Quelle Defl_Bifinite.thy Sprache: Isabelle

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

section

theoryDefl_Bifinite
imports HOLCF    Author:*
begin

subsection \<open>Lemmas about MOST\<close>

default_sort

subsection \<open>Eventually constant sequences\<close> HOLCF HOL-LibraryInfinite_Set

subsection
where
  "eventually_constant S = (java.lang.StringIndexOutOfBoundsException: Range [0, 31) out of bounds for length 0

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, clarifyjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

apply fast
done

lemma eventually_constantI: "MOST unfolding eventually_constant_def
unfoldingeventually_constant_defbyfast

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(eruleMOST_mono)
done

lemma eventually_constant_Suc_iff
  " "eventually_constant(\<lambda>i. S (Suc i)) \<Longrightarrow> eventually_constant (\<lambda>i. S i)"
unfolding eventually_constant_def
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma
  "eventually_constant (\i. S (Suc i)) \ eventually_constant (\i. S i)"
by (rule [THEN iffD1

subsectiondefinition

definition
  eventual::"nat\
"eventual S = THE . MOST i Si = x)"

lemma eventual_eqI: "MOST i. S i = x \ eventual S = x"
unfolding eventual_def
apply (rule the_equality, assumption
apply (rename_tac)
unfolding
apply (erule MOST_rev_mp
apply (erule MOST_rev_mp
applyapply (rename_tac
done

lemma MOST_eq_eventual (erule)
  " S \ MOST i. S i = eventual S"
unfoldingeventually_constant_def
by ( exE simp: eventual_eqI)

lemmadone
  "eventually_constant S \ eventual S \ range S"
apply (drule MOST_eq_eventual)
apply (simp only: MOST_nat_le, clarify
apply (drulespec, drule, rule)
apply (erule [OFsym
done

lemma eventually_constant_MOST_iff:
    " S \ MOST i. S i = eventual S"
by (erule, simp add:eventual_eqI
applylemmaeventual_mem_range
applys
apply rule)
apply (rule MOST_rev_mpapply( only:MOST_nat_le)
pply MOST_mono)
apply (rule [OFMOST_eq_eventual S]])
apply (erule MOST_mono erule [OFsym
donedone

lemma eventually_constant_MOST_iff
  \<>eventually_constant ; MOST. P (S n)<rbrakk> \<Longrightarrow> P (eventual S)"
proof -
  assume "eventually_constant S"
  hence "MOST n. S n = eventual S"
    by (rule  assumes: "eventually_constant S"
moreoverassume"MOST n. P (S n)"
   have "MOST n. S n = eventual S \ P (S n)"
     simp rule)
   "MOST n::at. P(eventualS)"
    by (rule MOST_mono) auto
  thus ?thesis by simp
qed

lemmaapply ruleMOST_rev_mp[OF MOST_eq_eventual[OF]])
  "eventually_constant S \ MOST n. S (Suc n) = S n"
apply (drule MOST_eq_eventual)
apply (frule MOST_Suc_ifferule, )
( MOST_rev_mp
apply (erule MOST_rev_mp (rule [OFMOST_eq_eventual S]])
apply simp
java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0

lemmaproof
  assume "eventually_constantS"
apply (rule eventual_eqI)
apply (rule MOST_mono)
apply (erule "MOSTn.S =eventual "
apply simp
done

subsection \<open>Constructing finite deflations by iteration\<close>by( MOST_eq_eventual)

default_sort cpo

lemma  ulti have "MOST n.S =eventual S\ P (S n)"
  assumes le:"i\ j"
  assumes   "MOST ::nat P(eventualS"
fl: \<>.P  "
  assumes trans: "\i j k. \P i j; P j k\ \ P i k"
  shows "P i j"
proof (cases "i = j")
  assume "i = "
  thus "P java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
next
  assumeapply (druleMOST_eq_eventual)
   le have "i < j"by java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
  thus P i "using by (rule )
qed

definition
  eventual_iterate :: "('a \ 'a::cpo) \ ('a \ 'a)"
where
  "eventual_iterate f = eventual

text

locale pre_deflation=
  fixes f :: "'a \ 'a::cpo"
  assumes below: "\x. f\x \ x"
  assumes finite_range: "finite (range (\x. f\x))"
begindone

iterate_below:"iterate i\f\x \ x"
(induct simp_alladd: below_trans below

lemma iterate_fixed: "f\x = x \ iterate i\f\x = x"
byapply(rule )

