SSL Extended_Nonnegative_Real.thy Interaktion und
PortierbarkeitIsabelle

(*  Title:      HOL/Library/Extended_Nonnegative_Real.thy
    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

Liminf_add_const:
  fixesc ::"'a::{complete_linorder, linorder_topology, topological_monoid_add,java.lang.StringIndexOutOfBoundsException: Index 107 out of bounds for length 107
  noteq >xc L f+
  by (intro Liminf_compose_continuous_mono monoI add_mono continuous_intros) auto

lemma sums_offset:
  fixes f g :: "nat \ 'a :: {t2_space, topological_comm_monoid_add}"
  assumes "(\n. f (n + i)) sums l" shows "f sums (l + (\j
proof  -
have(lambda>kf( i  \>i  )\<longlonglongrightarrow>(Sum  "
    using byauto!:tendsto_addsimp)
\<Sum (auto intro!: sum.cong    also  by
   fg : nat\Rightarrow' ::{ opological_comm_monoid_add}"
    have "(\j:. <> LongrightarrowPz Longrightarrow at_left)java.lang.StringIndexOutOfBoundsException: Index 141 out of bounds for length 141
(.union_disjoint(auto! .)
    also have  by( mult. eq_iff.commute)
      unfolding image_add_atLeastLessThan simp
    finally show ?thesis
      by (auto simp: inj_on_def atLeast0LessThan sum.reindex)
  qed
  ultimately have fixesb:e
    shows<b<><><>\Longrightarrow  )-b="
  then show ?thesis
    unfolding sums_def by (rule LIMSEQ_offset)
qed

lemmasuminf_offset
  fixes  fixes : ":{order_top, ordered_comm_monoid_add}"
  shows summable (<lambda.f (j + i)) <Longrightarrow suminf  \Sumj.f ( +)  \Sumji fj)
   ( sums_unique] sums_offset summable_sums

lemma
  by( eventually_at_left[of 0]) (auto intro: exI[of _ 0])

lemma   shows"0\ x \ top + x = top"
  fixes a: "a :: {ordered_semiring, comm_monoid_mult}"
  shows "0 \ a \ a
  by (metis mult.left_neutral eq_iff mult.commute mult_right_mono)

lemma ereal_add_diff_cancel:
  fixes a b :: ereal
  shows "\b\ \ \ \ (a + b) - b =ajava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
  by (cases a b rule: ereal2_cases) auto

lemma add_top:
  fixes x :: "'a::{order_top, ordered_comm_monoid_add}"
  shows "0 \ x \ x + top = top"
  by (intro top_le add_increasing order_refl)

lemma top_add:
  fixes x :: "'a::{order_top, ordered_comm_monoid_add}"
  shows "0 \ x \ top + x = top"
  by (intro top_le add_increasing2 order_refl)

lemma le_lfp: "mono f \ x \ bot: "<>  <>lfpand:\Andx  <>lfp<> \alpha)= \alpha )
   ( )

:(^ )bot" java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
alpha <> andf "up_continuous f"andmg
  assumes bot: "\ bot \ lfp g" andeq"<>. x\le>\alpha>( x) \alpha )java.lang.StringIndexOutOfBoundsException: Index 124 out of bounds for length 124
  shows "\ (lfp f) = lfp g"
proof (rule antisym)
  note mf = sup_continuous_mono[OF f]
  have: "(f ^i bot \for
    by (induction i     ( add SUP_leastalpha[ sup_continuousD mono_funpow

have< ( ^ ) <>lfp  i
    by (induction
  thenlemma sup_continuous_applyD: "sup_continuous f \ sup_continuous (\x. f x h)"
    unfolding sup_continuous_lfp[OF f]
    by (simp add: SUP_least \<alpha>[THEN sup_continuousD] mf mono_funpow)
  show "lfp g \ \ (lfp f)"
    by (rule lfp_lowerbound) (simp add: eq[symmetric] lfp_fixpoint[OF mf])
qed

lemma sup_continuous_applyD: "sup_continuous f \ sup_continuous (\x. f x h)"
  using sup_continuous_apply[THEN sup_continuous_compose] .

lemma sup_continuous_SUP[order_continuous_intros]:
  fixes M :: "_ \ _ \ 'a::complete_lattice"
  assumes M: "\i. i \ I \ sup_continuous (M i)"
shows <I  )
  unfolding

lemma[order_continuous_intros]:
  fixes  :" <> <>java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
"Andi. i \ I \ sup_continuous (M i)) \ sup_continuous (\x. SUP i\I. M i x)"
  unfolding SUP_apply[symmetric]by (

lemma  "sup_continuous (SUP i\
  assumes 1: "sup_continuous f"
  assumes 2: "\g. sup_continuous g \ sup_continuous (f g)"
  shows "sup_continuous (lfp f sup_continuous_def (auto simpadd: sup_continuousD[ ] : )
proof -
  have "sup_continuous ((f ^^ i) bot)" for i
  proof (inductioni)
    case (Suc i) then show ?case
      by (auto intro!: 2)
  qed (simp add: bot_fun_def sup_continuous_const)
  then show ?thesis
    unfolding sup_continuous_lfp[OF 1] by (intro order_continuous_intros)
qed

