src/HOLCF/Deflation.thy
 author huffman Mon May 11 08:28:09 2009 -0700 (2009-05-11) changeset 31095 b79d140f6d0b parent 31076 99fe356cbbc2 child 33503 3496616b2171 permissions -rw-r--r--
simplify fixrec proofs for mutually-recursive definitions; generate better fixpoint induction rules
```     1 (*  Title:      HOLCF/Deflation.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Continuous Deflations and Embedding-Projection Pairs *}
```
```     6
```
```     7 theory Deflation
```
```     8 imports Cfun
```
```     9 begin
```
```    10
```
```    11 defaultsort cpo
```
```    12
```
```    13 subsection {* Continuous deflations *}
```
```    14
```
```    15 locale deflation =
```
```    16   fixes d :: "'a \<rightarrow> 'a"
```
```    17   assumes idem: "\<And>x. d\<cdot>(d\<cdot>x) = d\<cdot>x"
```
```    18   assumes below: "\<And>x. d\<cdot>x \<sqsubseteq> x"
```
```    19 begin
```
```    20
```
```    21 lemma below_ID: "d \<sqsubseteq> ID"
```
```    22 by (rule below_cfun_ext, simp add: below)
```
```    23
```
```    24 text {* The set of fixed points is the same as the range. *}
```
```    25
```
```    26 lemma fixes_eq_range: "{x. d\<cdot>x = x} = range (\<lambda>x. d\<cdot>x)"
```
```    27 by (auto simp add: eq_sym_conv idem)
```
```    28
```
```    29 lemma range_eq_fixes: "range (\<lambda>x. d\<cdot>x) = {x. d\<cdot>x = x}"
```
```    30 by (auto simp add: eq_sym_conv idem)
```
```    31
```
```    32 text {*
```
```    33   The pointwise ordering on deflation functions coincides with
```
```    34   the subset ordering of their sets of fixed-points.
```
```    35 *}
```
```    36
```
```    37 lemma belowI:
```
```    38   assumes f: "\<And>x. d\<cdot>x = x \<Longrightarrow> f\<cdot>x = x" shows "d \<sqsubseteq> f"
```
```    39 proof (rule below_cfun_ext)
```
```    40   fix x
```
```    41   from below have "f\<cdot>(d\<cdot>x) \<sqsubseteq> f\<cdot>x" by (rule monofun_cfun_arg)
```
```    42   also from idem have "f\<cdot>(d\<cdot>x) = d\<cdot>x" by (rule f)
```
```    43   finally show "d\<cdot>x \<sqsubseteq> f\<cdot>x" .
```
```    44 qed
```
```    45
```
```    46 lemma belowD: "\<lbrakk>f \<sqsubseteq> d; f\<cdot>x = x\<rbrakk> \<Longrightarrow> d\<cdot>x = x"
```
```    47 proof (rule below_antisym)
```
```    48   from below show "d\<cdot>x \<sqsubseteq> x" .
```
```    49 next
```
```    50   assume "f \<sqsubseteq> d"
```
```    51   hence "f\<cdot>x \<sqsubseteq> d\<cdot>x" by (rule monofun_cfun_fun)
```
```    52   also assume "f\<cdot>x = x"
```
```    53   finally show "x \<sqsubseteq> d\<cdot>x" .
```
```    54 qed
```
```    55
```
```    56 end
```
```    57
```
```    58 lemma adm_deflation: "adm (\<lambda>d. deflation d)"
```
```    59 by (simp add: deflation_def)
```
```    60
```
```    61 lemma deflation_ID: "deflation ID"
```
```    62 by (simp add: deflation.intro)
```
```    63
```
```    64 lemma deflation_UU: "deflation \<bottom>"
```
```    65 by (simp add: deflation.intro)
```
```    66
```
```    67 lemma deflation_below_iff:
```
```    68   "\<lbrakk>deflation p; deflation q\<rbrakk> \<Longrightarrow> p \<sqsubseteq> q \<longleftrightarrow> (\<forall>x. p\<cdot>x = x \<longrightarrow> q\<cdot>x = x)"
```
```    69  apply safe
```
```    70   apply (simp add: deflation.belowD)
```
```    71  apply (simp add: deflation.belowI)
```
```    72 done
```
```    73
```
```    74 text {*
```
```    75   The composition of two deflations is equal to
```
```    76   the lesser of the two (if they are comparable).
