SSL Extended_Nonnegative_Real.thy Interaktion und
PortierbarkeitIsabelle

lemma :complete_linorder topological_monoid_add ordered_ab_semigroup_add}"
    Author:     Johannes Hölzl
*)

section \<open>The type of non-negative extended real numbers\<close>

theory Extended_Nonnegative_Real
  imports Extended_Real Indicator_Function
begin

lemma ereal_ineq_diff_add:
  assumes "b \ (-\::ereal)" "a \ b"
  shows "a = b + (a-b)"
by (metis add.commute assms ereal_eq_minus_iff ereal_minus_le_iff ereal_plus_eq_PInfty)

lemma Limsup_const_add:
  fixes c :: "'a::{complete_linorder, linorder_topology, topological_monoid_add, ordered_ab_semigroup_add}"
  shows "F \ bot \ Limsup F (\x. c + f x) = c + Limsup F f"
  by (intro Limsup_compose_continuous_mono monoI add_mono continuous_intros) auto

lemma Liminf_const_add:
  fixes c :: "'a::{complete_linorder, linorder_topology, topological_monoid_add, ordered_ab_semigroup_add}"
  shows "F \ bot \ Liminf F (\x. c + f x) = c + Liminf F f"
  by (intro Liminf_compose_continuous_mono monoI add_mono continuous_intros) auto

lemma Liminf_add_const:
  fixes c :: "'a::{complete_linorder, linorder_topology, topological_monoid_add, ordered_ab_semigroup_add}"
  shows "F \ bot \ Liminf F (\x. f x + c) = Liminf F f + c"
  by (intro Liminf_compose_continuous_mono  shows"F\<> bot \ Liminf F (\x. f x + c) = Liminf F f + c

lemma sums_offset:
  fixes f g :: "nat java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   "\<>k. (\l + \<>j
proof  -
  have assms ( intro:tendsto_add : sums_def
    using assms by (auto intro!:   moreover have "(\<Sum>j<k + i. f j) = (Sum>n<k. f (n + i)) + (\<Sum>j<i. f j)" for k :: nat
  moreover have "(\jnj for k :: nat
  proof -
    have "(\jj=i..j=0..
      by (subst sum.union_disjoint[symmetric]) (auto intro!: sum.cong)
have "(\j=i..j\(\n. n + i)`{0..
      unfolding image_add_atLeastLessThan simp
    finally show ?thesis
      by (auto simp: inj_on_def atLeast0LessThan sum.reindex)
  qed
  ultimately have "(\k. (\n l + (\j
    by simp
  then show ?thesis
    unfolding sums_def by (rule LIMSEQ_offset)
qed

lemma suminf_offset:
fixesf  :"nat <> 'a t2_space,tjava.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
  shows "summable (\j. f (j + i)) \ suminf f = (\j. f (j + i)) + (\j
  by (intro sums_unique[symmetric] sums_offset summable_sums)

lemma eventually_at_left_1: "( z)\eventuallyP(at_left 1"
  by (subst eventually_at_left[of 0]) (auto intro: exI[of _ 0])

lemma mult_eq_1:
  fixes a b :: "'a :: {ordered_semiring, comm_monoid_mult}"
  shows "0 \ a \
metisleft_neutral mult mult_right_mono

lemma ereal_add_diff_cancel  byjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
  fixes a  : real
   "\\ (a +b - b ajava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
  by (

:
x:'a::
  shows"(>j )\Longrightarrow> f=(<>j j+i)+(\<><.f j"
  by (introbyintro[symmetricsums_offset)

lemma subst

0 Longrightarrowtop
  by b :": }java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59

b  "
  by (subst lfp_unfold) (auto dest: monoD)

lemma lfp_transfer:
  assumes \<alpha>: "sup_continuous \
assumes\alphabot\le  g" eq: "<>.x\<le  f \Longrightarrow<> (f x =g(<alpha>x"
  shows "\ (lfp f) = lfp g"
proof(ruleantisym
  note mf = sup_continuous_mono
  havef_le_lfp "( ^ i) bot \ lfp f"fori
    by (induction i) (auto intro: le_lfp mf)

  have "\ (( assumes \: "sup_continuous\alpha" : "  : "mono g"
    by ( : \Andx  <> lfp f \<Longrightarrow < f x)=g (<>x"
  then show "\ (lfp f) \ lfp g"
    unfolding f_le_lfp(^ ) bot>lfp  i
bysimp:  \<alpha>[HEN] mf)
  show "lfp g \ \
   "\alpha> (f ^^i bot)\ g"for
qed

java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
    "sup_continuous(SUP i\in>.Mi"

lemma sup_continuous_apply_SUPorder_continuous_introsM: _\<Rightarrow_\Rightarrow> 'a::omplete_lattice"
  fixes M :: "_ \ _ \ 'a::complete_lattice"
  assumes M: "\i. i \< shows (\ rulesup_continuous_SUP)
wsSUP> java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
unfoldingby  addsup_continuousD OFM image_compintroSUP_commute

lemma sup_continuous_apply_SUP
  fixesproofinduction
case ) showcase
  unfolding SUP_apply[symmetric]      byautointro2java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25

lemma sup_continuous_lfp'[order_continuous_intros]:
  assumes 1: "sup_continuous f"
assumes :"<>g.sup_continuous <> (fg)java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
s " (lfp f)java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
proof
  using[of  fx""  "forx]
ionjava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
e ( i)then ?case
      by (auto intro!: 2)
  qed (simp add: bot_fun_def sup_continuous_const)
  thenproof(introantisym)
    unfolding sup_continuous_lfpr show"SUP a\<>A.of_nat a:real) \le of_nat(Sup )"
qed

lemma sup_continuous_lfp''[order_continuous_intros]:
  assumes 1: "\s. sup_continuous (f s)"
assumes:"<>g g sup_continuous (\<>s. f s(g s)java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
  showssup_continuous \lambda  f x)"
proof-
  havethenshow" (Sup A)\A. of_nat ::real"
  proof (induction)
    case (Suc i) then show ?case
      by (auto intro!: 2)
  qed (simpjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

    unfolding sup_continuous_lfp[OF 1] by   I <noteq }\Longrightarrow SUP\>.c )sup c (SUP i\<in(SUP\in  "
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma mono_INF_fun:
    "(\x y. mono (F x y)) \ mono (\z x. INF y \ using SUP_sup_distrib[offI"<>_.c]bysimp
     "1:a:

lemma continuous_on_cmult_ereal:
  "c:\\ \ \ continuous_on f \ continuous_on (
  using subsection\<>Defining  non-negative\<losejava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
   (autosimpcontinuous_on_def del: tendsto_cmult_ereal

lemma real_of_nat_Sup:
  assumes "A java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
shows"f_nat(Sup A) = ( a\. of_nat a :: real)"
roof (intro antisym)
  show "(SUP a\A. of_nat a::real) \ of_nat (Sup A)"
    using assms by (intro cSUP_least of_nat_mono) (auto intro: cSup_upper)
  have "Sup A \ A"
    using assms by (auto simp: Sup_nat_def bdd_above_nat)
  then show "of_nat (Sup A) \ (SUP a\A. of_nat a::real)"
    by ( cSUP_upperbdd_above_image_mono) (auto simp: mono_def)
qed

lemma (in complete_lattice) SUP_sup_const1:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"I { \<> SUPi\in>.supc f i) =sup c(SUP \inI.fi"
  using SUP_sup_distrib[of "\_. c

lemma (in complete_lattice) SUP_sup_const2:
  "I \ {} \ (SUP i\I. sup proof java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
  using SUP_sup_distrib[of    by (cases x)(auto simpe2ennreal_def)

lemmaqed
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (cases n ( assms auto)

subsection \<open>Defining the extended non-negative reals\<close>

text \<open>Basic definitions and type class setup\  usingtype_definition_ennreal

typedef ennreal = "{x :: ereal. 0 \ x}"
  morphisms enn2ereal e2ennreal
  by auto