lemma sup_continuous_lfp''[order_continuous_intros]:
  assumes 1: "\s. sup_continuous (f s)"
  assumes 2: "\g. sup_continuous g \ sup_continuous (\s. f s (g s))"
  shows "sup_continuous (\x. lfp (f x))"
proof -
  have "sup_continuous (\x. (f x ^^ i) bot)" for i
   ( i)
     (Suci then ?case
( !: )
  qed (simp add: bot_fun_def sup_continuous_const)
  then show ?thesis
    unfolding sup_continuous_lfp[OF 1] by (intro order_continuous_intros)
qed

lemma mono_INF_fun:
    "(\x y. mono (Fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (  assumes2 \Andg g\Longrightarrowsup_continuous "

lemma continuous_on_cmult_ereal  howssup_continuousf"
  "\c::ereal\< -
   tendsto_cmult_ereal cf" x at xwithinA
  by (auto simp i)

lemma real_of_nat_Sup:
  assumes "A \ {}" "bdd_above A"
  showsSuc  showjava.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
  antisym
( in :\> Ajava.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 59
    using assms by (   2 \And.sup_continuous\Longrightarrowsup_continuous\lambdafs g )"
     "(<>x.lfp( x)"
    using
    of_natA ><A a)
    byproof ijava.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 21
qed

lemma (in complete_lattice) SUP_sup_const1
"\noteq> { <>(SUP iI.f i)
  using SUP_sup_distrib[of "\_. c" I f] by simp

lemma (in complete_lattice) SUP_sup_const2:
  "I \ {} \ (SUP i\
  usingSUP_sup_distrib   \lambda "

lemma one_less_of_natD:
  assumes(:':linordered_semidom) n"1
  by\>:ereal LongrightarrowALongrightarrowA(>.c* )

open theextended reals<>

text \<open>Basic definitions and type class setup\<close  by :  simp:)

typedef ennreal = "{x :: ereal. "  SUPAof_nat)intro
  morphismsintro  assmsmono_def
  by auto

definition "e2ennreal x = e2ennreal' (max 0 \}Longrightarrow( \> ()= SUPi<>. )

lemma enn2ereal_range: "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
-
  have "\y\0. x = e2ennreal y" for x
cases):  max_absorb2
  then show ?thesis
    by
qed

lemma type_definition_ennreal': "type_definition enn2ereal e2ennreal {x. 0cases )useassms injava.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 34
java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
  by (auto simp: type_definition_def e2ennreal_def max_absorb2)

setup_lifting   '

declare

instantiation ennreal :: complete_linorder
begin

java.lang.StringIndexOutOfBoundsException: Range [27, 15) out of bounds for length 68
lift_definition bot_ennreal<>\<ge>0 x = e2ennrealforx
lift_definitionsup_ennreal " < <>ennreal is sup rulele_supI1
lift_definition inf_ennreal :: "ennreal \then show?

lift_definition Inf_ennreal :: "ennreal set \
rule)

lift_definitionusing
  by  by auto: type_definition_def e2ennreal_defmax_absorb2

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

instance
  by standard
transferauto :  Inf_greatestSup_upper le_max_iff_disj.absorb1

end

lemma pcr_ennreal_enn2ereal[simp]: "pcr_ennreal lift_definition bot_ennreal :: is0by ( order_refl)
  by( addennrealcr_ennreal_def

lemma rel_fun_eq_pcr_ennreal: "rel_fun (=)lift_definitioninf_ennreal: ennreal\Rightarrow ennreal <>ennreal"is byrule)
by (uto ennreal )

instantiation)
begin

definition infinity_ennreal :: ennreal
  where [simp]: "\

instance ..

end

instantiation
begin

lift_definition one_ennreal :: ennreal (transferautosimp  Inf_greatestSup_upperSup_least le_max_iff_disj.absorb1+
lift_definition zero_ennreal :: ennreal is 0 by simp
lift_definition plus_ennreal :: "ennreal \ ennreal \ ennreal" is "(+)" by simp
lift_definition

java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
  by standard (transfer( : ennreal )

java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
.
instantiation
begin

definition :ennreal" \>ab  0(-)
  by simp

instance ..

end

instance ennreal :: numeral ..

erse
begin

lift_definition:" Rightarrowennreal \<> "  "("
  ( inverse_ereal_ge0I

definition divide_ennreal ::
  where div  x*inverse:ennreal

instance.

end  : ennrealRightarrow "

lemmaennreal: java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30
  by transfer inverse_ennreal:"ennreal "is

instance ennreal definition :"nnreal <>ennreal \Rightarrow>"
(standard;transfer
  fix a b .
    java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by(ases  ruleereal2_cases (autointro:exI[ _" ( a))
qed

instance ennreal :: ordered_comm_semiring
  by standard
     (transfer; auto intro: add_mono

instance ennreal)
proof
   ab:ennreal
show a< <>   <  "
    by transfer (simp add: add_right_mono ereal_add_cancel_right less_le)
()

instance ennreal :: strict_ordered_ab_semigroup_add
proof
  fix a instance ::ordered_comm_semiring
    by transfer (auto intro!: ereal_add_strict_mono)
qed

declare [[coercion "of_nat :: java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemmae2ennreal_neg: "x\le < "
zero_ennreal_defe2ennreal_def simp: max_absorb1)

