author  huffman 
Sat, 09 Oct 2010 07:24:49 0700  
changeset 39987  8c2f449af35a 
parent 39986  38677db30cad 
child 39989  ad60d7311f43 
permissions  rwrr 
25903  1 
(* Title: HOLCF/Bifinite.thy 
2 
Author: Brian Huffman 

3 
*) 

4 

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header {* Bifinite domains *} 
25903  6 

7 
theory Bifinite 

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imports Algebraic Cprod Sprod Ssum Up Lift One Tr 
25903  9 
begin 
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39986  11 
subsection {* Class of bifinite domains *} 
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text {* 
39986  14 
We define a bifinite domain as a pcpo that is isomorphic to some 
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algebraic deflation over the universal domain. 
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*} 
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39986  18 
class bifinite = pcpo + 
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fixes emb :: "'a::pcpo \<rightarrow> udom" 
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fixes prj :: "udom \<rightarrow> 'a::pcpo" 
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fixes sfp :: "'a itself \<Rightarrow> sfp" 
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assumes ep_pair_emb_prj: "ep_pair emb prj" 
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assumes cast_SFP: "cast\<cdot>(sfp TYPE('a)) = emb oo prj" 
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24 

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syntax "_SFP" :: "type \<Rightarrow> sfp" ("(1SFP/(1'(_')))") 
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translations "SFP('t)" \<rightleftharpoons> "CONST sfp TYPE('t)" 
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39986  28 
interpretation bifinite: 
29 
pcpo_ep_pair "emb :: 'a::bifinite \<rightarrow> udom" "prj :: udom \<rightarrow> 'a::bifinite" 

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unfolding pcpo_ep_pair_def 
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by (rule ep_pair_emb_prj) 
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39986  33 
lemmas emb_inverse = bifinite.e_inverse 
34 
lemmas emb_prj_below = bifinite.e_p_below 

35 
lemmas emb_eq_iff = bifinite.e_eq_iff 

36 
lemmas emb_strict = bifinite.e_strict 

37 
lemmas prj_strict = bifinite.p_strict 

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39986  39 
subsection {* Bifinite domains have a countable compact basis *} 
33808  40 

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text {* 
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Eventually it should be possible to generalize this to an unpointed 
39986  43 
variant of the bifinite class. 
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*} 
33587  45 

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interpretation compact_basis: 
39986  47 
ideal_completion below Rep_compact_basis "approximants::'a::bifinite \<Rightarrow> _" 
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proof  
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obtain Y where Y: "\<forall>i. Y i \<sqsubseteq> Y (Suc i)" 
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and SFP: "SFP('a) = (\<Squnion>i. sfp_principal (Y i))" 
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by (rule sfp.obtain_principal_chain) 
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def approx \<equiv> "\<lambda>i. (prj oo cast\<cdot>(sfp_principal (Y i)) oo emb) :: 'a \<rightarrow> 'a" 
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53 
interpret sfp_approx: approx_chain approx 
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54 
proof (rule approx_chain.intro) 
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55 
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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58 
unfolding approx_def 
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by (simp add: lub_distribs Y SFP [symmetric] cast_SFP expand_cfun_eq) 
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60 
show "\<And>i. finite_deflation (approx i)" 
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61 
unfolding approx_def 
39986  62 
apply (rule bifinite.finite_deflation_p_d_e) 
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63 
apply (rule finite_deflation_cast) 
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64 
apply (rule sfp.compact_principal) 
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65 
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 SFP [symmetric] cast_SFP) 
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done 
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qed 
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(* FIXME: why does show ?thesis fail here? *) 
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show "ideal_completion below Rep_compact_basis (approximants::'a \<Rightarrow> _)" .. 
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qed 
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73 

