src/HOL/HOLCF/Map_Functions.thy
 author wenzelm Sun Nov 02 17:16:01 2014 +0100 (2014-11-02) changeset 58880 0baae4311a9f parent 42151 4da4fc77664b child 61169 4de9ff3ea29a permissions -rw-r--r--
```     1 (*  Title:      HOL/HOLCF/Map_Functions.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 section {* Map functions for various types *}
```
```     6
```
```     7 theory Map_Functions
```
```     8 imports Deflation
```
```     9 begin
```
```    10
```
```    11 subsection {* Map operator for continuous function space *}
```
```    12
```
```    13 default_sort cpo
```
```    14
```
```    15 definition
```
```    16   cfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'd)"
```
```    17 where
```
```    18   "cfun_map = (\<Lambda> a b f x. b\<cdot>(f\<cdot>(a\<cdot>x)))"
```
```    19
```
```    20 lemma cfun_map_beta [simp]: "cfun_map\<cdot>a\<cdot>b\<cdot>f\<cdot>x = b\<cdot>(f\<cdot>(a\<cdot>x))"
```
```    21 unfolding cfun_map_def by simp
```
```    22
```
```    23 lemma cfun_map_ID: "cfun_map\<cdot>ID\<cdot>ID = ID"
```
```    24 unfolding cfun_eq_iff by simp
```
```    25
```
```    26 lemma cfun_map_map:
```
```    27   "cfun_map\<cdot>f1\<cdot>g1\<cdot>(cfun_map\<cdot>f2\<cdot>g2\<cdot>p) =
```
```    28     cfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
```
```    29 by (rule cfun_eqI) simp
```
```    30
```
```    31 lemma ep_pair_cfun_map:
```
```    32   assumes "ep_pair e1 p1" and "ep_pair e2 p2"
```
```    33   shows "ep_pair (cfun_map\<cdot>p1\<cdot>e2) (cfun_map\<cdot>e1\<cdot>p2)"
```
```    34 proof
```
```    35   interpret e1p1: ep_pair e1 p1 by fact
```
```    36   interpret e2p2: ep_pair e2 p2 by fact
```
```    37   fix f show "cfun_map\<cdot>e1\<cdot>p2\<cdot>(cfun_map\<cdot>p1\<cdot>e2\<cdot>f) = f"
```
```    38     by (simp add: cfun_eq_iff)
```
```    39   fix g show "cfun_map\<cdot>p1\<cdot>e2\<cdot>(cfun_map\<cdot>e1\<cdot>p2\<cdot>g) \<sqsubseteq> g"
```
```    40     apply (rule cfun_belowI, simp)
```
```    41     apply (rule below_trans [OF e2p2.e_p_below])
```
```    42     apply (rule monofun_cfun_arg)
```
```    43     apply (rule e1p1.e_p_below)
```
```    44     done
```
```    45 qed
```
```    46
```
```    47 lemma deflation_cfun_map:
```
```    48   assumes "deflation d1" and "deflation d2"
```
```    49   shows "deflation (cfun_map\<cdot>d1\<cdot>d2)"
```
```    50 proof
```
```    51   interpret d1: deflation d1 by fact
```
```    52   interpret d2: deflation d2 by fact
```
```    53   fix f
```
```    54   show "cfun_map\<cdot>d1\<cdot>d2\<cdot>(cfun_map\<cdot>d1\<cdot>d2\<cdot>f) = cfun_map\<cdot>d1\<cdot>d2\<cdot>f"
```
```    55     by (simp add: cfun_eq_iff d1.idem d2.idem)
```
```    56   show "cfun_map\<cdot>d1\<cdot>d2\<cdot>f \<sqsubseteq> f"
```
```    57     apply (rule cfun_belowI, simp)
```
```    58     apply (rule below_trans [OF d2.below])
```
```    59     apply (rule monofun_cfun_arg)
```
```    60     apply (rule d1.below)
```
```    61     done
```
```    62 qed
```
```    63
```
```    64 lemma finite_range_cfun_map:
```
```    65   assumes a: "finite (range (\<lambda>x. a\<cdot>x))"
```
```    66   assumes b: "finite (range (\<lambda>y. b\<cdot>y))"
