SSL Defl_Bifinite.thy Sprache: Isabelle

(*  Title:      HOL/HOLCF/Library/Defl_Bifinite.thy
    Author:     Brian Huffman
*)

section

theory type
imports"."
begin

\<open>Lemmas about MOST\<close>

default_sort type

subsection \<open>Eventually constant sequences\<close>

definition
  eventually_constant :: "(nat \ 'a) \ bool"
where
  "eventually_constant S = (\x. MOST i. S i = x)"

lemma eventually_constant_MOST_MOST:
  "eventually_constant S \ (MOST m. MOST n. S n = S m)"
unfolding eventually_constant_def MOST_nat
apply safe
apply (rule_tac x=m in exI, clarify)
apply (rule_tac x=m in exI, clarify)
apply simp
apply fast
done

lemma eventually_constantI: "MOST i. S i = x \ eventually_constant S"
unfolding eventually_constant_def by fast

lemma eventually_constant_comp:
  "eventually_constant (\i. S i) \ eventually_constant (\i. f (S i))"
unfolding eventually_constant_def
apply (erule exE, rule_tac x="f x" in exI)
apply (erulejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
done

lemma eventually_constant_Suc_iff:
  "eventually_constant (\i. S (Suc i)) \ eventually_constant (\i. S i)"

by    java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

  , simp:

by (rule

subsection lemma eventually_constant_SucD: eventually_constant_Suc_iff ])

   :( > 'a) \<Rightarrow> 'a" where
  "eventual S (x. xjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

lemma y
eventual_def)
applyerule)
apply( y)
apply (subgoal_tac:
apply MOST_rev_mpeventually_constant
apply (erule
apply simperule, add
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

lemma  spec mp order_refl range_eqI ])
eventually_constant
unfolding eventually_constant_def
exE )

:
  "eventually_constant imp
apply (drule MOST_eq_eventual( iffI
simp , clarify
apply ( (erule, force MOST_rev_mp  [OF
apply( range_eqI ])
done

:"lbrakkS nS\java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
Seventually_constant
      MOST
applyultimatelyn
y
apply( iffIhencen:   java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
(    Sjava.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
apply ( MOST_monoforceapply(erule)
apply MOST_rev_mp  [OF
apply (erule
done

lemma MOST_eventual:
  "\eventually_constant S; MOST n. P (S n)\ \ P (eventual S)"
-
  assume java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
  hence" .S n =Sjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
     rule
  moreover assume "MOST
mately n=
    byassumes:
hencen.  )
    by (rule"Andi iijava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
  thusij"
qed

lemma eventually_constant_MOST_Suc_eq:
  "eventually_constant S
MOST_eq_eventual
apply (frulewithhavei"bysimp
apply (erule MOST_rev_mp" j steptransless_Suc_inductjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

\<open>A pre-deflation is like a deflation, but not idempotent.\<close> java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22

lemma i
   i, add [OF])
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
ruleMOST_mono

simp le_Suc_induct
done

subsection \<open>Constructing finite deflations by iteration\<close>

default_sort cpo

lemma le_Suc_induct:
  assumes lerule)
  assumes (1) )
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
   " (\i. iterate i\f\x) \ insert x (range (\x. f\x))"
  shows "P i j"
proof    "finite (insert x (range (\x. f\x)))"
  assume "i e"  j
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
nextjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
" java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
  with le java.lang.NullPointerException

qed

definition ?java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
eventual_iteratejava.lang.NullPointerException
where
  "eventual_iteratef=eventual\java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65

-

locale =
  by add)
  hence
       eventually_constant_MOST_MOST
begin

lemma iterate_below: "iterate i\f\x \ x"
by (induct MOST.

lemma
by( i )

   MOST.
apply (erule simp)
apply      Suc
apply (     simp)
apply
done  .

