author | wenzelm |
Wed, 13 Jan 2016 23:07:06 +0100 | |
changeset 62175 | 8ffc4d0e652d |
parent 58880 | 0baae4311a9f |
child 67312 | 0d25e02759b7 |
permissions | -rw-r--r-- |
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(* Title: HOL/HOLCF/Cpodef.thy |
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Author: Brian Huffman |
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*) |
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section \<open>Subtypes of pcpos\<close> |
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theory Cpodef |
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imports Adm |
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keywords "pcpodef" "cpodef" :: thy_goal |
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begin |
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subsection \<open>Proving a subtype is a partial order\<close> |
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text \<open> |
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A subtype of a partial order is itself a partial order, |
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if the ordering is defined in the standard way. |
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\<close> |
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setup \<open>Sign.add_const_constraint (@{const_name Porder.below}, NONE)\<close> |
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theorem typedef_po: |
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fixes Abs :: "'a::po \<Rightarrow> 'b::type" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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shows "OFCLASS('b, po_class)" |
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apply (intro_classes, unfold below) |
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apply (rule below_refl) |
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apply (erule (1) below_trans) |
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apply (rule type_definition.Rep_inject [OF type, THEN iffD1]) |
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apply (erule (1) below_antisym) |
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done |
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setup \<open>Sign.add_const_constraint (@{const_name Porder.below}, |
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SOME @{typ "'a::below \<Rightarrow> 'a::below \<Rightarrow> bool"})\<close> |
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subsection \<open>Proving a subtype is finite\<close> |
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lemma typedef_finite_UNIV: |
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fixes Abs :: "'a::type \<Rightarrow> 'b::type" |
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assumes type: "type_definition Rep Abs A" |
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shows "finite A \<Longrightarrow> finite (UNIV :: 'b set)" |
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proof - |
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assume "finite A" |
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hence "finite (Abs ` A)" by (rule finite_imageI) |
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thus "finite (UNIV :: 'b set)" |
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by (simp only: type_definition.Abs_image [OF type]) |
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qed |
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subsection \<open>Proving a subtype is chain-finite\<close> |
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lemma ch2ch_Rep: |
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assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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shows "chain S \<Longrightarrow> chain (\<lambda>i. Rep (S i))" |
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unfolding chain_def below . |
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theorem typedef_chfin: |
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fixes Abs :: "'a::chfin \<Rightarrow> 'b::po" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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shows "OFCLASS('b, chfin_class)" |
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apply intro_classes |
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apply (drule ch2ch_Rep [OF below]) |
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apply (drule chfin) |
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apply (unfold max_in_chain_def) |
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apply (simp add: type_definition.Rep_inject [OF type]) |
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done |
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subsection \<open>Proving a subtype is complete\<close> |
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text \<open> |
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A subtype of a cpo is itself a cpo if the ordering is |
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defined in the standard way, and the defining subset |
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is closed with respect to limits of chains. A set is |
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closed if and only if membership in the set is an |
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admissible predicate. |
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\<close> |
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lemma typedef_is_lubI: |
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assumes below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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shows "range (\<lambda>i. Rep (S i)) <<| Rep x \<Longrightarrow> range S <<| x" |
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unfolding is_lub_def is_ub_def below by simp |
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lemma Abs_inverse_lub_Rep: |
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fixes Abs :: "'a::cpo \<Rightarrow> 'b::po" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and adm: "adm (\<lambda>x. x \<in> A)" |
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shows "chain S \<Longrightarrow> Rep (Abs (\<Squnion>i. Rep (S i))) = (\<Squnion>i. Rep (S i))" |
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apply (rule type_definition.Abs_inverse [OF type]) |
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apply (erule admD [OF adm ch2ch_Rep [OF below]]) |
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apply (rule type_definition.Rep [OF type]) |
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done |
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theorem typedef_is_lub: |
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fixes Abs :: "'a::cpo \<Rightarrow> 'b::po" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and adm: "adm (\<lambda>x. x \<in> A)" |
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shows "chain S \<Longrightarrow> range S <<| Abs (\<Squnion>i. Rep (S i))" |
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proof - |
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assume S: "chain S" |
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hence "chain (\<lambda>i. Rep (S i))" by (rule ch2ch_Rep [OF below]) |
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hence "range (\<lambda>i. Rep (S i)) <<| (\<Squnion>i. Rep (S i))" by (rule cpo_lubI) |
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hence "range (\<lambda>i. Rep (S i)) <<| Rep (Abs (\<Squnion>i. Rep (S i)))" |
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by (simp only: Abs_inverse_lub_Rep [OF type below adm S]) |
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thus "range S <<| Abs (\<Squnion>i. Rep (S i))" |
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by (rule typedef_is_lubI [OF below]) |
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qed |
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lemmas typedef_lub = typedef_is_lub [THEN lub_eqI] |
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theorem typedef_cpo: |
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fixes Abs :: "'a::cpo \<Rightarrow> 'b::po" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and adm: "adm (\<lambda>x. x \<in> A)" |
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shows "OFCLASS('b, cpo_class)" |
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proof |
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fix S::"nat \<Rightarrow> 'b" assume "chain S" |
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hence "range S <<| Abs (\<Squnion>i. Rep (S i))" |
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by (rule typedef_is_lub [OF type below adm]) |
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thus "\<exists>x. range S <<| x" .. |
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qed |
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subsubsection \<open>Continuity of \emph{Rep} and \emph{Abs}\<close> |
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text \<open>For any sub-cpo, the @{term Rep} function is continuous.\<close> |
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theorem typedef_cont_Rep: |
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fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and adm: "adm (\<lambda>x. x \<in> A)" |
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shows "cont (\<lambda>x. f x) \<Longrightarrow> cont (\<lambda>x. Rep (f x))" |
