author | huffman |
Sun, 19 Dec 2010 06:34:41 -0800 | |
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parent 41286 | 3d7685a4a5ff |
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permissions | -rw-r--r-- |
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(* Title: HOLCF/Representable.thy |
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Author: Brian Huffman |
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*) |
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header {* Representable domains *} |
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theory Representable |
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imports Algebraic Map_Functions Countable |
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begin |
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subsection {* Class of representable domains *} |
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text {* |
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We define a ``domain'' as a pcpo that is isomorphic to some |
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algebraic deflation over the universal domain; this is equivalent |
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to being omega-bifinite. |
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A predomain is a cpo that, when lifted, becomes a domain. |
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*} |
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class predomain = cpo + |
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fixes liftdefl :: "('a::cpo) itself \<Rightarrow> udom defl" |
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fixes liftemb :: "'a\<^sub>\<bottom> \<rightarrow> udom" |
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fixes liftprj :: "udom \<rightarrow> 'a\<^sub>\<bottom>" |
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assumes predomain_ep: "ep_pair liftemb liftprj" |
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assumes cast_liftdefl: "cast\<cdot>(liftdefl TYPE('a::cpo)) = liftemb oo liftprj" |
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syntax "_LIFTDEFL" :: "type \<Rightarrow> logic" ("(1LIFTDEFL/(1'(_')))") |
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translations "LIFTDEFL('t)" \<rightleftharpoons> "CONST liftdefl TYPE('t)" |
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class "domain" = predomain + pcpo + |
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fixes emb :: "'a::cpo \<rightarrow> udom" |
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fixes prj :: "udom \<rightarrow> 'a::cpo" |
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fixes defl :: "'a itself \<Rightarrow> udom defl" |
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assumes ep_pair_emb_prj: "ep_pair emb prj" |
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assumes cast_DEFL: "cast\<cdot>(defl TYPE('a)) = emb oo prj" |
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syntax "_DEFL" :: "type \<Rightarrow> logic" ("(1DEFL/(1'(_')))") |
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translations "DEFL('t)" \<rightleftharpoons> "CONST defl TYPE('t)" |
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interpretation "domain": pcpo_ep_pair emb prj |
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unfolding pcpo_ep_pair_def |
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by (rule ep_pair_emb_prj) |
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lemmas emb_inverse = domain.e_inverse |
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lemmas emb_prj_below = domain.e_p_below |
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lemmas emb_eq_iff = domain.e_eq_iff |
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lemmas emb_strict = domain.e_strict |
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lemmas prj_strict = domain.p_strict |
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subsection {* Domains are bifinite *} |
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lemma approx_chain_ep_cast: |
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assumes ep: "ep_pair (e::'a::pcpo \<rightarrow> udom) (p::udom \<rightarrow> 'a)" |
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assumes cast_t: "cast\<cdot>t = e oo p" |
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shows "\<exists>(a::nat \<Rightarrow> 'a::pcpo \<rightarrow> 'a). approx_chain a" |
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proof - |
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interpret ep_pair e p by fact |
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obtain Y where Y: "\<forall>i. Y i \<sqsubseteq> Y (Suc i)" |
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and t: "t = (\<Squnion>i. defl_principal (Y i))" |
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by (rule defl.obtain_principal_chain) |
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def approx \<equiv> "\<lambda>i. (p oo cast\<cdot>(defl_principal (Y i)) oo e) :: 'a \<rightarrow> 'a" |
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have "approx_chain approx" |
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proof (rule approx_chain.intro) |
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show "chain (\<lambda>i. approx i)" |
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unfolding approx_def by (simp add: Y) |
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show "(\<Squnion>i. approx i) = ID" |
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unfolding approx_def |
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by (simp add: lub_distribs Y t [symmetric] cast_t cfun_eq_iff) |
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show "\<And>i. finite_deflation (approx i)" |
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unfolding approx_def |
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apply (rule finite_deflation_p_d_e) |
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apply (rule finite_deflation_cast) |
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apply (rule defl.compact_principal) |
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apply (rule below_trans [OF monofun_cfun_fun]) |
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apply (rule is_ub_thelub, simp add: Y) |
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apply (simp add: lub_distribs Y t [symmetric] cast_t) |
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done |
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qed |
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thus "\<exists>(a::nat \<Rightarrow> 'a \<rightarrow> 'a). approx_chain a" by - (rule exI) |
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qed |
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instance "domain" \<subseteq> bifinite |
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by default (rule approx_chain_ep_cast [OF ep_pair_emb_prj cast_DEFL]) |
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instance predomain \<subseteq> profinite |
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by default (rule approx_chain_ep_cast [OF predomain_ep cast_liftdefl]) |
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subsection {* Chains of approx functions *} |
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definition u_approx :: "nat \<Rightarrow> udom\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>" |
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where "u_approx = (\<lambda>i. u_map\<cdot>(udom_approx i))" |
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definition sfun_approx :: "nat \<Rightarrow> (udom \<rightarrow>! udom) \<rightarrow> (udom \<rightarrow>! udom)" |
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where "sfun_approx = (\<lambda>i. sfun_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))" |
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definition prod_approx :: "nat \<Rightarrow> udom \<times> udom \<rightarrow> udom \<times> udom" |