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subsection {* Type combinators *} 
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definition 
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sfp_fun1 :: 
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"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) \<Rightarrow> (sfp \<rightarrow> sfp)" 
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where 
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"sfp_fun1 approx f = 
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sfp.basis_fun (\<lambda>a. 
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sfp_principal (Abs_fin_defl 
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(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)))" 
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definition 
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sfp_fun2 :: 
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"(nat \<Rightarrow> 'a \<rightarrow> 'a) \<Rightarrow> ((udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom) \<rightarrow> ('a \<rightarrow> 'a)) 
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\<Rightarrow> (sfp \<rightarrow> sfp \<rightarrow> sfp)" 
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where 
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"sfp_fun2 approx f = 
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sfp.basis_fun (\<lambda>a. 
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sfp.basis_fun (\<lambda>b. 
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sfp_principal (Abs_fin_defl 
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94 
(udom_emb approx oo 
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f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx))))" 
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96 

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lemma cast_sfp_fun1: 
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98 
assumes approx: "approx_chain approx" 
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99 
assumes f: "\<And>a. finite_deflation a \<Longrightarrow> finite_deflation (f\<cdot>a)" 
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100 
shows "cast\<cdot>(sfp_fun1 approx f\<cdot>A) = udom_emb approx oo f\<cdot>(cast\<cdot>A) oo udom_prj approx" 
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101 
proof  
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102 
have 1: "\<And>a. finite_deflation 
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103 
(udom_emb approx oo f\<cdot>(Rep_fin_defl a) oo udom_prj approx)" 
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104 
apply (rule ep_pair.finite_deflation_e_d_p) 
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105 
apply (rule approx_chain.ep_pair_udom [OF approx]) 
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106 
apply (rule f, rule finite_deflation_Rep_fin_defl) 
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107 
done 
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108 
show ?thesis 
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109 
by (induct A rule: sfp.principal_induct, simp) 
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110 
(simp only: sfp_fun1_def 
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111 
sfp.basis_fun_principal 
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sfp.basis_fun_mono 
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sfp.principal_mono 
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Abs_fin_defl_mono [OF 1 1] 
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115 
monofun_cfun below_refl 
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116 
Rep_fin_defl_mono 
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cast_sfp_principal 
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Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1]) 
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119 
qed 
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120 

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lemma cast_sfp_fun2: 
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122 
assumes approx: "approx_chain approx" 
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assumes f: "\<And>a b. finite_deflation a \<Longrightarrow> finite_deflation b \<Longrightarrow> 
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124 
finite_deflation (f\<cdot>a\<cdot>b)" 
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125 
shows "cast\<cdot>(sfp_fun2 approx f\<cdot>A\<cdot>B) = 
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126 
udom_emb approx oo f\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj approx" 
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127 
proof  
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128 
have 1: "\<And>a b. finite_deflation (udom_emb approx oo 
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129 
f\<cdot>(Rep_fin_defl a)\<cdot>(Rep_fin_defl b) oo udom_prj approx)" 
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130 
apply (rule ep_pair.finite_deflation_e_d_p) 
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131 
apply (rule ep_pair_udom [OF approx]) 
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132 
apply (rule f, (rule finite_deflation_Rep_fin_defl)+) 
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133 
done 
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134 
show ?thesis 
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135 
by (induct A B rule: sfp.principal_induct2, simp, simp) 
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136 
(simp only: sfp_fun2_def 
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137 
sfp.basis_fun_principal 
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138 
sfp.basis_fun_mono 
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139 
sfp.principal_mono 
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140 
Abs_fin_defl_mono [OF 1 1] 
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141 
monofun_cfun below_refl 
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142 
Rep_fin_defl_mono 
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143 
cast_sfp_principal 
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144 
Abs_fin_defl_inverse [unfolded mem_Collect_eq, OF 1]) 
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145 
qed 
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146 

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subsection {* The universal domain is bifinite *} 
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39986  149 
instantiation udom :: bifinite 
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begin 
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151 

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152 
definition [simp]: 
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"emb = (ID :: udom \<rightarrow> udom)" 
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154 

