Longrightarrow> (r\<^sup>+\<^sup>+)\<inverse>\<inverse> x y"
apply (rule conversepI)
apply (erule tranclp_induct)
- apply (iprover dest: conversepD intro: tranclp_trans)+
+ apply (iprover dest: conversepD intro: tranclp_trans)+
done
lemmas trancl_converseD = tranclp_converseD [to_set]
@@ -493,7 +500,7 @@
lemma converse_tranclpE:
assumes major: "tranclp r x z"
and base: "r x z \<Longrightarrow> P"
- and step: "\<And> y. r x y \<Longrightarrow> tranclp r y z \<Longrightarrow> P"
+ and step: "\<And>y. r x y \<Longrightarrow> tranclp r y z \<Longrightarrow> P"
shows P
proof -
from tranclpD [OF major] obtain y where "r x y" and "rtranclp r y z"
@@ -582,7 +589,7 @@
apply (blast intro: rtrancl_into_trancl1 trancl_into_rtrancl trancl_trans)
apply (rule subsetI)
apply (blast intro: trancl_mono rtrancl_mono
- [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
+ [THEN [2] rev_subsetD] rtrancl_trancl_trancl rtrancl_into_trancl2)
done
lemma trancl_insert2:
@@ -655,26 +662,23 @@
lemma single_valued_confluent:
"single_valued r \<Longrightarrow> (x, y) \<in> r\<^sup>* \<Longrightarrow> (x, z) \<in> r\<^sup>* \<Longrightarrow> (y, z) \<in> r\<^sup>* \<or> (z, y) \<in> r\<^sup>*"
apply (erule rtrancl_induct)
- apply simp
+ apply simp
apply (erule disjE)
apply (blast elim:converse_rtranclE dest:single_valuedD)
- apply(blast intro:rtrancl_trans)
+ apply (blast intro:rtrancl_trans)
done
lemma r_r_into_trancl: "(a, b) \<in> R \<Longrightarrow> (b, c) \<in> R \<Longrightarrow> (a, c) \<in> R\<^sup>+"
by (fast intro: trancl_trans)
lemma trancl_into_trancl: "(a, b) \<in> r\<^sup>+ \<Longrightarrow> (b, c) \<in> r \<Longrightarrow> (a, c) \<in> r\<^sup>+"
- apply (induct rule: trancl_induct)
- apply (fast intro: r_r_into_trancl)
- apply (fast intro: r_r_into_trancl trancl_trans)
- done
+ by (induct rule: trancl_induct) (fast intro: r_r_into_trancl trancl_trans)+
lemma tranclp_rtranclp_tranclp: "r\<^sup>+\<^sup>+ a b \<Longrightarrow> r\<^sup>*\<^sup>* b c \<Longrightarrow> r\<^sup>+\<^sup>+ a c"
apply (drule tranclpD)
apply (elim exE conjE)
apply (drule rtranclp_trans, assumption)
- apply (drule rtranclp_into_tranclp2, assumption, assumption)
+ apply (drule (2) rtranclp_into_tranclp2)
done
lemmas trancl_rtrancl_trancl = tranclp_rtranclp_tranclp [to_set]
@@ -704,14 +708,14 @@
begin
primrec relpow :: "nat \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set"
-where
- "relpow 0 R = Id"
-| "relpow (Suc n) R = (R ^^ n) O R"
+ where
+ "relpow 0 R = Id"
+ | "relpow (Suc n) R = (R ^^ n) O R"
primrec relpowp :: "nat \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"
-where
- "relpowp 0 R = HOL.eq"
-| "relpowp (Suc n) R = (R ^^ n) OO R"
+ where
+ "relpowp 0 R = HOL.eq"
+ | "relpowp (Suc n) R = (R ^^ n) OO R"
end
@@ -740,10 +744,12 @@
hide_const (open) relpow
hide_const (open) relpowp
-lemma relpow_1 [simp]: "R ^^ 1 = R" for R :: "('a \<times> 'a) set"
+lemma relpow_1 [simp]: "R ^^ 1 = R"
+ for R :: "('a \<times> 'a) set"
by simp
-lemma relpowp_1 [simp]: "P ^^ 1 = P" for P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
+lemma relpowp_1 [simp]: "P ^^ 1 = P"
+ for P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
by (fact relpow_1 [to_pred])
lemma relpow_0_I: "(x, x) \<in> R ^^ 0"
@@ -777,22 +783,20 @@
by (fact relpow_Suc_E [to_pred])
lemma relpow_E:
- "(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
- \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P)
- \<Longrightarrow> P"
+ "(x, z) \<in> R ^^ n \<Longrightarrow>
+ (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P) \<Longrightarrow>
+ (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R ^^ m \<Longrightarrow> (y, z) \<in> R \<Longrightarrow> P) \<Longrightarrow> P"
by (cases n) auto
lemma relpowp_E:
- "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
- \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q)
- \<Longrightarrow> Q"
+ "(P ^^ n) x z \<Longrightarrow>
+ (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q) \<Longrightarrow>
+ (\<And>y m. n = Suc m \<Longrightarrow> (P ^^ m) x y \<Longrightarrow> P y z \<Longrightarrow> Q) \<Longrightarrow> Q"
by (fact relpow_E [to_pred])
lemma relpow_Suc_D2: "(x, z) \<in> R ^^ Suc n \<Longrightarrow> (\<exists>y. (x, y) \<in> R \<and> (y, z) \<in> R ^^ n)"
- apply (induct n arbitrary: x z)
- apply (blast intro: relpow_0_I elim: relpow_0_E relpow_Suc_E)
- apply (blast intro: relpow_Suc_I elim: relpow_0_E relpow_Suc_E)
- done
+ by (induct n arbitrary: x z)
+ (blast intro: relpow_0_I relpow_Suc_I elim: relpow_0_E relpow_Suc_E)+
lemma relpowp_Suc_D2: "(P ^^ Suc n) x z \<Longrightarrow> \<exists>y. P x y \<and> (P ^^ n) y z"
by (fact relpow_Suc_D2 [to_pred])
@@ -810,18 +814,21 @@
by (fact relpow_Suc_D2' [to_pred])
lemma relpow_E2:
- "(x, z) \<in> R ^^ n \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P)
- \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P)
- \<Longrightarrow> P"
- apply (cases n, simp)
+ "(x, z) \<in> R ^^ n \<Longrightarrow>