```
```    67   shows "finite (range (\<lambda>f. cfun_map\<cdot>a\<cdot>b\<cdot>f))"  (is "finite (range ?h)")
```
```    68 proof (rule finite_imageD)
```
```    69   let ?f = "\<lambda>g. range (\<lambda>x. (a\<cdot>x, g\<cdot>x))"
```
```    70   show "finite (?f ` range ?h)"
```
```    71   proof (rule finite_subset)
```
```    72     let ?B = "Pow (range (\<lambda>x. a\<cdot>x) \<times> range (\<lambda>y. b\<cdot>y))"
```
```    73     show "?f ` range ?h \<subseteq> ?B"
```
```    74       by clarsimp
```
```    75     show "finite ?B"
```
```    76       by (simp add: a b)
```
```    77   qed
```
```    78   show "inj_on ?f (range ?h)"
```
```    79   proof (rule inj_onI, rule cfun_eqI, clarsimp)
```
```    80     fix x f g
```
```    81     assume "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) = range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
```
```    82     hence "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) \<subseteq> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
```
```    83       by (rule equalityD1)
```
```    84     hence "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) \<in> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
```
```    85       by (simp add: subset_eq)
```
```    86     then obtain y where "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) = (a\<cdot>y, b\<cdot>(g\<cdot>(a\<cdot>y)))"
```
```    87       by (rule rangeE)
```
```    88     thus "b\<cdot>(f\<cdot>(a\<cdot>x)) = b\<cdot>(g\<cdot>(a\<cdot>x))"
```
```    89       by clarsimp
```
```    90   qed
```
```    91 qed
```
```    92
```
```    93 lemma finite_deflation_cfun_map:
```
```    94   assumes "finite_deflation d1" and "finite_deflation d2"
```
```    95   shows "finite_deflation (cfun_map\<cdot>d1\<cdot>d2)"
```
```    96 proof (rule finite_deflation_intro)
```
```    97   interpret d1: finite_deflation d1 by fact
```
```    98   interpret d2: finite_deflation d2 by fact
```
```    99   have "deflation d1" and "deflation d2" by fact+
```
```   100   thus "deflation (cfun_map\<cdot>d1\<cdot>d2)" by (rule deflation_cfun_map)
```
```   101   have "finite (range (\<lambda>f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f))"
```
```   102     using d1.finite_range d2.finite_range
```
```   103     by (rule finite_range_cfun_map)
```
```   104   thus "finite {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
```
```   105     by (rule finite_range_imp_finite_fixes)
```
```   106 qed
```
```   107
```
```   108 text {* Finite deflations are compact elements of the function space *}
```
```   109
```
```   110 lemma finite_deflation_imp_compact: "finite_deflation d \<Longrightarrow> compact d"
```
```   111 apply (frule finite_deflation_imp_deflation)
```
```   112 apply (subgoal_tac "compact (cfun_map\<cdot>d\<cdot>d\<cdot>d)")
```
```   113 apply (simp add: cfun_map_def deflation.idem eta_cfun)
```
```   114 apply (rule finite_deflation.compact)
```
```   115 apply (simp only: finite_deflation_cfun_map)
```
```   116 done
```
```   117
```
```   118 subsection {* Map operator for product type *}
```
```   119
```
```   120 definition
```