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where "prod_approx = (\<lambda>i. cprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))" |
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definition sprod_approx :: "nat \<Rightarrow> udom \<otimes> udom \<rightarrow> udom \<otimes> udom" |
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where "sprod_approx = (\<lambda>i. sprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))" |
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definition ssum_approx :: "nat \<Rightarrow> udom \<oplus> udom \<rightarrow> udom \<oplus> udom" |
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where "ssum_approx = (\<lambda>i. ssum_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))" |
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lemma approx_chain_lemma1: |
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assumes "m\<cdot>ID = ID" |
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assumes "\<And>d. finite_deflation d \<Longrightarrow> finite_deflation (m\<cdot>d)" |
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shows "approx_chain (\<lambda>i. m\<cdot>(udom_approx i))" |
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by (rule approx_chain.intro) |
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(simp_all add: lub_distribs finite_deflation_udom_approx assms) |
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lemma approx_chain_lemma2: |
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assumes "m\<cdot>ID\<cdot>ID = ID" |
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assumes "\<And>a b. \<lbrakk>finite_deflation a; finite_deflation b\<rbrakk> |
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\<Longrightarrow> finite_deflation (m\<cdot>a\<cdot>b)" |
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shows "approx_chain (\<lambda>i. m\<cdot>(udom_approx i)\<cdot>(udom_approx i))" |
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by (rule approx_chain.intro) |
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(simp_all add: lub_distribs finite_deflation_udom_approx assms) |
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|
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huffman
parents:
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changeset
|
121 |
lemma u_approx: "approx_chain u_approx" |
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huffman
parents:
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changeset
|
122 |
using u_map_ID finite_deflation_u_map |
768f7e264e2b
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huffman
parents:
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diff
changeset
|
123 |
unfolding u_approx_def by (rule approx_chain_lemma1) |
768f7e264e2b
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huffman
parents:
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diff
changeset
|
124 |
|
40592
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huffman
parents:
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changeset
|
125 |
lemma sfun_approx: "approx_chain sfun_approx" |
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huffman
parents:
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changeset
|
126 |
using sfun_map_ID finite_deflation_sfun_map |
f432973ce0f6
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huffman
parents:
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diff
changeset
|
127 |
unfolding sfun_approx_def by (rule approx_chain_lemma2) |
40484
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reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
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diff
changeset
|
128 |
|
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
129 |
lemma prod_approx: "approx_chain prod_approx" |
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huffman
parents:
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diff
changeset
|
130 |
using cprod_map_ID finite_deflation_cprod_map |
768f7e264e2b
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huffman
parents:
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diff
changeset
|
131 |
unfolding prod_approx_def by (rule approx_chain_lemma2) |
768f7e264e2b
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huffman
parents:
40086
diff
changeset
|
132 |
|
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
133 |
lemma sprod_approx: "approx_chain sprod_approx" |
768f7e264e2b
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huffman
parents:
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diff
changeset
|
134 |
using sprod_map_ID finite_deflation_sprod_map |
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
135 |
unfolding sprod_approx_def by (rule approx_chain_lemma2) |
768f7e264e2b
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huffman
parents:
40086
diff
changeset
|
136 |
|
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
137 |
lemma ssum_approx: "approx_chain ssum_approx" |
768f7e264e2b
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huffman
parents:
40086
diff
changeset
|
138 |
using ssum_map_ID finite_deflation_ssum_map |
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
139 |
unfolding ssum_approx_def by (rule approx_chain_lemma2) |
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
140 |
|
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
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diff
changeset
|
141 |
subsection {* Type combinators *} |
310f98585107
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huffman
parents:
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changeset
|
142 |
|
41286
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huffman
parents:
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diff
changeset
|
143 |
default_sort bifinite |
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
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diff
changeset
|
144 |
|
39985
310f98585107
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huffman
parents:
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diff
changeset
|
145 |
definition |
39989
ad60d7311f43
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huffman
parents:
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diff
changeset
|
146 |
defl_fun1 :: |
41287
029a6fc1bfb8
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huffman
parents:
41286
diff
changeset
|
147 |
"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) \<Rightarrow> (udom defl \<rightarrow> udom defl)" |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
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diff
changeset
|
148 |
where |
39989
ad60d7311f43
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huffman
parents:
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diff
changeset
|
149 |
"defl_fun1 approx f = |
ad60d7311f43
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huffman
parents:
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diff
changeset
|
150 |
defl.basis_fun (\<lambda>a. |
ad60d7311f43
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huffman
parents:
39987
diff
changeset
|
151 |
defl_principal (Abs_fin_defl |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
152 |
(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)))" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
153 |
|
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
154 |
definition |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
155 |
defl_fun2 :: |
39985
310f98585107
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huffman
parents:
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diff
changeset
|
156 |
"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) |
41287
029a6fc1bfb8
type 'defl' takes a type parameter again (cf. b525988432e9)
huffman
parents:
41286
diff
changeset
|
157 |
\<Rightarrow> (udom defl \<rightarrow> udom defl \<rightarrow> udom defl)" |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
158 |
where |
39989
ad60d7311f43
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huffman
parents:
39987
diff