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155 
definition [simp]: 
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"prj = (ID :: udom \<rightarrow> udom)" 
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definition 
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"sfp (t::udom itself) = (\<Squnion>i. sfp_principal (Abs_fin_defl (udom_approx i)))" 
33808  160 

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instance proof 
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show "ep_pair emb (prj :: udom \<rightarrow> udom)" 
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by (simp add: ep_pair.intro) 
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164 
next 
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show "cast\<cdot>SFP(udom) = emb oo (prj :: udom \<rightarrow> udom)" 
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166 
unfolding sfp_udom_def 
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apply (subst contlub_cfun_arg) 
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168 
apply (rule chainI) 
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169 
apply (rule sfp.principal_mono) 
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apply (simp add: below_fin_defl_def) 
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apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx) 
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apply (rule chainE) 
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173 
apply (rule chain_udom_approx) 
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apply (subst cast_sfp_principal) 
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apply (simp add: Abs_fin_defl_inverse finite_deflation_udom_approx) 
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done 
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qed 
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178 

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end 
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180 

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subsection {* Continuous function space is a bifinite domain *} 
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182 

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183 
definition 
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cfun_approx :: "nat \<Rightarrow> (udom \<rightarrow> udom) \<rightarrow> (udom \<rightarrow> udom)" 
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where 
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"cfun_approx = (\<lambda>i. cfun_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))" 
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187 

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lemma cfun_approx: "approx_chain cfun_approx" 
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proof (rule approx_chain.intro) 
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show "chain (\<lambda>i. cfun_approx i)" 
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191 
unfolding cfun_approx_def by simp 
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show "(\<Squnion>i. cfun_approx i) = ID" 
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193 
unfolding cfun_approx_def 
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by (simp add: lub_distribs cfun_map_ID) 
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show "\<And>i. finite_deflation (cfun_approx i)" 
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196 
unfolding cfun_approx_def 
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by (intro finite_deflation_cfun_map finite_deflation_udom_approx) 
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qed 
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199 

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definition cfun_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp" 
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where "cfun_sfp = sfp_fun2 cfun_approx cfun_map" 
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202 

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lemma cast_cfun_sfp: 
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"cast\<cdot>(cfun_sfp\<cdot>A\<cdot>B) = 
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udom_emb cfun_approx oo cfun_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj cfun_approx" 
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unfolding cfun_sfp_def 
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apply (rule cast_sfp_fun2 [OF cfun_approx]) 
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apply (erule (1) finite_deflation_cfun_map) 
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209 
done 
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210 

39986  211 
instantiation cfun :: (bifinite, bifinite) bifinite 
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212 
begin 
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213 

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214 
definition 
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"emb = udom_emb cfun_approx oo cfun_map\<cdot>prj\<cdot>emb" 
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216 

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217 
definition 
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"prj = cfun_map\<cdot>emb\<cdot>prj oo udom_prj cfun_approx" 
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219 

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220 
definition 
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"sfp (t::('a \<rightarrow> 'b) itself) = cfun_sfp\<cdot>SFP('a)\<cdot>SFP('b)" 
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222 

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223 
instance proof 
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224 
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)" 
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225 
unfolding emb_cfun_def prj_cfun_def 
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226 
using ep_pair_udom [OF cfun_approx] 
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227 
by (intro ep_pair_comp ep_pair_cfun_map ep_pair_emb_prj) 
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228 
next 
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229 
show "cast\<cdot>SFP('a \<rightarrow> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<rightarrow> 'b)" 
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230 
unfolding emb_cfun_def prj_cfun_def sfp_cfun_def cast_cfun_sfp 
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231 
by (simp add: cast_SFP oo_def expand_cfun_eq cfun_map_map) 
27402
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232 
qed 
25903  233 

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234 
end 
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235 

39986  236 
lemma SFP_cfun: 
237 
"SFP('a::bifinite \<rightarrow> 'b::bifinite) = cfun_sfp\<cdot>SFP('a)\<cdot>SFP('b)" 