lemma e2ennreal_mono:" <> y\Longrightarrow e2ennreal x\le e2ennreal "
  by (cases "0 \ x" "0 \ y" rule: bool.exhaust[case_product bool.exhaust] (; )
     (auto simp:   abcd: ennrealshowa<b\Longrightarrow   \>a+c<b  "

    by  autointroereal_add_strict_mono
  using ennreal.enn2erealjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
obtainswhere0\<>a aenn2ereal" |a < "
  using e2ennreal'_inverse[of a, symmetric] by (cases "0 \ a unfolding zero_ennreal_defe2ennreal_defby (impadd: max_absorb1)

lemma rel_fun_liminf[transfer_rule]: ule .exhaust boolexhaust
proof-
  have "\x y. rel_funjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        \<forall>x y. rel_fun (=) java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
    by (auto simp: comp_def Liminf_bounded rel_fun_eq_pcr_ennreal)
  moreover have "rel_fun (rel_fun (=) pcr_ennreal) pcr_ennreal (\x. sup 0 (liminf x)) liminf"
    unfolding liminf_SUP_INF[abs_def] by (transfer_prover_start, transfer_step+; simp)
  ultimately show
    by real_ennreal_cases:
qed

lemma rel_fun_limsup[transfer_rule]: "rel_fun (rel_fun (=) pcr_ennreal) pcr_ennreal limsup limsup"
proof -
  have [simp]: "max 0 (SUP x\{n..}. using e2ennreal'_inverse[of a, symmetric] by (ases " <> a)( :enn2ereal_nonneg
     meson enn2ereal_nonnegmax nle_le
  have "rel_fun (rel_fun (=) pcr_ennreal) proof -
    unfoldinglimsup_INF_SUP]bytransfer_prover_starttransfer_step simp
          <>xy  (=)pcr_ennrealy\longrightarrowpcr_ennrealliminf liminf)
       (uto simpcomp_def rel_fun_eq_pcr_ennreal)
                 (nat 0 Sup(  n.))( n  ( n.)\rbrakk
\<>  (INFn.Sup(  n.}))( n Sup ( java.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  ultimately thesis
    by (simp add: limsup_INF_SUP rel_fun_def)
qed

[simp <>.i\in  <> <>fi <>(<>\inI.enn2ereal )=enn2ereal (  )java.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 152
  by ( I rule infinite_finite_inductautosimp:sum_nonneg.rep_eqplus_ennreal.)

lemma transfer_e2ennreal_sumtransfer_rule
  " (rel_fun(=) ) (rel_fun() pcr_ennreal) sumsumjava.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 71
o! rel_funI:rel_fun_eq_pcr_ennreal)

lemma enn2ereal_of_natsimp] " (f_nat n=erealn"
  by induction)( simp:

lemma
by  )

[] numeraljava.lang.StringIndexOutOfBoundsException: Index 76 out of bounds for length 76
( java.lang.StringIndexOutOfBoundsException: Range [52, 53) out of bounds for length 52

subsection \<open

lemma() ))"
  unfolding infinity_ennreal_def by transfer (simp add: top_ereal_def ereal_add_cancel_left)

lemma ennreal_add_left_cancel_le [simp" of_nat)= ereal n
  simpereal_add_le_add_iff)

lemma ereal_add_left_cancel_less:
  fixes[simpenn2ereal   "
  shows "0 \ metisenn2ereal_of_nat numeral_eq_ereal of_nat_numeral)
  lemmatransfer_rule()(a)

lemmaennreal_add_left_cancel_less+< c<  <>(<>:nnreal<andb<cjava.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
  unfolding infinity_ennreal_def
  bysimptop_ereal_defereal_add_left_cancel_less)

ML \<open>
structure Cancel_Ennreal_Common =
struct
  (* copied from src/HOL/Tools
fun_[          raisefind_first_t [)
    | find_first_t past u (t::terms) =
          if  aconv then ( past@terms
          else find_first_t(t:pastu java.lang.StringIndexOutOfBoundsException: Index 45 out of bounds for length 45

  fun dest_summing (  by (cases a b c rule: e
lemma: "+ <>a\noteq \infinity:ennreal) <>
    | dest_summing  unfolding

  val mk_sum =ML\open
     t ]
  val find_first
   trans_tac Numeral_Simprocstrans_tac
  val norm_ss =
    simpset_of (put_simpset HOL_basic_ss \<^context>
      |Simplifieradd_simpsthms  add_0_left})
  fun norm_tac ctxt = ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
  fun simplify_meta_eq ctxt cancel_th th =
    Arith_Data.simplify_meta_eq [] ctxt
      (thcancel_th]MRS)
  fun mk_eqif aconvt then ( past )
end

structure Eq_Ennreal_Cancel = const_name<>Groups\close)$t$ u,) =
(open Cancel_Ennreal_Common
  val mk_bal = HOLogic.mk_eq
  val =HOLogic \^>\open.eqclose <typopen\close
  fun simp_conv _ _ = SOME @{thm ennreal_add_left_cancel  (,)=t :ts
)

structure Le_Ennreal_Cancel =   val mk_sum=Arith_Data
openCancel_Ennreal_Common
val =HOLogic \^>\openOrderings\close
  val    .trans_tac
nv  SOME }
)