```
```    77 *}
```
```    78
```
```    79 lemma deflation_below_comp1:
```
```    80   assumes "deflation f"
```
```    81   assumes "deflation g"
```
```    82   shows "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>(g\<cdot>x) = f\<cdot>x"
```
```    83 proof (rule below_antisym)
```
```    84   interpret g: deflation g by fact
```
```    85   from g.below show "f\<cdot>(g\<cdot>x) \<sqsubseteq> f\<cdot>x" by (rule monofun_cfun_arg)
```
```    86 next
```
```    87   interpret f: deflation f by fact
```
```    88   assume "f \<sqsubseteq> g" hence "f\<cdot>x \<sqsubseteq> g\<cdot>x" by (rule monofun_cfun_fun)
```
```    89   hence "f\<cdot>(f\<cdot>x) \<sqsubseteq> f\<cdot>(g\<cdot>x)" by (rule monofun_cfun_arg)
```
```    90   also have "f\<cdot>(f\<cdot>x) = f\<cdot>x" by (rule f.idem)
```
```    91   finally show "f\<cdot>x \<sqsubseteq> f\<cdot>(g\<cdot>x)" .
```
```    92 qed
```
```    93
```
```    94 lemma deflation_below_comp2:
```
```    95   "\<lbrakk>deflation f; deflation g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> g\<cdot>(f\<cdot>x) = f\<cdot>x"
```
```    96 by (simp only: deflation.belowD deflation.idem)
```
```    97
```
```    98
```
```    99 subsection {* Deflations with finite range *}
```
```   100
```
```   101 lemma finite_range_imp_finite_fixes:
```
```   102   "finite (range f) \<Longrightarrow> finite {x. f x = x}"
```
```   103 proof -
```
```   104   have "{x. f x = x} \<subseteq> range f"
```
```   105     by (clarify, erule subst, rule rangeI)
```
```   106   moreover assume "finite (range f)"
```
```   107   ultimately show "finite {x. f x = x}"
```
```   108     by (rule finite_subset)
```
```   109 qed
```
```   110
```
```   111 locale finite_deflation = deflation +
```
```   112   assumes finite_fixes: "finite {x. d\<cdot>x = x}"
```
```   113 begin
```
```   114
```
```   115 lemma finite_range: "finite (range (\<lambda>x. d\<cdot>x))"
```
```   116 by (simp add: range_eq_fixes finite_fixes)
```
```   117
```
```   118 lemma finite_image: "finite ((\<lambda>x. d\<cdot>x) ` A)"
```
```   119 by (rule finite_subset [OF image_mono [OF subset_UNIV] finite_range])
```
```   120
```
```   121 lemma compact: "compact (d\<cdot>x)"
```
```   122 proof (rule compactI2)
```
```   123   fix Y :: "nat \<Rightarrow> 'a"
```
```   124   assume Y: "chain Y"
```
```   125   have "finite_chain (\<lambda>i. d\<cdot>(Y i))"
```
```   126   proof (rule finite_range_imp_finch)
```
```   127     show "chain (\<lambda>i. d\<cdot>(Y i))"
```
```   128       using Y by simp
```
```   129     have "range (\<lambda>i. d\<cdot>(Y i)) \<subseteq> range (\<lambda>x. d\<cdot>x)"
```
```   130       by clarsimp
```
```   131     thus "finite (range (\<lambda>i. d\<cdot>(Y i)))"
```
```   132       using finite_range by (rule finite_subset)
```
```   133   qed
```
```   134   hence "\<exists>j. (\<Squnion>i. d\<cdot>(Y i)) = d\<cdot>(Y j)"
```
```   135     by (simp add: finite_chain_def maxinch_is_thelub Y)
```
```   136   then obtain j where j: "(\<Squnion>i. d\<cdot>(Y i)) = d\<cdot>(Y j)" ..