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by (rule sfp_cfun_def) 
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239 

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240 
subsection {* Cartesian product is a bifinite domain *} 
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241 

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242 
definition 
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243 
prod_approx :: "nat \<Rightarrow> udom \<times> udom \<rightarrow> udom \<times> udom" 
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244 
where 
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"prod_approx = (\<lambda>i. cprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))" 
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246 

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247 
lemma prod_approx: "approx_chain prod_approx" 
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248 
proof (rule approx_chain.intro) 
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249 
show "chain (\<lambda>i. prod_approx i)" 
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250 
unfolding prod_approx_def by simp 
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251 
show "(\<Squnion>i. prod_approx i) = ID" 
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252 
unfolding prod_approx_def 
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253 
by (simp add: lub_distribs cprod_map_ID) 
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254 
show "\<And>i. finite_deflation (prod_approx i)" 
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255 
unfolding prod_approx_def 
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256 
by (intro finite_deflation_cprod_map finite_deflation_udom_approx) 
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257 
qed 
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258 

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259 
definition prod_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp" 
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where "prod_sfp = sfp_fun2 prod_approx cprod_map" 
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261 

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262 
lemma cast_prod_sfp: 
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"cast\<cdot>(prod_sfp\<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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265 
unfolding prod_sfp_def 
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266 
apply (rule cast_sfp_fun2 [OF prod_approx]) 
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267 
apply (erule (1) finite_deflation_cprod_map) 
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268 
done 
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269 

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270 
instantiation prod :: (bifinite, bifinite) bifinite 
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271 
begin 
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272 

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273 
definition 
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"emb = udom_emb prod_approx oo cprod_map\<cdot>emb\<cdot>emb" 
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275 

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276 
definition 
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277 
"prj = cprod_map\<cdot>prj\<cdot>prj oo udom_prj prod_approx" 
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278 

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279 
definition 
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280 
"sfp (t::('a \<times> 'b) itself) = prod_sfp\<cdot>SFP('a)\<cdot>SFP('b)" 
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281 

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282 
instance proof 
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283 
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<times> 'b)" 
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move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

284 
unfolding emb_prod_def prj_prod_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

285 
using ep_pair_udom [OF prod_approx] 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

286 
by (intro ep_pair_comp ep_pair_cprod_map ep_pair_emb_prj) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

287 
next 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

288 
show "cast\<cdot>SFP('a \<times> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<times> 'b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

289 
unfolding emb_prod_def prj_prod_def sfp_prod_def cast_prod_sfp 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

290 
by (simp add: cast_SFP oo_def expand_cfun_eq cprod_map_map) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

291 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

292 

26962
c8b20f615d6c
use new class package for classes profinite, bifinite; remove approx class
huffman
parents:
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diff
changeset

293 
end 
39987
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

294 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

295 
lemma SFP_prod: 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
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parents:
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diff
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296 
"SFP('a::bifinite \<times> 'b::bifinite) = prod_sfp\<cdot>SFP('a)\<cdot>SFP('b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

297 
by (rule sfp_prod_def) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

298 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

299 
subsection {* Strict product is a bifinite domain *} 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

300 

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move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

301 
definition 
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move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

302 
sprod_approx :: "nat \<Rightarrow> udom \<otimes> udom \<rightarrow> udom \<otimes> udom" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

303 
where 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

304 
"sprod_approx = (\<lambda>i. sprod_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

305 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

306 
lemma sprod_approx: "approx_chain sprod_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

307 
proof (rule approx_chain.intro) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

308 
show "chain (\<lambda>i. sprod_approx i)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

309 
unfolding sprod_approx_def by simp 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

310 
show "(\<Squnion>i. sprod_approx i) = ID" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

311 
unfolding sprod_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

312 
by (simp add: lub_distribs sprod_map_ID) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

313 
show "\<And>i. finite_deflation (sprod_approx i)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

314 
unfolding sprod_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

315 
by (intro finite_deflation_sprod_map finite_deflation_udom_approx) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