```   121   prod_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<times> 'c \<rightarrow> 'b \<times> 'd"
```
```   122 where
```
```   123   "prod_map = (\<Lambda> f g p. (f\<cdot>(fst p), g\<cdot>(snd p)))"
```
```   124
```
```   125 lemma prod_map_Pair [simp]: "prod_map\<cdot>f\<cdot>g\<cdot>(x, y) = (f\<cdot>x, g\<cdot>y)"
```
```   126 unfolding prod_map_def by simp
```
```   127
```
```   128 lemma prod_map_ID: "prod_map\<cdot>ID\<cdot>ID = ID"
```
```   129 unfolding cfun_eq_iff by auto
```
```   130
```
```   131 lemma prod_map_map:
```
```   132   "prod_map\<cdot>f1\<cdot>g1\<cdot>(prod_map\<cdot>f2\<cdot>g2\<cdot>p) =
```
```   133     prod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
```
```   134 by (induct p) simp
```
```   135
```
```   136 lemma ep_pair_prod_map:
```
```   137   assumes "ep_pair e1 p1" and "ep_pair e2 p2"
```
```   138   shows "ep_pair (prod_map\<cdot>e1\<cdot>e2) (prod_map\<cdot>p1\<cdot>p2)"
```
```   139 proof
```
```   140   interpret e1p1: ep_pair e1 p1 by fact
```
```   141   interpret e2p2: ep_pair e2 p2 by fact
```
```   142   fix x show "prod_map\<cdot>p1\<cdot>p2\<cdot>(prod_map\<cdot>e1\<cdot>e2\<cdot>x) = x"
```
```   143     by (induct x) simp
```
```   144   fix y show "prod_map\<cdot>e1\<cdot>e2\<cdot>(prod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y"
```
```   145     by (induct y) (simp add: e1p1.e_p_below e2p2.e_p_below)
```
```   146 qed
```
```   147
```
```   148 lemma deflation_prod_map:
```
```   149   assumes "deflation d1" and "deflation d2"
```
```   150   shows "deflation (prod_map\<cdot>d1\<cdot>d2)"
```
```   151 proof
```
```   152   interpret d1: deflation d1 by fact
```
```   153   interpret d2: deflation d2 by fact
```
```   154   fix x
```
```   155   show "prod_map\<cdot>d1\<cdot>d2\<cdot>(prod_map\<cdot>d1\<cdot>d2\<cdot>x) = prod_map\<cdot>d1\<cdot>d2\<cdot>x"
```
```   156     by (induct x) (simp add: d1.idem d2.idem)
```
```   157   show "prod_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x"
```
```   158     by (induct x) (simp add: d1.below d2.below)
```
```   159 qed
```
```   160
```
```   161 lemma finite_deflation_prod_map:
```
```   162   assumes "finite_deflation d1" and "finite_deflation d2"
```
```   163   shows "finite_deflation (prod_map\<cdot>d1\<cdot>d2)"
```
```   164 proof (rule finite_deflation_intro)
```
```   165   interpret d1: finite_deflation d1 by fact
```
```   166   interpret d2: finite_deflation d2 by fact
```
```   167   have "deflation d1" and "deflation d2" by fact+
```
```   168   thus "deflation (prod_map\<cdot>d1\<cdot>d2)" by (rule deflation_prod_map)
```
```   169   have "{p. prod_map\<cdot>d1\<cdot>d2\<cdot>p = p} \<subseteq> {x. d1\<cdot>x = x} \<times> {y. d2\<cdot>y = y}"
```
```   170     by clarsimp
```
```   171   thus "finite {p. prod_map\<cdot>d1\<cdot>d2\<cdot>p = p}"
```
```   172     by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
```
```   173 qed
```
```   174
```
```   175 subsection {* Map function for lifted cpo *}
```
```   176
```
```   177 definition
```
```   178   u_map :: "('a \<rightarrow> 'b) \<rightarrow> 'a u \<rightarrow> 'b u"
```
```   179 where
```
```   180   "u_map = (\<Lambda> f. fup\<cdot>(up oo f))"