changeset
|
159 |
"defl_fun2 approx f = |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
160 |
defl.basis_fun (\<lambda>a. |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
161 |
defl.basis_fun (\<lambda>b. |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
162 |
defl_principal (Abs_fin_defl |
39985
310f98585107
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huffman
parents:
39974
diff
changeset
|
163 |
(udom_emb approx oo |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
164 |
f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx))))" |
33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
31113
diff
changeset
|
165 |
|
39989
ad60d7311f43
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huffman
parents:
39987
diff
changeset
|
166 |
lemma cast_defl_fun1: |
39985
310f98585107
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huffman
parents:
39974
diff
changeset
|
167 |
assumes approx: "approx_chain approx" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
168 |
assumes f: "\<And>a. finite_deflation a \<Longrightarrow> finite_deflation (f\<cdot>a)" |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
169 |
shows "cast\<cdot>(defl_fun1 approx f\<cdot>A) = udom_emb approx oo f\<cdot>(cast\<cdot>A) oo udom_prj approx" |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
170 |
proof - |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
171 |
have 1: "\<And>a. finite_deflation |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
172 |
(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
173 |
apply (rule ep_pair.finite_deflation_e_d_p) |
41286
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41285
diff
changeset
|
174 |
apply (rule ep_pair_udom [OF approx]) |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
175 |
apply (rule f, rule finite_deflation_Rep_fin_defl) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
176 |
done |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
177 |
show ?thesis |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
178 |
by (induct A rule: defl.principal_induct, simp) |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
179 |
(simp only: defl_fun1_def |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
180 |
defl.basis_fun_principal |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
181 |
defl.basis_fun_mono |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
182 |
defl.principal_mono |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
183 |
Abs_fin_defl_mono [OF 1 1] |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
184 |
monofun_cfun below_refl |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
185 |
Rep_fin_defl_mono |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
186 |
cast_defl_principal |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
187 |
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1]) |
33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
31113
diff
changeset
|
188 |
qed |
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
31113
diff
changeset
|
189 |
|
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
190 |
lemma cast_defl_fun2: |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
191 |
assumes approx: "approx_chain approx" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
192 |
assumes f: "\<And>a b. finite_deflation a \<Longrightarrow> finite_deflation b \<Longrightarrow> |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
193 |
finite_deflation (f\<cdot>a\<cdot>b)" |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
194 |
shows "cast\<cdot>(defl_fun2 approx f\<cdot>A\<cdot>B) = |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
195 |
udom_emb approx oo f\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj approx" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
196 |
proof - |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
197 |
have 1: "\<And>a b. finite_deflation (udom_emb approx oo |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
198 |
f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx)" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
199 |
apply (rule ep_pair.finite_deflation_e_d_p) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
200 |
apply (rule ep_pair_udom [OF approx]) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
201 |
apply (rule f, (rule finite_deflation_Rep_fin_defl)+) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
202 |
done |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
203 |
show ?thesis |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
204 |
by (induct A B rule: defl.principal_induct2, simp, simp) |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
205 |
(simp only: defl_fun2_def |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
206 |
defl.basis_fun_principal |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
207 |
defl.basis_fun_mono |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
208 |
defl.principal_mono |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
209 |
Abs_fin_defl_mono [OF 1 1] |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
210 |
monofun_cfun below_refl |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
211 |
Rep_fin_defl_mono |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
212 |
cast_defl_principal |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
213 |
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1]) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
214 |
qed |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
215 |
|
41287
029a6fc1bfb8
type 'defl' takes a type parameter again (cf. b525988432e9)
huffman
parents:
41286
diff
changeset
|
216 |
definition u_defl :: "udom defl \<rightarrow> udom defl" |
40484
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
217 |
where "u_defl = defl_fun1 u_approx u_map" |
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
218 |
|
41287
029a6fc1bfb8
type 'defl' takes a type parameter again (cf. b525988432e9)
huffman
parents:
41286
diff
changeset
|
219 |
definition sfun_defl :: "udom defl \<rightarrow> udom defl \<rightarrow> udom defl" |
40592
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
220 |
where "sfun_defl = defl_fun2 sfun_approx sfun_map" |
40484
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
221 |
|
41287
029a6fc1bfb8
type 'defl' takes a type parameter again (cf. b525988432e9)
huffman
parents:
41286
diff
changeset
|
222 |
definition prod_defl :: "udom defl \<rightarrow> udom defl \<rightarrow> udom defl" |
40484
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
223 |
where "prod_defl = defl_fun2 prod_approx cprod_map" |
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
224 |
|
41287
029a6fc1bfb8
type 'defl' takes a type parameter again (cf. b525988432e9)
huffman
parents:
41286
diff
changeset
|
225 |
definition sprod_defl :: "udom defl \<rightarrow> udom defl \<rightarrow> udom defl" |
40484
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
226 |
where "sprod_defl = defl_fun2 sprod_approx sprod_map" |
768f7e264e2b
reorganize Bifinite.thy; simplify some proofs related to bifinite class instances
huffman
parents:
40086
diff
changeset
|
227 |
|
41287
029a6fc1bfb8
type 'defl' takes a type parameter again (cf. b525988432e9)
huffman
parents:
41286
diff
changeset
|
228 |
definition ssum_defl :: "udom defl \<rightarrow> udom defl \<rightarrow> udom defl" |
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where "ssum_defl = defl_fun2 ssum_approx ssum_map" |
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|
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lemma cast_u_defl: |
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"cast\<cdot>(u_defl\<cdot>A) = |
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udom_emb u_approx oo u_map\<cdot>(cast\<cdot>A) oo udom_prj u_approx" |
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using u_approx finite_deflation_u_map |
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unfolding u_defl_def by (rule cast_defl_fun1) |
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|