316 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

317 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

318 
definition sprod_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp" 
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move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

319 
where "sprod_sfp = sfp_fun2 sprod_approx sprod_map" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

320 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

321 
lemma cast_sprod_sfp: 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

322 
"cast\<cdot>(sprod_sfp\<cdot>A\<cdot>B) = 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

323 
udom_emb sprod_approx oo 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

324 
sprod_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

325 
udom_prj sprod_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

326 
unfolding sprod_sfp_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

327 
apply (rule cast_sfp_fun2 [OF sprod_approx]) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

328 
apply (erule (1) finite_deflation_sprod_map) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

329 
done 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

330 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

331 
instantiation sprod :: (bifinite, bifinite) bifinite 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

332 
begin 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

333 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

334 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

335 
"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

336 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

337 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

338 
"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

339 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

340 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

341 
"sfp (t::('a \<otimes> 'b) itself) = sprod_sfp\<cdot>SFP('a)\<cdot>SFP('b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

342 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

343 
instance proof 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

344 
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<otimes> 'b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

345 
unfolding emb_sprod_def prj_sprod_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

346 
using ep_pair_udom [OF sprod_approx] 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

347 
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

348 
next 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

349 
show "cast\<cdot>SFP('a \<otimes> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<otimes> 'b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

350 
unfolding emb_sprod_def prj_sprod_def sfp_sprod_def cast_sprod_sfp 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

351 
by (simp add: cast_SFP oo_def expand_cfun_eq sprod_map_map) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

352 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

353 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

354 
end 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

355 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

356 
lemma SFP_sprod: 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

357 
"SFP('a::bifinite \<otimes> 'b::bifinite) = sprod_sfp\<cdot>SFP('a)\<cdot>SFP('b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

358 
by (rule sfp_sprod_def) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

359 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

360 
subsection {* Lifted cpo is a bifinite domain *} 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

361 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

362 
definition u_approx :: "nat \<Rightarrow> udom\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

363 
where "u_approx = (\<lambda>i. u_map\<cdot>(udom_approx i))" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

364 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

365 
lemma u_approx: "approx_chain u_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

366 
proof (rule approx_chain.intro) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

367 
show "chain (\<lambda>i. u_approx i)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

368 
unfolding u_approx_def by simp 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

369 
show "(\<Squnion>i. u_approx i) = ID" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

370 
unfolding u_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

371 
by (simp add: lub_distribs u_map_ID) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

372 
show "\<And>i. finite_deflation (u_approx i)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

373 
unfolding u_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

374 
by (intro finite_deflation_u_map finite_deflation_udom_approx) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

375 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

376 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

377 
definition u_sfp :: "sfp \<rightarrow> sfp" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

378 
where "u_sfp = sfp_fun1 u_approx u_map" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

379 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

380 
lemma cast_u_sfp: 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

381 
"cast\<cdot>(u_sfp\<cdot>A) = 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

382 
udom_emb u_approx oo u_map\<cdot>(cast\<cdot>A) oo udom_prj u_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

383 
unfolding u_sfp_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

384 
apply (rule cast_sfp_fun1 [OF u_approx]) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

385 
apply (erule finite_deflation_u_map) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

386 
done 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

387 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

388 
instantiation u :: (bifinite) bifinite 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

389 
begin 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

390 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

391 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

392 
"emb = udom_emb u_approx oo u_map\<cdot>emb" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

393 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

394 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

395 
"prj = u_map\<cdot>prj oo udom_prj u_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

396 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

397 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

398 
"sfp (t::'a u itself) = u_sfp\<cdot>SFP('a)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

399 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

400 
instance proof 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

401 
show "ep_pair emb (prj :: udom \<rightarrow> 'a u)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

402 
unfolding emb_u_def prj_u_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

403 
using ep_pair_udom [OF u_approx] 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

404 
by (intro ep_pair_comp ep_pair_u_map ep_pair_emb_prj) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