```
```   181
```
```   182 lemma u_map_strict [simp]: "u_map\<cdot>f\<cdot>\<bottom> = \<bottom>"
```
```   183 unfolding u_map_def by simp
```
```   184
```
```   185 lemma u_map_up [simp]: "u_map\<cdot>f\<cdot>(up\<cdot>x) = up\<cdot>(f\<cdot>x)"
```
```   186 unfolding u_map_def by simp
```
```   187
```
```   188 lemma u_map_ID: "u_map\<cdot>ID = ID"
```
```   189 unfolding u_map_def by (simp add: cfun_eq_iff eta_cfun)
```
```   190
```
```   191 lemma u_map_map: "u_map\<cdot>f\<cdot>(u_map\<cdot>g\<cdot>p) = u_map\<cdot>(\<Lambda> x. f\<cdot>(g\<cdot>x))\<cdot>p"
```
```   192 by (induct p) simp_all
```
```   193
```
```   194 lemma u_map_oo: "u_map\<cdot>(f oo g) = u_map\<cdot>f oo u_map\<cdot>g"
```
```   195 by (simp add: cfcomp1 u_map_map eta_cfun)
```
```   196
```
```   197 lemma ep_pair_u_map: "ep_pair e p \<Longrightarrow> ep_pair (u_map\<cdot>e) (u_map\<cdot>p)"
```
```   198 apply default
```
```   199 apply (case_tac x, simp, simp add: ep_pair.e_inverse)
```
```   200 apply (case_tac y, simp, simp add: ep_pair.e_p_below)
```
```   201 done
```
```   202
```
```   203 lemma deflation_u_map: "deflation d \<Longrightarrow> deflation (u_map\<cdot>d)"
```
```   204 apply default
```
```   205 apply (case_tac x, simp, simp add: deflation.idem)
```
```   206 apply (case_tac x, simp, simp add: deflation.below)
```
```   207 done
```
```   208
```
```   209 lemma finite_deflation_u_map:
```
```   210   assumes "finite_deflation d" shows "finite_deflation (u_map\<cdot>d)"
```
```   211 proof (rule finite_deflation_intro)
```
```   212   interpret d: finite_deflation d by fact
```
```   213   have "deflation d" by fact
```
```   214   thus "deflation (u_map\<cdot>d)" by (rule deflation_u_map)
```
```   215   have "{x. u_map\<cdot>d\<cdot>x = x} \<subseteq> insert \<bottom> ((\<lambda>x. up\<cdot>x) ` {x. d\<cdot>x = x})"
```
```   216     by (rule subsetI, case_tac x, simp_all)
```
```   217   thus "finite {x. u_map\<cdot>d\<cdot>x = x}"
```
```   218     by (rule finite_subset, simp add: d.finite_fixes)
```
```   219 qed
```
```   220
```
```   221 subsection {* Map function for strict products *}
```
```   222
```
```   223 default_sort pcpo
```
```   224
```
```   225 definition
```
```   226   sprod_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<otimes> 'c \<rightarrow> 'b \<otimes> 'd"
```
```   227 where
```
```   228   "sprod_map = (\<Lambda> f g. ssplit\<cdot>(\<Lambda> x y. (:f\<cdot>x, g\<cdot>y:)))"
```
```   229
```
```   230 lemma sprod_map_strict [simp]: "sprod_map\<cdot>a\<cdot>b\<cdot>\<bottom> = \<bottom>"
```
```   231 unfolding sprod_map_def by simp
```
```   232
```
```   233 lemma sprod_map_spair [simp]:
```
```   234   "x \<noteq> \<bottom> \<Longrightarrow> y \<noteq> \<bottom> \<Longrightarrow> sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)"
```
```   235 by (simp add: sprod_map_def)
```
```   236
```
```   237 lemma sprod_map_spair':
```
```   238   "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> g\<cdot>\<bottom> = \<bottom> \<Longrightarrow> sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)"
```
```   239 by (cases "x = \<bottom> \<or> y = \<bottom>") auto