+ (n = 0 \<Longrightarrow> x = z \<Longrightarrow> P) \<Longrightarrow>
+ (\<And>y m. n = Suc m \<Longrightarrow> (x, y) \<in> R \<Longrightarrow> (y, z) \<in> R ^^ m \<Longrightarrow> P) \<Longrightarrow> P"
+ apply (cases n)
+ apply simp
apply (rename_tac nat)
- apply (cut_tac n=nat and R=R in relpow_Suc_D2', simp, blast)
+ apply (cut_tac n=nat and R=R in relpow_Suc_D2')
+ apply simp
+ apply blast
done
lemma relpowp_E2:
- "(P ^^ n) x z \<Longrightarrow> (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q)
- \<Longrightarrow> (\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q)
- \<Longrightarrow> Q"
+ "(P ^^ n) x z \<Longrightarrow>
+ (n = 0 \<Longrightarrow> x = z \<Longrightarrow> Q) \<Longrightarrow>
+ (\<And>y m. n = Suc m \<Longrightarrow> P x y \<Longrightarrow> (P ^^ m) y z \<Longrightarrow> Q) \<Longrightarrow> Q"
by (fact relpow_E2 [to_pred])
lemma relpow_add: "R ^^ (m + n) = R^^m O R^^n"
@@ -848,7 +855,8 @@
proof (cases p)
case (Pair x y)
with assms have "(x, y) \<in> R\<^sup>*" by simp
- then have "(x, y) \<in> (\<Union>n. R ^^ n)" proof induct
+ then have "(x, y) \<in> (\<Union>n. R ^^ n)"
+ proof induct
case base
show ?case by (blast intro: relpow_0_I)
next
@@ -869,7 +877,8 @@
proof (cases p)
case (Pair x y)
with assms have "(x, y) \<in> R ^^ n" by simp
- then have "(x, y) \<in> R\<^sup>*" proof (induct n arbitrary: x y)
+ then have "(x, y) \<in> R\<^sup>*"
+ proof (induct n arbitrary: x y)
case 0
then show ?case by simp
next
@@ -905,10 +914,12 @@
apply (clarsimp simp: relcomp_unfold)
apply fastforce
apply clarsimp
- apply (case_tac n, simp)
+ apply (case_tac n)
+ apply simp
apply clarsimp
apply (drule relpow_imp_rtrancl)
- apply (drule rtrancl_into_trancl1) apply auto
+ apply (drule rtrancl_into_trancl1)
+ apply auto
done
lemma tranclp_power: "(P\<^sup>+\<^sup>+) x y \<longleftrightarrow> (\<exists>n > 0. (P ^^ n) x y)"
@@ -954,7 +965,8 @@
lemma relpow_finite_bounded1:
fixes R :: "('a \<times> 'a) set"
assumes "finite R" and "k > 0"
- shows "R^^k \<subseteq> (\<Union>n\<in>{n. 0 < n \<and> n \<le> card R}. R^^n)" (is "_ \<subseteq> ?r")
+ shows "R^^k \<subseteq> (\<Union>n\<in>{n. 0 < n \<and> n \<le> card R}. R^^n)"
+ (is "_ \<subseteq> ?r")
proof -
have "(a, b) \<in> R^^(Suc k) \<Longrightarrow> \<exists>n. 0 < n \<and> n \<le> card R \<and> (a, b) \<in> R^^n" for a b k
proof (induct k arbitrary: b)
@@ -1050,8 +1062,7 @@
shows "R^^k \<subseteq> (UN n:{n. n \<le> card R}. R^^n)"
apply (cases k)
apply force
- using relpow_finite_bounded1[OF assms, of k]
- apply auto
+ apply (use relpow_finite_bounded1[OF assms, of k] in auto)
done
lemma rtrancl_finite_eq_relpow: "finite R \<Longrightarrow> R\<^sup>* = (\<Union>n\<in>{n. n \<le> card R}. R^^n)"
@@ -1076,20 +1087,26 @@
fixes R :: "('a \<times> 'a) set"
assumes "finite R"
shows "n > 0 \<Longrightarrow> finite (R^^n)"
- apply (induct n)
- apply simp
- apply (case_tac n)
- apply (simp_all add: assms)
- done
+proof (induct n)
+ case 0
+ then show ?case by simp
+next
+ case (Suc n)
+ then show ?case by (cases n) (use assms in simp_all)
+qed
lemma single_valued_relpow:
fixes R :: "('a \<times> 'a) set"
shows "single_valued R \<Longrightarrow> single_valued (R ^^ n)"
- apply (induct n arbitrary: R)
- apply simp_all
- apply (rule single_valuedI)
- apply (fast dest: single_valuedD elim: relpow_Suc_E)
- done
+proof (induct n arbitrary: R)
+ case 0
+ then show ?case by simp
+next
+ case (Suc n)
+ show ?case
+ by (rule single_valuedI)
+ (use Suc in \<open>fast dest: single_valuedD elim: relpow_Suc_E\<close>)
+qed
subsection \<open>Bounded transitive closure\<close>
@@ -1101,7 +1118,6 @@
proof
show "R \<subseteq> ntrancl 0 R"
unfolding ntrancl_def by fastforce
-next
have "0 < i \<and> i \<le> Suc 0 \<longleftrightarrow> i = 1" for i
by auto
then show "ntrancl 0 R \<le> R"
@@ -1110,31 +1126,30 @@
lemma ntrancl_Suc [simp]: "ntrancl (Suc n) R = ntrancl n R O (Id \<union> R)"
proof
- {
- fix a b
- assume "(a, b) \<in> ntrancl (Suc n) R"
- from this obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
+ have "(a, b) \<in> ntrancl n R O (Id \<union> R)" if "(a, b) \<in> ntrancl (Suc n) R" for a b
+ proof -
+ from that obtain i where "0 < i" "i \<le> Suc (Suc n)" "(a, b) \<in> R ^^ i"
unfolding ntrancl_def by auto
- have "(a, b) \<in> ntrancl n R O (Id \<union> R)"
+ show ?thesis
proof (cases "i = 1")
case True
from this \<open>(a, b) \<in> R ^^ i\<close> show ?thesis
- unfolding ntrancl_def by auto
+ by (auto simp: ntrancl_def)
next
case False
- from this \<open>0 < i\<close> obtain j where j: "i = Suc j" "0 < j"
+ with \<open>0 < i\<close> obtain j where j: "i = Suc j" "0 < j"
by (cases i) auto
- from this \<open>(a, b) \<in> R ^^ i\<close> obtain c where c1: "(a, c) \<in> R ^^ j" and c2:"(c, b) \<in> R"
+ with \<open>(a, b) \<in> R ^^ i\<close> obtain c where c1: "(a, c) \<in> R ^^ j" and c2: "(c, b) \<in> R"
by auto
from c1 j \<open>i \<le> Suc (Suc n)\<close> have "(a, c) \<in> ntrancl n R"
- unfolding ntrancl_def by fastforce
- from this c2 show ?thesis by fastforce