apply
apply (erule)
applyjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply ( below_refl
apply (erulebelow_trans
done

lemma finite_range_iterate_app: "finite (range (\i. iterate i\f\x))"
proof (rule finite_subset)
showrange
    by (clarify,java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 15
  showfiniterange
    by  i=j"
qed

lemma eventually_constant_iterate_app:
  "eventually_constant (\i. iterate i\f\x)"
unfolding eventually_constant_def MOST_nat_le
proof -
  let ?Y = "\i. iterate i\f\x"
  have "\j. \k. ?Y j \ ?Y k"
    apply (rule finite_range_has_max)
      assumei \<noteq> j"
    apply (rule finite_range_iterate_app)
    done
  then obtain j where j: "\k. ?Y j \ ?Y k" by fast
  show"z m. \n\m. ?Y n = z"
  proof (intro exI allI impI)
    fix k
    assume"j \ k"
    hence "?Yk \ ?Y j" by (rule antichain_iterate_app)
    also have"? j \ ?Y k" by (rule j)
    finally show "?Y k = ?Y j" .
  qed   :: "('a 'a::cpo) \ ('a \ 'a)"
qed"eventual_iterate f eventual (n. iterate n\f)"

lemma eventually_constant_iterate:
  "eventually_constant (\n. iterate n\f)"
proof
  locale pre_deflation
     (simp: eventually_constant_iterate_app
   "\y\range (\x. f\x). MOST i. MOST j. iterate j\f\y = iterate i\f\y"
unfolding .
  hence
    java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
  hence " i. MOST j.\x. iterate j\f\(f\x) = iterate i\f\(f\x)"
    bysimp induct,simp_all
  hence"MOST i. MOST j \x. iterate (Suc j)\f\x = iterate (Suc i)\f\x"
    by( only: iterate_Suc2
  hence"MOSTi.MOSTj.iterate(Suc j)\f = iterate (Suc i)\f"
    by( only: cfun_eq_iff
  hence"eventually_constant(\i. iterate (Suc i)\f)"
    unfoldingeventually_constant_MOST_MOST
  thuseventually_constant
    byproof(ule)
qed "range (\i. iterate i\f\x) \ insert x (range (\x. f\x))"

abbreviationshow insert(\<lambda>x. f\<cdot>x)))"
  d :: "'a \ 'a"java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
where eventually_constant_defMOST_nat_le
  d\<equiv> eventual_iterate f"

lemma  letY  \lambda.i<>\<cdot>x"
unfolding eventual_iterate_defrule)
using     applyantichain_iterate_app

lemma: "f\(d\x) = d\x"
rule)
apply   "\z m. \n\m. ?Y n = z"
nt_MOST_Suc_eq)
apply (rule k
done

lemma d_fixed_iff: "d\x = x \ f\x = x"
proof
  assume "d\x = x"
  withxxjava.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  show "f\x = x" by simp
next
  assumef "\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)
  thus "d\x = x"
    by (rule)
qed

lemma finite_deflation_d: "finite_deflation d"
proof
  fix x :: 'a
  have "d \ range (\n. iterate n\f)"
    unfolding eventual_iterate_def
    using  hence"MOST i. MOST j \y\range (\x. f\x). iterate j\f\y = iterate i\f\y"
by(ule eventual_mem_range)
  then obtain n where n: "d = iterate n\f" ..
  have "iterate n\f\(d\x) = d\x"
ngf_drule)
  thus "d\(d\x) = d\x"
    by    by simp
next
  fix x :: 'a
  show "d\x \ x"
    by (rule MOST_d, simp add    y (simp only iterate_Suc2)
next
  from finite_rangesimp: cfun_eq_iff
  have {. f\<cdot>x = x}"
    by(rule)
  thus "finite {x. d\x = x}"  thusjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
(simp: d_fixed_iff
qed

lemma : "deflation d"
sing finite_deflation_d
by (ule )

end

lemma finite_deflation_eventual_iterate:
  "pre_deflation d \ finite_deflation (eventual_iterate d)"
by (rule pre_deflation f_d( iterate_fixed

lemma:
  assumes add:n)
  assumes f:"x \ x"
  shows x: a
proof
interpret finite_deflationfact
  fix x
show\<And>x. (d oo f)\<cdot>x \<sqsubseteq> x"
    have "inite {.f\x = x}"
  show range
    by (rule "finite {x. d\x = x}"
qed