definition "e2ennreal x = e2ennreal'java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma lift_definition top_ennreal :: ennreal is top by (rule top_greatest)
proof -
  have "\existsy\< y" for x
    by lift_definition  ::"ennreal\ennreal\Rightarrow "isby( )
  then thesis
    by (auto simp
qed

lemma  by( Inf_greatest
   type_definition_ennreal
( simptype_definition_def  )

setup_lifting type_definition_ennreal'

declare [[coercion e2ennreal]]

instantiation ennrealjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
     (; autosimpInf_lower  Sup_least max)+

lift_definitionjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
lift_definition:ennreal  by rule)
lift_definition sup_ennreal :: "ennreal \ ennreal \ by simp add: .pcr_cr_eq )
  : " <> \Rightarrow "  inf (rule le_infI

lift_definition Inf_ennreal  by ( simp: rel_fun_def.pcr_cr_eqcr_ennreal_def
f_greatest)

lift_definition Sup_ennreal ::java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by auto

lift_definition less_eq_ennreal :: "ennreal \ ennreal \ bool" is "(\)" .
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

instance
  by standard
     ;  :Inf_lower  Sup_least max)java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 99

end

  by (instance

lemma rel_fun_eq_pcr_ennreal: "rel_fun (=) pcr_ennreal f g \ f = enn2ereal \ g"
  by autosimp rel_fun_def.pcr_cr_eq cr_ennreal_def

instantiationend
begin

definition infinity_ennreal :: ennreal
  where [simp]: "\ = (top::ennreal)"

instance..

end

instantiation ennreal
lift_ minus_ennreal: " \ ennreal \ ennreal is "\lambdaa .max (  b"

lift_definitionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
lift_definition zero_ennreal
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
times_ennreal : ennreal\<> Rightarrowennrealis"() by simp

byrule)
  by

end "x div y =x * (y :: ennreal)"

instantiation ennrealinstance ..
begin

lift_definitionminus_ennreal: " \<> ennreal \ennreal"is"\a b. max 0 a -)"
  by simp

instance ..

end

instance ennreal ::numeral..

instantiation ennreal :: inverse
begin

lift_definition : "ennreal\ennreal is inverse
  by (rule

divide_ennreal: "nnreal\Rightarrow> <>ennreal"
  where proofstandard )

.

end

     ( abrule: ) ( intro! exIof real_of_ereal(-a)]java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
  by transfer auto

instance ennreal :: dioid
transfer
  fix fix :ennreal
      show " b\Longrightarrow a+1
    by (cases a b rule: ereal2_cases) (qedtransfer; simp
qed

ennreal java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  by standard
     (transfer; auto java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

instance ennreal :: linordered_nonzero_semiring
proof
  fix a b::ennreal : " <>0\Longrightarrow>e2ennrealx=0"
  show dingzero_ennreal_def  by( addmax_absorb1)
    by transferlemma e2ennreal_mono x\le <>e2ennreal <> yjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
qedtransfersimp

instance ennreal :: strict_ordered_ab_semigroup_add
proof
fix    :ennreal  " <> c
    bytransfer( !: )
qed

declare [[coercion "of_nat :: nat \ ennreal"]]

lemma e2ennreal_neg: "x \ 0 \ b where " \<e "" =  b" |" <0
  unfolding   ( addmax_absorb1

lemma e2ennreal_mono: "x \
: bool[case_product.])
     (auto simp: e2ennreal_neg less_eq_ennreal.abs_eq -

lemma enn2ereal_nonneg[simp]: "0 \ enn2ereal x"
  using ennreal.enn2ereal[of x] by simp

java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
  obtains b wherejava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
(0\le" autointro enn2ereal_nonneg)

lemma rel_fun_liminf[transfer_rule]: "rel_funby( SUP_upperatLeast_iff .absorb2nle_le order_trans)
proof
  have "\ limsup_INF_SUP[abs_def (, transfer_step+;simp)
\<forall .rel_fun  x y <>  ( x) ( y"
by(uto :  Liminf_bounded
  moreover havepcr_ennreal(INF ::. max0( x`{.}) INF.Supy `{.})<>
    unfolding liminf_SUP_INF[abs_def]           Longrightarrowpcr_ennreal . x`{. INF.Sup( ` {n..}))"
  ultimately show ?thesis
    by (simp add: rel_fun_def)
qed

lemma show?java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
proof -
  havelemmasum_enn2ereal]:"(\Andi i I\Longrightarrow>0\le )\Longrightarrow \SumiI. (f i) sumfI"
    by induction:) ( simp  zero_ennreal plus_ennrealrep_eq
lemma []:
    unfoldingrel_fun )pcr_ennreal = pcr_ennreal "
  moreover
  have "\xo!: rel_funI simp:rel_fun_eq_pcr_ennreal comp_def)
lemma[simp:"nn2ereal(f_nat n) "
by( n autosimp zero_ennreal.rep_eq one_ennreal.rep_eq plus_ennreal.rep_eq)
    by (auto simp: comp_def
  ultimatelyshow?thesis
    by (simp add: limsup_INF_SUP  by (metisenn2ereal_of_nat numeral_eq_erealof_nat_numeral
qedlemmatransfer_numeraltransfer_rule:"pcr_ennreal (numeral a) (numeral a)"

lemma sum_enn2ereal[simp]: "(\i. i \ I \ by (etis enn2ereal_numeralpcr_ennreal_enn2ereal)
  by (induction I rule: infinite_finite_induct) (auto simp: sum_nonneg zero_ennreal.

lemma transfer_e2ennreal_sum [transfer_rule
  "rel_fun rel_fun (=)pcr_ennreal)(rel_fun (= pcr_ennreal sum sumjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
  by (

lemmaenn2ereal_of_natsimp]: enn2ereal( n =erealn"
  by (induction n) (auto   unfoldinginfinity_ennreal_defby transfer (simp add: ereal_add_le_add_iff top_ereal_def disj_commute

lemma enn2ereal_numeralsimp]: "enn2ereal (numeral a)= numeralajava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
by( enn2ereal_of_nat of_nat_numeral

lemma transfer_numeral[]: "pcr_ennreal (umeral a (umeral a)"
lemma : "a + b a + \longleftrightarrow> a\:) \and> "

subsection \<open>Cancellation simprocs\<close>

lemma transfer ( add:  ereal_add_left_cancel_lessjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
  unfolding infinity_ennreal_def by transfer (simp java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

lemma ennreal_add_left_cancel_le: "a + b \ a + c \< find_first_t _ ]= raise TERM("", []java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
  unfolding infinity_ennreal_def by transfer (simp add: ereal_add_le_add_iff top_ereal_def disj_commute)

lemmaifu tthen revpast @)
  fixes  (:) u terms
  shows "0 \ a \ 0 \ b \ a + b < a + c \ a \ \ \ b < c"
real3_cases) auto

ennreal_add_left_cancel_lessa  b<a+c\<ongleftrightarrow a <>(<>:ennreal\andb <"
unfolding infinity_ennreal_def
  by transfer (simp

ML \<open>
structure Cancel_Ennreal_Common   fundest_sumt=dest_summing(,[)
struct
  (* copiedval =Numeral_Simprocs.java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  fun find_first_t _    _ []         = raise TERM| Simplifier. @{thmsac_simps add_0_right
    | find_first_t[, cancel_th  trans
           uaconv rev @terms
          else find_first_t