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lemma cast_sfun_defl: |
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"cast\<cdot>(sfun_defl\<cdot>A\<cdot>B) = |
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udom_emb sfun_approx oo sfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj sfun_approx" |
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using sfun_approx finite_deflation_sfun_map |
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unfolding sfun_defl_def by (rule cast_defl_fun2) |
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242 |
|
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lemma cast_prod_defl: |
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"cast\<cdot>(prod_defl\<cdot>A\<cdot>B) = udom_emb prod_approx oo |
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cprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj prod_approx" |
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using prod_approx finite_deflation_cprod_map |
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unfolding prod_defl_def by (rule cast_defl_fun2) |
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|
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lemma cast_sprod_defl: |
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"cast\<cdot>(sprod_defl\<cdot>A\<cdot>B) = |
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udom_emb sprod_approx oo |
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sprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo |
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udom_prj sprod_approx" |
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using sprod_approx finite_deflation_sprod_map |
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unfolding sprod_defl_def by (rule cast_defl_fun2) |
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|
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lemma cast_ssum_defl: |
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"cast\<cdot>(ssum_defl\<cdot>A\<cdot>B) = |
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udom_emb ssum_approx oo ssum_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj ssum_approx" |
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260 |
using ssum_approx finite_deflation_ssum_map |
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261 |
unfolding ssum_defl_def by (rule cast_defl_fun2) |
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|
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subsection {* Lemma for proving domain instances *} |
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264 |
|
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text {* |
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A class of domains where @{const liftemb}, @{const liftprj}, |
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and @{const liftdefl} are all defined in the standard way. |
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268 |
*} |
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|
40497 | 270 |
class liftdomain = "domain" + |
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assumes liftemb_eq: "liftemb = udom_emb u_approx oo u_map\<cdot>emb" |
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assumes liftprj_eq: "liftprj = u_map\<cdot>prj oo udom_prj u_approx" |
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assumes liftdefl_eq: "liftdefl TYPE('a::cpo) = u_defl\<cdot>DEFL('a)" |
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274 |
|
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text {* Temporarily relax type constraints. *} |
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276 |
|
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277 |
setup {* |
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278 |
fold Sign.add_const_constraint |
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[ (@{const_name defl}, SOME @{typ "'a::pcpo itself \<Rightarrow> udom defl"}) |
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, (@{const_name emb}, SOME @{typ "'a::pcpo \<rightarrow> udom"}) |
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, (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::pcpo"}) |
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, (@{const_name liftdefl}, SOME @{typ "'a::pcpo itself \<Rightarrow> udom defl"}) |
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, (@{const_name liftemb}, SOME @{typ "'a::pcpo u \<rightarrow> udom"}) |
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, (@{const_name liftprj}, SOME @{typ "udom \<rightarrow> 'a::pcpo u"}) ] |
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285 |
*} |
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286 |
|
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287 |
default_sort pcpo |
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288 |
|
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289 |
lemma liftdomain_class_intro: |
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assumes liftemb: "(liftemb :: 'a u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
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291 |
assumes liftprj: "(liftprj :: udom \<rightarrow> 'a u) = u_map\<cdot>prj oo udom_prj u_approx" |
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292 |
assumes liftdefl: "liftdefl TYPE('a) = u_defl\<cdot>DEFL('a)" |
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293 |
assumes ep_pair: "ep_pair emb (prj :: udom \<rightarrow> 'a)" |
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assumes cast_defl: "cast\<cdot>DEFL('a) = emb oo (prj :: udom \<rightarrow> 'a)" |
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shows "OFCLASS('a, liftdomain_class)" |
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296 |
proof |
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297 |
show "ep_pair liftemb (liftprj :: udom \<rightarrow> 'a u)" |
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298 |
unfolding liftemb liftprj |
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299 |
by (intro ep_pair_comp ep_pair_u_map ep_pair ep_pair_udom u_approx) |
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300 |
show "cast\<cdot>LIFTDEFL('a) = liftemb oo (liftprj :: udom \<rightarrow> 'a u)" |
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301 |
unfolding liftemb liftprj liftdefl |
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302 |
by (simp add: cfcomp1 cast_u_defl cast_defl u_map_map) |
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303 |
next |
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304 |
qed fact+ |
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305 |
|
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306 |
text {* Restore original type constraints. *} |
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307 |
|
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308 |
setup {* |
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309 |
fold Sign.add_const_constraint |
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310 |
[ (@{const_name defl}, SOME @{typ "'a::domain itself \<Rightarrow> udom defl"}) |
40497 | 311 |
, (@{const_name emb}, SOME @{typ "'a::domain \<rightarrow> udom"}) |
312 |
, (@{const_name prj}, SOME @{typ "udom \<rightarrow> 'a::domain"}) |
|
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313 |
, (@{const_name liftdefl}, SOME @{typ "'a::predomain itself \<Rightarrow> udom defl"}) |
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314 |
, (@{const_name liftemb}, SOME @{typ "'a::predomain u \<rightarrow> udom"}) |
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, (@{const_name liftprj}, SOME @{typ "udom \<rightarrow> 'a::predomain u"}) ] |
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316 |
*} |
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317 |
|
40506 | 318 |
subsection {* Class instance proofs *} |
319 |
||
320 |
subsubsection {* Universal domain *} |
|
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321 |
|
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instantiation udom :: liftdomain |