+ by (fastforce simp: ntrancl_def)
+ with c2 show ?thesis by fastforce
qed
- }
+ qed
then show "ntrancl (Suc n) R \<subseteq> ntrancl n R O (Id \<union> R)"
by auto
show "ntrancl n R O (Id \<union> R) \<subseteq> ntrancl (Suc n) R"
- unfolding ntrancl_def by fastforce
+ by (fastforce simp: ntrancl_def)
qed
lemma [code]: "ntrancl (Suc n) r = (let r' = ntrancl n r in r' \<union> r' O r)"
@@ -1169,9 +1184,7 @@
qed
lemma acyclic_insert [iff]: "acyclic (insert (y, x) r) \<longleftrightarrow> acyclic r \<and> (x, y) \<notin> r\<^sup>*"
- apply (simp add: acyclic_def trancl_insert)
- apply (blast intro: rtrancl_trans)
- done
+ by (simp add: acyclic_def trancl_insert) (blast intro: rtrancl_trans)
lemma acyclic_converse [iff]: "acyclic (r\<inverse>) \<longleftrightarrow> acyclic r"
by (simp add: acyclic_def trancl_converse)
@@ -1179,9 +1192,8 @@
lemmas acyclicP_converse [iff] = acyclic_converse [to_pred]
lemma acyclic_impl_antisym_rtrancl: "acyclic r \<Longrightarrow> antisym (r\<^sup>*)"
- apply (simp add: acyclic_def antisym_def)
- apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
- done
+ by (simp add: acyclic_def antisym_def)
+ (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
(* Other direction:
acyclic = no loops
@@ -1197,7 +1209,6 @@
subsection \<open>Setup of transitivity reasoner\<close>
ML \<open>
-
structure Trancl_Tac = Trancl_Tac
(
val r_into_trancl = @{thm trancl.r_into_trancl};
--- a/src/HOL/Wellfounded.thy Fri Aug 05 17:36:38 2016 +0200
+++ b/src/HOL/Wellfounded.thy Fri Aug 05 18:14:34 2016 +0200
@@ -187,9 +187,7 @@
text \<open>Well-foundedness of subsets\<close>
lemma wf_subset: "wf r \<Longrightarrow> p \<subseteq> r \<Longrightarrow> wf p"
- apply (simp add: wf_eq_minimal)
- apply fast
- done
+ by (simp add: wf_eq_minimal) fast
lemmas wfP_subset = wf_subset [to_pred]
@@ -198,11 +196,12 @@
lemma wf_empty [iff]: "wf {}"
by (simp add: wf_def)
-lemma wfP_empty [iff]:
- "wfP (\<lambda>x y. False)"
+lemma wfP_empty [iff]: "wfP (\<lambda>x y. False)"
proof -
- have "wfP bot" by (fact wf_empty [to_pred bot_empty_eq2])
- then show ?thesis by (simp add: bot_fun_def)
+ have "wfP bot"
+ by (fact wf_empty [to_pred bot_empty_eq2])
+ then show ?thesis
+ by (simp add: bot_fun_def)
qed
lemma wf_Int1: "wf r \<Longrightarrow> wf (r \<inter> r')"
@@ -217,9 +216,10 @@
shows "wf R"
proof (rule wfI_pf)
fix A assume "A \<subseteq> R `` A"
- then have "A \<subseteq> (R ^^ n) `` A" by (induct n) force+
- with \<open>wf (R ^^ n)\<close>
- show "A = {}" by (rule wfE_pf)
+ then have "A \<subseteq> (R ^^ n) `` A"
+ by (induct n) force+
+ with \<open>wf (R ^^ n)\<close> show "A = {}"
+ by (rule wfE_pf)
qed
text \<open>Well-foundedness of \<open>insert\<close>.\<close>
@@ -257,7 +257,7 @@
apply (case_tac "\<exists>p. f p \<in> Q")
apply (erule_tac x = "{p. f p \<in> Q}" in allE)
apply (fast dest: inj_onD)
-apply blast
+ apply blast
done
@@ -379,7 +379,7 @@
qed
qed
-lemma wf_comp_self: "wf R = wf (R O R)" \<comment> \<open>special case\<close>
+lemma wf_comp_self: "wf R \<longleftrightarrow> wf (R O R)" \<comment> \<open>special case\<close>
by (rule wf_union_merge [where S = "{}", simplified])
@@ -390,12 +390,13 @@
lemma qc_wf_relto_iff:
assumes "R O S \<subseteq> (R \<union> S)\<^sup>* O R" \<comment> \<open>R quasi-commutes over S\<close>
shows "wf (S\<^sup>* O R O S\<^sup>*) \<longleftrightarrow> wf R"
- (is "wf ?S \<longleftrightarrow> _")
+ (is "wf ?S \<longleftrightarrow> _")
proof
show "wf R" if "wf ?S"
proof -
have "R \<subseteq> ?S" by auto
- with that show "wf R" using wf_subset by auto
+ with wf_subset [of ?S] that show "wf R"
+ by auto
qed
next
show "wf ?S" if "wf R"
@@ -607,10 +608,7 @@
qed
lemma accp_downwards_aux: "r\<^sup>*\<^sup>* b a \<Longrightarrow> accp r a \<longrightarrow> accp r b"
- apply (erule rtranclp_induct)
- apply blast
- apply (blast dest: accp_downward)
- done
+ by (erule rtranclp_induct) (blast dest: accp_downward)+
theorem accp_downwards: "accp r a \<Longrightarrow> r\<^sup>*\<^sup>* b a \<Longrightarrow> accp r b"
by (blast dest: accp_downwards_aux)
@@ -691,7 +689,7 @@
text \<open>Inverse Image\<close>
lemma wf_inv_image [simp,intro!]: "wf r \<Longrightarrow> wf (inv_image r f)"
- for f :: "'a \<Rightarrow> 'b"
+ for f :: "'a \<Rightarrow> 'b"
apply (simp add: inv_image_def wf_eq_minimal)
apply clarify
apply (subgoal_tac "\<exists>w::'b. w \<in> {w. \<exists>x::'a. x \<in> Q \<and> f x = w}")
@@ -776,7 +774,7 @@
done
lemma trans_finite_psubset: "trans finite_psubset"
- by (auto simp add: finite_psubset_def less_le trans_def)
+ by (auto simp: finite_psubset_def less_le trans_def)
lemma in_finite_psubset[simp]: "(A, B) \<in> finite_psubset \<longleftrightarrow> A \<subset> B \<and> finite B"
unfolding finite_psubset_def by auto
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Pure/Isar/document_structure.scala Fri Aug 05 18:14:34 2016 +0200
@@ -0,0 +1,210 @@
+/* Title: Pure/Isar/document_structure.scala
+ Author: Makarius
+
+Overall document structure.
