src/HOLCF/Domain_Aux.thy
 author huffman Wed Nov 10 17:56:08 2010 -0800 (2010-11-10) changeset 40502 8e92772bc0e8 parent 40327 1dfdbd66093a child 40503 4094d788b904 permissions -rw-r--r--
move map functions to new theory file Map_Functions; add theory file Plain_HOLCF
```     1 (*  Title:      HOLCF/Domain_Aux.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Domain package support *}
```
```     6
```
```     7 theory Domain_Aux
```
```     8 imports Map_Functions Fixrec
```
```     9 uses
```
```    10   ("Tools/Domain/domain_take_proofs.ML")
```
```    11 begin
```
```    12
```
```    13 subsection {* Continuous isomorphisms *}
```
```    14
```
```    15 text {* A locale for continuous isomorphisms *}
```
```    16
```
```    17 locale iso =
```
```    18   fixes abs :: "'a \<rightarrow> 'b"
```
```    19   fixes rep :: "'b \<rightarrow> 'a"
```
```    20   assumes abs_iso [simp]: "rep\<cdot>(abs\<cdot>x) = x"
```
```    21   assumes rep_iso [simp]: "abs\<cdot>(rep\<cdot>y) = y"
```
```    22 begin
```
```    23
```
```    24 lemma swap: "iso rep abs"
```
```    25   by (rule iso.intro [OF rep_iso abs_iso])
```
```    26
```
```    27 lemma abs_below: "(abs\<cdot>x \<sqsubseteq> abs\<cdot>y) = (x \<sqsubseteq> y)"
```
```    28 proof
```
```    29   assume "abs\<cdot>x \<sqsubseteq> abs\<cdot>y"
```
```    30   then have "rep\<cdot>(abs\<cdot>x) \<sqsubseteq> rep\<cdot>(abs\<cdot>y)" by (rule monofun_cfun_arg)
```
```    31   then show "x \<sqsubseteq> y" by simp
```
```    32 next
```
```    33   assume "x \<sqsubseteq> y"
```
```    34   then show "abs\<cdot>x \<sqsubseteq> abs\<cdot>y" by (rule monofun_cfun_arg)
```
```    35 qed
```
```    36
```
```    37 lemma rep_below: "(rep\<cdot>x \<sqsubseteq> rep\<cdot>y) = (x \<sqsubseteq> y)"
```
```    38   by (rule iso.abs_below [OF swap])
```
```    39
```
```    40 lemma abs_eq: "(abs\<cdot>x = abs\<cdot>y) = (x = y)"
```
```    41   by (simp add: po_eq_conv abs_below)
```
```    42
```
```    43 lemma rep_eq: "(rep\<cdot>x = rep\<cdot>y) = (x = y)"
```
```    44   by (rule iso.abs_eq [OF swap])
```
```    45
```
```    46 lemma abs_strict: "abs\<cdot>\<bottom> = \<bottom>"
```
```    47 proof -
```
```    48   have "\<bottom> \<sqsubseteq> rep\<cdot>\<bottom>" ..
```
```    49   then have "abs\<cdot>\<bottom> \<sqsubseteq> abs\<cdot>(rep\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
```
```    50   then have "abs\<cdot>\<bottom> \<sqsubseteq> \<bottom>" by simp
```
```    51   then show ?thesis by (rule UU_I)
```
```    52 qed
```
```    53
```
```    54 lemma rep_strict: "rep\<cdot>\<bottom> = \<bottom>"
```
```    55   by (rule iso.abs_strict [OF swap])
```
```    56
```
```    57 lemma abs_defin': "abs\<cdot>x = \<bottom> \<Longrightarrow> x = \<bottom>"
```
```    58 proof -
```
```    59   have "x = rep\<cdot>(abs\<cdot>x)" by simp
```
```    60   also assume "abs\<cdot>x = \<bottom>"
```
```    61   also note rep_strict
```
```    62   finally show "x = \<bottom>" .