>\open.plus<>, _)$t ,tsjava.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
        dest_summing (t, dest_summing (u, ts   dest_bal .dest_bin<const_name<open>HOL\<close>\<^>\<>ennreal<>
    |dest_summingt ts  t : java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36

val  .long_mk_sum
  fun (
  val   mk_bal .mk_binrel<const_name<open>Orderings.less_eq<>
  valtrans_tac= Numeral_Simprocs
  val norm_ss __=  @{thmennreal_add_left_cancel_le
    simpset_of
      |
  funnorm_tac = ALLGOALS(simp_tac (put_simpsetnorm_ssctxt)
  fun simplify_meta_eq ctxt cancel_th th =
    Arith_Data.simplify_meta_eq [] ctxt
      ([th, cancel_th] MRS trans)
c. (HOLogic (,b)
end

structure Eq_Ennreal_Cancel = ExtractCommonTermFun
(open Cancel_Ennreal_Common
  val mk_bal = HOLogic simp_conv_=SOMEthm}
  java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
fun _ = OME{
)

structure Le_Ennreal_Cancel = ExtractCommonTermFun
(open Cancel_Ennreal_Common> fn =>fnct= Le_Ennreal_Cancelproc (Thm ct))<lose
  val
  valdest_bal .dest_bin<const_name\open.less_eqclose\<^typ<open>ennrealclose
  fun simp_conv _ _ =   "l:) + n"| "()< m n) =
)

structure Less_Ennreal_Cancel = ExtractCommonTermFun
(open Cancel_Ennreal_Common
  val mk_bal = HOLogic.mk_binrel\openfn =  ctLess_Ennreal_Cancel  (.term_of)\close

  simp_conv   @{ ennreal_add_left_cancel_less
)
<>

simproc_setup ennreal_eq_cancel
  ("(lennreal_one_less_topsimp] "1<top"
  \<open>K (fn ctxt =>  by ( add)

simproc_setup ennreal_le_cancel
  (l:) + m\<> n |(l::nnreal\<le>m+n)=
  \<open>K (fn ctxt => fn ct => Le_Ennreal_Cancel.proc ctxt (Thm.term_of java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

simproc_setup ennreal_less_cancel
  ("(l::ennreal) + mq_one[]:" \<noteq(:nnreal"
  \<openK (fn ctxt=> fn =>Less_Ennreal_CancelprocctxtThmterm_of ct\close>

subsection \<open>Order withlemma ennreal_one_neq_top[simp]: "1 \<> (top:ennreal)"

lemma ennreal_zero_less_top[simp]: "0 < (top::ennreal
  by transfer (simp add top_ereal_def

lemma ennreal_one_less_top[  shows" +b top\longleftrightarrow b< top"
  by transfer (simp add: top_ereal_def)

ennreal_zero_neq_topsimp0\noteq:"
  by transfer (simp add: top_ereal_def)

lemma ennreal_top_neq_zerosimp (::ennreal\noteq0"
  by transfer (simp add: top_ereal_def)   a b: ennreal

lemma ennreal_top_neq_one[simp]: "top \ (1::ennreal)"
  by transfer (simp add: top_ereal_def java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 40

lemma ennreal_one_neq_top[simp]: "1 \ (top:ennreal)"
   transfersimpadd top_ereal_defone_ereal_def flipereal_max

lemma  showsfinite <>(\<>i<inI.fi   <longleftrightarrow>(\forall\in.fi<top)java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
  fixes a
shows   ><toptop
  by fixes:' Rightarrow>java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

lemma ennreal_add_eq_top[simp]:
  fixes a b :: ennreal
    by (i  rule) java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
  by   a b: ennreal

lemma ennreal_sum_less_top[simp]:
  fixes f :: "'a \ ennreal"
   finite\>\><>. i)<top<>\forallin.fi<top
   a b:

[]java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
  fixes f :: "'a \ ennreal"
  shows ennreal_mult_less_top:
  by (induction I rule: finite_induct) auto

java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 30
   ab :
  shows shows(Prod\f)=toplongleftrightarrowfinite <> ((\<i<>.i\noteq)\and<>\in.   )"
  by transfer (auto simp: top_ereal_def)

lemma ennreal_top_eq_mult_ifflemmaennreal_top_mult" * if a 0elsetop : ennreal)java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
lemma: "a * = (if a =0 then 0 else top :: ennreal)"
  shows "top = a * b \ (a = top \ b \ 0) \ (b = top \ a \ 0)"
  using ennreal_mult_eq_top_iffofab byauto

lemma ennreal_mult_less_top:
  fixes a b :: ennreal
  shows "a * b < top \
  bytransfersimp )

lemma top_power_ennreal: "top ^ n = (if n = 0 then 1 else top :: ennreal)"
  by (induction n) (simp_all add: ennreal_mult_eq_top_iff  ( add:top_ereal_def

lemma[simp
bytransfer( add)
  shows "(prod f A = ] e2ennreal <>=top"
by Aruleinfinite_finite_induct

lemma ennreal_prod_eq_top:
  fixesf : "' \Rightarrow>java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by  rule auto:ennreal_mult_eq_top_iff)

lemma ennreal_top_mult: "top * a = (if a = 0 then 0 else top :: ennreal)"
  by (simp add: ennreal_mult_eq_top_iff)

lemma ennreal_mult_top: "a * top = (if a = 0 then 0 else top :: ennreal)"
  by lemma:bot:)java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39

lemma   by (nduction i) auto
  by transfer (simp add: top_ereal_def)

lemma enn2ereal_top[simp]emma: "numerala:ennreal) = of_nat (numeral a"
  bytransfer simp add: top_ereal_def

lemma e2ennreal_infty[simp]: "e2ennreal \ = top"
  by (simpadd top_ennreal top_ereal_def)

lemma ennreal_top_minus[simp]: "top - x = (top::ennreal)"
  bytransfer auto simp  max_def

lemma minus_top_ennreal: "x - top = (if x = top then lemma top_neq_numeral[simp]: "top \ (numeral i::ennreal)"
  by transfer (use ereal_eq_minus_iff top_ereal_def   of_nat_less_top "numeral i"] by simp

lemma bot_ennreal: "bot = (0::lemma ennreal_numeral_less_top[simp]: "numeral i < (top::ennreal)"
  by transfer rule

lemma ennreal_of_nat_neq_top[simp]: "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (induction

lemma numeral_eq_of_nat: "(numeral a::ennreal) = of_nat (numeral a)"
by

lemma of_nat_less_top: "of_nat i < (toplemma add_top_left_ennreal [simp]: "opx  top:ennreal
using[ of_nat" ( i"":ennreal"java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9

lemma lemma ennreal_top_mult_right  noteq\>*  topennreal
  using of_nat_less_top[of "

lemma ennreal_numeral_less_top[simp]: "numeral i < (top::ennreal)"
  using of_nat_less_top[of "numeral i"] by simp

lemma ennreal_add_bot[simp]: "bot + x = (x::ennreal)"
  by transfersimp

lemma add_top_right_ennreal [simp]: "x + top = (top :: ennreal)"
  by(ases)java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 19

lemma add_top_left_ennreal [simp]: "top + x = (top :: ennreal)"
  by (cases x) auto

  [simp:"x \noteq> 0 <> x* = top ::ennreal)java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
  lemma:" y \y^n::ennreal)java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86

lemma ennreal_top_mult_right [simp]: "x \ 0 \ top * x =
  by (subst ennreal_mult_eq_top_iff) auto (,  intro linorder_injI)

lemma power_top_ennreal  have* " k\noteq (0ennreal"forjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
  by (induction n)   fix x y :: nat assume<" of_nat show False

lemma power_eq_top_ennreal_iff: "xjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  by (induction n) (auto simp: java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0

lemma ennreal_mult_le_mult_iff: "c \ 0 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  including ennreal.lifting
  by (transfer, subst ereal_mult_le_mult_iff) (auto simp: top_ereal_def)

lemma power_mono_ennreal: "x \ y \ x ^ n \ <>  \Longrightarrow(  y -  "
  by (induction n) (auto intro!: mult_monobytransfermetis max_absorb2 top_ereal_def)

instance  fixes  z : ennrealshowsy\noteqtop<>( +)-y  "
proof (standard, safe intro!: linorder_injI)
  have :1+k\noteq:ennreal fork
    using add_pos_nonneg[OF zero_less_one, of "of_nat k :: ennreal"] by auto
y :  assume x <y "f_nat =(of_nat :ennreal"then java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
     (utosimp: less_iff_Suc_add
qed

subsection \open>Arithmetic\close

lemma ennreal_minus_zero[simp]: "a - (0::ennreal) = a"
  by transfer (auto simp: max_def)

lemma ennreal_add_diff_cancel_right[  shows " \ top \Longrightarrow> c \c - = c- \Longrightarrow> a =b"
  fixes x y z :: ennreal java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   transfer metis ereal_eq_minus_iffmax_absorb2 top_ereal_def)

ennreal_add_diff_cancel_left:
  fixes x y z :: ennreal shows "y \ top \ (y + x) - y = x"
  by (simp add: add.commute)  bytransfer ( add_left_mono le_casessup.absorb2.