" (\i. iterate i\f)"
( finite_subset
  show
    by (clarify,
   "finite(insert x (range (\x. f\x)))"
    by (simp add: finite_range)
qed

lemma eventually_constant_iterate_app:
  "eventually_constant (\i. iterate i\f\x)"
unfoldingeventually_constant_def
proof " \ eventual_iterate f"
  ?Y="<>i. iterate \cdotf
  have "\j. \k. ?Y j \ ?Y k"
    apply ( finite_range_has_max
(erule antichain_iterate_app)
    apply (
    done f_d
  thenapply ( MOST_d
show
  proof (apply (rule eventually_consta)
    fix
    java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 4
    hence "? f_d [where x=]
    also have "?Y j \ ?Y k" by (rule j)

  qed :fjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
qed

lemma eventually_constant_iterate:
  "eventually_constant (\n. iterate n\f)"
proof -
  have "java.lang.StringIndexOutOfBoundsException: Range [0, 87) out of bounds for length 54
    by (simp ALL_MOST
  by( MOST_d
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  .
    by (simp     (uleeventual_mem_rangejava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
  hence  by ( iterate_fixed
bysimp
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
bsimp: iterate_Suc2
  hence "MOST
    by ( only )
  hence  "finitex. f\x = x}"
    unfolding ( finite_range_imp_finite_fixes
   "eventually_constant (i. iterate i\f)"
    by (rule eventually_constant_SucD)
qed

abbreviation
  d :: "'a \ 'a"
where
  "d \ eventual_iterate f"

lemma MOST_d: "MOST n. P (iterate n\f) \ P d"
unfolding eventual_iterate_def
using eventually_constant_iterate by (rule MOST_eventual)

lemma f_d: "f\(d\x) = d\x"
apply (rule MOST_d)
apply (subst iterate_Suc [symmetric])
apply (rule eventually_constant_MOST_Suc_eq)
apply (rule eventually_constant_iterate_app)
done

lemma d_fixed_iff: "d\x = x \ f\x = x"
proof
  assume "d\x = x"
  with f_d [where x=x]
  show "f\x = x" by simp
next
  assume f: "f\x = x"
  have "\n. iterate n\f\x = x"
    by (rule allI, rule nat.induct, simp, simp add: f)
  hence "MOST n. iterate n\f\x = x"
    by (rule ALL_MOST)
  thus "d\x = x"
    by (rule MOST_d)
qed

lemma finite_deflation_d: "finite_deflation d"
proof
      by add)
  have
    unfolding
    using eventually_constant_iteratedeflation_d
using
  then obtain(ulefinite_deflation_imp_deflation
  java.lang.StringIndexOutOfBoundsException: Range [0, 1) out of bounds for length 0
    using by rule)
  thus pre_deflation_oo
imp add n)
nextassumes f:"<>x.f\x \ x"
  fix : '
  show   d: d by fact
   "\x. (d oo f)\x \ x"
next
  from finite_range
  have "inite x f\x = x}"
    by  "finite( (\x. (d oo f)\x))"
  thusfinite
    by (simp java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
qed

lemma proof-
using d  d by
le)

end

lemma:
  "re_deflation d \ finite_deflation (eventual_iterate d)"
by     \<open>finite_deflation d\<close> f

lemma pre_deflation_oo:
assumesd
  assumesshow?
  showsoo
proof
et d: finite_deflationdby java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  fix x
  show ".( oo f) \ x"
         ( below_antisym
   "finite ( (\x. (d oo f)\x))"
    byapply( substrule)
qed

lemma eventual_iterate_oo_fixed_iff:
  assumes "finite_deflation d"
  assumes f: "\x. f\x \ x"
  shows "ventual_iterate (d oo f\x = x \ d\x = x \ f\x = x"
proof :
  interpret d: finite_deflation A: "ventually_constant "
  let  " oo "
  interpretassumesbelow: \<n  n
using
    by (rule pre_deflation_oo)
  let A have "OST n.A n Ajava.lang.StringIndexOutOfBoundsException: Range [40, 41) out of bounds for length 40
  show ?thesis
     ( MOST_mono substr below
    with "eventual \ eventual B"
    apply saferule)
    apply (erule
    apply (rule
applyrule)
    apply ( assumes :"re_deflationf and g: " g" and " java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
    applyerule, rule dbelow
     simp
    done
qed

lemma ( pre_deflation [OF)
  assumes A: "eventually_constant A"
  assumescont2cont_eventual_iterate_oo
  assumes: "<>n.A n \ B n"
  showseventual
proof -
  from  haveMOST= "
    by (( cont
   have" n. eventual B n"
    by (rule MOST_mono) (erule subst, rule below)
  with B show "eventual \ eventual B"
    by     ( monofunI
qed

lemma eventual_iterate_mono:
  assumes      ( pre_deflation_oo d below
  shows f\<sqsubseteq> eventual_iterate g"
unfolding monofun_cfun_arg
apply (rule eventual_mono)
apply    apply erule [OF])
apply ( pre_deflation [OF)
apply (rule
donefixY :" \ 'b"