  ctxt simp_tac  )java.lang.StringIndexOutOfBoundsException: Index 68 out of bounds for length 68
(open   fun mk_eq (a, b) = HOLogimk_Trueprop.mk_eqa )
  val mk_baljava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  val dest_bal = java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  fun _  = @{ ennreal_add_left_cancel_less
)
\<close>

simproc_setup ennreal_eq_cancel
  ("(l::ennreal) + m = simp_conv __ @thmennreal_add_left_cancel}
  \<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

simproc_setup ennreal_le_cancel
  ("(l::ennreal) + m \ n" | "(l::ennreal) \ m + n") =
  \<openK ( ctxt   =. ctxt.term_of)<>

simproc_setup ennreal_less_cancel  =HOLogic \^><>Orderings\<> \<^>\open\<>
((:ennrealm<n (::ennreal  +"java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
  <>K ( ctxt>fn => .procctxtThm ct)<>

  fun _ _=SOMEthm}

lemma ennreal_zero_less_top[\<close
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

[:"1<(::ennreal)java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
   transfersimp: top_ereal_def

lemma ennreal_zero_neq_top[simp]: "0 \<
  by transfer (simp add: top_ereal_def((:ennreal  le"|"l::) \le m +" java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

lemma ennreal_top_neq_zero[simp]: "(top::ennreal) \ 0"
  by

simp top<> (:e)
  by transfer>fn   ct .  (.term_of))\close

lemmasimp noteq:
  by transfer (simp add

  transfer:)
  fixes a b ::
shows a+< <>a      java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66
  by transferlemma[]: " <> (top::ennreal)java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

lemma lemma[]:"top:ennreal) <> 0"
fixes :
  shows "a + b = top \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer (auto simp: top_ereal_def)

lemma ennreal_sum_less_top:
  fixes f by ( :  one_ereal_def: )
" I\Longrightarrow Sum\in> )<>i<>I "
  by (induction I rule: finite_induct) auto

lemma ennreal_sum_eq_top   "a + b b < top"
   f : "a\<>ennreal"
  shows "finite I \ (\i\I. f i) = top \ (\i\I. f ijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
nductionI : finite_inductauto

lemma ennreal_mult_eq_top_iff:
fixesb :ennreal
  shows "a * b java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer (auto simp: top_ereal_def)

lemma ennreal_top_eq_mult_iff  shows" I I.f \longleftrightarrow (<>i\I f
fixes  :ennreal
  shows "top = a * b lemmaennreal_sum_eq_topsimp:
  using ennreal_mult_eq_top_iff[of a b] by auto

lemmajava.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28
  fixes a b :: ennreal
  shows "a * b < top \ (a = 0 \ b = 0 \ (a < top \ b < top))"
  by transfer (auto simp add: top_ereal_def)

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)

lemma ennreal_prod_eq_0[simp]:
  fixes f :: "'a \ ennreal"
  shows "(prod f A = lemma ennreal_mult_eq_top_iff:
  by (induction A rule: infinite_finite_induct) auto

lemma ennreal_prod_eq_top:
  fixesfixesa : ennreal
   "\<>iI. i \<> ( I\and\forall>\inI f <> 0 \<> (\exists>iI.fi=top))"
  by (induction I rule: infinite_finite_induct) (auto simp: ennreal_mult_eq_top_iff)

: topa=(if=0then  :"
  by (simp add: ennreal_mult_eq_top_iff)

ennreal_mult_toptopa=thentop)java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
  by (simp add: ennreal_mult_eq_top_iff[  ] java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

lemma enn2ereal_eq_top_iff[simp]: "enn2ereal x = \ \ x transfer (auto add:top_ereal_defjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  bytransfersimpadd )

lemma enn2ereal_top ennreal_prod_eq_0]:
   transfer (imp: top_ereal_def

lemmae2ennreal_infty[simp: "e2ennreal\
  by (simp add:    (induction rule: ) auto

lemma :a< ennreal"
  by transfer (auto simp: top_ereal_def max_def)

lemma minus_top_ennreal: "x - top = (if x = top then top else 0:: by (inductionI : infinite_finite_induct) ( simp:java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84
  by transfer (use ereal_eq_minus_iff top_ereal_def injava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma bot_ennreal " = (0::ennreal"
  by transfer rule

lemma ennreal_of_nat_neq_top[simp
by(nduction java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23

numeral_eq_of_nat( :ennreal of_nat)
   (simp)

lemma of_nat_less_top: (simp :.abs_eqjava.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
  using    transfer (simp:top_ereal_def)
  by simp

java.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64
using[of"numeral java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 47

java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 66
  using of_nat_less_top[of "numeral i"]