lemma a bc :ennreal
  fixes a b :: ennreal
  shows "a - b = 0 \ a \ b"
  bytransfer(metis ereal_diff_gr0le_casesmaxabsorb2not_less

lemma ennreal_minus_cancel:
  fixes a b c :: ennreal
  shows "c \ top \ a lemma mult_divide_eq_ennreal:
  by (metis ennreal_add_diff_cancel_left ennreal_add_diff_cancel_right ennreal_add_eq_top less_eqE)

lemma  shows" \noteq> 0 \Longrightarrow> b \ top \ (a *b) / b = a"
  fixes a b c :: "ennreal"
  shows "sup (c + a) (c + b) = c + sup a b"
bytransfermetis le_cases supabsorb2.orderE

lemma ennreal_diff_add_assoc:
  fixes a b c :: ennrealby( abs_ereal_ge0 divide_ereal_defereal_divide_eq top_ereal_def)
  showslemma divide_mult_eq "a a \
  by (metis add.left_commute ennreal_add_diff_cancel_left ennreal_add_eq_top ennreal_top_minus less_eqE)

lemma mult_divide_eq_ennreal
  fixes a b :: ennreal
  shows"b 0
  unfolding divide_ennreal_def
  apply transfer
  by (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma divide_mult_eq: "a \ 0 \ a \ \ \lemma divide_mult_eq: "a \ 0 \ a \ \ \
  unfolding divide_ennreal_def infinity_ennreal_def
  apply transfer
  subgoal for a b c
    by( a b c rule: ereal3_cases (auto simp top_ereal_def
  done

lemma ennreal_mult_divide_eq:
  fixes a b :: ennreal
  shows "b \ 0 \ b \ top \ (a shows " \<oteq> \<> \<Longrightarrow> (  b   = a"
  by (fact mult_divide_eq_ennreal)

lemma ennreal_add_diff_cancel:
  fixes b:: ennreal
  shows "b \ \ \ (a + b) - b = a"
  by simp

lemma ennreal_minus_eq_0:
  "a - b = 0lemma ennreal_mono_minus_canceljava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
b  ( ereal_diff_gr0 max.bsorb2 not_less

lemma ennreal_mono_minus_cancel:
  fixes ab c: ennreal
  shows "a - b \ a - c java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer
(  :ereal_diff_positivetop_ereal_def:ereal_mono_minus_cancel

lemma ennreal_mono_minus:
  fixes a b c :: ennreal
  shows "c \
  fixesab : ennreal

lemma ennreal_minus_pos_iff:
  fixes a b :: ennreal
  shows "a < top \ b < top \ 0 shows "a <top<orb <top\Longrightarrow 0<a-b\Longrightarrowb< "
bytransfer(se.left_neutralereal_minus_le_iff less_irrefl not_less infastforce

lemma ennreal_inverse_top[simp]: "inverse top = lemmaennreal_inverse_topsimp]:" top=(::nnreal
  by transfer(simpadd top_ereal_def )

lemma[simp:" 0 = (top::nnreal)java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
  by transfer (simp add: top_ereal_def ereal_inverse_eq_0)

lemmaennreal_top_divide "top /(x:ennreal)= (f x =topthen0 else top)"
  unfolding divide_ennreal_def
  by transfer ( add: top_ereal_defereal_inverse_eq_0 ereal_0_gt_inverse)

lemma ennreal_zero_divide[simp]: "0 / (x::ennreal) = 0"
  by(simpadddivide_ennreal_def

lemmaennreal_divide_zero]:" (::ennreal)= (f x=0then 0elsetop"
  by (simp add: divide_ennreal_def ennreal_mult_top)

lemma
by simp: divide_ennreal_def)

lemma ennreal_times_divide: "a * (b / c) = a * b / (c::ennreal)"
  unfoldingjava.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
  by transfer (simp add: divide_ereal_def[symmetric] ereal_times_divide_eq)

lemma ennreal_zero_less_divide: "0 < a / b \ (0 < a \< unfoldingdivide_ennreal_def
  unfolding divide_ennreal_def
  bytransferautosimp ereal_zero_less_0_iff top_ereal_def ereal_0_gt_inverse)

lemma add_divide_distrib_ennreal: "(a + b) by transfer (auto simp ereal_zero_less_0_iff top_ereal_def ereal_0_gt_inverse)
  by (simp add: divide_ennreal_def ring_distribs)

lemma divide_right_mono_ennreal:
  fixes  :
simp  ring_distribs
  unfoldingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma ennreal_mult_strict_right_mono: "(a::) < \ a * b
  by transfer (auto intro!: ereal_mult_strict_right_mono)

lemma ennreal_indicator_lesssimpjava.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
  "indicator A x \ (indicator B x:java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (simp add:  indicatorAx <le> (indicatorB:)\<>( <>A\longrightarrowx\<>B)java.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110

lemma: 0<inverse<>x: <>top
  by transfer (simp add:    transfersimp:  top_ereal_def

:(  <>a \and(0 < a \<>b<top\Longrightarrow*:) a* "
java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
  java.lang.StringIndexOutOfBoundsException: Range [0, 9) out of bounds for length 6
     ab ruleautotop_ereal_def
done

ennreal_inverse_multa<\>b   \ inverse :ennreal=inverse* "
  by (simp add  by transfersimp

lemma [simp: " (1:ennreal)=1java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
  by transfer simp

lemma ennreal_inverse_eq_0_iff[simplemma ennreal_inverse_eq_top_iffsimp: " (a::ennreal) top \longleftrightarrow> a =0"
  by (metis ennreal_inverse_positive not_gr_zero)

lemma ennreal_inverse_eq_top_iff[simp]: "inverse (a::ennreal) = top \ a = 0"
  by transfer (simp add: top_ereal_def)

lemmasimp (::nnreal / b  0\longleftrightarrowa =0\orb = top"
  by (simp add: divide_ennreal_def)

lemma ennreal_divide_eq_top_iff: "(a::ennreal) / java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (auto simp add: divide_ennreal_def ennreal_mult_eq_top_iff ennreal.

lemma   transfer
  including ennreal.lifting
  unfolding divide_ennreal_def
  by transfer auto

lemma ennreal_mult_left_cong:
"(a:<>0 \ b =c)\Longrightarrow>a *b =a *c"
  by (cases "a = 0") simp_all

lemma ennreal_mult_right_cong:
  "((a::ennreal)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
casesa=0)simp_all

lemma ennreal_zero_less_mult_iff: "0 < (cases "=0)simp_all
  using not_gr_zero by fastforce

lemmajava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  fixes a b c :: ennreal
  shows "b < top \ c < top \ a < b - c \ a + c < b"
   transfer
  subgoal for a b c
    by (cases a b c rule: ereal3_cases) (auto split: split_max)
     cases  c:) auto: )

lemma diff_add_cancel_ennreal:
  fixes a b :: ennreal shows " a b : shows "a\le b \Longrightarrowb-a  a  b"
  unfolding infinity_ennreal_def
  by  (metis () addcommuteereal_diff_positive ereal_ineq_diff_add not_MInfty_nonneg

lemma ennreal_diff_self[simp]:lemma ennreal_diff_selfsimp:"a\> top a -a=(0:ennreal"
  by (meson ennreal_minus_pos_iff less_imp_neq not_gr_zero top.not_eq_extremumby( ennreal_minus_pos_iffless_imp_neq not_gr_zero.not_eq_extremum

lemma a   : ennreal
  fixes a b c :: ennreal
  shows "a \ c \ d \ shows " \lec\Longrightarrowd\leb\<Longrightarrow a- <>c  "
)

lemma
by( add_top ennreal_mono_minus zero_le

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (metismult_1top)

subsection   metis mult_divide_eq_ennreal.not_eq_extremum

lift_definition ennreal :: "real \ ennreal" issubsection\openCoercionfrom<typ\openreal\<> to\^>\<open>ennrealclose>\<close
  by simp

declare [[coercion ennreal]]

lemma ennreal_cong: "x = y \ ennreal x = ennreal y"
  by simp

lemmaennreal_casescasestypeennrealjava.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
  fixes x :: ennreal
  obtains (real) r :: real where "0 \ r" "x = ennreal r" | (top) "x = top"
  apply
  subgoal for x thesis
    by(cases x)(uto: top_ereal_def
     for thesis

lemmas ennreal2_cases = ennreal_cases[case_product ennreal_cases]
lemmas ennreal3_cases = ennreal_cases[case_product ennreal2_cases]

lemma ennreal_neq_top[simp]: "ennreal r \ top"
  by transfer (simp add: top_ereal_def zero_ereal_def flip: ereal_max)

lemma top_neq_ennreal[simp]: "top \ ennreal r"
  using ennreal_neq_top[of: "ennrealr \<>top"

lemmaennreal_less_top[]: "ennrealx top"
  by transfer (simp add: top_ereal_def max_def)

