author | wenzelm |
Tue, 13 Sep 2005 22:19:23 +0200 | |
changeset 17339 | ab97ccef124a |
parent 16933 | 91ded127f5f7 |
child 17585 | f12d7ac88eb4 |
permissions | -rw-r--r-- |
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(* Title: HOLCF/Up.thy |
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ID: $Id$ |
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Author: Franz Regensburger and Brian Huffman |
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Lifting. |
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*) |
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header {* The type of lifted values *} |
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theory Up |
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imports Cfun Sum_Type Datatype |
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begin |
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defaultsort cpo |
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subsection {* Definition of new type for lifting *} |
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datatype 'a u = Ibottom | Iup 'a |
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consts |
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Ifup :: "('a \<rightarrow> 'b::pcpo) \<Rightarrow> 'a u \<Rightarrow> 'b" |
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primrec |
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"Ifup f Ibottom = \<bottom>" |
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"Ifup f (Iup x) = f\<cdot>x" |
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subsection {* Ordering on type @{typ "'a u"} *} |
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instance u :: (sq_ord) sq_ord .. |
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defs (overloaded) |
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less_up_def: |
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"(op \<sqsubseteq>) \<equiv> (\<lambda>x y. case x of Ibottom \<Rightarrow> True | Iup a \<Rightarrow> |
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(case y of Ibottom \<Rightarrow> False | Iup b \<Rightarrow> a \<sqsubseteq> b))" |
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lemma minimal_up [iff]: "Ibottom \<sqsubseteq> z" |
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by (simp add: less_up_def) |
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lemma not_Iup_less [iff]: "\<not> Iup x \<sqsubseteq> Ibottom" |
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by (simp add: less_up_def) |
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lemma Iup_less [iff]: "(Iup x \<sqsubseteq> Iup y) = (x \<sqsubseteq> y)" |
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by (simp add: less_up_def) |
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subsection {* Type @{typ "'a u"} is a partial order *} |
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lemma refl_less_up: "(x::'a u) \<sqsubseteq> x" |
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by (simp add: less_up_def split: u.split) |
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lemma antisym_less_up: "\<lbrakk>(x::'a u) \<sqsubseteq> y; y \<sqsubseteq> x\<rbrakk> \<Longrightarrow> x = y" |
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apply (simp add: less_up_def split: u.split_asm) |
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apply (erule (1) antisym_less) |
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done |
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lemma trans_less_up: "\<lbrakk>(x::'a u) \<sqsubseteq> y; y \<sqsubseteq> z\<rbrakk> \<Longrightarrow> x \<sqsubseteq> z" |
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apply (simp add: less_up_def split: u.split_asm) |
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apply (erule (1) trans_less) |
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done |
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instance u :: (cpo) po |
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by intro_classes |
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(assumption | rule refl_less_up antisym_less_up trans_less_up)+ |
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subsection {* Type @{typ "'a u"} is a cpo *} |
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lemma is_lub_Iup: |
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"range S <<| x \<Longrightarrow> range (\<lambda>i. Iup (S i)) <<| Iup x" |
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apply (rule is_lubI) |
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apply (rule ub_rangeI) |
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apply (subst Iup_less) |
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apply (erule is_ub_lub) |
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apply (case_tac u) |
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apply (drule ub_rangeD) |
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apply simp |
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apply simp |
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apply (erule is_lub_lub) |
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apply (rule ub_rangeI) |
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apply (drule_tac i=i in ub_rangeD) |
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apply simp |
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done |
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text {* Now some lemmas about chains of @{typ "'a u"} elements *} |
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lemma up_lemma1: "z \<noteq> Ibottom \<Longrightarrow> Iup (THE a. Iup a = z) = z" |
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by (case_tac z, simp_all) |
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lemma up_lemma2: |
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"\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> \<Longrightarrow> Y (i + j) \<noteq> Ibottom" |
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apply (erule contrapos_nn) |
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apply (drule_tac x="j" and y="i + j" in chain_mono3) |
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apply (rule le_add2) |
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apply (case_tac "Y j") |
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apply assumption |
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apply simp |
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done |
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lemma up_lemma3: |
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"\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> \<Longrightarrow> Iup (THE a. Iup a = Y (i + j)) = Y (i + j)" |
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by (rule up_lemma1 [OF up_lemma2]) |
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lemma up_lemma4: |
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"\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> \<Longrightarrow> chain (\<lambda>i. THE a. Iup a = Y (i + j))" |
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apply (rule chainI) |
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apply (rule Iup_less [THEN iffD1]) |
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apply (subst up_lemma3, assumption+)+ |
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apply (simp add: chainE) |
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done |
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lemma up_lemma5: |
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"\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> \<Longrightarrow> |
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(\<lambda>i. Y (i + j)) = (\<lambda>i. Iup (THE a. Iup a = Y (i + j)))" |
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by (rule ext, rule up_lemma3 [symmetric]) |
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lemma up_lemma6: |
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"\<lbrakk>chain Y; Y j \<noteq> Ibottom\<rbrakk> |
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\<Longrightarrow> range Y <<| Iup (\<Squnion>i. THE a. Iup a = Y(i + j))" |
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apply (rule_tac j1 = j in is_lub_range_shift [THEN iffD1]) |
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apply assumption |
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apply (subst up_lemma5, assumption+) |
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apply (rule is_lub_Iup) |
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apply (rule thelubE [OF _ refl]) |
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apply (erule (1) up_lemma4) |
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done |
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lemma up_chain_cases: |
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"chain Y \<Longrightarrow> |
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(\<exists>A. chain A \<and> lub (range Y) = Iup (lub (range A)) \<and> |
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(\<exists>j. \<forall>i. Y (i + j) = Iup (A i))) \<or> (Y = (\<lambda>i. Ibottom))" |
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apply (rule disjCI) |
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apply (simp add: expand_fun_eq) |
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apply (erule exE, rename_tac j) |
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apply (rule_tac x="\<lambda>i. THE a. Iup a = Y (i + j)" in exI) |