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apply (erule cont_apply [OF _ _ cont_const]) |
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apply (rule contI) |
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apply (simp only: typedef_lub [OF type below adm]) |
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apply (simp only: Abs_inverse_lub_Rep [OF type below adm]) |
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apply (rule cpo_lubI) |
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apply (erule ch2ch_Rep [OF below]) |
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done |
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text \<open> |
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For a sub-cpo, we can make the @{term Abs} function continuous |
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only if we restrict its domain to the defining subset by |
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composing it with another continuous function. |
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\<close> |
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theorem typedef_cont_Abs: |
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fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo" |
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fixes f :: "'c::cpo \<Rightarrow> 'a::cpo" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and adm: "adm (\<lambda>x. x \<in> A)" (* not used *) |
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and f_in_A: "\<And>x. f x \<in> A" |
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shows "cont f \<Longrightarrow> cont (\<lambda>x. Abs (f x))" |
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unfolding cont_def is_lub_def is_ub_def ball_simps below |
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by (simp add: type_definition.Abs_inverse [OF type f_in_A]) |
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subsection \<open>Proving subtype elements are compact\<close> |
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theorem typedef_compact: |
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fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and adm: "adm (\<lambda>x. x \<in> A)" |
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shows "compact (Rep k) \<Longrightarrow> compact k" |
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proof (unfold compact_def) |
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have cont_Rep: "cont Rep" |
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by (rule typedef_cont_Rep [OF type below adm cont_id]) |
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assume "adm (\<lambda>x. Rep k \<notsqsubseteq> x)" |
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with cont_Rep have "adm (\<lambda>x. Rep k \<notsqsubseteq> Rep x)" by (rule adm_subst) |
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thus "adm (\<lambda>x. k \<notsqsubseteq> x)" by (unfold below) |
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qed |
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subsection \<open>Proving a subtype is pointed\<close> |
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text \<open> |
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A subtype of a cpo has a least element if and only if |
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the defining subset has a least element. |
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\<close> |
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theorem typedef_pcpo_generic: |
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fixes Abs :: "'a::cpo \<Rightarrow> 'b::cpo" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and z_in_A: "z \<in> A" |
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and z_least: "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x" |
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shows "OFCLASS('b, pcpo_class)" |
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apply (intro_classes) |
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apply (rule_tac x="Abs z" in exI, rule allI) |
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apply (unfold below) |
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apply (subst type_definition.Abs_inverse [OF type z_in_A]) |
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apply (rule z_least [OF type_definition.Rep [OF type]]) |
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done |
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text \<open> |
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As a special case, a subtype of a pcpo has a least element |
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if the defining subset contains @{term \<bottom>}. |
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\<close> |
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theorem typedef_pcpo: |
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fixes Abs :: "'a::pcpo \<Rightarrow> 'b::cpo" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and bottom_in_A: "\<bottom> \<in> A" |
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shows "OFCLASS('b, pcpo_class)" |
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by (rule typedef_pcpo_generic [OF type below bottom_in_A], rule minimal) |
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subsubsection \<open>Strictness of \emph{Rep} and \emph{Abs}\<close> |
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text \<open> |
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For a sub-pcpo where @{term \<bottom>} is a member of the defining |
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subset, @{term Rep} and @{term Abs} are both strict. |
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\<close> |
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theorem typedef_Abs_strict: |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and bottom_in_A: "\<bottom> \<in> A" |
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shows "Abs \<bottom> = \<bottom>" |
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apply (rule bottomI, unfold below) |
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apply (simp add: type_definition.Abs_inverse [OF type bottom_in_A]) |
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done |
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theorem typedef_Rep_strict: |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and bottom_in_A: "\<bottom> \<in> A" |
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shows "Rep \<bottom> = \<bottom>" |
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apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst]) |
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apply (rule type_definition.Abs_inverse [OF type bottom_in_A]) |
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done |
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theorem typedef_Abs_bottom_iff: |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and bottom_in_A: "\<bottom> \<in> A" |
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shows "x \<in> A \<Longrightarrow> (Abs x = \<bottom>) = (x = \<bottom>)" |
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apply (rule typedef_Abs_strict [OF type below bottom_in_A, THEN subst]) |
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apply (simp add: type_definition.Abs_inject [OF type] bottom_in_A) |
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done |
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theorem typedef_Rep_bottom_iff: |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and bottom_in_A: "\<bottom> \<in> A" |
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shows "(Rep x = \<bottom>) = (x = \<bottom>)" |
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apply (rule typedef_Rep_strict [OF type below bottom_in_A, THEN subst]) |
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apply (simp add: type_definition.Rep_inject [OF type]) |
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done |
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subsection \<open>Proving a subtype is flat\<close> |
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theorem typedef_flat: |
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fixes Abs :: "'a::flat \<Rightarrow> 'b::pcpo" |
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assumes type: "type_definition Rep Abs A" |
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and below: "op \<sqsubseteq> \<equiv> \<lambda>x y. Rep x \<sqsubseteq> Rep y" |
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and bottom_in_A: "\<bottom> \<in> A" |
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shows "OFCLASS('b, flat_class)" |
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apply (intro_classes) |
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apply (unfold below) |
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apply (simp add: type_definition.Rep_inject [OF type, symmetric]) |
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apply (simp add: typedef_Rep_strict [OF type below bottom_in_A]) |
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apply (simp add: ax_flat) |
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done |
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subsection \<open>HOLCF type definition package\<close> |
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ML_file "Tools/cpodef.ML" |
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end |