```
```   137
```
```   138   assume "d\<cdot>x \<sqsubseteq> (\<Squnion>i. Y i)"
```
```   139   hence "d\<cdot>(d\<cdot>x) \<sqsubseteq> d\<cdot>(\<Squnion>i. Y i)"
```
```   140     by (rule monofun_cfun_arg)
```
```   141   hence "d\<cdot>x \<sqsubseteq> (\<Squnion>i. d\<cdot>(Y i))"
```
```   142     by (simp add: contlub_cfun_arg Y idem)
```
```   143   hence "d\<cdot>x \<sqsubseteq> d\<cdot>(Y j)"
```
```   144     using j by simp
```
```   145   hence "d\<cdot>x \<sqsubseteq> Y j"
```
```   146     using below by (rule below_trans)
```
```   147   thus "\<exists>j. d\<cdot>x \<sqsubseteq> Y j" ..
```
```   148 qed
```
```   149
```
```   150 end
```
```   151
```
```   152
```
```   153 subsection {* Continuous embedding-projection pairs *}
```
```   154
```
```   155 locale ep_pair =
```
```   156   fixes e :: "'a \<rightarrow> 'b" and p :: "'b \<rightarrow> 'a"
```
```   157   assumes e_inverse [simp]: "\<And>x. p\<cdot>(e\<cdot>x) = x"
```
```   158   and e_p_below: "\<And>y. e\<cdot>(p\<cdot>y) \<sqsubseteq> y"
```
```   159 begin
```
```   160
```
```   161 lemma e_below_iff [simp]: "e\<cdot>x \<sqsubseteq> e\<cdot>y \<longleftrightarrow> x \<sqsubseteq> y"
```
```   162 proof
```
```   163   assume "e\<cdot>x \<sqsubseteq> e\<cdot>y"
```
```   164   hence "p\<cdot>(e\<cdot>x) \<sqsubseteq> p\<cdot>(e\<cdot>y)" by (rule monofun_cfun_arg)
```
```   165   thus "x \<sqsubseteq> y" by simp
```
```   166 next
```
```   167   assume "x \<sqsubseteq> y"
```
```   168   thus "e\<cdot>x \<sqsubseteq> e\<cdot>y" by (rule monofun_cfun_arg)
```
```   169 qed
```
```   170
```
```   171 lemma e_eq_iff [simp]: "e\<cdot>x = e\<cdot>y \<longleftrightarrow> x = y"
```
```   172 unfolding po_eq_conv e_below_iff ..
```
```   173
```
```   174 lemma p_eq_iff:
```
```   175   "\<lbrakk>e\<cdot>(p\<cdot>x) = x; e\<cdot>(p\<cdot>y) = y\<rbrakk> \<Longrightarrow> p\<cdot>x = p\<cdot>y \<longleftrightarrow> x = y"
```
```   176 by (safe, erule subst, erule subst, simp)
```
```   177
```
```   178 lemma p_inverse: "(\<exists>x. y = e\<cdot>x) = (e\<cdot>(p\<cdot>y) = y)"
```
```   179 by (auto, rule exI, erule sym)
```
```   180
```
```   181 lemma e_below_iff_below_p: "e\<cdot>x \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> p\<cdot>y"
```
```   182 proof
```
```   183   assume "e\<cdot>x \<sqsubseteq> y"
```
```   184   then have "p\<cdot>(e\<cdot>x) \<sqsubseteq> p\<cdot>y" by (rule monofun_cfun_arg)
```
```   185   then show "x \<sqsubseteq> p\<cdot>y" by simp
```
```   186 next
```
```   187   assume "x \<sqsubseteq> p\<cdot>y"
```
```   188   then have "e\<cdot>x \<sqsubseteq> e\<cdot>(p\<cdot>y)" by (rule monofun_cfun_arg)
```
```   189   then show "e\<cdot>x \<sqsubseteq> y" using e_p_below by (rule below_trans)
```
```   190 qed
```
```   191
```
```   192 lemma compact_e_rev: "compact (e\<cdot>x) \<Longrightarrow> compact x"
```
```   193 proof -
```
```   194   assume "compact (e\<cdot>x)"
```
```   195   hence "adm (\<lambda>y. \<not> e\<cdot>x \<sqsubseteq> y)" by (rule compactD)
```
```   196   hence "adm (\<lambda>y. \<not> e\<cdot>x \<sqsubseteq> e\<cdot>y)" by (rule adm_subst [OF cont_Rep_CFun2])
```
```   197   hence "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" by simp