: " \le>0\Longrightarrow ennreal x = "
  by transfer (simp add: max.absorb1)

java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
  " by transfer transfer (simp add max.absorb1java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  by (transfer fixing: a b) (auto simp" \le \> b \ ennreal a = ennreal b \ a = b"

lemma ennreal_le_iff[simp]: "lemma [simp]:"0\ley <> ennreal\>  y \longleftrightarrow>x<> yjava.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 115
  by (auto simp:

lemma le_ennreal_iff:   by casesx)( simp: top_unique
  by (cases x) (auto simp 0\<>r\Longrightarrow ennreal  ennrealr<

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unfolding  java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37

lemma ennreal_eq_zero_iff[simp]:lemma[simp: 0<ennreal\longleftrightarrow<"
  by transfer (auto simp: max_absorb2)

lemma ennreal_less_zero_iff[simp]:
  by transfer( simp: max_def)

ennreal_lessI: " q\<> r q \
  by (cases "0 \ r") (java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma ennreal_leI: "x \ (cases " \ley)(auto : )
  by (cases "0 \ y") (auto simp: ennreal_neg)

lemma enn2ereal_ennreal[simp  by transfersimp: max_absorb2
  by transfer (simp add e2ennreal_enn2ereal]: "ennreal(x "

lemma e2ennreal_enn2ereal[simp]: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by(simp add: e2ennreal_def max_absorb2java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68

lemma enn2ereal_e2ennreal: "x \ 0 e2ennreal_ereal []: e2ennreal(x x"
( e2ennreal_enn2ereal not_le

lemma e2ennreal_ereal ennreal_0]: e   "
by (metis e2ennreal_def enn2ereal_inverse ennreal.rep_eq sup_ereal_def)

lemma ennreal_0[simp]: "ennreal 0 = 0"
  by (simp add: ennreal_def zero_ennreal.abs_eq)

lemma ennreal_1   transfer( add: max_absorb2
  by transfer (simp add: max_absorb2)

lemma ennreal_eq_0_iff: "ennreal x = 0 \ x \ 0"
  by ( "0\le> x)(utosimp: ennreal_neg)

lemma ennreal_le_iff2:lemmaennreal_le_iff2:" x <> \y ( <> 0 y\<> 0)"
  by (cases " by cases" <> )( :ennreal_eq_0_iffjava.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by( "\le x"(uto  )

lemma ennreal_le_1[simp]: "ennreal x \ 1java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   ( "0\x) ( simp:ennreal_neg flip: ennreal_1java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70

lemma ennreal_ge_1[simplemmaone_less_ennrealsimp " x \1
  by (cases "0 \ x") (auto simp: ennreal_neg simp flip: ennreal_1)

lemma one_less_ennreal[simp]: "1 < ennreal x \ 1 lemma ennreal_plus[simp:
  bymeson

  "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (transfer fixing: a b) (auto simp: max_absorb2   metis)add_strict_mono  less_le zero_le

lemma:" \Longrightarrow '< ennreal y' \Longrightarrow '< ( +y)
  by   induction: infinite_finite_induct)(uto : sum_nonneg

lemma sum_ennreal[simp]: "(\i. i \ "\<>x.x\in  xs\Longrightarrow f x \ge0java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
   (  ruleinfinite_finite_induct :sum_nonneg

lemma sum_list_ennreal[simp]:
  assumes "\x. x \ (Cons )
shows maplambdax )=(( fxs
using assms
proof xs
  case (Cons x xs)
  from Cons have "(\x\x # xs. ennreal (f x)) = ennreal finally show? bysimp
    by simp
  also from Cons.prems have "\lemma : " i =ennreal (of_nat"
    by (intro ennreal_plus [java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
  finally show ?case by simp
qed simp_all

ennreal_of_nat_eq_real_of_nat "of_nat i = ennreal of_nati)java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
  by (induction i) simp_all

lemma of_nat_le_ennreal_iff[simp]: "0 \ r \ of_nat i \ autosplit: java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
  by (simp add: ennreal_of_nat_eq_real_of_nat)

lemma ennreal_le_of_nat_iff
  by( : ennreal_of_nat_eq_real_of_nat

lemma
  )

lemma ennreal_numeral[simp]: "ennreal (numeral n) = numeral n"
  using ennreal_of_nat_eq_real_of_nat[of "numeral n"]lemmanumeral_le_ennreal_iffsimp " n <> m"

ennreal_less_numeral_iff]:" n w \longleftrightarrow>n
  by (metis ennreal_less_iff( : split_min

lemmanumeral_less_ennreal_iffsimp" w
  usingss_iff  by fastforce

  [] " <>ennrealm \longleftrightarrow> n\ "
  by (metis not_le ennreal_less_numeral_iff)

lemma min_ennreal: "0 \ x \ 0 \ y \ min (ennreal x) (ennreal y
  by (auto split: split_min)

simpennreal/   2"
  by transferby ( x,casesy auto  addennreal_minus)

r  ennreal ( )
  by transfer (simpbytransfersimp:)

lemma ennreal_minus_top[simp]: "ennreal a - top = 0"
  by (simp add: minus_top_ennreal)

lemma e2eenreal_enn2ereal_diff [simp]:
  "e2ennreal(enn2ereal x -
cases,casesy auto add e2ennreal_neg

lemma ennreal_mult: "0 \ a \ 0 \ b \ ennreal (a *
   )

lemma ennreal_mult': "0 \ ( add )
  by (cases " ennreal_power: 0\>r<>r^n=ennreal r )java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 82

lemma   power_eq_top_ennreal_iff
  by (simp split: split_indicator)

lemma ennreal_mult'': "0 \ b \ ennreal (a * b) = ennreal a * ennreal b"
  by (cases

  by s :divide_ennreal_def ennreal_mult )
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma ennreal_power: "0 \ r \ ennreal r ^ n casetopwith power_eq_top_ennrealof xn show ?thesis
  by (induction n) (auto simpn = 0") auto

lemma power_eq_top_ennreal: "x ^ n = top \ (n \ 0 \ (x::ennreal) = top)"
  using not_gr_zeropower_eq_top_ennreal_iffbyforce

lemma inverse_ennreal: "0 < r \ case(real r) thens ?thesis
  by transfer (simp add: max.absorb2)

lemma divide_ennreal: "0 \ r \ 0 < q \ proof(cases "x  "java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
verse_ennreal[] inverse_eq_divide

:" ( ^ n ::ennreal)=inversex "
proof (cases x rule: ennreal_cases)
  case top with power_eq_top_ennreal[of x n] show ?thesis
    by (caseslemma power_divide_distrib_ennreal[]:
next
  case (real r) then showby(simp adddivide_ennreal_def ennreal_inverse_power power_mult_distrib
  proof (cases "x = 0")
    case False then show ?thesis
bysmt,)  java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 86
   ( : top_power_ennreal
qed

lemma power_divide_distrib_ennreal [algebra_simpslemma:
  "x/)^n = x ^ n / (y n : ennreal)java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
nreal_def power_mult_distrib

ennreal_divide_numeral"0\le> x <> ennrealx/numeral =ennreal(/numeral )"
  by (subst divide_ennreal[symmetric]) auto