+*/
+
+package isabelle
+
+
+import scala.collection.mutable
+import scala.annotation.tailrec
+
+
+object Document_Structure
+{
+ /** general structure **/
+
+ sealed abstract class Document { def length: Int }
+ case class Block(name: String, text: String, body: List[Document]) extends Document
+ { val length: Int = (0 /: body)(_ + _.length) }
+ case class Atom(length: Int) extends Document
+
+ private def is_theory_command(keywords: Keyword.Keywords, name: String): Boolean =
+ keywords.kinds.get(name) match {
+ case Some(kind) => Keyword.theory(kind) && !Keyword.theory_end(kind)
+ case None => false
+ }
+
+
+
+ /** context blocks **/
+
+ def parse_blocks(
+ syntax: Outer_Syntax,
+ node_name: Document.Node.Name,
+ text: CharSequence): List[Document] =
+ {
+ def is_plain_theory(command: Command): Boolean =
+ is_theory_command(syntax.keywords, command.span.name) &&
+ !command.span.is_begin && !command.span.is_end
+
+
+ /* stack operations */
+
+ def buffer(): mutable.ListBuffer[Document] = new mutable.ListBuffer[Document]
+
+ var stack: List[(Command, mutable.ListBuffer[Document])] =
+ List((Command.empty, buffer()))
+
+ def open(command: Command) { stack = (command, buffer()) :: stack }
+
+ def close(): Boolean =
+ stack match {
+ case (command, body) :: (_, body2) :: _ =>
+ body2 += Block(command.span.name, command.source, body.toList)
+ stack = stack.tail
+ true
+ case _ =>
+ false
+ }
+
+ def flush() { if (is_plain_theory(stack.head._1)) close() }
+
+ def result(): List[Document] =
+ {
+ while (close()) { }
+ stack.head._2.toList
+ }
+
+ def add(command: Command)
+ {
+ if (command.span.is_begin || is_plain_theory(command)) { flush(); open(command) }
+ else if (command.span.is_end) { flush(); close() }
+
+ stack.head._2 += Atom(command.source.length)
+ }
+
+
+ /* result structure */
+
+ val spans = syntax.parse_spans(text)
+ spans.foreach(span => add(Command(Document_ID.none, node_name, Command.no_blobs, span)))
+ result()
+ }
+
+
+
+ /** section headings **/
+
+ trait Item
+ {
+ def name: String = ""
+ def source: String = ""
+ def heading_level: Option[Int] = None
+ }
+
+ object No_Item extends Item
+
+ class Sections(keywords: Keyword.Keywords)
+ {
+ private def buffer(): mutable.ListBuffer[Document] = new mutable.ListBuffer[Document]
+
+ private var stack: List[(Int, Item, mutable.ListBuffer[Document])] =
+ List((0, No_Item, buffer()))
+
+ @tailrec private def close(level: Int => Boolean)
+ {
+ stack match {
+ case (lev, item, body) :: (_, _, body2) :: _ if level(lev) =>
+ body2 += Block(item.name, item.source, body.toList)
+ stack = stack.tail
+ close(level)
+ case _ =>
+ }
+ }
+
+ def result(): List[Document] =
+ {
+ close(_ => true)
+ stack.head._3.toList
+ }
+
+ def add(item: Item)
+ {
+ item.heading_level match {
+ case Some(i) =>
+ close(_ > i)
+ stack = (i + 1, item, buffer()) :: stack
+ case None =>
+ }
+ stack.head._3 += Atom(item.source.length)
+ }
+ }
+
+
+ /* outer syntax sections */
+
+ private class Command_Item(keywords: Keyword.Keywords, command: Command) extends Item
+ {
+ override def name: String = command.span.name
+ override def source: String = command.source
+ override def heading_level: Option[Int] =
+ {
+ name match {
+ case Thy_Header.CHAPTER => Some(0)
+ case Thy_Header.SECTION => Some(1)
+ case Thy_Header.SUBSECTION => Some(2)
+ case Thy_Header.SUBSUBSECTION => Some(3)
+ case Thy_Header.PARAGRAPH => Some(4)
+ case Thy_Header.SUBPARAGRAPH => Some(5)
+ case _ if is_theory_command(keywords, name) => Some(6)
+ case _ => None
+ }
+ }
+ }
+
+ def parse_sections(
+ syntax: Outer_Syntax,
+ node_name: Document.Node.Name,
+ text: CharSequence): List[Document] =
+ {
+ val sections = new Sections(syntax.keywords)
+
+ for { span <- syntax.parse_spans(text) } {
+ sections.add(
+ new Command_Item(syntax.keywords,
+ Command(Document_ID.none, node_name, Command.no_blobs, span)))
+ }
+ sections.result()
+ }
+
+
+ /* ML sections */
+
+ private class ML_Item(token: ML_Lex.Token, level: Option[Int]) extends Item
+ {
+ override def source: String = token.source
+ override def heading_level: Option[Int] = level
+ }
+
+ def parse_ml_sections(SML: Boolean, text: CharSequence): List[Document] =
+ {
+ val sections = new Sections(Keyword.Keywords.empty)
+ val nl = new ML_Item(ML_Lex.Token(ML_Lex.Kind.SPACE, "\n"), None)
+
+ var context: Scan.Line_Context = Scan.Finished
+ for (line <- Library.separated_chunks(_ == '\n', text)) {
+ val (toks, next_context) = ML_Lex.tokenize_line(SML, line, context)
+ val level =
+ toks.filterNot(_.is_space) match {
+ case List(tok) if tok.is_comment =>
+ val s = tok.source
+ if (s.startsWith("(**** ") && s.endsWith(" ****)")) Some(0)
+ else if (s.startsWith("(*** ") && s.endsWith(" ***)")) Some(1)
+ else if (s.startsWith("(** ") && s.endsWith(" **)")) Some(2)
+ else if (s.startsWith("(* ") && s.endsWith(" *)")) Some(3)
+ else None
+ case _ => None