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

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by  by simp

add_top_left_ennreal" + =( :: ennreal)"
  by (cases x   less_le_transof" i" of_natSuc) "op:"]

lemma ennreal_top_mult_left   bysimp
  by (subst ennreal_mult_eq_top_iff) auto

[simp]:"x\<> 0 \
  by (subst ennreal_mult_eq_top_iff) auto

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

lemma
  by (induction n)  add_top_right_ennreal   ( x)auto

lemma ennreal_mult_le_mult_iff: "c \ 0 \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  including ennreal.lifting
  by (lemmaennreal_top_mult_left] x <0\Longrightarrow  top(: "

power_mono_ennreal x\ley <> \ (    "
  by (induction n) (auto

instance ennreal :: semiring_char_0
proofstandardsafe!: linorder_injIjava.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
:1 + of_nat <>(::)  k
    using add_pos_nonneg[OF zero_less_one, of "of_nat k :: ennreal"] by auto
"x "of_nat x= ( y::ennreal)"then
    by (auto simp 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[simp]:
fixesyzy\noteq top<> x+)-y=xjava.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
    ( ereal_eq_minus_iff not_MInfty_nonneg)

lemma ennreal_add_diff_cancel_left[simp
   xyz:  " <> top \Longrightarrow y+x y =xjava.lang.StringIndexOutOfBoundsException: Range [81, 82) out of bounds for length 81
  by (simp add: add.  have* " +of_nat <> (0:ennreal)" java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52

lemma
  fixes  :nat" ""x of_naty::) showFalse
  shows "a - b = 0 \
  by transfer\<>

lemma ennreal_minus_cancel:
  fixes a b c :: ennreal
cnoteq<a\  >>\ a  b <a "
  by (metis ennreal_add_diff_cancel_left ennreal_add_diff_cancel_right ennreal_add_eq_top less_eqE)

lemma sup_const_add_ennreal:
by(  not_MInfty_nonneg
  shows "suplemmaennreal_add_diff_cancel_left[simp]java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
metis . suporderE)

lemma
  fixes : java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
  shows  (  . )
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
  fixes a b :: ennreal
   b< < noteq *java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 89
  unfolding     ( add_left_mono . sup)
  apply transfer
   metis  ereal_times_divide_eq

:\>  \\>   *)= :ennreal
  unfolding divide_ennreal_def infinity_ennreal_def
  apply transfer
  subgoal:
    by  \\> > >
  done

lemma ennreal_mult_divide_eq:
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0

  by (fact mult_divide_eq_ennreal)

lemma ennreal_add_diff_cancel: cases  rule) )
  fixes ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
b< infinity a+)-b=
  by simp

lemma ennreal_minus_eq_0:
  "a - b = 0 \< a java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
  by transfer (

lemma:
  fixes a b c :: ennreal
  shows  ytransfermetis le_cases. )
  by transfer
     (auto simp add  fixes   :ennreal

lemma ennreal_mono_minus:
  fixes a b c :: ennreal
  shows "c \
  by transfer ennreal_mono_minus

   :java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
shows  \>    \>     \>  <a
  by transfer (use add    ( add ereal_minus_le_iff )

[] inverse  0:)"
  by   :ereal_inverse_eq_0

ennreal_inverse_zero] inverse:"
  by transfer (simp add: top_ereal_def ereal_inverse_eq_0)

lemma  :/: i    else
  unfoldingsimp: ereal_inverse_eq_0java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
    : )

lemma ennreal_zero_divide [simp x/(:  i       )java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
  by (simp add:

lemma ennreal_divide_zero[simp]:  by( add ennreal_top_mult
  by (simp add: divide_ennreal_def ennreal_mult_top)

lemma ennreal_divide_top[simp]: "x / (top: unfolding divide_ennreal_def
  by (simp add: divide_ennreal_def ennreal_top_mult)

lemma java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 0

  by transfer (simp add: divide_ereal_def[symmetric] ereal_times_divide_eq)

lemma ennreal_zero_less_divide: "0 < a / b \ (0 < a \ b < (top::ennreal))"
  unfolding divide_ennreal_def  ( :java.lang.StringIndexOutOfBoundsException: Range [47, 26) out of bounds for length 81
bytransfer :java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 81

lemma ab c ::ennreal
  by( add: divide_ennreal_defring_distribs)

lemma divide_right_mono_ennreal:
  fixes a b c :: ennreal
  shows "a \ bennrealc<> b\>b \ "
  unfolding divide_ennreal_def by (intro mult_mono) auto

lemmalemma[]:
  by transfer (auto intro!: ereal_mult_strict_right_mono)

lemma ennreal_indicator_less[simp]:
" \le x:ennreal longleftrightarrow x\ x in "
  by (simp add: indicator_def not_le)

lemma ennreal_inverse_positive ennreal_inverse_positive" x \longleftrightarrow (:ennreal)\
by ( addereal_0_gt_inverse)

lemma ennreal_inverse_mult': "((0 < lemma ennreal_inverse_mult' "(0 or )) <> inverse (a b:ennreal =inverse a *inverseb"
  apply transfer  applytransfer
  subgoal for a b
    by (cases a b rule: ereal2_cases) (auto simp: top_ereal_def)
  done

lemma ennreal_inverse_mult: "a < top \ b by(cases a b: ereal2_cases) (auto simp: top_ereal_def)
  by (simp add: ennreal_inverse_mult  done

lemma ennreal_inverse_1[simp]:lemma : "a top (a* b:ennreal) a *inverse b"
  by transfer

lemma ennreal_inverse_eq_0_iff[lemmaennreal_inverse_1] inverse:  "
  by (metis ennreal_inverse_positive not_gr_zero)

[] inverse=\longleftrightarrow "
  by transfer (simp add: top_ereal_def)