```
```   198   thus "compact x" by (rule compactI)
```
```   199 qed
```
```   200
```
```   201 lemma compact_e: "compact x \<Longrightarrow> compact (e\<cdot>x)"
```
```   202 proof -
```
```   203   assume "compact x"
```
```   204   hence "adm (\<lambda>y. \<not> x \<sqsubseteq> y)" by (rule compactD)
```
```   205   hence "adm (\<lambda>y. \<not> x \<sqsubseteq> p\<cdot>y)" by (rule adm_subst [OF cont_Rep_CFun2])
```
```   206   hence "adm (\<lambda>y. \<not> e\<cdot>x \<sqsubseteq> y)" by (simp add: e_below_iff_below_p)
```
```   207   thus "compact (e\<cdot>x)" by (rule compactI)
```
```   208 qed
```
```   209
```
```   210 lemma compact_e_iff: "compact (e\<cdot>x) \<longleftrightarrow> compact x"
```
```   211 by (rule iffI [OF compact_e_rev compact_e])
```
```   212
```
```   213 text {* Deflations from ep-pairs *}
```
```   214
```
```   215 lemma deflation_e_p: "deflation (e oo p)"
```
```   216 by (simp add: deflation.intro e_p_below)
```
```   217
```
```   218 lemma deflation_e_d_p:
```
```   219   assumes "deflation d"
```
```   220   shows "deflation (e oo d oo p)"
```
```   221 proof
```
```   222   interpret deflation d by fact
```
```   223   fix x :: 'b
```
```   224   show "(e oo d oo p)\<cdot>((e oo d oo p)\<cdot>x) = (e oo d oo p)\<cdot>x"
```
```   225     by (simp add: idem)
```
```   226   show "(e oo d oo p)\<cdot>x \<sqsubseteq> x"
```
```   227     by (simp add: e_below_iff_below_p below)
```
```   228 qed
```
```   229
```
```   230 lemma finite_deflation_e_d_p:
```
```   231   assumes "finite_deflation d"
```
```   232   shows "finite_deflation (e oo d oo p)"
```
```   233 proof
```
```   234   interpret finite_deflation d by fact
```
```   235   fix x :: 'b
```
```   236   show "(e oo d oo p)\<cdot>((e oo d oo p)\<cdot>x) = (e oo d oo p)\<cdot>x"
```
```   237     by (simp add: idem)
```
```   238   show "(e oo d oo p)\<cdot>x \<sqsubseteq> x"
```
```   239     by (simp add: e_below_iff_below_p below)
```
```   240   have "finite ((\<lambda>x. e\<cdot>x) ` (\<lambda>x. d\<cdot>x) ` range (\<lambda>x. p\<cdot>x))"
```
```   241     by (simp add: finite_image)
```
```   242   hence "finite (range (\<lambda>x. (e oo d oo p)\<cdot>x))"
```
```   243     by (simp add: image_image)
```
```   244   thus "finite {x. (e oo d oo p)\<cdot>x = x}"
```
```   245     by (rule finite_range_imp_finite_fixes)
```
```   246 qed
```
```   247
```
```   248 lemma deflation_p_d_e:
```
```   249   assumes "deflation d"
```
```   250   assumes d: "\<And>x. d\<cdot>x \<sqsubseteq> e\<cdot>(p\<cdot>x)"
```
```   251   shows "deflation (p oo d oo e)"
```
```   252 proof -
```
```   253   interpret d: deflation d by fact
```
```   254   {
```
```   255     fix x
```
```   256     have "d\<cdot>(e\<cdot>x) \<sqsubseteq> e\<cdot>x"
```
```   257       by (rule d.below)
```
```   258     hence "p\<cdot>(d\<cdot>(e\<cdot>x)) \<sqsubseteq> p\<cdot>(e\<cdot>x)"
```
```   259       by (rule monofun_cfun_arg)
```
```   260     hence "(p oo d oo e)\<cdot>x \<sqsubseteq> x"
```
```   261       by simp
```
```   262   }
```
```   263   note p_d_e_below = this
```
```   264   show ?thesis
```
```   265   proof
```
```   266     fix x