```
```   240
```
```   241 lemma sprod_map_ID: "sprod_map\<cdot>ID\<cdot>ID = ID"
```
```   242 unfolding sprod_map_def by (simp add: cfun_eq_iff eta_cfun)
```
```   243
```
```   244 lemma sprod_map_map:
```
```   245   "\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow>
```
```   246     sprod_map\<cdot>f1\<cdot>g1\<cdot>(sprod_map\<cdot>f2\<cdot>g2\<cdot>p) =
```
```   247      sprod_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
```
```   248 apply (induct p, simp)
```
```   249 apply (case_tac "f2\<cdot>x = \<bottom>", simp)
```
```   250 apply (case_tac "g2\<cdot>y = \<bottom>", simp)
```
```   251 apply simp
```
```   252 done
```
```   253
```
```   254 lemma ep_pair_sprod_map:
```
```   255   assumes "ep_pair e1 p1" and "ep_pair e2 p2"
```
```   256   shows "ep_pair (sprod_map\<cdot>e1\<cdot>e2) (sprod_map\<cdot>p1\<cdot>p2)"
```
```   257 proof
```
```   258   interpret e1p1: pcpo_ep_pair e1 p1 unfolding pcpo_ep_pair_def by fact
```
```   259   interpret e2p2: pcpo_ep_pair e2 p2 unfolding pcpo_ep_pair_def by fact
```
```   260   fix x show "sprod_map\<cdot>p1\<cdot>p2\<cdot>(sprod_map\<cdot>e1\<cdot>e2\<cdot>x) = x"
```
```   261     by (induct x) simp_all
```
```   262   fix y show "sprod_map\<cdot>e1\<cdot>e2\<cdot>(sprod_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y"
```
```   263     apply (induct y, simp)
```
```   264     apply (case_tac "p1\<cdot>x = \<bottom>", simp, case_tac "p2\<cdot>y = \<bottom>", simp)
```
```   265     apply (simp add: monofun_cfun e1p1.e_p_below e2p2.e_p_below)
```
```   266     done
```
```   267 qed
```
```   268
```
```   269 lemma deflation_sprod_map:
```
```   270   assumes "deflation d1" and "deflation d2"
```
```   271   shows "deflation (sprod_map\<cdot>d1\<cdot>d2)"
```
```   272 proof
```
```   273   interpret d1: deflation d1 by fact
```
```   274   interpret d2: deflation d2 by fact
```
```   275   fix x
```
```   276   show "sprod_map\<cdot>d1\<cdot>d2\<cdot>(sprod_map\<cdot>d1\<cdot>d2\<cdot>x) = sprod_map\<cdot>d1\<cdot>d2\<cdot>x"
```
```   277     apply (induct x, simp)
```
```   278     apply (case_tac "d1\<cdot>x = \<bottom>", simp, case_tac "d2\<cdot>y = \<bottom>", simp)
```
```   279     apply (simp add: d1.idem d2.idem)
```
```   280     done
```
```   281   show "sprod_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x"
```
```   282     apply (induct x, simp)
```
```   283     apply (simp add: monofun_cfun d1.below d2.below)
```
```   284     done
```
```   285 qed
```
```   286
```
```   287 lemma finite_deflation_sprod_map:
```
```   288   assumes "finite_deflation d1" and "finite_deflation d2"
```
```   289   shows "finite_deflation (sprod_map\<cdot>d1\<cdot>d2)"
```
```   290 proof (rule finite_deflation_intro)
```
```   291   interpret d1: finite_deflation d1 by fact
```
```   292   interpret d2: finite_deflation d2 by fact
```
```   293   have "deflation d1" and "deflation d2" by fact+
```
```   294   thus "deflation (sprod_map\<cdot>d1\<cdot>d2)" by (rule deflation_sprod_map)
```
```   295   have "{x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq> insert \<bottom>
```
```   296         ((\<lambda>(x, y). (:x, y:)) ` ({x. d1\<cdot>x = x} \<times> {y. d2\<cdot>y = y}))"