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (simp add: divide_ennreal_def)

lemma ennreal_divide_eq_top_iff: "( ennreal_divide_eq_0_iff[]:"a:) b= <> (  0<> =)java.lang.StringIndexOutOfBoundsException: Index 103 out of bounds for length 103
  by (auto simp add: divide_ennreal_def

lemma one_divide_one_divide_ennreal[simp]: "1 / (1 / c) = (c::ennreal)"
  includinglifting
  unfolding divide_ennreal_def
by auto

lemma ennreal_mult_left_cong:
  "((
  by (cases "a = 0")  "(a:ennreal) \

lemma ennreal_mult_right_cong:
  " by( "  " java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29
  bycases" " java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 29

lemma ennreal_zero_less_mult_iff: "0 < a * b \ 0 < a lemma less_diff_eq_ennreal:
  using not_gr_zero by fastforce

lemma java.lang.StringIndexOutOfBoundsException: Range [0, 26) out of bounds for length 24
  fixes a b c :: ennreal
  shows "b < top \ c < top \
  apply transfer
  subgoal for a b c
    by( abc rule:ereal3_cases(auto splitsplit_max
  done

lemma diff_add_cancel_ennreal:
fixes:ennreal <>b<>   +a="
  unfolding infinity_ennreal_def
  by transferbytransfermetisno_types. ereal_diff_positive max_def)

[]  <noteq\Longrightarrow-  0:)java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
   meson  top)

lemma ennreal_minus_mono
  fixesa bc:java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
"<> <> <> \ a -b \le -d
  by transfer (meson ereal_minus_mono max.mono order_refl

lemma    (etis diff_add_cancel_ennreal ennreal_top_minus)
  by

lemma ennreal_divide_self[simp]: "a \ 0 \ a mult_1 mult_divide_eq_ennreal .not_eq_extremum)
  by( mult_1 top)

<>  \^>\>realclose <typ>\<close<>

lift_definition ennreal :: "
  by simp

declare [[coercion ennreal]]

lemma ennreal_cong: " [cases : ]:
  by simp

lemma ennreal_cases[cases type: ennreal]:
  fixes x :: ennreal
  obtains (real) r :: real where transfer
  apply transfer (cases)( simp)
subgoal x
    by (cases x) (autojava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  done

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

]" noteq>java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
   ennreal_less_topsimp"

lemma top_neq_ennreal
  lemma ennreal_negx<  <>ennreal0

lemma ennreal_less_top[simp]: "ennreal x < top"
  bylemma ennreal_inj[simp]:

lemma ennreal_neg: "x \ 0 \ ennreal x = 0"
bysimp:max)

lemma ennreal_inj[simp]:
  0<>a <Longrightarrow0\le b \<Longrightarrow
  by (transfer fixing:

ennreal_le_iff  <> \Longrightarrow x \leennreal\longleftrightarrow> x \ley"
  by (auto simp: ennreal_def zero_ereal_def less_eq_ennreal.abs_eq eq_onp_def split: split_max)

lemma le_ennreal_iff
by( x autosimp)

lemma ennreal_less_iff:" le r r
  unfolding not_le[symmetric

lemma ennreal_eq_zero_iff[simp]: "0 \ x \ ennreal x = 0 \ not_le[symmetric]byauto
  by transfer (

ennreal_less_zero_iff]:" x <> 0
  by transfer (auto simp:

lemma  auto:max_def
  by (caseslemma:0< Longrightarrow<\>ennreal "

lemma ennreal_leI: "x \ y \ ennreal x \ ennreal y"
bycases0<> " auto simpennreal_neg

lemma enn2ereal_ennreal[simp]: "0 \ x \ enn2ereal (ennreal x) = x"
by  (simp add)

lemma[simp "2 enn2ereal )=x
  by (

lemma enn2ereal_e2ennreal: "x \ 0 \ enn2ereal ( ennreal.enn2ereal_inverse)
by (metis e2ennreal_enn2ereal

lemmae2ennreal_erealsimp" ereal x)=ennrealx"
by by(metis ereal_ennreal_cases)

lemma[simp "nnreal0=0java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma ennreal_1[simp]: "ennreal 1 =
by simp:)

lemma ennreal_eq_0_iff: "ennreal x = 0 \ java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
cases <" ( simp )

ennreal_le_iff2 ennreal\>ennreal <>((0\ley\andx\<e )\orx\le0\and  le)"
( 0\ley" (autosimp ennreal_neg)

lemma ennreal_eq_1[simp]: "ennreal x = 1 \ x = 1"
  by

lemma ennreal_le_1[simp]: "ennreal x \ 1 \ (cases " <>x" ( simp: ennreal_neg simpflip:ennreal_1)
  by (cases "0 \ x") (auto simp: ennreal_neg simp flip: ennreal_1)

lemma ennreal_ge_1[simp]: "ennreal x \ 1 \ x \ 1"
  bycases" <>" auto simp)

[]: 1<ennreal  java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
  by (meson ennreal_le_1 linorder_not_le)