```
```   267     show "(p oo d oo e)\<cdot>x \<sqsubseteq> x"
```
```   268       by (rule p_d_e_below)
```
```   269   next
```
```   270     fix x
```
```   271     show "(p oo d oo e)\<cdot>((p oo d oo e)\<cdot>x) = (p oo d oo e)\<cdot>x"
```
```   272     proof (rule below_antisym)
```
```   273       show "(p oo d oo e)\<cdot>((p oo d oo e)\<cdot>x) \<sqsubseteq> (p oo d oo e)\<cdot>x"
```
```   274         by (rule p_d_e_below)
```
```   275       have "p\<cdot>(d\<cdot>(d\<cdot>(d\<cdot>(e\<cdot>x)))) \<sqsubseteq> p\<cdot>(d\<cdot>(e\<cdot>(p\<cdot>(d\<cdot>(e\<cdot>x)))))"
```
```   276         by (intro monofun_cfun_arg d)
```
```   277       hence "p\<cdot>(d\<cdot>(e\<cdot>x)) \<sqsubseteq> p\<cdot>(d\<cdot>(e\<cdot>(p\<cdot>(d\<cdot>(e\<cdot>x)))))"
```
```   278         by (simp only: d.idem)
```
```   279       thus "(p oo d oo e)\<cdot>x \<sqsubseteq> (p oo d oo e)\<cdot>((p oo d oo e)\<cdot>x)"
```
```   280         by simp
```
```   281     qed
```
```   282   qed
```
```   283 qed
```
```   284
```
```   285 lemma finite_deflation_p_d_e:
```
```   286   assumes "finite_deflation d"
```
```   287   assumes d: "\<And>x. d\<cdot>x \<sqsubseteq> e\<cdot>(p\<cdot>x)"
```
```   288   shows "finite_deflation (p oo d oo e)"
```
```   289 proof -
```
```   290   interpret d: finite_deflation d by fact
```
```   291   show ?thesis
```
```   292   proof (intro_locales)
```
```   293     have "deflation d" ..
```
```   294     thus "deflation (p oo d oo e)"
```
```   295       using d by (rule deflation_p_d_e)
```
```   296   next
```
```   297     show "finite_deflation_axioms (p oo d oo e)"
```
```   298     proof
```
```   299       have "finite ((\<lambda>x. d\<cdot>x) ` range (\<lambda>x. e\<cdot>x))"
```
```   300         by (rule d.finite_image)
```
```   301       hence "finite ((\<lambda>x. p\<cdot>x) ` (\<lambda>x. d\<cdot>x) ` range (\<lambda>x. e\<cdot>x))"
```
```   302         by (rule finite_imageI)
```
```   303       hence "finite (range (\<lambda>x. (p oo d oo e)\<cdot>x))"
```
```   304         by (simp add: image_image)
```
```   305       thus "finite {x. (p oo d oo e)\<cdot>x = x}"
```
```   306         by (rule finite_range_imp_finite_fixes)
```
```   307     qed
```
```   308   qed
```
```   309 qed
```
```   310
```
```   311 end
```
```   312
```
```   313 subsection {* Uniqueness of ep-pairs *}
```
```   314
```
```   315 lemma ep_pair_unique_e_lemma:
```
```   316   assumes "ep_pair e1 p" and "ep_pair e2 p"
```
```   317   shows "e1 \<sqsubseteq> e2"
```
```   318 proof (rule below_cfun_ext)
```
```   319   interpret e1: ep_pair e1 p by fact
```
```   320   interpret e2: ep_pair e2 p by fact
```
```   321   fix x
```
```   322   have "e1\<cdot>(p\<cdot>(e2\<cdot>x)) \<sqsubseteq> e2\<cdot>x"
```
```   323     by (rule e1.e_p_below)
```
```   324   thus "e1\<cdot>x \<sqsubseteq> e2\<cdot>x"
```
```   325     by (simp only: e2.e_inverse)
```
```   326 qed
```
```   327
```
```   328 lemma ep_pair_unique_e:
```
```   329   "\<lbrakk>ep_pair e1 p; ep_pair e2 p\<rbrakk> \<Longrightarrow> e1 = e2"
```
```   330 by (fast intro: below_antisym elim: ep_pair_unique_e_lemma)
```
```   331
```
```   332 lemma ep_pair_unique_p_lemma:
```