+ }
+ if (level.isDefined && context == Scan.Finished && next_context == Scan.Finished)
+ toks.foreach(tok => sections.add(new ML_Item(tok, if (tok.is_comment) level else None)))
+ else
+ toks.foreach(tok => sections.add(new ML_Item(tok, None)))
+
+ sections.add(nl)
+ context = next_context
+ }
+ sections.result()
+ }
+}
--- a/src/Pure/Isar/keyword.scala Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/Isar/keyword.scala Fri Aug 05 18:14:34 2016 +0200
@@ -87,6 +87,8 @@
val proof_close = qed + PRF_CLOSE
val proof_enclose = Set(PRF_BLOCK, NEXT_BLOCK, QED_BLOCK, PRF_CLOSE)
+ val close_structure = Set(NEXT_BLOCK, QED_BLOCK, PRF_CLOSE, THY_END)
+
/** keyword tables **/
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/Pure/Isar/line_structure.scala Fri Aug 05 18:14:34 2016 +0200
@@ -0,0 +1,67 @@
+/* Title: Pure/Isar/line_structure.scala
+ Author: Makarius
+
+Line-oriented document structure, e.g. for fold handling.
+*/
+
+package isabelle
+
+
+object Line_Structure
+{
+ val init = Line_Structure()
+}
+
+sealed case class Line_Structure(
+ improper: Boolean = true,
+ command: Boolean = false,
+ depth: Int = 0,
+ span_depth: Int = 0,
+ after_span_depth: Int = 0,
+ element_depth: Int = 0)
+{
+ def update(keywords: Keyword.Keywords, tokens: List[Token]): Line_Structure =
+ {
+ val improper1 = tokens.forall(_.is_improper)
+ val command1 = tokens.exists(_.is_begin_or_command)
+
+ val command_depth =
+ tokens.iterator.filter(_.is_proper).toStream.headOption match {
+ case Some(tok) =>
+ if (keywords.is_command(tok, Keyword.close_structure))
+ Some(after_span_depth - 1)
+ else None
+ case None => None
+ }
+
+ val depth1 =
+ if (tokens.exists(tok =>
+ keywords.is_before_command(tok) ||
+ !tok.is_end && keywords.is_command(tok, Keyword.theory))) element_depth
+ else if (command_depth.isDefined) command_depth.get
+ else if (command1) after_span_depth
+ else span_depth
+
+ val (span_depth1, after_span_depth1, element_depth1) =
+ ((span_depth, after_span_depth, element_depth) /: tokens) {
+ case (depths @ (x, y, z), tok) =>
+ if (tok.is_begin) (z + 2, z + 1, z + 1)
+ else if (tok.is_end) (z + 1, z - 1, z - 1)
+ else if (tok.is_command) {
+ val depth0 = element_depth
+ if (keywords.is_command(tok, Keyword.theory_goal)) (depth0 + 2, depth0 + 1, z)
+ else if (keywords.is_command(tok, Keyword.theory)) (depth0 + 1, depth0, z)
+ else if (keywords.is_command(tok, Keyword.proof_open)) (y + 2, y + 1, z)
+ else if (keywords.is_command(tok, Set(Keyword.PRF_BLOCK))) (y + 2, y + 1, z)
+ else if (keywords.is_command(tok, Set(Keyword.QED_BLOCK))) (y - 1, y - 2, z)
+ else if (keywords.is_command(tok, Set(Keyword.PRF_CLOSE))) (y, y - 1, z)
+ else if (keywords.is_command(tok, Keyword.proof_close)) (y + 1, y - 1, z)
+ else if (keywords.is_command(tok, Keyword.qed_global)) (depth0 + 1, depth0, z)
+ else depths
+ }
+ else depths
+ }
+
+ Line_Structure(improper1, command1, depth1, span_depth1, after_span_depth1, element_depth1)
+ }
+}
--- a/src/Pure/Isar/outer_syntax.scala Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/Isar/outer_syntax.scala Fri Aug 05 18:14:34 2016 +0200
@@ -8,7 +8,6 @@
import scala.collection.mutable
-import scala.annotation.tailrec
object Outer_Syntax
@@ -42,35 +41,6 @@
result += '"'
result.toString
}
-
-
- /* line-oriented structure */
-
- object Line_Structure
- {
- val init = Line_Structure()
- }
-
- sealed case class Line_Structure(
- improper: Boolean = true,
- command: Boolean = false,
- depth: Int = 0,
- span_depth: Int = 0,
- after_span_depth: Int = 0,
- element_depth: Int = 0)
-
-
- /* overall document structure */
-
- sealed abstract class Document { def length: Int }
- case class Document_Block(name: String, text: String, body: List[Document]) extends Document
- {
- val length: Int = (0 /: body)(_ + _.length)
- }
- case class Document_Atom(command: Command) extends Document
- {
- def length: Int = command.length
- }
}
final class Outer_Syntax private(
@@ -155,59 +125,6 @@
/** parsing **/
- /* line-oriented structure */
-
- private val close_structure =
- Set(Keyword.NEXT_BLOCK, Keyword.QED_BLOCK, Keyword.PRF_CLOSE, Keyword.THY_END)
-
- def line_structure(tokens: List[Token], structure: Outer_Syntax.Line_Structure)
- : Outer_Syntax.Line_Structure =
- {
- val improper1 = tokens.forall(_.is_improper)
- val command1 = tokens.exists(_.is_begin_or_command)
-
- val command_depth =
- tokens.iterator.filter(_.is_proper).toStream.headOption match {
- case Some(tok) =>
- if (keywords.is_command(tok, close_structure))
- Some(structure.after_span_depth - 1)
- else None
- case None => None
- }
-
- val depth0 = structure.element_depth
- val depth1 =
- if (tokens.exists(tok =>
- keywords.is_before_command(tok) ||
- !tok.is_end && keywords.is_command(tok, Keyword.theory))) depth0
- else if (command_depth.isDefined) command_depth.get
- else if (command1) structure.after_span_depth