```
```    63 qed
```
```    64
```
```    65 lemma rep_defin': "rep\<cdot>z = \<bottom> \<Longrightarrow> z = \<bottom>"
```
```    66   by (rule iso.abs_defin' [OF swap])
```
```    67
```
```    68 lemma abs_defined: "z \<noteq> \<bottom> \<Longrightarrow> abs\<cdot>z \<noteq> \<bottom>"
```
```    69   by (erule contrapos_nn, erule abs_defin')
```
```    70
```
```    71 lemma rep_defined: "z \<noteq> \<bottom> \<Longrightarrow> rep\<cdot>z \<noteq> \<bottom>"
```
```    72   by (rule iso.abs_defined [OF iso.swap]) (rule iso_axioms)
```
```    73
```
```    74 lemma abs_bottom_iff: "(abs\<cdot>x = \<bottom>) = (x = \<bottom>)"
```
```    75   by (auto elim: abs_defin' intro: abs_strict)
```
```    76
```
```    77 lemma rep_bottom_iff: "(rep\<cdot>x = \<bottom>) = (x = \<bottom>)"
```
```    78   by (rule iso.abs_bottom_iff [OF iso.swap]) (rule iso_axioms)
```
```    79
```
```    80 lemma casedist_rule: "rep\<cdot>x = \<bottom> \<or> P \<Longrightarrow> x = \<bottom> \<or> P"
```
```    81   by (simp add: rep_bottom_iff)
```
```    82
```
```    83 lemma compact_abs_rev: "compact (abs\<cdot>x) \<Longrightarrow> compact x"
```
```    84 proof (unfold compact_def)
```
```    85   assume "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> y)"
```
```    86   with cont_Rep_cfun2
```
```    87   have "adm (\<lambda>y. \<not> abs\<cdot>x \<sqsubseteq> abs\<cdot>y)" by (rule adm_subst)
```
```    88   then show "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" using abs_below by simp
```
```    89 qed
```
```    90
```
```    91 lemma compact_rep_rev: "compact (rep\<cdot>x) \<Longrightarrow> compact x"
```
```    92   by (rule iso.compact_abs_rev [OF iso.swap]) (rule iso_axioms)
```
```    93
```
```    94 lemma compact_abs: "compact x \<Longrightarrow> compact (abs\<cdot>x)"
```
```    95   by (rule compact_rep_rev) simp
```
```    96
```
```    97 lemma compact_rep: "compact x \<Longrightarrow> compact (rep\<cdot>x)"
```
```    98   by (rule iso.compact_abs [OF iso.swap]) (rule iso_axioms)
```
```    99
```
```   100 lemma iso_swap: "(x = abs\<cdot>y) = (rep\<cdot>x = y)"
```
```   101 proof
```
```   102   assume "x = abs\<cdot>y"
```
```   103   then have "rep\<cdot>x = rep\<cdot>(abs\<cdot>y)" by simp
```
```   104   then show "rep\<cdot>x = y" by simp
```
```   105 next
```
```   106   assume "rep\<cdot>x = y"
```
```   107   then have "abs\<cdot>(rep\<cdot>x) = abs\<cdot>y" by simp
```
```   108   then show "x = abs\<cdot>y" by simp
```
```   109 qed
```
```   110
```
```   111 end
```
```   112
```
```   113
```
```   114 subsection {* Proofs about take functions *}
```
```   115
```
```   116 text {*
```
```   117   This section contains lemmas that are used in a module that supports
```
```   118   the domain isomorphism package; the module contains proofs related
```
```   119   to take functions and the finiteness predicate.