```   333   assumes "ep_pair e p1" and "ep_pair e p2"
```
```   334   shows "p1 \<sqsubseteq> p2"
```
```   335 proof (rule below_cfun_ext)
```
```   336   interpret p1: ep_pair e p1 by fact
```
```   337   interpret p2: ep_pair e p2 by fact
```
```   338   fix x
```
```   339   have "e\<cdot>(p1\<cdot>x) \<sqsubseteq> x"
```
```   340     by (rule p1.e_p_below)
```
```   341   hence "p2\<cdot>(e\<cdot>(p1\<cdot>x)) \<sqsubseteq> p2\<cdot>x"
```
```   342     by (rule monofun_cfun_arg)
```
```   343   thus "p1\<cdot>x \<sqsubseteq> p2\<cdot>x"
```
```   344     by (simp only: p2.e_inverse)
```
```   345 qed
```
```   346
```
```   347 lemma ep_pair_unique_p:
```
```   348   "\<lbrakk>ep_pair e p1; ep_pair e p2\<rbrakk> \<Longrightarrow> p1 = p2"
```
```   349 by (fast intro: below_antisym elim: ep_pair_unique_p_lemma)
```
```   350
```
```   351 subsection {* Composing ep-pairs *}
```
```   352
```
```   353 lemma ep_pair_ID_ID: "ep_pair ID ID"
```
```   354 by default simp_all
```
```   355
```
```   356 lemma ep_pair_comp:
```
```   357   assumes "ep_pair e1 p1" and "ep_pair e2 p2"
```
```   358   shows "ep_pair (e2 oo e1) (p1 oo p2)"
```
```   359 proof
```
```   360   interpret ep1: ep_pair e1 p1 by fact
```
```   361   interpret ep2: ep_pair e2 p2 by fact
```
```   362   fix x y
```
```   363   show "(p1 oo p2)\<cdot>((e2 oo e1)\<cdot>x) = x"
```
```   364     by simp
```
```   365   have "e1\<cdot>(p1\<cdot>(p2\<cdot>y)) \<sqsubseteq> p2\<cdot>y"
```
```   366     by (rule ep1.e_p_below)
```
```   367   hence "e2\<cdot>(e1\<cdot>(p1\<cdot>(p2\<cdot>y))) \<sqsubseteq> e2\<cdot>(p2\<cdot>y)"
```
```   368     by (rule monofun_cfun_arg)
```
```   369   also have "e2\<cdot>(p2\<cdot>y) \<sqsubseteq> y"
```
```   370     by (rule ep2.e_p_below)
```
```   371   finally show "(e2 oo e1)\<cdot>((p1 oo p2)\<cdot>y) \<sqsubseteq> y"
```
```   372     by simp
```
```   373 qed
```
```   374
```
```   375 locale pcpo_ep_pair = ep_pair +
```
```   376   constrains e :: "'a::pcpo \<rightarrow> 'b::pcpo"
```
```   377   constrains p :: "'b::pcpo \<rightarrow> 'a::pcpo"
```
```   378 begin
```
```   379
```
```   380 lemma e_strict [simp]: "e\<cdot>\<bottom> = \<bottom>"
```
```   381 proof -
```
```   382   have "\<bottom> \<sqsubseteq> p\<cdot>\<bottom>" by (rule minimal)
```
```   383   hence "e\<cdot>\<bottom> \<sqsubseteq> e\<cdot>(p\<cdot>\<bottom>)" by (rule monofun_cfun_arg)
```
```   384   also have "e\<cdot>(p\<cdot>\<bottom>) \<sqsubseteq> \<bottom>" by (rule e_p_below)
```
```   385   finally show "e\<cdot>\<bottom> = \<bottom>" by simp
```
```   386 qed
```
```   387
```
```   388 lemma e_defined_iff [simp]: "e\<cdot>x = \<bottom> \<longleftrightarrow> x = \<bottom>"
```
```   389 by (rule e_eq_iff [where y="\<bottom>", unfolded e_strict])
```
```   390
```
```   391 lemma e_defined: "x \<noteq> \<bottom> \<Longrightarrow> e\<cdot>x \<noteq> \<bottom>"
```
```   392 by simp
```
```   393
```
```   394 lemma p_strict [simp]: "p\<cdot>\<bottom> = \<bottom>"
```
```   395 by (rule e_inverse [where x="\<bottom>", unfolded e_strict])
```
```   396
```
```   397 lemmas stricts = e_strict p_strict
```
```   398
```
```   399 end
```
```   400
```
```   401 end
```