405 
next 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

406 
show "cast\<cdot>SFP('a u) = emb oo (prj :: udom \<rightarrow> 'a u)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

407 
unfolding emb_u_def prj_u_def sfp_u_def cast_u_sfp 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

408 
by (simp add: cast_SFP oo_def expand_cfun_eq u_map_map) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

409 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

410 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

411 
end 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

412 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

413 
lemma SFP_u: "SFP('a::bifinite u) = u_sfp\<cdot>SFP('a)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

414 
by (rule sfp_u_def) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

415 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

416 
subsection {* Lifted countable types are bifinite domains *} 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

417 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

418 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

419 
lift_approx :: "nat \<Rightarrow> 'a::countable lift \<rightarrow> 'a lift" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

420 
where 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

421 
"lift_approx = (\<lambda>i. FLIFT x. if to_nat x < i then Def x else \<bottom>)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

422 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

423 
lemma chain_lift_approx [simp]: "chain lift_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

424 
unfolding lift_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

425 
by (rule chainI, simp add: FLIFT_mono) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

426 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

427 
lemma lub_lift_approx [simp]: "(\<Squnion>i. lift_approx i) = ID" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

428 
apply (rule ext_cfun) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

429 
apply (simp add: contlub_cfun_fun) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

430 
apply (simp add: lift_approx_def) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

431 
apply (case_tac x, simp) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

432 
apply (rule thelubI) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

433 
apply (rule is_lubI) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

434 
apply (rule ub_rangeI, simp) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

435 
apply (drule ub_rangeD) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

436 
apply (erule rev_below_trans) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

437 
apply simp 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

438 
apply (rule lessI) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

439 
done 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

440 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

441 
lemma finite_deflation_lift_approx: "finite_deflation (lift_approx i)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

442 
proof 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

443 
fix x 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

444 
show "lift_approx i\<cdot>x \<sqsubseteq> x" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

445 
unfolding lift_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

446 
by (cases x, simp, simp) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

447 
show "lift_approx i\<cdot>(lift_approx i\<cdot>x) = lift_approx i\<cdot>x" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

448 
unfolding lift_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

449 
by (cases x, simp, simp) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

450 
show "finite {x::'a lift. lift_approx i\<cdot>x = x}" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

451 
proof (rule finite_subset) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

452 
let ?S = "insert (\<bottom>::'a lift) (Def ` to_nat ` {..<i})" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

453 
show "{x::'a lift. lift_approx i\<cdot>x = x} \<subseteq> ?S" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

454 
unfolding lift_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

455 
by (rule subsetI, case_tac x, simp, simp split: split_if_asm) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

456 
show "finite ?S" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

457 
by (simp add: finite_vimageI) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

458 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

459 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

460 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

461 
lemma lift_approx: "approx_chain lift_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

462 
using chain_lift_approx lub_lift_approx finite_deflation_lift_approx 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

463 
by (rule approx_chain.intro) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

464 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

465 
instantiation lift :: (countable) bifinite 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

466 
begin 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

467 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

468 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

469 
"emb = udom_emb lift_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

470 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

471 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

472 
"prj = udom_prj lift_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

473 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

474 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

475 
"sfp (t::'a lift itself) = 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

476 
(\<Squnion>i. sfp_principal (Abs_fin_defl (emb oo lift_approx i oo prj)))" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

477 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

478 
instance proof 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

479 
show ep: "ep_pair emb (prj :: udom \<rightarrow> 'a lift)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

480 
unfolding emb_lift_def prj_lift_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

481 
by (rule ep_pair_udom [OF lift_approx]) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

482 
show "cast\<cdot>SFP('a lift) = emb oo (prj :: udom \<rightarrow> 'a lift)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

483 
unfolding sfp_lift_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

484 
apply (subst contlub_cfun_arg) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

485 
apply (rule chainI) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

486 
apply (rule sfp.principal_mono) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

487 
apply (simp add: below_fin_defl_def) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