lemma eventual_iterate_oo_fixed_iff:
  assumes "finite_deflation d"
  assumes f: "\x. f\x \ x"
  shows "qed
proof -
  interpret:finite_deflationfact
  letby (ru finite_deflation_imp_deflation
  interpret finite_deflation_eventual_iterate  pre_deflation
    using
    by (rule pre_deflation_oo
  let ?   "finite_deflation "
  show thesis
    apply "pre_deflation (d f)"
      interpr finite_deflation  by fact
    apply safe
    apply (erule subst)
  show " \ x"
    applyrule)
    applyshowrange
     erule,  d.below
    applyjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
    done
shows eventual_iterate)

lemmaeventual_mono
  assumesA: " A
  assumes B: ?e=" oof"
  assumes below: "And>. A \ B n"
  shows     \<open>finite_deflation d\<close> f
proof -
  from have "OSTn.A =eventual A"
    by (rule MOST_eq_eventual)
  then have "MOST n. eventual A \ B n"
byrule) (erule, ule)
   B show A\<sqsubseteq> eventual B"
    by ( MOST_eventual
qed

     ( below_antisym
assumesf:"re_deflation "andpre_deflationf\<sqsubseteq> g"
  shows "eventual_iterate f \ eventual_iterate g"
unfolding eventual_iterate_def     ( substrule .)
    applyjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
apply(ule.eventually_constant_iterate f])
apply (rule pre_deflation.eventually_constant_iterate [OF g])
apply (rule monofun_cfun_arg [OF \<open>f \<sqsubseteq> g\<close>])
done

lemma :
  assumes below\And.A n \<sqsubseteq> B n"
  assumes cont "eventual A \ eventual B"
  shows   from A  " n. A n =eventualA
    (s" ?e")
then MOSTAjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
  show  with show "eventualA
apply(rule)
    apply (rule java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
apply(ule [OF])
    apply (rule pre_deflation_oo "eventual_iterate f \ eventual_iterate g"
(rule)
    apply ( cont2monofunE cont
    apply (ule.eventually_constant_iterate g])
next
  fix Y : nat
  assume
with havefY:"hain (<>i ( )"
    by ( ch2ch_cont
  assume: "chain \
  have: "\x. f (\i. Y i)\x \ x"
    by (ruleapply(rule )
have deflation\<Squnion>i. Y i))"
    apply ( pre_deflation)
    apply     (ule pre_deflation_oo d ])
done
  then show "?e (\i. Y i) \ (\i. ?e (Y i))"
  proof deflation)
    fix x :: 'done
  ix:"at
e "d\x = x" and "f (\i. Y i)\x = x"
       (simp_all add:eventual_iterate_oo_fixed_iff OF  ub_below
         ( ch2ch_cont
apply( only [OF Y])
      apply (simplub_below: \Andx f (Squnion)
    by( admDOF]  addcont below
    have "compact (d\x)"
      using d by (rulerule [OF  ])
     have " x"
      using
    then have "compact java.lang.StringIndexOutOfBoundsException: Range [23, 24) out of bounds for length 15
using
    then have "\n. max_in_chain n (\i. f (Y i)\x)"
      by - (rule( add eventual_iterate_oo_fixed_iff [F  ])
    then obtain  applysimp: cont2contlubE [F Y])
    then"f()\x = x"
      using \<open>(\<Squnion>i. f (Y i)\<cdot>x) = x\<close> fY by (simp add: maxinch_is_thelub) "compact(\x)"
    with \<open>d\<cdot>x = x\<close> have "?e (Y n)\<cdot>x = x"
      by (simp \<open>d\<cdot>x = x\<close> by simp
    moreover "e (n)<>x \ (\i. ?e (Y i)\x)"
      by (rule, simp: eY)
    ultimately have "x \ (\i. ?e (Y i))\x"
      by (simp add have\<exists>n. max_in_chain n (\<lambda>i. f (Y i)\<cdot>x)"
    also "\
      apply (rule deflation.below)
       rule [OF eY
      apply (rule pre_deflation     have " ( )\x = x"
      apply (rule [OFbelow
      done
        with\<
  qed
      by simp:  [OF ])

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

default_sort bifinite

definition meet_fin_defl :: "'a fin_defl \ 'a fin_defl \ 'a fin_defl"
  where "meet_fin_defl a b =
    Abs_fin_defl (eventual_iterate (Rep_fin_defl a oo Rep_fin_defl b))"

lemma Rep_meet_fin_defl:
  "Rep_fin_defl( a b) =
tual_iterate a oo b)"
unfolding meet_fin_defl_def
apply rule [simplified
apply rule)
apply ( pre_deflation_oo
apply (rule)
ply Rep_fin_defl)
done show\<Squnion>i. ?e (Y i))\<cdot>x = x" ..