- else structure.span_depth
-
- val (span_depth1, after_span_depth1, element_depth1) =
- ((structure.span_depth, structure.after_span_depth, structure.element_depth) /: tokens) {
- case (depths @ (x, y, z), tok) =>
- if (tok.is_begin) (z + 2, z + 1, z + 1)
- else if (tok.is_end) (z + 1, z - 1, z - 1)
- else if (tok.is_command) {
- if (keywords.is_command(tok, Keyword.theory_goal)) (depth0 + 2, depth0 + 1, z)
- else if (keywords.is_command(tok, Keyword.theory)) (depth0 + 1, depth0, z)
- else if (keywords.is_command(tok, Keyword.proof_open)) (y + 2, y + 1, z)
- else if (keywords.is_command(tok, Set(Keyword.PRF_BLOCK))) (y + 2, y + 1, z)
- else if (keywords.is_command(tok, Set(Keyword.QED_BLOCK))) (y - 1, y - 2, z)
- else if (keywords.is_command(tok, Set(Keyword.PRF_CLOSE))) (y, y - 1, z)
- else if (keywords.is_command(tok, Keyword.proof_close)) (y + 1, y - 1, z)
- else if (keywords.is_command(tok, Keyword.qed_global)) (depth0 + 1, depth0, z)
- else depths
- }
- else depths
- }
-
- Outer_Syntax.Line_Structure(
- improper1, command1, depth1, span_depth1, after_span_depth1, element_depth1)
- }
-
-
/* command spans */
def parse_spans(toks: List[Token]): List[Command_Span.Span] =
@@ -257,72 +174,4 @@
def parse_spans(input: CharSequence): List[Command_Span.Span] =
parse_spans(Token.explode(keywords, input))
-
-
- /* overall document structure */
-
- def heading_level(command: Command): Option[Int] =
- {
- val name = command.span.name
- name match {
- case Thy_Header.CHAPTER => Some(0)
- case Thy_Header.SECTION => Some(1)
- case Thy_Header.SUBSECTION => Some(2)
- case Thy_Header.SUBSUBSECTION => Some(3)
- case Thy_Header.PARAGRAPH => Some(4)
- case Thy_Header.SUBPARAGRAPH => Some(5)
- case _ =>
- keywords.kinds.get(name) match {
- case Some(kind) if Keyword.theory(kind) && !Keyword.theory_end(kind) => Some(6)
- case _ => None
- }
- }
- }
-
- def parse_document(node_name: Document.Node.Name, text: CharSequence):
- List[Outer_Syntax.Document] =
- {
- /* stack operations */
-
- def buffer(): mutable.ListBuffer[Outer_Syntax.Document] =
- new mutable.ListBuffer[Outer_Syntax.Document]
-
- var stack: List[(Int, Command, mutable.ListBuffer[Outer_Syntax.Document])] =
- List((0, Command.empty, buffer()))
-
- @tailrec def close(level: Int => Boolean)
- {
- stack match {
- case (lev, command, body) :: (_, _, body2) :: rest if level(lev) =>
- body2 += Outer_Syntax.Document_Block(command.span.name, command.source, body.toList)
- stack = stack.tail
- close(level)
- case _ =>
- }
- }
-
- def result(): List[Outer_Syntax.Document] =
- {
- close(_ => true)
- stack.head._3.toList
- }
-
- def add(command: Command)
- {
- heading_level(command) match {
- case Some(i) =>
- close(_ > i)
- stack = (i + 1, command, buffer()) :: stack
- case None =>
- }
- stack.head._3 += Outer_Syntax.Document_Atom(command)
- }
-
-
- /* result structure */
-
- val spans = parse_spans(text)
- spans.foreach(span => add(Command(Document_ID.none, node_name, Command.no_blobs, span)))
- result()
- }
}
--- a/src/Pure/Isar/proof_context.ML Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/Isar/proof_context.ML Fri Aug 05 18:14:34 2016 +0200
@@ -1013,11 +1013,12 @@
| retrieve pick context (Facts.Named ((xname, pos), sel)) =
let
val thy = Context.theory_of context;
- fun immediate thm = {name = xname, dynamic = false, thms = [Thm.transfer thy thm]};
+ fun immediate thms = {name = xname, dynamic = false, thms = map (Thm.transfer thy) thms};
val {name, dynamic, thms} =
(case xname of
- "" => immediate Drule.dummy_thm
- | "_" => immediate Drule.asm_rl
+ "" => immediate [Drule.dummy_thm]
+ | "_" => immediate [Drule.asm_rl]
+ | "nothing" => immediate []
| _ => retrieve_generic context (xname, pos));
val thms' =
if dynamic andalso Config.get_generic context dynamic_facts_dummy
--- a/src/Pure/ML/ml_lex.scala Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/ML/ml_lex.scala Fri Aug 05 18:14:34 2016 +0200
@@ -66,6 +66,8 @@
{
def is_keyword: Boolean = kind == Kind.KEYWORD
def is_delimiter: Boolean = is_keyword && !Symbol.is_ascii_identifier(source)
+ def is_space: Boolean = kind == Kind.SPACE
+ def is_comment: Boolean = kind == Kind.COMMENT
}
--- a/src/Pure/PIDE/command_span.scala Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/PIDE/command_span.scala Fri Aug 05 18:14:34 2016 +0200
@@ -32,6 +32,9 @@
def position: Position.T =
kind match { case Command_Span(_, pos) => pos case _ => Position.none }
+ def is_begin: Boolean = content.exists(_.is_begin)
+ def is_end: Boolean = content.exists(_.is_end)
+
def length: Int = (0 /: content)(_ + _.source.length)
def compact_source: (String, Span) =
--- a/src/Pure/build-jars Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/build-jars Fri Aug 05 18:14:34 2016 +0200
@@ -49,7 +49,9 @@
General/url.scala
General/word.scala
General/xz_file.scala
+ Isar/document_structure.scala
Isar/keyword.scala
+ Isar/line_structure.scala