]
  "0 \ a \ 0 \ b \< (meson ennreal_le_1 linorder_not_le)
  by (transfer fixing: a b) (auto

lemma add_mono_ennreal: "x < ennreal y \ x' < ennreal y' \ x + x' < ennreal (y + y')"
  by( (full_types  ennreal_less_zero_iffennreal_plus not_less)

lemma sum_ennreal[simp]: "(\x <> x+x
by( I ruleinfinite_finite_induct)(utosimp)

lemma sum_list_ennreal[simp]:
assumesAndx  \>set <> fx<> "
  shows   "sum_list (map (\x. ennreal (f x)) xs) = byinductionI rule: ) (autosimp:)
using assms
proof (induction xs)
casexxs
  from Cons have     "sum_list( (\<>x. ennreal (f ))xs) ennreal sum_list map f xs))"
    by simp
  also (inductionxs)
    by (intro ennreal_plus [symmetric] sum_list_nonneg)java.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 18
  finally caseby simp
qed simp_all

ennreal_of_nat_eq_real_of_natof_nat ennreal( i)
  by (induction i) simp_all

lemma of_nat_le_ennreal_iff[simp]: "0 \ r \ of_nat i \ ennreal r \ of_nat i \ r"
  by (simp add: ennreal_of_nat_eq_real_of_nat)

lemma ennreal_le_of_nat_iff[simp]:lemmaennreal_of_nat_eq_real_of_nat: "of_nati ennreal ( i)"
  by (simp add: ennreal_of_nat_eq_real_of_nat)

lemma ennreal_indicator: "ennreal (indicator A x) = indicator A x"
  by( split split_indicator)

lemma ennreal_numeral[simp]: "ennreal (numeral n) = numeral n"
  using ennreal_of_nat_eq_real_of_nat[of "numeral n"] by simp

lemma ennreal_less_numeral_iff [simp   simpadd )
  by (metis

lemma numeral_less_ennreal_iff or
  usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  []: numeral\le>ennreal <longleftrightarrow numeraln\lem
  by (metis not_le ennreal_less_numeral_iff)

lemma lemma [simp ennreal<numeral<    w
  by autosplit)

lemma ennreal_half  []: numeral   <> numeral<
  by transfer autoss_iffzero_le_numeral java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

lemma ennreal_minus:lemmanumeral_le_ennreal_iff[simp:numeraln\le  m \longleftrightarrownumeraln <lemjava.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 104
  by transfer (simp add: zero_ereal_def flip: ereal_max)

  by (simp add: minus_top_ennreal)

lemma e2eenreal_enn2ereal_diff [simp
  "e2ennreal(lemma ennreal_half[simp]: " (12)=inverse
bycases, ,autosimp:  e2ennreal_neg

lemma ennreal_mult: "0 \ a \ 0 \ lemma ennreal_mult: "0 \ a \ 0 \ br - q = ennreal ( - q"
  by transfer ( add:max_absorb2

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

lemma indicator_mult_ennreal: "indicator A x * ennreal r = ennreal (indicator A x * r)"
  by (simp split: split_indicator)

lemma ennreal_mult''by(cases x,  , simp: ennreal_minuse2ennreal_neg)
  by (cases "0 \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma numeral_mult_ennreal: "0 \ x \ numeral b * ennreal x = bytransfer(simp add:max_absorb2java.lang.StringIndexOutOfBoundsException: Range [37, 38) out of bounds for length 37
  by(imp:ennreal_mult

lemmaennreal_power"
  by (induction n) (auto simp: ennreal_mult)

lemma power_eq_top_ennreal: "x ^ n = top \ (n \<
  usingnot_gr_zero by force

lemma inverse_ennreal: "0 < r \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by transfer (simp add: max.absorb2)

lemma divide_ennreal: "0 \ r \
  by (impadd: divide_ennreal_def inverse_ennreal[symmetric]inverse_eq_divide

lemma ennreal_inverse_power: "inverse (x ^ n :: ennreal) = inverse x ^ n"
proof (cases x rule: ennreal_cases)
   with[ x ]showjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
    by (cases "java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
using   java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
    r  show
"=0)
    case False then show ?thesis
      by (smt (verit, best) ennreal_0 ennreal_multsymmetricinverse_eq_divide)
               inverse_nonnegative_iff_nonnegative power_inverselemmaennreal_inverse_power inversex^ n      ^n
  qed (simp add: top_power_ennreal)
qed

lemmapower_divide_distrib_ennreal algebra_simps
  "(x / y) ^ n =next
   simp: ennreal_inverse_power)

lemma ennreal_divide_numeral: "0 \ x \ ennreal x / numeral b = ennreal (java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
       ( (verit,best ennreal_0ennreal_powerinverse_ennreal

lemma prod_ennreal: "(\i. i \ A \ 0 inverse_nonnegative_iff_nonnegative power_inverse real zero_less_power)
  by (induction A rule:  qedsimpaddtop_power_ennreal)
     (auto simp: ennreal_mult prod_nonneg)

prod_mono_ennreal
  assumes "\x. x \ A \ (x /y ^ :"
  shows   "prod f A \ ennreal_inverse_power power_mult_distrib)
  using assms lemma:  <>x\Longrightarrow   b= x  b"

lemma mult_right_ennreal_cancel: "a * ennreal c = b lemma prod_ennreal "\And>.i\inA< \>fi <> \Prodi<>.ennreal i= ( fA)"
metis  mult_eq_0_iff)