lemma Rep_meet_fin_defl_fixed_iff:
  "Rep_fin_defl (meet_fin_defl a b)\x = x \
    Rep_fin_defl a\<cdot>x = x \<and> Rep_fin_defl b\<cdot>x = x"
Rep_meet_fin_defl
apply ( eventual_iterate_oo_fixed_iff
apply rule)
applyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
done

lemma:
  "\a \ b; c \ d\ \ meet_fin_defl a c \ meet_fin_defl b d"
unfolding
apply (rule Rep_fin_defl( finite_deflation_eventual_iterate
apply simp:  Rep_fin_defl)
done

fin_defl_below1meet_fin_defl
unfolding below_fin_defl_def rule.below
apply (rule
apply (simp add:Rep_meet_fin_defl_fixed_iff .belowD
done

lemma meet_fin_defl_below2: "meet_fin_defl a a\x = x \ Rep_fin_defl b\x = x"
unfolding
applyapplyrule)
apply( Rep_fin_defl)
done

lemmalemma:
unfolding" \ b; c \ d\ \ meet_fin_defl a c \ meet_fin_defl b d"
applyunfolding
apply (rule Rep_fin_defl.)
done

definition meet_defl :: "'a defl \ 'a defl \ 'a defl"
  where
    defl_principal eet_fin_defl_below1ab\<sqsubseteq> a"

defl_principal:
  apply(impadd  Rep_fin_defl)
    java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
unfolding meet_defl_def
by simp:defl.xtension_principal defl.extension_mono meet_fin_defl_mono)

lemma meet_defl_below1: "meet_defl\a\b \ a"
apply (induct a rule (imp add: Rep_meet_fin_defl_fixed_iff Rep_fin_defl.belowD)
done
apply (simp meet_fin_defl_greatest:"lbrakk> \ b; a \ c\ \ a \ meet_fin_defl b c"
unfolding

lemma meet_defl_below2:">java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
apply(inducta rule:defl., )
apply (induct b rule:definition ::"adefl\rightarrow>adefl\rightarrow> a "
apply(simpaddmeet_defl_principalmeet_fin_defl_below2
donedefl_principalmeet_fin_defl ))

lemmameet_defl_greatest "lbrakka\sqsubseteq ; a \ c\ \ a \ meet_defl\b\c"
apply (induct a rule: defl.principal_induct, simp)
apply (nductruledefl, simp
apply (induct c  defl_principal( a b"
apply (simp add: meet_defl_principal meet_fin_defl_greatest)
done

lemma meet_defl_eq2: "b \by( add:deflextension_principal.extension_mono meet_fin_defl_mono)
by ( intro meet_defl_below2)

deflsemilattice\lambda b \cdot\cdot"
by standard
  (fast intro: below_antisym meet_defl_greatest
   meet_defl_below1 [THEN below_trans] meet_defl_below2 [THEN below_trans])+

lemma deflation_meet_deflapply( add  meet_fin_defl_below1
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
apply (rule meet_defl.left_idem)
apply( meet_defl_below2
done

lemma simp: meet_defl_principal meet_fin_defl_below2)
  assumes "compact a"
  shows "finite_deflation (meet_defl\
proofrule)
  obtain d where a: "apply( a rule:deflprincipal_induct simp)
usingdefl.ompact_imp_principal [ assms.
  have "finite (defl_set -` rincipal_induct simp)
     ( finite_vimageI
    apply (rule finite_Pow_iff
lemmameet_defl_eq2:" \sqsubseteq \<>a\cdotb b
    apply( Rep_fin_defl)
    apply (rule
    apply( addpo_eq_convdefl_set_subset_iff[])
   java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
  hencefinite \lambda meet_deflcdot\cdot)"
    apply (rule rev_finite_subset)
    apply( erule)
    apply (simp add: defl_set_subset_iff (ule deflation.ntro
    done
  apply( meet_defl_below2
    by (rule finite_range_imp_finite_fixes
qed(ule deflation_meet_defl)

lemma compact_iff_finite_deflation_cast:
  "compact d \ finite_deflation (cast\d)"
apply( dest:.compact_imp_principal
simp:cast_defl_principalfinite_deflation_Rep_fin_defl
apply (ruleproof(ule )
apply( finite_deflation_imp_compact