   cont fY:"hain\lambda. f Yi)
  assumesrule)
  assumes eY(<>.e(i)"
  shows "cont (\x. eventual_iterate (d oo f x))"
    (is "cont ?e")
proof lub_below
  show "monofun ?e"
    apply rule monofunI
  have"deflation (?e (\i. Y i))"
    apply (rulerule.deflation_d
    apply(ule  [OFbelow
    apply    java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
    apply (erule (rule.belowI

next
f Y ::" \ 'b"
      henc
  with contby(simp_all add:eventual_iterate_oo_fixed_iff[Fdl])
    byrule)
         simp: cont2contlubE cont
  have lub_below: "<>x.f (i. Y i\x \ x"
rule [ _ Y] simp: , rule)
  have "deflation (?e (\i. Y i))"
    apply (rule pre_deflation.deflation_d)
    apply ( pre_deflation_ooOFdlub_below
    done
thencompact
  proof (rule deflation \<open>d\<cdot>x = x\<close> by simp
    fix x :: 'a
    assume "?e (\i. Y i)\x = x"
    hence       \<open>(\<Squnion>i. f (Y i)\<cdot>x) = x\<close> by simp
      by simp_all: eventual_iterate_oo_fixed_iff [F dlub_below
    hence "(\i. f (Y i)\x) = x"
      ( onlycont2contlubE [ cont
      apply (simp only:     have " Y n)\x = x"
      done
    have (
      using d by (rule finite_deflation.compact)
    then have "compact x"
      using\<open>d\<cdot>x = x\<close> by simp
    then have? ( )<dot
      using \<open>(\<Squnion>i. f (Y i)\<cdot>x) = x\<close> by simp is_ub_thelub add eYjava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    then "
      by - (rule compact_imp_max_in_chain, simp      have(<>.eY i)<> \<sqsubseteq> x"
    then obtainapply( admD adm_deflation])
then f Yn\<cdot>x = x"
      using \<open>(\<Squnion>i. f (Y i)\<cdot>x) = x\<close> fY by (simp add: maxinch_is_thelub) pre_deflation_oo d ])
    with open>d\<cdot>x = x\<close> have "?e (Y n)\<cdot>x = x"
         by ( add eventual_iterate_oo_fixed_iff dbelow
subsection
      by (rule is_ub_thelub
    ultimately meet_fin_defl
      by (Rep_fin_defl Rep_fin_defl
    unfolding meet_fin_defl_def
      apply (rule( Abs_fin_defl_inverse])
      apply (apply( finite_deflation_eventual_iterate
apply rule)
      apply (rule pre_deflation_oo finite_deflation_Rep_fin_defl
ap (rule.below
    finally "(\i. ?e (Y i))\x = x" ..
  qed
qed

subsection \<open>Intersection of algebraic deflations\<close>

default_sort bifinite

definition unfolding
  whererule)
    Abs_fin_defl (apply( finite_deflation_Rep_fin_defl

lemma Rep_meet_fin_defl:
  "Rep_fin_defl (meet_fin_defl a b) =
    eventual_iterate (Rep_fin_defl a oo Rep_fin_defl b)"
unfolding meet_fin_defl_def meet_fin_defl_mono
apply (rule Abs_fin_defl_inverse [simplified below_fin_defl_def
apply rule)
apply (rule pre_deflation_oo( addRep_meet_fin_defl_fixed_iff.belowD
apply (rule: " a b \ a"
apply ( Rep_fin_defl)
done

lemma Rep_meet_fin_defl_fixed_iffsimp add:Rep_meet_fin_defl_fixed_iffRep_fin_defl)
  "
    Rep_fin_defl
unfolding Rep_meet_fin_defl
apply (rule below_fin_defl_def
( finite_deflation_Rep_fin_defl
rule.below
done

meet_fin_defl_mono
  "<> \ b; c \ d\ \ meet_fin_defl a c \ meet_fin_defl b d"
below_fin_defl_def
apply(rule Rep_fin_defl.belowI
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
done

lemmam: "meet_fin_defl
unfolding below_fin_defl_def
apply (apply (rule
(imp add:Rep_meet_fin_defl_fixed_iff.belowD
done

lemma meet_fin_defl_below2: "meet_fin_defl a b \ b"
unfolding below_fin_defl_def
apply (by( add deflextension_principal defl
apply (imp add