Isar/outer_syntax.scala
Isar/parse.scala
Isar/token.scala
--- a/src/Pure/pure_thy.ML Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/pure_thy.ML Fri Aug 05 18:14:34 2016 +0200
@@ -223,7 +223,6 @@
\ (CONST Pure.term :: 'a itself => prop) (CONST Pure.type :: 'a itself)"),
(Binding.make ("conjunction_def", @{here}),
prop "(A &&& B) == (!!C::prop. (A ==> B ==> C) ==> C)")] #> snd
- #> Global_Theory.add_thmss [((Binding.make ("nothing", @{here}), []), [])] #> snd
#> fold (fn (a, prop) =>
snd o Thm.add_axiom_global (Binding.make (a, @{here}), prop)) Proofterm.equality_axms);
--- a/src/Pure/term.ML Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/term.ML Fri Aug 05 18:14:34 2016 +0200
@@ -181,12 +181,12 @@
for resolution.*)
type indexname = string * int;
-(* Types are classified by sorts. *)
+(*Types are classified by sorts.*)
type class = string;
type sort = class list;
type arity = string * sort list * sort;
-(* The sorts attached to TFrees and TVars specify the sort of that variable *)
+(*The sorts attached to TFrees and TVars specify the sort of that variable.*)
datatype typ = Type of string * typ list
| TFree of string * sort
| TVar of indexname * sort;
@@ -293,15 +293,15 @@
| dest_funT T = raise TYPE ("dest_funT", [T], []);
-(* maps [T1,...,Tn]--->T to the list [T1,T2,...,Tn]*)
+(*maps [T1,...,Tn]--->T to the list [T1,T2,...,Tn]*)
fun binder_types (Type ("fun", [T, U])) = T :: binder_types U
| binder_types _ = [];
-(* maps [T1,...,Tn]--->T to T*)
+(*maps [T1,...,Tn]--->T to T*)
fun body_type (Type ("fun", [_, U])) = body_type U
| body_type T = T;
-(* maps [T1,...,Tn]--->T to ([T1,T2,...,Tn], T) *)
+(*maps [T1,...,Tn]--->T to ([T1,T2,...,Tn], T)*)
fun strip_type T = (binder_types T, body_type T);
@@ -361,11 +361,11 @@
in ((a, T) :: a', t') end
| strip_abs t = ([], t);
-(* maps (x1,...,xn)t to t *)
+(*maps (x1,...,xn)t to t*)
fun strip_abs_body (Abs(_,_,t)) = strip_abs_body t
| strip_abs_body u = u;
-(* maps (x1,...,xn)t to [x1, ..., xn] *)
+(*maps (x1,...,xn)t to [x1, ..., xn]*)
fun strip_abs_vars (Abs(a,T,t)) = (a,T) :: strip_abs_vars t
| strip_abs_vars u = [] : (string*typ) list;
@@ -381,18 +381,18 @@
in strip end;
-(* maps (f, [t1,...,tn]) to f(t1,...,tn) *)
+(*maps (f, [t1,...,tn]) to f(t1,...,tn)*)
val list_comb : term * term list -> term = Library.foldl (op $);
-(* maps f(t1,...,tn) to (f, [t1,...,tn]) ; naturally tail-recursive*)
+(*maps f(t1,...,tn) to (f, [t1,...,tn]) ; naturally tail-recursive*)
fun strip_comb u : term * term list =
let fun stripc (f$t, ts) = stripc (f, t::ts)
| stripc x = x
in stripc(u,[]) end;
-(* maps f(t1,...,tn) to f , which is never a combination *)
+(*maps f(t1,...,tn) to f , which is never a combination*)
fun head_of (f$t) = head_of f
| head_of u = u;
@@ -580,11 +580,11 @@
val propT : typ = Type ("prop",[]);
-(* maps !!x1...xn. t to t *)
+(*maps !!x1...xn. t to t*)
fun strip_all_body (Const ("Pure.all", _) $ Abs (_, _, t)) = strip_all_body t
| strip_all_body t = t;
-(* maps !!x1...xn. t to [x1, ..., xn] *)
+(*maps !!x1...xn. t to [x1, ..., xn]*)
fun strip_all_vars (Const ("Pure.all", _) $ Abs (a, T, t)) = (a, T) :: strip_all_vars t
| strip_all_vars t = [];
--- a/src/Pure/thm.ML Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Pure/thm.ML Fri Aug 05 18:14:34 2016 +0200
@@ -1319,7 +1319,7 @@
prop = prop'})
end;
-(* Replace all TFrees not fixed or in the hyps by new TVars *)
+(*Replace all TFrees not fixed or in the hyps by new TVars*)
fun varifyT_global' fixed (Thm (der, {cert, maxidx, shyps, hyps, tpairs, prop, ...})) =
let
val tfrees = fold Term.add_tfrees hyps fixed;
@@ -1339,7 +1339,7 @@
val varifyT_global = #2 o varifyT_global' [];
-(* Replace all TVars by TFrees that are often new *)
+(*Replace all TVars by TFrees that are often new*)
fun legacy_freezeT (Thm (der, {cert, shyps, hyps, tpairs, prop, ...})) =
let
val prop1 = attach_tpairs tpairs prop;
@@ -1635,7 +1635,7 @@
and nlift = Logic.count_prems (strip_all_body Bi) + (if eres_flg then ~1 else 0)
val cert = join_certificate2 (state, orule);
val context = make_context [state, orule] opt_ctxt cert;
- (** Add new theorem with prop = '[| Bs; As |] ==> C' to thq **)
+ (*Add new theorem with prop = '[| Bs; As |] ==> C' to thq*)
fun addth A (As, oldAs, rder', n) ((env, tpairs), thq) =
let val normt = Envir.norm_term env;
(*perform minimal copying here by examining env*)
--- a/src/Tools/jEdit/src/Isabelle.props Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Tools/jEdit/src/Isabelle.props Fri Aug 05 18:14:34 2016 +0200
@@ -87,7 +87,8 @@
mode.isabelle.folding=isabelle
mode.isabelle.sidekick.parser=isabelle
mode.isabelle.sidekick.showStatusWindow.label=true
-mode.ml.sidekick.parser=isabelle
+mode.isabelle-ml.sidekick.parser=isabelle-ml
+mode.sml.sidekick.parser=isabelle-sml
sidekick.parser.isabelle.label=Isabelle
mode.bibtex.folding=sidekick
mode.bibtex.sidekick.parser=bibtex