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323 |
begin |
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324 |
|
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325 |
definition [simp]: |
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326 |
"emb = (ID :: udom \<rightarrow> udom)" |
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327 |
|
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328 |
definition [simp]: |
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329 |
"prj = (ID :: udom \<rightarrow> udom)" |
25903 | 330 |
|
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331 |
definition |
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332 |
"defl (t::udom itself) = (\<Squnion>i. defl_principal (Abs_fin_defl (udom_approx i)))" |
33808 | 333 |
|
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334 |
definition |
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335 |
"(liftemb :: udom u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
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336 |
|
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337 |
definition |
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338 |
"(liftprj :: udom \<rightarrow> udom u) = u_map\<cdot>prj oo udom_prj u_approx" |
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339 |
|
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340 |
definition |
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341 |
"liftdefl (t::udom itself) = u_defl\<cdot>DEFL(udom)" |
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342 |
|
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343 |
instance |
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344 |
using liftemb_udom_def liftprj_udom_def liftdefl_udom_def |
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add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents:
40493
diff
changeset
|
345 |
proof (rule liftdomain_class_intro) |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
346 |
show "ep_pair emb (prj :: udom \<rightarrow> udom)" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
347 |
by (simp add: ep_pair.intro) |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
348 |
show "cast\<cdot>DEFL(udom) = emb oo (prj :: udom \<rightarrow> udom)" |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
349 |
unfolding defl_udom_def |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
350 |
apply (subst contlub_cfun_arg) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
351 |
apply (rule chainI) |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
352 |
apply (rule defl.principal_mono) |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
353 |
apply (simp add: below_fin_defl_def) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
354 |
apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
355 |
apply (rule chainE) |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
356 |
apply (rule chain_udom_approx) |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
357 |
apply (subst cast_defl_principal) |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
358 |
apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx) |
33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
31113
diff
changeset
|
359 |
done |
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
31113
diff
changeset
|
360 |
qed |
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
31113
diff
changeset
|
361 |
|
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
362 |
end |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
363 |
|
40506 | 364 |
subsubsection {* Lifted cpo *} |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
365 |
|
40494
db8a09daba7b
add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents:
40493
diff
changeset
|
366 |
instantiation u :: (predomain) liftdomain |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
367 |
begin |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
368 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
369 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
370 |
"emb = liftemb" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
371 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
372 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
373 |
"prj = liftprj" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
374 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
375 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
376 |
"defl (t::'a u itself) = LIFTDEFL('a)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
377 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
378 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
379 |
"(liftemb :: 'a u u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
380 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
381 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
382 |
"(liftprj :: udom \<rightarrow> 'a u u) = u_map\<cdot>prj oo udom_prj u_approx" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
383 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
384 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
385 |
"liftdefl (t::'a u itself) = u_defl\<cdot>DEFL('a u)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
386 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
387 |
instance |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
388 |
using liftemb_u_def liftprj_u_def liftdefl_u_def |
40494
db8a09daba7b
add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents:
40493
diff
changeset
|
389 |
proof (rule liftdomain_class_intro) |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
390 |
show "ep_pair emb (prj :: udom \<rightarrow> 'a u)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
391 |
unfolding emb_u_def prj_u_def |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
392 |
by (rule predomain_ep) |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
393 |
show "cast\<cdot>DEFL('a u) = emb oo (prj :: udom \<rightarrow> 'a u)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
394 |
unfolding emb_u_def prj_u_def defl_u_def |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
395 |
by (rule cast_liftdefl) |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
396 |
qed |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
397 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
398 |
end |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
399 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
400 |
lemma DEFL_u: "DEFL('a::predomain u) = LIFTDEFL('a)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
401 |
by (rule defl_u_def) |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
402 |
|
40592
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
403 |
subsubsection {* Strict function space *} |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
404 |
|
40592
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
405 |
instantiation sfun :: ("domain", "domain") liftdomain |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
406 |
begin |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
407 |
|
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
408 |
definition |
40592
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
409 |
"emb = udom_emb sfun_approx oo sfun_map\<cdot>prj\<cdot>emb" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
410 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
411 |
definition |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
412 |
"prj = sfun_map\<cdot>emb\<cdot>prj oo udom_prj sfun_approx" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
413 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
414 |
definition |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
415 |
"defl (t::('a \<rightarrow>! 'b) itself) = sfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
416 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
417 |
definition |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
418 |
"(liftemb :: ('a \<rightarrow>! 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
419 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