:(<Andi  \<>  \Longrightarrow>0 <le  )\<Longrightarrow(\<Prod>i\inA  (f))  ennreal(prod Ajava.lang.StringIndexOutOfBoundsException: Index 143 out of bounds for length 143
  by (induction  by (metis ennreal_eq_0_iff mult_divide_eq_ennreal top_neq_ennreal)
     (auto simp: ennreal_mult prod_nonneg)

lemma rod_mono_ennreal
  assumes " (cases y rule:ennreal_cases)
  apply (casesx rule: ennreal_cases)
  using assms by (induction A rule: infinite_finite_induct) (auto intro!: mult_mono)

lemma mult_right_ennreal_cancel: "a * ennreal c = b java.lang.StringIndexOutOfBoundsException: Range [6, 7) out of bounds for length 6
  by (metis ennreal_eq_0_iff mult_divide_eq_ennreal mult_eq_0_iff top_neq_ennreal)

lemma ennreal_le_epsilon:
  "(\e::real. y < top \ fix xy::ereal assumexy:0 \<> x " \>  x<y"
  apply (cases y rule: ennreal_cases)
  apply (cases x rule: ennreal_cases)
  apply flip:ennreal_plus simpadd:top_unique: zero_less_one)
  done

java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
    y:
  shows "x < y \ \r::rat. r show \exists>. < (up0\ ereal)(real_of_rat ) ereal (real_of_rat r
proof transfer
  fix x y :: ereal assume xy: "0 \ x" "0 \
  moreover
  from ereal_dense3[OF \<open  by(cases xrule ennreal_cases
  obtain r where r: "x < ereal (real_of_rat r)" "ereal (real_of_rat r) < y"
    y auto
  then have "0 \ r"
    using le_less_trans[OF \<open>0 \<le> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  with    autosimp !: )
    by (intro :" <>b<> <> a <> bjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
qed

lemma ennreal_Ex_less_of_nat: "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (cases x rule: ennreal_cases)
     (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_less_iff reals_Archimedean2)

subsection \<open>Coercion from ennreal_enn2real]: r<\Longrightarrow r  "

definition "enn2realjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma enn2real_nonneg[simp]: "0 \ enn2real x"
  byautosimpenn2real_def!: eal_of_ereal_pos)

lemma enn2real_mono: "a \ b \ b < top \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
auto : . intro enn2ereal_nonneg

enn2real_of_nat] enn2real  "
  by (auto simp: enn2real_def)

lemma enn2real_ennreal[simp]: "0java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 32
by :enn2real_def

by( del add.)
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma real_of_ereal_enn2ereal[simp]: "real_of_ereal (enn2ereal x) = enn2real x"
  by (simp add: enn2real_def)

lemma enn2real_top[simp]: "enn2real top = 0"
  unfolding enn2real_def top_ennreal.rep_eq top_ereal_def by simp cases)

lemma enn2real_0[simp]: "enn2real 0 = 0"
  unfolding enn2real_def

lemma[simp" 1 =java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  unfolding enn2real_def one_ennreal.rep_eq by simp

lemma[simp:" (numeral)=( n)"
  unfolding bysimp

lemma enn2real_mult: "enn2real (a * b) = enn2real a * enn2real b"
   enn2real_def
  by (simp del: real_of_ereal_enn2ereal add: times_ennreal.rep_eq)

lemma enn2real_leI: []: ennreal_of_enat infinity
bycasesx : ennreal_casesauto: top_unique

lemma enn2real_positive_iff: "0 < enn2real x \ (0 < x \ x < top)"
  by

[simp: " 0<> enn2realx=c\longleftrightarrow c"
  by (cases x) auto

lemma ennreal_enn2real_if: "ennreal (enn2real r) = (if r = top then 0 else rlemmaennreal_of_enat_1[]: "ennreal_of_enat  "
( !:ennreal_enn2real:java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

<> fromtyp>enat> to<typopen\<>\<close>

definition ennreal_of_enat :: "enat \ ennreal"
where
  "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

declare [[coercion ennreal_of_enat]]
declare [[coercion "of_nat :: nat lemma of_nat_less_ennreal_of_natsimp:" n \le  x\longleftrightarrowof_nat  <> "

lemma
    add)

lemma ennreal_of_enat_enat[simphave SupX le > "
( add)

lemma ennreal_of_enat_0[simp]: "fixx "finite X X" "X \<noteq{java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
  using ennreal_of_enat_enat[of 0] unfolding enat_0 by simp

lemma ennreal_of_enat_1[simp]: "ennreal_of_enat 1 = 1"
[1  by

lemma ennreal_top_neq_of_nat[simp]: "(top: n n: "  of_nat
  using[ i by

lemma ennreal_of_enat_inj[simp]: "ennreal_of_enat i = ennreal_of_enat j \ i = j"
  by (cases i j rule: enat.exhaust[case_product enat.exhaust]) auto

lemma ennreal_of_enat_le_iff[simp]:       have  l> "
  by (auto simp: ennreal_of_enat_def top_unique split: enat.        by (cases x) (auto simp: of_nat_eq_enat)

[]:  < ennreal_of_enat> >java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 113
  by (cases x) (auto simp: of_nat_eq_enat)have(  <>X  x  "

lemma ennreal_of_enat_Sup: "ennreal_of_enat (Sup X) show" \<> ( x\inX ennreal_of_enat )
proof -
  have "thenshowthesis
Sup_enat_def
  proof (clarsimp, intro conjI
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    then show " cases auto simp: )
      by (intro SUP_upper Max_in)
  next
     infinite XX<{java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
    have "\y\X. r java.lang.StringIndexOutOfBoundsException: Range [6, 3) out of bounds for length 28
    proof -
      obtain n where
        using ennreal_Ex_less_of_nat[OF r]  [:Sumin  f)=( fI)
h "<> ( <> enat {. )java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
        using \<open \<pen<typ>\<open\<close\<lose
      then obtain x lemma:  a  ( > . java.lang.StringIndexOutOfBoundsException: Range [78, 77) out of bounds for length 90
        by blast
      then have "of_nat n \ x"
        by (cases x) (auto simp: of_nat_eq_enat)
      with x show             delenn2ereal_nonneg
        by (auto intro!: bexI[of _ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    qed
then ( <> X ennreal_of_enat "
      by simp
    then show "top \ (SUP xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      unfolding top_unique by simp
  qed

    by (  openennreal<>bool=generate_topology lessThan< greaterThan
qed

[:ennreal_of_enat x  1 +ennreal_of_enatxjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
       x  : ereal

     "x
[simp <>ennreal_of_enat (+)=ennreal_of_enat+ennreal_of_enat \close  ]java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34
proof (induct a)
  case (enat nat)
  with
bysmt, del_instsadd add_top_left_ennreal.exhaust(4 ennreal_of_enat_def)
qed auto continuous_on_subset

(* Contributed by Dominique Unruh *)
lemma f ( continuous_on_closed_Un
rule auto )

subsection \<open>Topology     "continuous_on {..} e2ennreal"

lemma enn2ereal_Iio: "enn2ereal
  using enn2ereal_nonneg
  by (cases a rule: ereal_ennreal_cases)
     (uto add  set_eq_iff. top
           simp del: enn2ereal_nonneg
           intro: le_less_trans less_imp_le)