--- a/src/Tools/jEdit/src/isabelle_sidekick.scala Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Tools/jEdit/src/isabelle_sidekick.scala Fri Aug 05 18:14:34 2016 +0200
@@ -110,7 +110,8 @@
class Isabelle_Sidekick_Structure(
name: String,
- node_name: Buffer => Option[Document.Node.Name])
+ node_name: Buffer => Option[Document.Node.Name],
+ parse: (Outer_Syntax, Document.Node.Name, CharSequence) => List[Document_Structure.Document])
extends Isabelle_Sidekick(name)
{
override def parser(buffer: Buffer, syntax: Outer_Syntax, data: SideKickParsedData): Boolean =
@@ -118,11 +119,11 @@
def make_tree(
parent: DefaultMutableTreeNode,
offset: Text.Offset,
- documents: List[Outer_Syntax.Document])
+ documents: List[Document_Structure.Document])
{
(offset /: documents) { case (i, document) =>
document match {
- case Outer_Syntax.Document_Block(name, text, body) =>
+ case Document_Structure.Block(name, text, body) =>
val range = Text.Range(i, i + document.length)
val node =
new DefaultMutableTreeNode(
@@ -137,24 +138,39 @@
node_name(buffer) match {
case Some(name) =>
- make_tree(data.root, 0, syntax.parse_document(name, JEdit_Lib.buffer_text(buffer)))
+ make_tree(data.root, 0, parse(syntax, name, JEdit_Lib.buffer_text(buffer)))
true
- case None => false
+ case None =>
+ false
}
}
}
+class Isabelle_Sidekick_Default extends
+ Isabelle_Sidekick_Structure("isabelle",
+ PIDE.resources.theory_node_name, Document_Structure.parse_sections _)
-class Isabelle_Sidekick_Default extends
- Isabelle_Sidekick_Structure("isabelle", PIDE.resources.theory_node_name)
-
+class Isabelle_Sidekick_Context extends
+ Isabelle_Sidekick_Structure("isabelle-context",
+ PIDE.resources.theory_node_name, Document_Structure.parse_blocks _)
class Isabelle_Sidekick_Options extends
- Isabelle_Sidekick_Structure("isabelle-options", _ => Some(Document.Node.Name("options")))
-
+ Isabelle_Sidekick_Structure("isabelle-options",
+ _ => Some(Document.Node.Name("options")), Document_Structure.parse_sections _)
class Isabelle_Sidekick_Root extends
- Isabelle_Sidekick_Structure("isabelle-root", _ => Some(Document.Node.Name("ROOT")))
+ Isabelle_Sidekick_Structure("isabelle-root",
+ _ => Some(Document.Node.Name("ROOT")), Document_Structure.parse_sections _)
+
+class Isabelle_Sidekick_ML extends
+ Isabelle_Sidekick_Structure("isabelle-ml",
+ buffer => Some(PIDE.resources.node_name(buffer)),
+ (_, _, text) => Document_Structure.parse_ml_sections(false, text))
+
+class Isabelle_Sidekick_SML extends
+ Isabelle_Sidekick_Structure("isabelle-sml",
+ buffer => Some(PIDE.resources.node_name(buffer)),
+ (_, _, text) => Document_Structure.parse_ml_sections(true, text))
class Isabelle_Sidekick_Markup extends Isabelle_Sidekick("isabelle-markup")
--- a/src/Tools/jEdit/src/services.xml Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Tools/jEdit/src/services.xml Fri Aug 05 18:14:34 2016 +0200
@@ -17,9 +17,18 @@
<SERVICE NAME="isabelle" CLASS="sidekick.SideKickParser">
new isabelle.jedit.Isabelle_Sidekick_Default();
</SERVICE>
+ <SERVICE NAME="isabelle-context" CLASS="sidekick.SideKickParser">
+ new isabelle.jedit.Isabelle_Sidekick_Context();
+ </SERVICE>
<SERVICE NAME="isabelle-markup" CLASS="sidekick.SideKickParser">
new isabelle.jedit.Isabelle_Sidekick_Markup();
</SERVICE>
+ <SERVICE NAME="isabelle-ml" CLASS="sidekick.SideKickParser">
+ new isabelle.jedit.Isabelle_Sidekick_ML();
+ </SERVICE>
+ <SERVICE NAME="isabelle-sml" CLASS="sidekick.SideKickParser">
+ new isabelle.jedit.Isabelle_Sidekick_SML();
+ </SERVICE>
<SERVICE NAME="isabelle-news" CLASS="sidekick.SideKickParser">
new isabelle.jedit.Isabelle_Sidekick_News();
</SERVICE>
--- a/src/Tools/jEdit/src/token_markup.scala Fri Aug 05 17:36:38 2016 +0200
+++ b/src/Tools/jEdit/src/token_markup.scala Fri Aug 05 18:14:34 2016 +0200
@@ -178,7 +178,7 @@
object Line_Context
{
def init(mode: String): Line_Context =
- new Line_Context(mode, Some(Scan.Finished), Outer_Syntax.Line_Structure.init)
+ new Line_Context(mode, Some(Scan.Finished), Line_Structure.init)
def prev(buffer: JEditBuffer, line: Int): Line_Context =
if (line == 0) init(JEdit_Lib.buffer_mode(buffer))
@@ -202,7 +202,7 @@
class Line_Context(
val mode: String,
val context: Option[Scan.Line_Context],
- val structure: Outer_Syntax.Line_Structure)
+ val structure: Line_Structure)
extends TokenMarker.LineContext(new ParserRuleSet(mode, "MAIN"), null)
{
def get_context: Scan.Line_Context = context.getOrElse(Scan.Finished)
@@ -406,7 +406,7 @@
case (Some(ctxt), Some(syntax)) if syntax.has_tokens =>
val (tokens, ctxt1) = Token.explode_line(syntax.keywords, line, ctxt)
- val structure1 = syntax.line_structure(tokens, structure)
+ val structure1 = structure.update(syntax.keywords, tokens)
val styled_tokens =
tokens.map(tok => (Rendering.token_markup(syntax, tok), tok.source))
(styled_tokens, new Line_Context(line_context.mode, Some(ctxt1), structure1))