420 |
definition |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
421 |
"(liftprj :: udom \<rightarrow> ('a \<rightarrow>! 'b) u) = u_map\<cdot>prj oo udom_prj u_approx" |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
422 |
|
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
423 |
definition |
40592
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
424 |
"liftdefl (t::('a \<rightarrow>! 'b) itself) = u_defl\<cdot>DEFL('a \<rightarrow>! 'b)" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
425 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
426 |
instance |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
427 |
using liftemb_sfun_def liftprj_sfun_def liftdefl_sfun_def |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
428 |
proof (rule liftdomain_class_intro) |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
429 |
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow>! 'b)" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
430 |
unfolding emb_sfun_def prj_sfun_def |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
431 |
using ep_pair_udom [OF sfun_approx] |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
432 |
by (intro ep_pair_comp ep_pair_sfun_map ep_pair_emb_prj) |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
433 |
show "cast\<cdot>DEFL('a \<rightarrow>! 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow>! 'b)" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
434 |
unfolding emb_sfun_def prj_sfun_def defl_sfun_def cast_sfun_defl |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
435 |
by (simp add: cast_DEFL oo_def sfun_eq_iff sfun_map_map) |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
436 |
qed |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
437 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
438 |
end |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
439 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
440 |
lemma DEFL_sfun: |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
441 |
"DEFL('a::domain \<rightarrow>! 'b::domain) = sfun_defl\<cdot>DEFL('a)\<cdot>DEFL('b)" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
442 |
by (rule defl_sfun_def) |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
443 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
444 |
subsubsection {* Continuous function space *} |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
445 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
446 |
instantiation cfun :: (predomain, "domain") liftdomain |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
447 |
begin |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
448 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
449 |
definition |
40830 | 450 |
"emb = emb oo encode_cfun" |
40592
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
451 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
452 |
definition |
40830 | 453 |
"prj = decode_cfun oo prj" |
40592
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
454 |
|
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
455 |
definition |
40830 | 456 |
"defl (t::('a \<rightarrow> 'b) itself) = DEFL('a u \<rightarrow>! 'b)" |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
457 |
|
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
458 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
459 |
"(liftemb :: ('a \<rightarrow> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
460 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
461 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
462 |
"(liftprj :: udom \<rightarrow> ('a \<rightarrow> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
463 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
464 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
465 |
"liftdefl (t::('a \<rightarrow> 'b) itself) = u_defl\<cdot>DEFL('a \<rightarrow> 'b)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
466 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
467 |
instance |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
468 |
using liftemb_cfun_def liftprj_cfun_def liftdefl_cfun_def |
40494
db8a09daba7b
add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents:
40493
diff
changeset
|
469 |
proof (rule liftdomain_class_intro) |
40592
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
470 |
have "ep_pair encode_cfun decode_cfun" |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
471 |
by (rule ep_pair.intro, simp_all) |
f432973ce0f6
move strict function type into main HOLCF; instance cfun :: (predomain, domain) domain
huffman
parents:
40506
diff
changeset
|
472 |
thus "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)" |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
473 |
unfolding emb_cfun_def prj_cfun_def |
40830 | 474 |
using ep_pair_emb_prj by (rule ep_pair_comp) |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
475 |
show "cast\<cdot>DEFL('a \<rightarrow> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)" |
40830 | 476 |
unfolding emb_cfun_def prj_cfun_def defl_cfun_def |
477 |
by (simp add: cast_DEFL cfcomp1) |
|
27402
253a06dfadce
reuse proofs from Deflation.thy; clean up proof of finite_range_cfun_lemma
huffman
parents:
27310
diff
changeset
|
478 |
qed |
25903 | 479 |
|
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39974
diff
changeset
|
480 |
end |
33504
b4210cc3ac97
map functions for various types, with ep_pair/deflation/finite_deflation lemmas
huffman
parents:
31113
diff
changeset
|
481 |
|
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
482 |
lemma DEFL_cfun: |
40830 | 483 |
"DEFL('a::predomain \<rightarrow> 'b::domain) = DEFL('a u \<rightarrow>! 'b)" |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
484 |
by (rule defl_cfun_def) |
39972
4244ff4f9649
add lemmas finite_deflation_imp_compact, cast_below_cast_iff
Brian Huffman <brianh@cs.pdx.edu>
parents:
37678
diff
changeset
|
485 |
|
40506 | 486 |
subsubsection {* Strict product *} |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
487 |
|
40497 | 488 |
instantiation sprod :: ("domain", "domain") liftdomain |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
489 |
begin |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
490 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
491 |
definition |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
492 |
"emb = udom_emb sprod_approx oo sprod_map\<cdot>emb\<cdot>emb" |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
493 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
494 |
definition |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
495 |
"prj = sprod_map\<cdot>prj\<cdot>prj oo udom_prj sprod_approx" |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
496 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
497 |
definition |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
498 |
"defl (t::('a \<otimes> 'b) itself) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)" |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
499 |
|
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
500 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
501 |
"(liftemb :: ('a \<otimes> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
502 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
503 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
504 |
"(liftprj :: udom \<rightarrow> ('a \<otimes> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
505 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
506 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
507 |
"liftdefl (t::('a \<otimes> 'b) itself) = u_defl\<cdot>DEFL('a \<otimes> 'b)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
508 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
509 |
instance |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
510 |
using liftemb_sprod_def liftprj_sprod_def liftdefl_sprod_def |
40494
db8a09daba7b
add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents:
40493
diff
changeset
|
511 |
proof (rule liftdomain_class_intro) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