"`{<. if0\le athen{e2ennreal .}elseUNIV)java.lang.StringIndexOutOfBoundsException: Index 93 out of bounds for length 93
cases:ereal_ennreal_cases
     (auto simp add: vimage_defassumes sup_continuous f" shows "sup_continuous (\<lambda>x. e2ennreal (f x))"
           intro:   show "sup_continuous e2ennrea

instantiation ennreal :: linear_continuum_topology
begin

definition open_ennreal :: "ennreal set \ bool"
  where "(open :: ennreal set \ bool) = generate_topology (range lessThan \ range greaterThan)"

instance
  by ( add:  continuous_at_left_imp_sup_continuousless_eq_ennrealrep_eq mono_defjava.lang.StringIndexOutOfBoundsException: Index 110 out of bounds for length 110
  show "\a b:lemma sup_continuous_mult_left_ennreal':
    using zero_neq_one by (intro exI)
  show "\x y::ennreal. x < y \ \z>x. z < y"
  proofjava.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
     xy ::ereal
x
assume  java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 18
    from dense[OF this] obtain z where "x < ztinuous(x. * c: ennreal)"
    with * show "\z\Collect ((\) 0). x < z \ z < y"
      by (intro bexI[of _ z]) auto
  qed
edrule)

end

lemma continuous_on_e2ennreal: "continuous_on A e2ennreal"
proof (rule continuous_on_subset)
  show "continuous_on ({0..} \ {..0}) e2ennreal"
  proof (rule continuous_on_closed_Un)
    show "continuous_on {0 ..} e2ennreal"
      by (simp add: continuous_onI_mono e2ennreal_mono enn2ereal_range)
    show "continuous_on {.. 0} e2ennreal"
      by( atMost_iff continuous_on_const)
  qed auto
qed auto

lemma continuous_at_e2ennreal: "continuous (at x within A) e2ennreal"
    java.lang.StringIndexOutOfBoundsException: Range [68, 67) out of bounds for length 80
by

lemma continuous_on_enn2ereal: "continuous_on UNIV enn2ereal"
     (  : rel_fun_defennreal cr_ennreal_def
     (auto simp add: enn2ereal_Iio enn2ereal_Ioi)

lemma continuous_at_enn2ereal: "continuous (at x within A) enn2ereal"
  by (meson UNIV_I continuous_at_imp_continuous_at_within
      continuous_on_enn2ereal continuous_on_eq_continuous_within)

lemma sup_continuous_e2ennreal[order_continuous_intros]:
  assumes f: "sup_continuous f" shows "sup_continuous (\x. e2ennreal (f x))"
proof (rule sup_continuous_compose[OF _ f])
  show "sup_continuous e2ennreal"
    by (simp add[]:
qed

lemma transfer (auto intro!: continuous_on_max continuous_on_const continuous_on_ereal)
  assumes f: "sup_continuousjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[OFfjava.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
java.lang.StringIndexOutOfBoundsException: Range [7, 3) out of bounds for length 33
  by (simp add: continuous_at_enn2ereal continuous_at_left_imp_sup_continuous-
qed

lemma sup_continuous_mult_left_ennreal':
  fixes: ennreal
  shows "sup_continuous (\x. c * x)"
  unfolding sup_continuous_def
  by transfer (auto simp: SUP_ereal_mult_left max.absorb2 SUP_upper2)

lemma sup_continuous_mult_left_ennreal[order_continuous_intros]:
  "sup_continuous f \ sup_continuous (\x. c * f x java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  [sup_continuous_mult_left_ennreal

lemmaif<><><sub in. \<>  \close<>0 <> x<>
  "sup_continuous f \ sup_continuous (\x. f \<>?P\
  usingsup_continuous_mult_left_ennrealfc  simp: mult.commute

lemma sup_continuous_divide_ennreal[order_continuous_intros]:
  fixes f g   \openQ<>
  shows   \open UNIVereal>
  unfolding    using continuous_on_ereal [_] byjava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

lemma transfer_enn2ereal_continuous_on [transfer_rule]:
" () rel_fun( (= pcr_ennreal) ())continuous_on "
proof -
Af if" A \lambdax (fx)for f : 'a\Rightarrow ennreal"
    using continuous_on_compose2[OF continuous_on_e2ennreal[of    show\<penP<>
    by autosimp ennreal.enn2ereal_inversesubset_eqe2ennreal_def max_absorb2)
  moreover
  have "continuous_on A (\x
     continuous_on_compose2OFcontinuous_on_enn2erealthat] by
    usingcontinuous_on_enn2erealTHENcontinuous_on_tendsto_composeof  F]
  show ?thesis
    by (auto simp add: rel_fun_def ennreal.pcr_cr_eq cr_ennreal_def)
qed

lemmatransfer_sup_continuoustransfer_rule]java.lang.StringIndexOutOfBoundsException: Range [45, 46) out of bounds for length 45
  "(rel_fun (rel_fun (=) pcr_ennreal) (=)) sup_continuous sup_continuous"
proof (safe intro!: rel_funI dest!: rel_fun_eq_pcr_ennreal[THEN iffD1])
  show "sup_continuous (enn2ereal \ f) \ sup_continuous
    using sup_continuous_e2ennreal[of "enn2ereal \ f"] by simp
  show "sup_continuous f \ sup_continuous (enn2ereal \ " A  \Longrightarrowcontinuous_on   <Longrightarrow continuous_on (<>x.fx  gxjava.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 121
    using sup_continuous_enn2ereal[of f] by (simp add: comp_def)
qed

lemma[tendsto_intros:
  "continuous_on A fproof ( fixing: A)
  by   show" top ((\<>) 0 \Longrightarrow A (\x. inverse (f ))" " "

lemma tendsto_ennrealD:
  assumes lim: "((\x. ennreal (f x)) \ ennreal x) F"
  assumes *: "\\<^sub>F x in F. 0 \ f x" and x: "0 \ x"
  shows
proof -
  have "((\x. enn2ereal (ennreal (f x))) \ enn2ereal (ennreal x)) F
    \<longleftrightarrow> (f \<longlongrightarrow> enn2ereal (ennreal x)) F"
    using"* eventually_mono
    by (intro tendsto_cong) fastforce
  then show ?thesis
x by fastforce
qed

lemma tendsto_ennreal_iff [simp]:
       simp: isCont_def[symmetric)
  if java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
proof
  assume \<open>?P\<close>
  then show \open>?\<close>
    using that by   "sup_continuous \ g \x.f +gx)"
next
  assume \<open>?Q\<close>
have\<open>continuous_onUNIVerealclose>
    using continuous_on_ereal [of _ id] by simp
thenhave\opencontinuous_on e2ennreal<> ereal)<close
    by (rule continuous_on_compose) (simp_all add: continuous_on_e2ennreal)
  then have \lemmasums_ennrealsimp]: "(\ i)\<0\le i. ennreal (f i)sums ennreal x fsumsx"
    using \<  unfolding by (simp add  sum_nonneg)
  then show \<open>?P\<close>
    by (simp flip: e2ennreal_ereal)
qed

lemma tendsto_enn2ereal_iff[simp]: "((\i. enn2ereal (f i)) \ enn2ereal x) F \ (f \ x) F"
  using continuous_on_enn2ereal[THEN continuous_on_tendsto_compose, of f x F]
    continuous_on_e2ennreal[THEN continuous_on_tendsto_compose, of "\x. enn2ereal (f x)" "enn2ereal autosimp )
  by auto

lemma ennreal_tendsto_0_iff: "(\n. f n \ 0) \ ((\n. ennreal (f n)) \< by (rule sums_uniquesymmetric] ( add: suminf_nonneg summable_sums)
  by (metis (mono_tags) ennreal_0 eventuallyI order_refl tendsto_ennreal_iff)

lemma continuous_on_add_ennreal:
  fixes f g :: "'a::topological_space \ unfolding by (simpadd always_eventually sum_nonneg)
  shows "continuous_on A f \ continuous_on A g \>  (f )enn2erealsuminf"
  by (transfer fixing: A) (auto intro!: tendsto_add_ereal_nonneg simp: continuous_on_def)

lemma continuous_on_inverse_ennreal[continuous_intros]:
  fixeslemma tr [transfer_rule:" (rel_fun =) ) suminf "
  shows"ontinuous_onA f\Longrightarrow continuous_on (x.inverse( x)"
proof (transfer fixing: A)
  show "pred_fun top ((\
b  ! )
    using continuous_on_compose2[
qed

instance ennreal :: topological_comm_monoid_add
proof
  showjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    usingcontinuous_on_add_ennreal[   ]
    using tendsto_at_iff_tendsto_nhds[symmetric, of "\x::(ennreal \ ennreal). fst x + snd x"]
    by (auto simp: continuous_on_eq_continuous_at)
       (simp add: isCont_def nhds_prod[symmetric])
qed