```
```   297     by (rule subsetI, case_tac x, auto simp add: spair_eq_iff)
```
```   298   thus "finite {x. sprod_map\<cdot>d1\<cdot>d2\<cdot>x = x}"
```
```   299     by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
```
```   300 qed
```
```   301
```
```   302 subsection {* Map function for strict sums *}
```
```   303
```
```   304 definition
```
```   305   ssum_map :: "('a \<rightarrow> 'b) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> 'a \<oplus> 'c \<rightarrow> 'b \<oplus> 'd"
```
```   306 where
```
```   307   "ssum_map = (\<Lambda> f g. sscase\<cdot>(sinl oo f)\<cdot>(sinr oo g))"
```
```   308
```
```   309 lemma ssum_map_strict [simp]: "ssum_map\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"
```
```   310 unfolding ssum_map_def by simp
```
```   311
```
```   312 lemma ssum_map_sinl [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)"
```
```   313 unfolding ssum_map_def by simp
```
```   314
```
```   315 lemma ssum_map_sinr [simp]: "x \<noteq> \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)"
```
```   316 unfolding ssum_map_def by simp
```
```   317
```
```   318 lemma ssum_map_sinl': "f\<cdot>\<bottom> = \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinl\<cdot>x) = sinl\<cdot>(f\<cdot>x)"
```
```   319 by (cases "x = \<bottom>") simp_all
```
```   320
```
```   321 lemma ssum_map_sinr': "g\<cdot>\<bottom> = \<bottom> \<Longrightarrow> ssum_map\<cdot>f\<cdot>g\<cdot>(sinr\<cdot>x) = sinr\<cdot>(g\<cdot>x)"
```
```   322 by (cases "x = \<bottom>") simp_all
```
```   323
```
```   324 lemma ssum_map_ID: "ssum_map\<cdot>ID\<cdot>ID = ID"
```
```   325 unfolding ssum_map_def by (simp add: cfun_eq_iff eta_cfun)
```
```   326
```
```   327 lemma ssum_map_map:
```
```   328   "\<lbrakk>f1\<cdot>\<bottom> = \<bottom>; g1\<cdot>\<bottom> = \<bottom>\<rbrakk> \<Longrightarrow>
```
```   329     ssum_map\<cdot>f1\<cdot>g1\<cdot>(ssum_map\<cdot>f2\<cdot>g2\<cdot>p) =
```
```   330      ssum_map\<cdot>(\<Lambda> x. f1\<cdot>(f2\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
```
```   331 apply (induct p, simp)
```
```   332 apply (case_tac "f2\<cdot>x = \<bottom>", simp, simp)
```
```   333 apply (case_tac "g2\<cdot>y = \<bottom>", simp, simp)
```
```   334 done
```
```   335
```
```   336 lemma ep_pair_ssum_map:
```
```   337   assumes "ep_pair e1 p1" and "ep_pair e2 p2"
```
```   338   shows "ep_pair (ssum_map\<cdot>e1\<cdot>e2) (ssum_map\<cdot>p1\<cdot>p2)"
```
```   339 proof
```
```   340   interpret e1p1: pcpo_ep_pair e1 p1 unfolding pcpo_ep_pair_def by fact
```
```   341   interpret e2p2: pcpo_ep_pair e2 p2 unfolding pcpo_ep_pair_def by fact
```
```   342   fix x show "ssum_map\<cdot>p1\<cdot>p2\<cdot>(ssum_map\<cdot>e1\<cdot>e2\<cdot>x) = x"
```
```   343     by (induct x) simp_all
```
```   344   fix y show "ssum_map\<cdot>e1\<cdot>e2\<cdot>(ssum_map\<cdot>p1\<cdot>p2\<cdot>y) \<sqsubseteq> y"
```
```   345     apply (induct y, simp)
```
```   346     apply (case_tac "p1\<cdot>x = \<bottom>", simp, simp add: e1p1.e_p_below)
```
```   347     apply (case_tac "p2\<cdot>y = \<bottom>", simp, simp add: e2p2.e_p_below)
```