512 |
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<otimes> 'b)" |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
513 |
unfolding emb_sprod_def prj_sprod_def |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
514 |
using ep_pair_udom [OF sprod_approx] |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
515 |
by (intro ep_pair_comp ep_pair_sprod_map ep_pair_emb_prj) |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
516 |
next |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
517 |
show "cast\<cdot>DEFL('a \<otimes> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<otimes> 'b)" |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
518 |
unfolding emb_sprod_def prj_sprod_def defl_sprod_def cast_sprod_defl |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
39989
diff
changeset
|
519 |
by (simp add: cast_DEFL oo_def cfun_eq_iff sprod_map_map) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
520 |
qed |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
521 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
522 |
end |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
523 |
|
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
524 |
lemma DEFL_sprod: |
40497 | 525 |
"DEFL('a::domain \<otimes> 'b::domain) = sprod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)" |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
526 |
by (rule defl_sprod_def) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
527 |
|
40830 | 528 |
subsubsection {* Cartesian product *} |
529 |
||
530 |
instantiation prod :: (predomain, predomain) predomain |
|
531 |
begin |
|
532 |
||
533 |
definition |
|
534 |
"liftemb = emb oo encode_prod_u" |
|
535 |
||
536 |
definition |
|
537 |
"liftprj = decode_prod_u oo prj" |
|
538 |
||
539 |
definition |
|
540 |
"liftdefl (t::('a \<times> 'b) itself) = DEFL('a\<^sub>\<bottom> \<otimes> 'b\<^sub>\<bottom>)" |
|
541 |
||
542 |
instance proof |
|
543 |
have "ep_pair encode_prod_u decode_prod_u" |
|
544 |
by (rule ep_pair.intro, simp_all) |
|
545 |
thus "ep_pair liftemb (liftprj :: udom \<rightarrow> ('a \<times> 'b) u)" |
|
546 |
unfolding liftemb_prod_def liftprj_prod_def |
|
547 |
using ep_pair_emb_prj by (rule ep_pair_comp) |
|
548 |
show "cast\<cdot>LIFTDEFL('a \<times> 'b) = liftemb oo (liftprj :: udom \<rightarrow> ('a \<times> 'b) u)" |
|
549 |
unfolding liftemb_prod_def liftprj_prod_def liftdefl_prod_def |
|
550 |
by (simp add: cast_DEFL cfcomp1) |
|
551 |
qed |
|
552 |
||
553 |
end |
|
554 |
||
555 |
instantiation prod :: ("domain", "domain") "domain" |
|
556 |
begin |
|
557 |
||
558 |
definition |
|
559 |
"emb = udom_emb prod_approx oo cprod_map\<cdot>emb\<cdot>emb" |
|
560 |
||
561 |
definition |
|
562 |
"prj = cprod_map\<cdot>prj\<cdot>prj oo udom_prj prod_approx" |
|
563 |
||
564 |
definition |
|
565 |
"defl (t::('a \<times> 'b) itself) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)" |
|
566 |
||
567 |
instance proof |
|
568 |
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<times> 'b)" |
|
569 |
unfolding emb_prod_def prj_prod_def |
|
570 |
using ep_pair_udom [OF prod_approx] |
|
571 |
by (intro ep_pair_comp ep_pair_cprod_map ep_pair_emb_prj) |
|
572 |
next |
|
573 |
show "cast\<cdot>DEFL('a \<times> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<times> 'b)" |
|
574 |
unfolding emb_prod_def prj_prod_def defl_prod_def cast_prod_defl |
|
575 |
by (simp add: cast_DEFL oo_def cfun_eq_iff cprod_map_map) |
|
576 |
qed |
|
577 |
||
578 |
end |
|
579 |
||
580 |
lemma DEFL_prod: |
|
581 |
"DEFL('a::domain \<times> 'b::domain) = prod_defl\<cdot>DEFL('a)\<cdot>DEFL('b)" |
|
582 |
by (rule defl_prod_def) |
|
583 |
||
584 |
lemma LIFTDEFL_prod: |
|
585 |
"LIFTDEFL('a::predomain \<times> 'b::predomain) = DEFL('a u \<otimes> 'b u)" |
|
586 |
by (rule liftdefl_prod_def) |
|
587 |
||
41034 | 588 |
subsubsection {* Unit type *} |
589 |
||
590 |
instantiation unit :: liftdomain |
|
591 |
begin |
|
592 |
||
593 |
definition |
|
594 |
"emb = (\<bottom> :: unit \<rightarrow> udom)" |
|
595 |
||
596 |
definition |
|
597 |
"prj = (\<bottom> :: udom \<rightarrow> unit)" |
|
598 |
||
599 |
definition |
|
600 |
"defl (t::unit itself) = \<bottom>" |
|
601 |
||
602 |
definition |
|
603 |
"(liftemb :: unit u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
|
604 |
||
605 |
definition |
|
606 |
"(liftprj :: udom \<rightarrow> unit u) = u_map\<cdot>prj oo udom_prj u_approx" |
|
607 |
||
608 |
definition |
|
609 |
"liftdefl (t::unit itself) = u_defl\<cdot>DEFL(unit)" |
|
610 |
||
611 |
instance |
|
612 |
using liftemb_unit_def liftprj_unit_def liftdefl_unit_def |
|
613 |
proof (rule liftdomain_class_intro) |
|
614 |
show "ep_pair emb (prj :: udom \<rightarrow> unit)" |
|
615 |
unfolding emb_unit_def prj_unit_def |
|
616 |
by (simp add: ep_pair.intro) |
|
617 |
next |
|
618 |
show "cast\<cdot>DEFL(unit) = emb oo (prj :: udom \<rightarrow> unit)" |
|
619 |
unfolding emb_unit_def prj_unit_def defl_unit_def by simp |
|
620 |
qed |
|
621 |
||
622 |
end |
|
623 |
||
40506 | 624 |
subsubsection {* Discrete cpo *} |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
625 |
|
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
626 |
instantiation discr :: (countable) predomain |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
627 |
begin |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
628 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
629 |
definition |
41286
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41285
diff
changeset
|
630 |
"(liftemb :: 'a discr u \<rightarrow> udom) = udom_emb discr_approx" |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
631 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
632 |
definition |
41286
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41285
diff
changeset
|
633 |
"(liftprj :: udom \<rightarrow> 'a discr u) = udom_prj discr_approx" |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
634 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
635 |
definition |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
636 |
"liftdefl (t::'a discr itself) = |
41286
3d7685a4a5ff
reintroduce 'bifinite' class, now with existentially-quantified approx function (cf. b525988432e9)
huffman
parents:
41285
diff
changeset
|
637 |
(\<Squnion>i. defl_principal (Abs_fin_defl (liftemb oo discr_approx i oo (liftprj::udom \<rightarrow> 'a discr u))))" |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
638 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
639 |
instance proof |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
640 |
show "ep_pair liftemb (liftprj :: udom \<rightarrow> 'a discr u)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
641 |
unfolding liftemb_discr_def liftprj_discr_def |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
642 |
by (rule ep_pair_udom [OF discr_approx]) |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
643 |
show "cast\<cdot>LIFTDEFL('a discr) = liftemb oo (liftprj :: udom \<rightarrow> 'a discr u)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
644 |
unfolding liftemb_discr_def liftprj_discr_def liftdefl_discr_def |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
645 |
apply (subst contlub_cfun_arg) |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
646 |
apply (rule chainI) |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
647 |
apply (rule defl.principal_mono) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
648 |
apply (simp add: below_fin_defl_def) |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
649 |
apply (simp add: Abs_fin_defl_inverse |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
650 |
ep_pair.finite_deflation_e_d_p [OF ep_pair_udom [OF discr_approx]] |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
651 |
approx_chain.finite_deflation_approx [OF discr_approx]) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
652 |
apply (intro monofun_cfun below_refl) |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
653 |
apply (rule chainE) |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
654 |