lemmameet_fin_defl_greatest:"<>\ b; a \ c\ \ a \ meet_fin_defl b c"
unfolding below_fin_defl_def
apply (rule Rep_fin_defl.belowI)lemma meet_defl_below2:meet_defl\<cdot>a\<cdot>b \<sqsubseteq>b"
apply (simp add: Rep_meet_fin_defl_fixed_iff Rep_fin_defl.belowD)
done ( a rule:defl.rincipal_inductsimp

meet_defl:' defl <>'  \rightarrow' defl
apply  : meet_defl_principal )
     ( ab)"

lemma : "<> <>b
  apply ( b : .principal_induct)
   defl_principal meet_fin_defl b)
unfolding meet_defl_def
simp . deflmeet_fin_defl_mono

by (ast: below_antisym meet_defl_greatest
apply (induct a rule:  "a .meet_deflabjava.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
apply (induct b rule: defl.principal_induct, simp)
(imp:meet_defl_principal)
done

lemma meet_defl_below2: "meet_defl\a\b \ b"
apply (induct a rule: defl.principal_induct rule)
apply (induct b rule
apply( addmeet_defl_principal meet_fin_defl_below2java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
done

lemma meet_defl_greatest: "\a proof (rule finite_deflation_intro
inductrule .,simp
apply (induct b rule: defl     defl.ompact_imp_principalOF] .
,simp
apply (simp add: meet_defl_principalapplyrule)
done

meet_defl_eq2:b <>a\Longrightarrow meet_deflcdot<> ="
    apply rule.finite_fixes

interpretation simp: po_eq_conv defl_set_subset_iff symmetric
done
  (fast intro: below_antisym meet_defl_greatest
   meet_defl_below1 [THEN below_trans] meet_defl_below2 [THEN    " (range(<>b.meet_defl\a<>b)"

lemma deflation_meet_defl: "deflation (larsimp, rev_subsetD
apply(ule deflation.)
apply (rule
rule)
done

lemma finite_deflation_meet_defl (ule deflation_meet_defljava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
  assumes safe!:defl)
  shows "finite_deflationapply (imp add )
(ulefinite_deflation_intro
   erule)
    done
  have "finite (defl_setlemma:
    apply  compact<  defl_set"
    apply ( finite_Pow_iff [HEN iffD2]
    finite_deflation_defdeflation_cast )
    apply (rule Rep_fin_defl.finite_fixes
    apply ( injI)
    apply(imp add  defl_set_subset_iffsymmetric]
apply ( rev_finite_subset
hencefiniterange<> \cdot<>)
    apply (rule rev_finite_subset
    lemma :" b \Longrightarrow>compact(\<>a\<>b)
    apply(imp add:defl_set_subset_iff)
    done
subsection \openChain approx on deflations<>
    by
qed (rule deflation_meet_defl)

lemma compact_iff_finite_deflation_cast:
  "compact d \
apply (safe destdefinition defl_approx ::nat<Rightarrow deflrightarrowadefl
apply (wheredefl_approxi  meet_defl<>defl_principal( ))java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
  ave:"hain \. ( approx i))
apply (erule finite_deflation_imp_compact     (ule)
done

lemma compact_iff_finite_defl_set:
  "compact d \ finite (defl_set d)"
bysimp: compact_iff_finite_deflation_cast java.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
  finite_deflation_defdeflation_cast)

lemma     done
compact_iff_finite_defl_set
apply (erule rev_finite_subset    unfoldingdefl_approx_defby simp add chain1
( :defl_set_subset_iffjava.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
done

compact_meet_defl2 " b \Longrightarrow (eet_defl<>\
by (subst meet_defl.commute, rule compact_meet_defl1)

subsection \<open>Chain of approx functions on algebraic deflations\<close>

context    applsimp: contlub_cfun_funchain)
begin

definition defl_approx :: "nat \ 'a defl \ 'a defl"
  where"defl_approx i = meet_defl\defl_principal (Abs_fin_defl (approx i)))"