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apply (rule conjI) |
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apply (simp add: up_lemma4) |
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apply (rule conjI) |
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apply (simp add: up_lemma6 [THEN thelubI]) |
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apply (rule_tac x=j in exI) |
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apply (simp add: up_lemma3) |
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done |
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lemma cpo_up: "chain (Y::nat \<Rightarrow> 'a u) \<Longrightarrow> \<exists>x. range Y <<| x" |
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apply (frule up_chain_cases, safe) |
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apply (rule_tac x="Iup (lub (range A))" in exI) |
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apply (erule_tac j1="j" in is_lub_range_shift [THEN iffD1]) |
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apply (simp add: is_lub_Iup thelubE) |
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apply (rule_tac x="Ibottom" in exI) |
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apply (rule lub_const) |
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done |
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|
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instance u :: (cpo) cpo |
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by intro_classes (rule cpo_up) |
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|
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subsection {* Type @{typ "'a u"} is pointed *} |
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|
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lemma least_up: "EX x::'a u. ALL y. x\<sqsubseteq>y" |
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apply (rule_tac x = "Ibottom" in exI) |
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apply (rule minimal_up [THEN allI]) |
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done |
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|
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instance u :: (cpo) pcpo |
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by intro_classes (rule least_up) |
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|
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text {* for compatibility with old HOLCF-Version *} |
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lemma inst_up_pcpo: "\<bottom> = Ibottom" |
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by (rule minimal_up [THEN UU_I, symmetric]) |
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|
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subsection {* Continuity of @{term Iup} and @{term Ifup} *} |
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|
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text {* continuity for @{term Iup} *} |
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|
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lemma cont_Iup: "cont Iup" |
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apply (rule contI) |
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apply (rule is_lub_Iup) |
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apply (erule thelubE [OF _ refl]) |
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done |
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|
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text {* continuity for @{term Ifup} *} |
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|
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lemma cont_Ifup1: "cont (\<lambda>f. Ifup f x)" |
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by (induct x, simp_all) |
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|
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lemma monofun_Ifup2: "monofun (\<lambda>x. Ifup f x)" |
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apply (rule monofunI) |
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apply (case_tac x, simp) |
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apply (case_tac y, simp) |
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apply (simp add: monofun_cfun_arg) |
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done |
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|
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lemma cont_Ifup2: "cont (\<lambda>x. Ifup f x)" |
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apply (rule contI) |
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apply (frule up_chain_cases, safe) |
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apply (rule_tac j1="j" in is_lub_range_shift [THEN iffD1]) |
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apply (erule monofun_Ifup2 [THEN ch2ch_monofun]) |
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apply (simp add: cont_cfun_arg) |
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apply (simp add: thelub_const lub_const) |
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done |
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|
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subsection {* Continuous versions of constants *} |
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|
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constdefs |
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up :: "'a \<rightarrow> 'a u" |
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"up \<equiv> \<Lambda> x. Iup x" |
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|
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fup :: "('a \<rightarrow> 'b::pcpo) \<rightarrow> 'a u \<rightarrow> 'b" |
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"fup \<equiv> \<Lambda> f p. Ifup f p" |
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|
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translations |
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"case l of up\<cdot>x \<Rightarrow> t" == "fup\<cdot>(LAM x. t)\<cdot>l" |
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|
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text {* continuous versions of lemmas for @{typ "('a)u"} *} |
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|
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lemma Exh_Up: "z = \<bottom> \<or> (\<exists>x. z = up\<cdot>x)" |
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apply (induct z) |
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apply (simp add: inst_up_pcpo) |
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apply (simp add: up_def cont_Iup) |
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done |
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|
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lemma up_eq [simp]: "(up\<cdot>x = up\<cdot>y) = (x = y)" |
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by (simp add: up_def cont_Iup) |
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|
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lemma up_inject: "up\<cdot>x = up\<cdot>y \<Longrightarrow> x = y" |
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by simp |
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|
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lemma up_defined [simp]: " up\<cdot>x \<noteq> \<bottom>" |
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by (simp add: up_def cont_Iup inst_up_pcpo) |
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|
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lemma not_up_less_UU [simp]: "\<not> up\<cdot>x \<sqsubseteq> \<bottom>" |
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by (simp add: eq_UU_iff [symmetric]) |
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|
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lemma up_less [simp]: "(up\<cdot>x \<sqsubseteq> up\<cdot>y) = (x \<sqsubseteq> y)" |
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by (simp add: up_def cont_Iup) |
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232 |
|
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lemma upE: "\<lbrakk>p = \<bottom> \<Longrightarrow> Q; \<And>x. p = up\<cdot>x \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" |
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apply (case_tac p) |
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apply (simp add: inst_up_pcpo) |
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apply (simp add: up_def cont_Iup) |
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done |
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|
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lemma fup1 [simp]: "fup\<cdot>f\<cdot>\<bottom> = \<bottom>" |
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by (simp add: fup_def cont_Ifup1 cont_Ifup2 inst_up_pcpo) |
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|
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lemma fup2 [simp]: "fup\<cdot>f\<cdot>(up\<cdot>x) = f\<cdot>x" |
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by (simp add: up_def fup_def cont_Iup cont_Ifup1 cont_Ifup2) |
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|
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lemma fup3 [simp]: "fup\<cdot>up\<cdot>x = x" |
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by (rule_tac p=x in upE, simp_all) |
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|
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248 |
end |