488 
apply (simp add: Abs_fin_defl_inverse finite_deflation_lift_approx 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

489 
ep_pair.finite_deflation_e_d_p [OF ep]) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

490 
apply (intro monofun_cfun below_refl) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

491 
apply (rule chainE) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

492 
apply (rule chain_lift_approx) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

493 
apply (subst cast_sfp_principal) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

494 
apply (simp add: Abs_fin_defl_inverse finite_deflation_lift_approx 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

495 
ep_pair.finite_deflation_e_d_p [OF ep] lub_distribs) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

496 
done 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

497 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

498 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

499 
end 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

500 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

501 
subsection {* Strict sum is a bifinite domain *} 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

502 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

503 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

504 
ssum_approx :: "nat \<Rightarrow> udom \<oplus> udom \<rightarrow> udom \<oplus> udom" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

505 
where 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

506 
"ssum_approx = (\<lambda>i. ssum_map\<cdot>(udom_approx i)\<cdot>(udom_approx i))" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

507 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

508 
lemma ssum_approx: "approx_chain ssum_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

509 
proof (rule approx_chain.intro) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

510 
show "chain (\<lambda>i. ssum_approx i)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

511 
unfolding ssum_approx_def by simp 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

512 
show "(\<Squnion>i. ssum_approx i) = ID" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

513 
unfolding ssum_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

514 
by (simp add: lub_distribs ssum_map_ID) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

515 
show "\<And>i. finite_deflation (ssum_approx i)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

516 
unfolding ssum_approx_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

517 
by (intro finite_deflation_ssum_map finite_deflation_udom_approx) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

518 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

519 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

520 
definition ssum_sfp :: "sfp \<rightarrow> sfp \<rightarrow> sfp" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

521 
where "ssum_sfp = sfp_fun2 ssum_approx ssum_map" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

522 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

523 
lemma cast_ssum_sfp: 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

524 
"cast\<cdot>(ssum_sfp\<cdot>A\<cdot>B) = 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

525 
udom_emb ssum_approx oo ssum_map\<cdot>(cast\<cdot>A)\<cdot>(cast\<cdot>B) oo udom_prj ssum_approx" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

526 
unfolding ssum_sfp_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

527 
apply (rule cast_sfp_fun2 [OF ssum_approx]) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

528 
apply (erule (1) finite_deflation_ssum_map) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

529 
done 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

530 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

531 
instantiation ssum :: (bifinite, bifinite) bifinite 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

532 
begin 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

533 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

534 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

535 
"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

536 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

537 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

538 
"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

539 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

540 
definition 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

541 
"sfp (t::('a \<oplus> 'b) itself) = ssum_sfp\<cdot>SFP('a)\<cdot>SFP('b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

542 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

543 
instance proof 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

544 
show "ep_pair emb (prj :: udom \<rightarrow> 'a \<oplus> 'b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

545 
unfolding emb_ssum_def prj_ssum_def 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

546 
using ep_pair_udom [OF ssum_approx] 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

547 
by (intro ep_pair_comp ep_pair_ssum_map ep_pair_emb_prj) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

548 
next 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

549 
show "cast\<cdot>SFP('a \<oplus> 'b) = emb oo (prj :: udom \<rightarrow> 'a \<oplus> 'b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

550 
unfolding emb_ssum_def prj_ssum_def sfp_ssum_def cast_ssum_sfp 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

551 
by (simp add: cast_SFP oo_def expand_cfun_eq ssum_map_map) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

552 
qed 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

553 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

554 
end 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

555 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

556 
lemma SFP_ssum: 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
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diff
changeset

557 
"SFP('a::bifinite \<oplus> 'b::bifinite) = ssum_sfp\<cdot>SFP('a)\<cdot>SFP('b)" 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

558 
by (rule sfp_ssum_def) 
8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

559 

8c2f449af35a
move all bifinite class instances to Bifinite.thy
huffman
parents:
39986
diff
changeset

560 
end 