lemma defl_approx: "approx_chain defl_approx"
proof (rule approx_chain.intro)
  have chain1: "chain (\i. defl_principal (Abs_fin_defl (approx i)))"
    apply (rule chainI)
    apply (rule cast_below_imp_below
    apply (    apply (simpadd contlub_cfun_arg chain1)
    apply (    apply (simpadd: cast_defl_principal Abs_fin_defl_inverse)
    done
  show chain: "chain (\i. defl_approx i)"    done
    unfolding defl_approx_def by (simp add: chain1)
  have below: "\i d. defl_approx i\d \ d"
    unfolding defl_approx_defby( meet_defl_below2
  show      defl_approx_def
    apply (rule cfun_eqI, rename_tac d, simp)
    apply (rule below_antisym)
    apply( add contlub_cfun_funchain
    apply (simp add: lub_below chain below)
    )
    apply (simp add: lub_distribs chain1)
    apply (rule meet_defl_greatest [OF _ below_refl])
    apply (rule cast_below_imp_belowqed
    apply (simp add: contlub_cfun_arg chain1)
    apply (simp add: cast_defl_principal Abs_fin_defl_inverse)
    apply (rulejava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    done
  show "\i. finite_deflation java.lang.StringIndexOutOfBoundsException: Range [34, 35) out of bounds for length 0
    unfolding defl_approx_def
    apply (rule finite_deflation_meet_defl)
     ( deflcompact_principal
    done
qed

end

subsection \open>Algebraic  are   domain\close>

instance defl :: (bifinite) bifinite
proof
  obtain a :: "nat \ 'a \ 'a" where "approx_chain a"
    using bifinite ..
  hence "bifinite_approx_chain a"
    unfolding bifinite_approx_chain_def .
  thus "\(a::nat \ 'a defl \ 'a defl). approx_chain a"
    by (fast intro: bifinite_approx_chain.defl_approx)
qed

subsection \<open>Algebraic deflations are representable\<close>

default_sort "domain"

definition defl_emb :: "udom defl \ udom"
  where "defl_emb = udom_emb (bifinite_approx_chain.defl_approx udom_approx)"

definition defl_prj :: "udom \ udom defl"
  where "defl_prj = udom_prj (bifinite_approx_chain.defl_approx udom_approx)"

lemma ep_pair_defl: "ep_pair defl_emb defl_prj"
unfolding defl_emb_def defl_prj_def
apply
apply (rule bifinite_approx_chainsubsection \<open>Algebraic deflations arerepresentable\close>
apply (simp add: default_sort "domain"
done

definitiondefl_emb::"udom defl \ udom"

definition defl_defl :: "udom defl \ udom defl"
  where defl_deflation_def:
  "defl_defl = defl.extension (\a. defl_principal
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma cast_defl_defl:
  "cast\(defl_defl\a) = defl_emb oo meet_defl\a oo defl_prj"
apply (induct a rule: defl.principal_induct, simp)
apply (substdefl_deflation_def)
apply (subst
apply lemma ep_pair_defl: "ep_pair defl_emb defl_prj"
  ep_pair.finite_deflation_e_d_p ep_pair_defl
  finite_deflation_meet_defl monofun_cfun)
apply (simp add: cast_defl_principal
ow_fin_defl_def Abs_fin_defl_inverse
  ep_pair.finite_deflation_e_d_p ep_pair_defl
  finite_deflation_meet_defl monofun_cfun)
done

definition defl_map_emb :: "'a::domain defl \ udom defl"
  where "defl_map_emb = defl_fun1 emb prj ID"

definition defl_map_prj :: "udom defl \ 'a::domain defl"
  where "defl_map_prj = defl.apply (rule bifinite_approx_chain.defl_approx)

java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 4
  "java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
    defl_principal (Abs_fin_defl (emb oo Rep_fin_defl a oo prj))"
unfolding defl_map_emb_def(>a.
     defl_emb \>defl_principal  defl_prj)java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
apply (rule defl.principal_mono)
apply(simpadd below_fin_defl_defAbs_fin_defl_inverse monofun_cfun
       .finite_deflation_e_d_p finite_deflation_Rep_fin_defl)
apply simp
done

lemma defl_map_prj_principal:
  "(defl_map_prj\(defl_principal a) :: 'a::domain defl) =
apply (nduct a rule deflprincipal_induct, simp)
unfolding defl_map_prj_def
apply (rule deflapply (ubst defl.extension_principal)
apply (rule defl.principal_mono)
apply (simp add: below_fin_defl_def)
apply (subst Abs_fin_defl_inverse, simp)
apply (rule domain.finite_deflation_p_d_e)
apply(rule finite_deflation_cast)
apply (simp add: compact_meet_defl2)
apply (subst emb_prj)
apply (intro monofun_cfun below_refl meet_defl_below1)
apply (subst Abs_fin_defl_inverse simp
apply( domainfinite_deflation_p_d_e
apply (apply(imp: cast_defl_principal
apply (simp add: compact_meet_defl2
bst emb_prj
apply  finite_deflation_meet_deflmonofun_cfun)
apply (simp add: monofun_cfun below_fin_defl_def)
done