```   348     done
```
```   349 qed
```
```   350
```
```   351 lemma deflation_ssum_map:
```
```   352   assumes "deflation d1" and "deflation d2"
```
```   353   shows "deflation (ssum_map\<cdot>d1\<cdot>d2)"
```
```   354 proof
```
```   355   interpret d1: deflation d1 by fact
```
```   356   interpret d2: deflation d2 by fact
```
```   357   fix x
```
```   358   show "ssum_map\<cdot>d1\<cdot>d2\<cdot>(ssum_map\<cdot>d1\<cdot>d2\<cdot>x) = ssum_map\<cdot>d1\<cdot>d2\<cdot>x"
```
```   359     apply (induct x, simp)
```
```   360     apply (case_tac "d1\<cdot>x = \<bottom>", simp, simp add: d1.idem)
```
```   361     apply (case_tac "d2\<cdot>y = \<bottom>", simp, simp add: d2.idem)
```
```   362     done
```
```   363   show "ssum_map\<cdot>d1\<cdot>d2\<cdot>x \<sqsubseteq> x"
```
```   364     apply (induct x, simp)
```
```   365     apply (case_tac "d1\<cdot>x = \<bottom>", simp, simp add: d1.below)
```
```   366     apply (case_tac "d2\<cdot>y = \<bottom>", simp, simp add: d2.below)
```
```   367     done
```
```   368 qed
```
```   369
```
```   370 lemma finite_deflation_ssum_map:
```
```   371   assumes "finite_deflation d1" and "finite_deflation d2"
```
```   372   shows "finite_deflation (ssum_map\<cdot>d1\<cdot>d2)"
```
```   373 proof (rule finite_deflation_intro)
```
```   374   interpret d1: finite_deflation d1 by fact
```
```   375   interpret d2: finite_deflation d2 by fact
```
```   376   have "deflation d1" and "deflation d2" by fact+
```
```   377   thus "deflation (ssum_map\<cdot>d1\<cdot>d2)" by (rule deflation_ssum_map)
```
```   378   have "{x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x} \<subseteq>
```
```   379         (\<lambda>x. sinl\<cdot>x) ` {x. d1\<cdot>x = x} \<union>
```
```   380         (\<lambda>x. sinr\<cdot>x) ` {x. d2\<cdot>x = x} \<union> {\<bottom>}"
```
```   381     by (rule subsetI, case_tac x, simp_all)
```
```   382   thus "finite {x. ssum_map\<cdot>d1\<cdot>d2\<cdot>x = x}"
```
```   383     by (rule finite_subset, simp add: d1.finite_fixes d2.finite_fixes)
```
```   384 qed
```
```   385
```
```   386 subsection {* Map operator for strict function space *}
```
```   387
```
```   388 definition
```
```   389   sfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow>! 'c) \<rightarrow> ('b \<rightarrow>! 'd)"
```
```   390 where
```
```   391   "sfun_map = (\<Lambda> a b. sfun_abs oo cfun_map\<cdot>a\<cdot>b oo sfun_rep)"
```
```   392
```
```   393 lemma sfun_map_ID: "sfun_map\<cdot>ID\<cdot>ID = ID"
```
```   394   unfolding sfun_map_def
```
```   395   by (simp add: cfun_map_ID cfun_eq_iff)
```
```   396
```
```   397 lemma sfun_map_map:
```
```   398   assumes "f2\<cdot>\<bottom> = \<bottom>" and "g2\<cdot>\<bottom> = \<bottom>" shows
```
```   399   "sfun_map\<cdot>f1\<cdot>g1\<cdot>(sfun_map\<cdot>f2\<cdot>g2\<cdot>p) =
```
```   400     sfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
```
```   401 unfolding sfun_map_def
```
```   402 by (simp add: cfun_eq_iff strictify_cancel assms cfun_map_map)
```
```   403
```
```   404 lemma ep_pair_sfun_map:
```
```   405   assumes 1: "ep_pair e1 p1"
```
```   406   assumes 2: "ep_pair e2 p2"
```