apply (rule chain_discr_approx) |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
655 |
apply (subst cast_defl_principal) |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
656 |
apply (simp add: Abs_fin_defl_inverse |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
657 |
ep_pair.finite_deflation_e_d_p [OF ep_pair_udom [OF discr_approx]] |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
658 |
approx_chain.finite_deflation_approx [OF discr_approx]) |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
659 |
apply (simp add: lub_distribs) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
660 |
done |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
661 |
qed |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
662 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
663 |
end |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
664 |
|
40506 | 665 |
subsubsection {* Strict sum *} |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
666 |
|
40497 | 667 |
instantiation ssum :: ("domain", "domain") liftdomain |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
668 |
begin |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
669 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
670 |
definition |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
671 |
"emb = udom_emb ssum_approx oo ssum_map\<cdot>emb\<cdot>emb" |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
672 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
673 |
definition |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
674 |
"prj = ssum_map\<cdot>prj\<cdot>prj oo udom_prj ssum_approx" |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
675 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
676 |
definition |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
677 |
"defl (t::('a \<oplus> 'b) itself) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)" |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
678 |
|
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
679 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
680 |
"(liftemb :: ('a \<oplus> 'b) u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
681 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
682 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
683 |
"(liftprj :: udom \<rightarrow> ('a \<oplus> 'b) u) = u_map\<cdot>prj oo udom_prj u_approx" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
684 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
685 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
686 |
"liftdefl (t::('a \<oplus> 'b) itself) = u_defl\<cdot>DEFL('a \<oplus> 'b)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
687 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
688 |
instance |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
689 |
using liftemb_ssum_def liftprj_ssum_def liftdefl_ssum_def |
40494
db8a09daba7b
add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents:
40493
diff
changeset
|
690 |
proof (rule liftdomain_class_intro) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
691 |
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<oplus> 'b)" |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
692 |
unfolding emb_ssum_def prj_ssum_def |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
693 |
using ep_pair_udom [OF ssum_approx] |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
694 |
by (intro ep_pair_comp ep_pair_ssum_map ep_pair_emb_prj) |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
695 |
show "cast\<cdot>DEFL('a \<oplus> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<oplus> 'b)" |
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
696 |
unfolding emb_ssum_def prj_ssum_def defl_ssum_def cast_ssum_defl |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
39989
diff
changeset
|
697 |
by (simp add: cast_DEFL oo_def cfun_eq_iff ssum_map_map) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
698 |
qed |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
699 |
|
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
700 |
end |
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
701 |
|
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
702 |
lemma DEFL_ssum: |
40497 | 703 |
"DEFL('a::domain \<oplus> 'b::domain) = ssum_defl\<cdot>DEFL('a)\<cdot>DEFL('b)" |
39989
ad60d7311f43
renamed type and constant 'sfp' to 'defl'; replaced syntax SFP('a) with DEFL('a)
huffman
parents:
39987
diff
changeset
|
704 |
by (rule defl_ssum_def) |
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
705 |
|
40506 | 706 |
subsubsection {* Lifted HOL type *} |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
707 |
|
40494
db8a09daba7b
add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents:
40493
diff
changeset
|
708 |
instantiation lift :: (countable) liftdomain |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
709 |
begin |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
710 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
711 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
712 |
"emb = emb oo (\<Lambda> x. Rep_lift x)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
713 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
714 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
715 |
"prj = (\<Lambda> y. Abs_lift y) oo prj" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
716 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
717 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
718 |
"defl (t::'a lift itself) = DEFL('a discr u)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
719 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
720 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
721 |
"(liftemb :: 'a lift u \<rightarrow> udom) = udom_emb u_approx oo u_map\<cdot>emb" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
722 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
723 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
724 |
"(liftprj :: udom \<rightarrow> 'a lift u) = u_map\<cdot>prj oo udom_prj u_approx" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
725 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
726 |
definition |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
727 |
"liftdefl (t::'a lift itself) = u_defl\<cdot>DEFL('a lift)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
728 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
729 |
instance |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
730 |
using liftemb_lift_def liftprj_lift_def liftdefl_lift_def |
40494
db8a09daba7b
add class liftdomain, for bifinite domains where DEFL('a u) = u_defl('a)
huffman
parents:
40493
diff
changeset
|
731 |
proof (rule liftdomain_class_intro) |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
732 |
note [simp] = cont_Rep_lift cont_Abs_lift Rep_lift_inverse Abs_lift_inverse |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
733 |
have "ep_pair (\<Lambda>(x::'a lift). Rep_lift x) (\<Lambda> y. Abs_lift y)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
734 |
by (simp add: ep_pair_def) |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
735 |
thus "ep_pair emb (prj :: udom \<rightarrow> 'a lift)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
736 |
unfolding emb_lift_def prj_lift_def |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
737 |
using ep_pair_emb_prj by (rule ep_pair_comp) |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
738 |
show "cast\<cdot>DEFL('a lift) = emb oo (prj :: udom \<rightarrow> 'a lift)" |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
739 |
unfolding emb_lift_def prj_lift_def defl_lift_def cast_DEFL |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
740 |
by (simp add: cfcomp1) |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
741 |
qed |
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
742 |
|
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset
|
743 |
end |
40491
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
744 |
|
6de5839e2fb3
add 'predomain' class: unpointed version of bifinite
huffman
parents:
40484
diff
changeset
|
745 |
end |