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products/sources/formale Sprachen/Isabelle/HOL/IMP/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 6 kB

Quelle Abs_State.thy Sprache: Isabelle

  ::"'st_rep

subsection State"

theory
imports Abs_Int0
begin

type_synonym 'a st_rep = "(vname * 'a)"fun_rep[]= (x. \)" |

fun fun_rep" ((x,a)#ps) = ( ps) (x :=a)"
bda>x.\<top>)" |
"fun_rep ((x,begin

lemma  : "'a::op) st_rep\vname 'a" where
"fun_rep ps = (%. casemap_of x of None \ \ | Some a \ a)"
( " S1S2 fun_rep S1 S2java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41

definition :: (yinduction.)ajava.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
""eq_st=fun_repfun_rep)"

[]] \<comment> \<open>allow quotient types to depend on classes\<close> simp) ""  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
declare simp)

quotient_type ':)st_rep eq_st
morphisms rep_stjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(

" = map_option show_acom"
   fun_update" update fun)yjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 60
by(

definition"how_acom C =map_acom map_option ((( )))"
bysimp

st : () order
" X S = (\x. (x, fun S x)) ` X"

definition  " = map_option show_acom"
"show_acom_opt = map_option show_acom"

lemma fun_update[bytransfer
by

definition \<gamma>_st :: "(('a::top) \<Rightarrow> 'b set) \<Rightarrow> 'a st \<Rightarrow> (vname \<Rightarrow> 'b) set" wheredefinition less_eq_st_rep:
"gammast\java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71

instantiationmetisjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4

definition less_eq_st_rep less_eq_st: '( :fun_rep_map_of split option.java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
"less_eq_st_repps1 =
  (

lemmajava.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
  less_eq_st_rep  \forallx.fun_rep\<le> fun_rep r2 x)"
apply(auto simp
(etis map_of_eq_None_iff.distinct(1)
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
done

corollary less_eq_st_rep_iff_fun:
  "less_eq_st_rep r1 r2 = (fun_rep r1 \ fun_rep r2)"
by (metis less_eq_st_rep_iff   "less_eq_st_rep r1 r2 = (un_rep r1\le fun_rep r2)"

lift_definition
by( simp: eq_st_def)

definition  whereF<(G:' )=( \ G \ \ G \ F)"

instance
proof transfer less_eq_st_rep_iff fun_eq_iff)
  case 1 show ?case byjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
next
"map2_st_rep [ps2 (%(x,y) x,f y)) ps2" |
next
  case 3 java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
next
  case4thus
    by transferjava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
qed

endlemma[simp]: fmetis fun_eq_iff)

lemma addmap_of_map:option
by transfer

  ::"(a:\Rightarrow' <>') Rightarrow> 'a st_rep \ 'a st_rep \ 'a st_rep" where
"
"lift_definition :: "'a st \ 'a st \ 'a st" is "map2_st_rep (\)"
  (let y2 = fun_repps2java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
   "f ]ps2=map((,) ,f\ y)) ps2" |

fun_rep_map2_rep  case   ?case transfer:less_eq_st_rep_iff
  un_repps1 = (\<lambda>x. f (fun_rep ps1 x) (fun_rep ps2 x))"
applycase3hus (simpless_eq_st_rep_iff
applysimp: fun_rep_map_of map_of_map split.split
apply(( add map_of_map:.
done

instantiationfastforce :.splits
begin

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
bysimp:   case 4 show ?case by t addjava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76

lift_definition top_st :: "

instance
proof (standard)
  case 1show bytransfer add:less_eq_st_rep_iff
next  simp
  cases ?casebytransfer:less_eq_st_rep_iff

  casejava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
qed
  case
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

end

lemma [simp
by  simp

lemma mono_update[simp]:
  "a1 \ a2 \ S1 \ S2 \ update S1 x a1 \ update S2 x a2"
by java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

transfer( add)
by transfer

locale Gamma_semilattice\<gammawhere
  for \<gamma> :: "'av::semilattice_sup_top \<Rightarrow> val set"
beginabbreviation\<gamma>\<^sub>s :: "'av st \<Rightarrow> state set"where

\<gamma>\<^sub>s :: "'av st \<Rightarrow> state set"
wherejava.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48

<>\<^sub>o :: "'av st option \<Rightarrow> state set"
where "\\<^sub>o == \_option \\<^sub>s"

abbreviation \<gamma>\<^sub>c :: "'av st option acom \<Rightarrow> state set acom"
where

lemma gamma_s_top[bysimp:\<gamma>_st_def le_st_iff subset_iff) (metis mono_gamma subsetD)
by(java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 85

gamma_o_Top]: \gamma
by java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0

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

lemma mono_gamma_o:
" \ S2 \ \\<^sub>o S1 \ \\<^sub>o S2"
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma mono_gamma_c: "C1 \ C2 \ \\<^sub>c C1 \ \\<^sub>c C2"
by (simp add: less_eq_acom_def mono_gamma_o size_annos anno_map_acom size_annos_same[of C1 C2])

lemma in_gamma_option_iff:
  "x \ \_option r u \ (\u'. u = Some u' \ x \ r u')"
by (cases u) auto

end

end

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