```   407   shows "ep_pair (sfun_map\<cdot>p1\<cdot>e2) (sfun_map\<cdot>e1\<cdot>p2)"
```
```   408 proof
```
```   409   interpret e1p1: pcpo_ep_pair e1 p1
```
```   410     unfolding pcpo_ep_pair_def by fact
```
```   411   interpret e2p2: pcpo_ep_pair e2 p2
```
```   412     unfolding pcpo_ep_pair_def by fact
```
```   413   fix f show "sfun_map\<cdot>e1\<cdot>p2\<cdot>(sfun_map\<cdot>p1\<cdot>e2\<cdot>f) = f"
```
```   414     unfolding sfun_map_def
```
```   415     apply (simp add: sfun_eq_iff strictify_cancel)
```
```   416     apply (rule ep_pair.e_inverse)
```
```   417     apply (rule ep_pair_cfun_map [OF 1 2])
```
```   418     done
```
```   419   fix g show "sfun_map\<cdot>p1\<cdot>e2\<cdot>(sfun_map\<cdot>e1\<cdot>p2\<cdot>g) \<sqsubseteq> g"
```
```   420     unfolding sfun_map_def
```
```   421     apply (simp add: sfun_below_iff strictify_cancel)
```
```   422     apply (rule ep_pair.e_p_below)
```
```   423     apply (rule ep_pair_cfun_map [OF 1 2])
```
```   424     done
```
```   425 qed
```
```   426
```
```   427 lemma deflation_sfun_map:
```
```   428   assumes 1: "deflation d1"
```
```   429   assumes 2: "deflation d2"
```
```   430   shows "deflation (sfun_map\<cdot>d1\<cdot>d2)"
```
```   431 apply (simp add: sfun_map_def)
```
```   432 apply (rule deflation.intro)
```
```   433 apply simp
```
```   434 apply (subst strictify_cancel)
```
```   435 apply (simp add: cfun_map_def deflation_strict 1 2)
```
```   436 apply (simp add: cfun_map_def deflation.idem 1 2)
```
```   437 apply (simp add: sfun_below_iff)
```
```   438 apply (subst strictify_cancel)
```
```   439 apply (simp add: cfun_map_def deflation_strict 1 2)
```
```   440 apply (rule deflation.below)
```
```   441 apply (rule deflation_cfun_map [OF 1 2])
```
```   442 done
```
```   443
```
```   444 lemma finite_deflation_sfun_map:
```
```   445   assumes 1: "finite_deflation d1"
```
```   446   assumes 2: "finite_deflation d2"
```
```   447   shows "finite_deflation (sfun_map\<cdot>d1\<cdot>d2)"
```
```   448 proof (intro finite_deflation_intro)
```
```   449   interpret d1: finite_deflation d1 by fact
```
```   450   interpret d2: finite_deflation d2 by fact
```
```   451   have "deflation d1" and "deflation d2" by fact+
```
```   452   thus "deflation (sfun_map\<cdot>d1\<cdot>d2)" by (rule deflation_sfun_map)
```
```   453   from 1 2 have "finite_deflation (cfun_map\<cdot>d1\<cdot>d2)"
```
```   454     by (rule finite_deflation_cfun_map)
```
```   455   then have "finite {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
```
```   456     by (rule finite_deflation.finite_fixes)
```
```   457   moreover have "inj (\<lambda>f. sfun_rep\<cdot>f)"
```
```   458     by (rule inj_onI, simp add: sfun_eq_iff)
```
```   459   ultimately have "finite ((\<lambda>f. sfun_rep\<cdot>f) -` {f. cfun_map\<cdot>d1\<cdot>d2\<cdot>f = f})"
```
```   460     by (rule finite_vimageI)
```
```   461   then show "finite {f. sfun_map\<cdot>d1\<cdot>d2\<cdot>f = f}"
```
```   462     unfolding sfun_map_def sfun_eq_iff
```
```   463     by (simp add: strictify_cancel
```
```   464          deflation_strict `deflation d1` `deflation d2`)
```
```   465 qed
```
```   466
```
```   467 end
```