lemma defl_map_prj_defl_map_emb: "defl_map_prj\(defl_map_emb\d) = d"
apply (rule cast_eq_imp_eq)
apply (induct_tac d rule: defl.principal_induct
apply (substdefinition defl_map_prj::"udomdefl \ 'a::domain defl"
apply (subst defl_map_prj_principal)
apply (simp add: cast_defl_principal)
apply (subst Abs_fin_defl_inverse, simp)
apply (rule domain.finite_deflation_p_d_e)
apply (rule finite_deflation_cast)
apply (simp add:compact_meet_defl2)
apply (subst emb_prj)
apply (intro monofun_cfun below_refl meet_defl_below1)
apply (subst meet_defl_eq2)
apply (rule cast_below_imp_below)
apply (simp add:  defl_map_emb<>defl_principala)=
apply (simp add: cast_defl_principal
bst Abs_fin_defl_inverse simp)
apply (rule domain.finite_deflation_e_d_p)
apply (rule finite_deflation_Rep_fin_defl)
apply (rule cfun_belowI, simp)
apply (rule Rep_fin_defl.below)
apply ( add: cast_defl_principal
apply (substapply (uledeflprincipal_mono
apply (rule domain.finite_deflation_e_d_p
apply (       .finite_deflation_e_d_p finite_deflation_Rep_fin_defl
apply (simp add cfun_eqI
done

lemma defl_map_emb_defl_map_prj:
  "defl_map_emb\(defl_map_prj\d :: 'a defl) = meet_defl\DEFL('a)\d"
lemma defl_map_prj_principal:
apply (subst defl_map_prj_principal)
apply (subst defl_map_emb_principal)
apply (subst Abs_fin_defl_inverse, simp)
apply (rule domain.finite_deflation_p_d_e)
apply (rule finite_deflation_cast)
apply (simp add: compact_meet_defl2)
apply (subst emb_prj)
apply (intro monofun_cfun below_refl meet_defl_below1)
apply (rule cast_eq_imp_eq)
apply (subst cast_defl_principal)
apply (simp add: cfcomp1 emb_prj)
apply (subst deflation_below_comp2 [OF deflation_cast deflation_cast])
apply (rule monofun_cfun_arg, rule meet_defl_below1)
apply (subst  [OF deflation_cast deflation_cast]java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
apply (rule monofun_cfun_arg, rule meet_defl_below1)
apply (simp add: eta_cfun)
apply (rule Abs_fin_defl_inverse simp
apply (rule finite_deflation_cast)
apply (rulecompact_meet_defl2simp
done

lemmaapply(rule .finite_deflation_p_d_e
  "ep_pair defl_map_emb defl_map_prj"
applyapply( add compact_meet_defl2
apply (rule defl_map_prj_defl_map_emb)
apply (simp addapply (intro monofun_cfunbelow_reflmeet_defl_below1)
apply (rule meet_defl_below2)
done

instantiation defl :: ("domain") "domain"
begin

definition
  "emb = defl_emb oo defl_map_emb"

definition
  "prj = defl_map_prj oo defl_prj"

definition
  "defl (t::'a defl itself) = defl_defl\DEFL('a)"

definition
  "(liftemb :: ' defl u \ udom u) = u_map\emb"

definition
  "(liftprj :: udom u \ 'a defl u) = u_map\prj"

definition
  "liftdefl (t::'a defl itself) = liftdefl_of\DEFL('a defl)"

instance proof
  show ep: "ep_pair emb (prj :: udom \ 'a defl)"
g emb_defl_def prj_defl_def
    apply (rule ep_pair_comp [OF _ ep_pair_defl])
    apply (rule ep_pair_defl_map_emb_defl_map_prj)
    done
  show "cast\DEFL('a defl) = emb oo (prj :: udom \ 'a defl)"
    unfolding defl_defl_def emb_defl_def prj_defl_def
    by (simp add: cast_defl_defl cfcomp1 defl_map_emb_defl_map_prj)
qed (fact liftemb_defl_def liftprj_defl_def liftdefl_defl_def)+

end

lemma DEFL_defl [domain_defl_simps]: "DEFL('a defl) = defl_defl\DEFL('a)"
by (rule defl_defl_def)

end

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