```
```   120 *}
```
```   121
```
```   122 lemma deflation_abs_rep:
```
```   123   fixes abs and rep and d
```
```   124   assumes abs_iso: "\<And>x. rep\<cdot>(abs\<cdot>x) = x"
```
```   125   assumes rep_iso: "\<And>y. abs\<cdot>(rep\<cdot>y) = y"
```
```   126   shows "deflation d \<Longrightarrow> deflation (abs oo d oo rep)"
```
```   127 by (rule ep_pair.deflation_e_d_p) (simp add: ep_pair.intro assms)
```
```   128
```
```   129 lemma deflation_chain_min:
```
```   130   assumes chain: "chain d"
```
```   131   assumes defl: "\<And>n. deflation (d n)"
```
```   132   shows "d m\<cdot>(d n\<cdot>x) = d (min m n)\<cdot>x"
```
```   133 proof (rule linorder_le_cases)
```
```   134   assume "m \<le> n"
```
```   135   with chain have "d m \<sqsubseteq> d n" by (rule chain_mono)
```
```   136   then have "d m\<cdot>(d n\<cdot>x) = d m\<cdot>x"
```
```   137     by (rule deflation_below_comp1 [OF defl defl])
```
```   138   moreover from `m \<le> n` have "min m n = m" by simp
```
```   139   ultimately show ?thesis by simp
```
```   140 next
```
```   141   assume "n \<le> m"
```
```   142   with chain have "d n \<sqsubseteq> d m" by (rule chain_mono)
```
```   143   then have "d m\<cdot>(d n\<cdot>x) = d n\<cdot>x"
```
```   144     by (rule deflation_below_comp2 [OF defl defl])
```
```   145   moreover from `n \<le> m` have "min m n = n" by simp
```
```   146   ultimately show ?thesis by simp
```
```   147 qed
```
```   148
```
```   149 lemma lub_ID_take_lemma:
```
```   150   assumes "chain t" and "(\<Squnion>n. t n) = ID"
```
```   151   assumes "\<And>n. t n\<cdot>x = t n\<cdot>y" shows "x = y"
```
```   152 proof -
```
```   153   have "(\<Squnion>n. t n\<cdot>x) = (\<Squnion>n. t n\<cdot>y)"
```
```   154     using assms(3) by simp
```
```   155   then have "(\<Squnion>n. t n)\<cdot>x = (\<Squnion>n. t n)\<cdot>y"
```
```   156     using assms(1) by (simp add: lub_distribs)
```
```   157   then show "x = y"
```
```   158     using assms(2) by simp
```
```   159 qed
```
```   160
```
```   161 lemma lub_ID_reach:
```
```   162   assumes "chain t" and "(\<Squnion>n. t n) = ID"
```
```   163   shows "(\<Squnion>n. t n\<cdot>x) = x"
```
```   164 using assms by (simp add: lub_distribs)
```
```   165
```
```   166 lemma lub_ID_take_induct:
```
```   167   assumes "chain t" and "(\<Squnion>n. t n) = ID"
```
```   168   assumes "adm P" and "\<And>n. P (t n\<cdot>x)" shows "P x"
```
```   169 proof -
```
```   170   from `chain t` have "chain (\<lambda>n. t n\<cdot>x)" by simp
```
```   171   from `adm P` this `\<And>n. P (t n\<cdot>x)` have "P (\<Squnion>n. t n\<cdot>x)" by (rule admD)
```
```   172   with `chain t` `(\<Squnion>n. t n) = ID` show "P x" by (simp add: lub_distribs)
```
```   173 qed
```
```   174
```
```   175
```
```   176 subsection {* Finiteness *}
```
```   177
```
```   178 text {*
```
```   179   Let a ``decisive'' function be a deflation that maps every input to
```
```   180   either itself or bottom.  Then if a domain's take functions are all
```
```   181   decisive, then all values in the domain are finite.