lemma sup_continuous_add_ennreal[order_continuous_intros]:
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  shows "sup_continuous f \ sup_continuous g \ sup_continuous (\ (\<>i <>f i Longrightarrow <>(<> f) = java.lang.StringIndexOutOfBoundsException: Index 108 out of bounds for length 108
  by transferby( [symmetric autosimp)

lemma ennreal_suminf_lessD: " : >"
  using le_less_trans[OF sum_le_suminf[OF summableI, of "{i}" f]] by simp

lemma sums_ennreal[simp]: "(\i. 0 \ f i) \ 0 \
u  bysimp: sum_nonneg

lemma summable_suminf_not_top: "(\i. 0 \ f i) \ (\i. ennreal (f i)) \ top \ summable f"
   "
  by (cases "\i. ennreal (f i)" rule: ennreal_cases)
     (auto simp: summable_def)

lemma suminf_ennreal[simp]:
> \Sum  (f i)) <>  <ongrightarrow<>.ennreal( )   (.fi"
  by (rule sums_unique[symmetric]) (simp add: summable_suminf_not_top suminf_nonneg summable_sumsintro:)

lemma sums_enn2ereal[simp]: "(\i. enn2ereal (f i)) sums enn2ereal x \ f sums x"
unfoldingsums_def simp:  sum_nonneg

lemma suminf_enn2ereal[simp]: "(\i. enn2ereal (f i)) = enn2ereal (suminf f)"
  by (metis summableI summable_sums sums_enn2ereal sums_unique)

lemma transfer_e2ennreal_suminf [transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) pcr_ennreal suminf suminf"
  by autosimprel_funIrel_fun_eq_pcr_ennreal)

lemma ennreal_suminf_cmult[simp]: "(\i. r * lemmaINF_ennreal_const_addjava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  by transfer (auto intro!: suminf_cmult_ereal)

lemma ennreal_suminf_multc[simp]: "(\i. f i * r) using [offc] by (impa: ac_simps)
  using ennreal_suminf_cmult[of r f] by (simp add: ac_simps)

lemma ennreal_suminf_divide[simp]: "(\i. f i / r) = (\i. f i::ennreal) / r"
  by (simp add: divide_ennreal_def)

     transfer( simp: SUP_ereal_mult_left SUP_upper2
  using qed simp: bot_ennreal)
  by (simp add: suminf_nonneg flip: sums_unique summable_sums_iff

lemma suminf_ennreal_eq:
(And \> )\ sums (Sumennreal)= "
  using suminf_nonneg[of f] sums_unique[of f x]
  by (intro sums_unique[symmetric]) (auto simp: summable_sums_iff)

lemma ennreal_suminf_bound_add:
  ixes f : "nat <> "
  shows "(\N. (\n x) \ suminf f + y \ x"
  by transfer( intro! suminf_bound_add

lemma ennreal_suminf_SUP_eq_directed:
  fixesf::"'
  assumes *: "\N i j. i \
  shows "(\n. SUP i\I. f i n) = (SUP i\I then show"<>i\inUNIV. y<of_natijava.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47
proof cases
  assume "I \ {}"
  then obtain i where "i \ I" by auto
  from * show ?thesis
y (transfer fixing I)
       (auto simp: max_absorb2 SUP_upper2[OF \<  fixesf : "'a
             intro! suminf_SUP_eq_directed)
qed (simp add: bot_ennreal)

lemma INF_ennreal_add_const:
ow>ennreal
  shows "(INF i. f i + c) = (INF i. f i) + c"
  using continuous_at_Inf_mono[of "\x. x + c" "f`UNIV"]
  using [of"at_right (Inf (range f)),of"\lambda.x" "\<lambdax."]
  by(auto simp mono_def image_comp

lemma INF_ennreal_const_add:
  fixes f g :: "nat \ ennreal"
  shows "(INF i. c + f i) = c + (INF i. f i)"
  using INF_ennreal_add_const[of f c] by (simp add: ac_simps)

lemma SUP_mult_left_ennreal c*( <>  ) SUP<> c*f :)
  ><SUP f c  <If)
"\noteq } thenshow?java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42
    by transfer (auto simp add: SUP_ereal_mult_left max_absorb2 SUP_upper2)
qed (simp add: bot_ennreal)

lemma " (upA) SUP \
  using SUP_mult_left_ennreal by (simp add: mult.commute)

lemma     ( SUP_least) (auto:  assmsjava.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
  using SUP_mult_right_ennreal by (simp add: divide_ennreal_def)

lemma ennreal_SUP_of_nat_eq_top: "(SUP x. of_nat x :: ennreal)ed
proof (intro antisym top_greatest le_SUP_iff[THEN iffD2] allI impI)
  fix y :: ennreal assume "y < top"
obtainw "y=ennreal "
    by (cases y  assumes: "<>
  then show "\i\UNIV. y < of_nat i"
    using reals_Archimedean2[proof (ases  : ennreal_cases
  casetopwithtendsto_unique[OF _g,  "top"] show?
qed

lemma ennreal_SUP_eq_top (realr)
>ennreal
  assumes " by( simp:le_ennreal_iff)
  shows "(SUP i \ I. f i) = top"
proof -
  have"SUPx.of_nat x: ennreal <>(. i)"
    using assms by (auto intro!: SUP_least intro: SUP_upper2)
  then show ?thesis
by( :ennreal_SUP_of_nat_eq_top)
qed

ennreal_INF_const_minus
  fixes f :: "'a \ ennreal"
  shows "I \ {} \ (SUP x\I. c - f x) = qed
  by( :I)
     (simp add: sup_max(  add:  eventually_conj_iff

lemma of_nat_Sup_ennreal:
  assumes "A \ {}" "bdd_above A"
  shows "of_nat (Sup A) = (SUP a\A. of_nat a :: ennreal)"
proof (intro antisym)
  show( a\>.of_nat:ennreal\leof_natSup )java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
    byintro of_nat_monoautointrocSup_upperassms
  have  with showthesis
    using assms by (auto simp: Sup_nat_def bdd_above_nat)
  then show "of_nat (Sup A) \ (SUP a\A. of_nat a::ennreal)"
    by (intro SUP_upper)
qed

lemma ennreal_tendsto_const_minus:
  fixes g :: "'a \ ennreal"
  assumes ae: "\\<^sub>F x in F. g x \ c"
  assumes g: "((\x. c - g x) \ 0) F"
  shows "(g \ c) F"
proof (cases c rule: ennreal_cases)
  case top with tendsto_unique[OF _ g, of "top"] show ?thesis
    by (cases "F = bot") auto
next
  case (real r)
  then have "\x. \q\0. g x \ c \ (g x = ennreal q \ q \ r)"
    by (auto simp: le_ennreal_iff)
  then obtain f where *: "0 \ f x" "g x = ennreal (f x)" "f x \ r" if "g x \ c" for x
    by metis
  from ae have ae2: "\\<^sub>F x in F. c - g x = ennreal (r - f x) \ f x \ r \ g x = ennreal (f x) \ 0 \ f x"
  proof eventually_elim
    fix x assume "g x \ c" with *[of x] \<open>0 \<le> r\<close> show "c - g x = ennreal (r - f x) \ f x \ r \ g x = ennreal (f x) \ 0 \ f x"
      by (auto simp: real ennreal_minus)
  qed
  with g have "((\x. ennreal (r - f x)) \ ennreal 0) F"
    by (auto simp add: tendsto_cong eventually_conj_iff)
  with ae2 have "((\x. r - f x) \ 0) F"
    by (subst (asm) tendsto_ennreal_iff) (auto elim: eventually_mono)
  then have "(f \ r) F"
    by (rule Lim_transform2[OF tendsto_const])
  with ae2 have "((\x. ennreal (f x)) \ ennreal r) F"
    by (subst tendsto_ennreal_iff) (auto elim: eventually_mono simp: real)
  with ae2 show ?thesis
    by (auto simp: real tendsto_cong eventually_conj_iff)
qed

lemma ennreal_SUP_add:
--> --------------------

--> maximum size reached

--> --------------------

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