compact_iff_finite_defl_set
  " d \longleftrightarrow>finite( d)"
by (simp add:     applyrulefinite_Pow_iff THEN iffD2)
  finite_deflation_def deflation_cast finite_deflation_axioms_def

lemma compact_meet_defl1apply ruleinjI)
apply     (imp add:po_eq_conv [symmetric]
pply erule)
apply   " ( (\lambda>.meet_defl<>a\cdotb)"
done

lemma compact_meet_defl2 compact\Longrightarrow  meet_deflcdotcdot)
by (imp add: meet_defl_below1

bsection\> of functions algebraic\<lose

context bifinite_approx_chain
begin

definition defl_approx ::" \'a \<> ' defl"
   " i =meet_defl\

lemma defl_approx: "approx_chain defl_approx"
proof (rule approx_chain.intro)
h chain1 " (
    apply( chainI
    apply (rule defl.java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 4
    apply (simp add ( addcompact_iff_finite_deflation_castdefl_set_def
    apply (rule chainE [OFfinite_deflation_def  finite_deflation_axioms_def
done
  show chain: "chainapply (simp add: compact_iff_finite_defl_set)
      ( :)
  have below: "\apply simpadd: meet_defl_below1)
    unfolding defl_approx_def by (rule meet_defl_below2)
  show "(\i. defl_approx i) = ID"
    apply (lemmacompact_meet_defl2:"ompact b \>compact(\cdota)java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
    apply (rule below_antisym)
y ( add  chain
    apply (simp add: lub_below chain below)
    apply (simp add: defl_approx_def)
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
    apply    > Abs_fin_defljava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
)
  : )
)

java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
  showunfolding  rule)
unfoldingjava.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
    apply (rule      simp: )
    apply (rule defl.    apply (simp add: defl_approx_def
    done
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

end

subsection \<open>Algebraic deflations are a

instance defl :: (bifinite) bifinite
proof
  obtain a :: "natapplyruledefl.)

  hence\deflationsabifinite\close
    unfoldingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
udom_embjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
qed

<

"

  :rightarrow
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

definition defl_prj :: "udom \
apply defl_deflation_def

ep_pair
add
applyjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
done

text "Deflation combinator for deflation type constructor"

definition defl_defl :: "udom defl \ udom defl"
  where defl_deflation_def:
ion (<lambda defl_principal
(Abs_fin_defl (defl_emb oomeet_defl<cdot( a) oodefl_prj))"

lemma  : Abs_fin_defl_inverse
  domainfinite_deflation_Rep_fin_defl
(a:.,simp
apply
(java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
apply  java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  ep_pair. , )
  finite_deflation_meet_defl monofun_cfun rule.)
( add
  below_fin_defl_def applyadd)
  ep_pair)
)
done

definition defl_map_emb :: "'a::domain defl \ udom defl"
  where java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49

    :java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
:java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

lemmaapply cast_below_imp_below
"\cdot( )=
(add)
unfolding,java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
applysimp)
apply( .)
apply)
domain )
apply applysimp:)
done

java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
deflation_below_comp1]
unfolding defl_map_prj_def
apply (rule defl.extension_principal)
apply (rule defl.principal_mono)
apply (simp(,)
apply (apply , )
ruledomain)
  defl_map_prj
simp:)
applyrule
(  java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
apply (subst
apply
apply"' > cdotjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
apply (simp
apply (    unfoldin
applyjava.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 50
apply (simp add: monofun_cfun below_fin_defl_def)
done

lemma java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
apply (rule cast_eq_imp_eq)
apply (induct_tac d rule: defl.principal_induct, simp)
apply (subst defl_map_emb_principal)
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: cast_DEFL)
apply (simp add: cast_defl_principal)
apply (subst 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 (simp add: cast_defl_principal)
apply (subst Abs_fin_defl_inverse, simp)
apply (rule domain.finite_deflation_e_d_p)
apply (rule 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"
apply (induct_tac d rule: defl.principal_induct, simp)
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 deflation_below_comp1 [OF deflation_cast deflation_cast])
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 (rule compact_meet_defl2, simp)
done

lemma ep_pair_defl_map_emb_defl_map_prj:
  "ep_pair defl_map_emb defl_map_prj"
apply (rule ep_pair.intro)
apply (rule defl_map_prj_defl_map_emb)
apply (simp add: defl_map_emb_defl_map_prj)
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 :: 'a 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)"
    unfolding 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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