```
```   182 *}
```
```   183
```
```   184 definition
```
```   185   decisive :: "('a::pcpo \<rightarrow> 'a) \<Rightarrow> bool"
```
```   186 where
```
```   187   "decisive d \<longleftrightarrow> (\<forall>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>)"
```
```   188
```
```   189 lemma decisiveI: "(\<And>x. d\<cdot>x = x \<or> d\<cdot>x = \<bottom>) \<Longrightarrow> decisive d"
```
```   190   unfolding decisive_def by simp
```
```   191
```
```   192 lemma decisive_cases:
```
```   193   assumes "decisive d" obtains "d\<cdot>x = x" | "d\<cdot>x = \<bottom>"
```
```   194 using assms unfolding decisive_def by auto
```
```   195
```
```   196 lemma decisive_bottom: "decisive \<bottom>"
```
```   197   unfolding decisive_def by simp
```
```   198
```
```   199 lemma decisive_ID: "decisive ID"
```
```   200   unfolding decisive_def by simp
```
```   201
```
```   202 lemma decisive_ssum_map:
```
```   203   assumes f: "decisive f"
```
```   204   assumes g: "decisive g"
```
```   205   shows "decisive (ssum_map\<cdot>f\<cdot>g)"
```
```   206 apply (rule decisiveI, rename_tac s)
```
```   207 apply (case_tac s, simp_all)
```
```   208 apply (rule_tac x=x in decisive_cases [OF f], simp_all)
```
```   209 apply (rule_tac x=y in decisive_cases [OF g], simp_all)
```
```   210 done
```
```   211
```
```   212 lemma decisive_sprod_map:
```
```   213   assumes f: "decisive f"
```
```   214   assumes g: "decisive g"
```
```   215   shows "decisive (sprod_map\<cdot>f\<cdot>g)"
```
```   216 apply (rule decisiveI, rename_tac s)
```
```   217 apply (case_tac s, simp_all)
```
```   218 apply (rule_tac x=x in decisive_cases [OF f], simp_all)
```
```   219 apply (rule_tac x=y in decisive_cases [OF g], simp_all)
```
```   220 done
```
```   221
```
```   222 lemma decisive_abs_rep:
```
```   223   fixes abs rep
```
```   224   assumes iso: "iso abs rep"
```
```   225   assumes d: "decisive d"
```
```   226   shows "decisive (abs oo d oo rep)"
```
```   227 apply (rule decisiveI)
```
```   228 apply (rule_tac x="rep\<cdot>x" in decisive_cases [OF d])
```
```   229 apply (simp add: iso.rep_iso [OF iso])
```
```   230 apply (simp add: iso.abs_strict [OF iso])
```
```   231 done
```
```   232
```
```   233 lemma lub_ID_finite:
```
```   234   assumes chain: "chain d"
```
```   235   assumes lub: "(\<Squnion>n. d n) = ID"
```
```   236   assumes decisive: "\<And>n. decisive (d n)"
```
```   237   shows "\<exists>n. d n\<cdot>x = x"
```
```   238 proof -
```
```   239   have 1: "chain (\<lambda>n. d n\<cdot>x)" using chain by simp
```
```   240   have 2: "(\<Squnion>n. d n\<cdot>x) = x" using chain lub by (rule lub_ID_reach)
```
```   241   have "\<forall>n. d n\<cdot>x = x \<or> d n\<cdot>x = \<bottom>"
```
```   242     using decisive unfolding decisive_def by simp
```
```   243   hence "range (\<lambda>n. d n\<cdot>x) \<subseteq> {x, \<bottom>}"
```
```   244     by auto
```
```   245   hence "finite (range (\<lambda>n. d n\<cdot>x))"
```
```   246     by (rule finite_subset, simp)
```
```   247   with 1 have "finite_chain (\<lambda>n. d n\<cdot>x)"
```
```   248     by (rule finite_range_imp_finch)
```
```   249   then have "\<exists>n. (\<Squnion>n. d n\<cdot>x) = d n\<cdot>x"
```
```   250     unfolding finite_chain_def by (auto simp add: maxinch_is_thelub)
```
```   251   with 2 show "\<exists>n. d n\<cdot>x = x" by (auto elim: sym)
```
```   252 qed
```
```   253
```
```   254 lemma lub_ID_finite_take_induct:
```
```   255   assumes "chain d" and "(\<Squnion>n. d n) = ID" and "\<And>n. decisive (d n)"
```
```   256   shows "(\<And>n. P (d n\<cdot>x)) \<Longrightarrow> P x"
```
```   257 using lub_ID_finite [OF assms] by metis
```
```   258
```
```   259 subsection {* ML setup *}
```
```   260
```
```   261 use "Tools/Domain/domain_take_proofs.ML"
```
```   262
```
```   263 setup Domain_Take_Proofs.setup
```
```   264
```
```   265 end
```