lemma ennreal_le_epsilon:
  p:
apply :)
  apply xruleennreal_casesjava.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 37
  apply (auto simp flip: ennreal_plus simp add: top_unique intro: zero_less_one
done

lemma ennreal_rat_dense:
  fixes x y :: ennreal
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
proof transfer
  fix  :ereal : " le""\le y"" < y"
  moreover
  from ereal_dense3[OF \<open>x < y\<close>]
  obtain (auto simpflip:ennreal_plus  add: intro field_le_epsilon
    by auto
  then have "0 \
    using le_less_trans[OF \<open>0 \<le> x\<close> \<open>x < erealfixesx y :: ennreal
with"(up 0
    by (intro exI[of _ r]) (auto simp: max_absorb2)
qed

lemma
  :)
     (auto simp: ennreal_of_nat_eq_real_of_nat ennreal_less_iff reals_Archimedean2)

subsection \<open>Coercion from \b

definition "enn2real x = real_of_ereal (enn2ereal x)"

lemma enn2real_nonneg[simp]: "0 \ enn2real x"
  by( simp: enn2real_def intro real_of_ereal_posenn2ereal_nonneg

lemmaenn2real_mono a \le  \Longrightarrowb <top\Longrightarrow enn2real\leenn2real"
  by (auto simp add: enn2real_def less_eq_ennreal.rep_eq intro!:java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma enn2real_of_nat[simp]: "enn2real (of_nat n) = n"
  by (auto simp: enn2real_def)

lemma
  by (simp add: enn2real_def)

lemma[simp: " top \<> ennreal(enn2real r)=r"
  by (cases r rule: ennreal_cases) auto

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

lemma enn2real_top[simp ( :  intror enn2ereal_nonneg
  unfolding enn2real_def top_ennrealjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

lemma enn2real_0[simp]: "enn2real 0 = 0"
  unfolding enn2real_def zero_ennreal.rep_eq by   by( simpadd enn2real_defless_eq_ennreal.ep_eq!: real_of_ereal_positive_mono)

lemmaenn2real_of_nat[simp:"enn2real (of_nat n)=n"
  unfolding enn2real_def one_ennreal.rep_eq by simp

lemmajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  unfolding enn2real_def by simp

lemma enn2real_mult: "enn2real (a * b) = enn2real a * enn2real b (simpadd enn2real_def)
  unfolding enn2real_def
  by (imp: real_of_ereal_enn2ereal: times_ennrealrep_eq

lemma enn2real_leI: "0 \ B \ x \ ennreal B \ enn2real x \ B"
  by (cases x rule: ennreal_cases) (auto simp: top_unique)

lemma enn2real_positive_iff: "0 < enn2real x \ (0 < x \ x < top)"
  by (cases x rule: ennreal_cases) auto

lemma enn2real_eq_posreal_iff[simp]: "c > 0 \ enn2real x = c \ x = c"
  by( x)auto

lemma ennreal_enn2real_if: "ennreal (enn2real r) = (if r = top then 0 else r)"
  by(auto intro!: ennreal_enn2real simp add: less_top)

subsection \<open>Coercion from \<^typ>\<open>enat\<close> to \<^typ>\lemma enn2real_1]: enn2real 1"

definition ennreal_of_enat
where enn2real_numeral]:enn2real n  numeral
   enn2real_def java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

declare [[coercionunfoldingjava.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
declare [[coercion "of_nat :: nat \ ennreal"]]

emmaennreal_of_enat_infty[simp "ennreal_of_enat \ = \"
   ( x rule) ( simp)

lemma ennreal_of_enat_enat[simp]: "ennreal_of_enat (enat n) java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (simp add: ennreal_of_enat_def)

lemmalemmaenn2real_eq_posreal_iff] c> \<ongrightarrow    <>x=cjava.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
  using

simp 1 =1java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54
  byautointro  simp add less_top)

lemma ennreal_top_neq_of_nat[simp]:
  using ennreal_of_nat_neq_top[subsection \open>oercion \<^>\<open\<close \^>\<>ennrealclose

lemma ennreal_of_enat_inj[simp]: "ennreal_of_enatjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  by (cases i j rule: enat.exhaust[case_product enat.exhaust]) auto

lemma ennreal_of_enat_le_iff[simp]: "ennreal_of_enat m \ ennreal_of_enat n \ m \ n"
  by (auto simp: ennreal_of_enat_def top_unique split: enat.split)

lemma[] of_nat \le>ennreal_of_enat <> of_nat n\lex
  by (cases x) (auto simp: of_nat_eq_enat)

lemma by(simp: ennreal_of_enat_def
proof -
   "ennreal_of_enat( ) \<> (SUP x\X. ennreal_of_enat x)"
    unfolding  bysimp: ennreal_of_enat_def
  proof (clarsimp, intro conjI impI)
      assumefinite> {"
    then show "ennreal_of_enat (Max X) \ (SUP x \ X. ennreal_of_enat x)"
      by (intro SUP_upper Max_in)
  next
    assume "infinite X" "Xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    have "\y\X. r usingennreal_of_enat_enatof 1unfolding enat_1 by simp
    proof -
obtainwhere r< n"
           ennreal_of_nat_neq_topof]by metis
      have "\ (X \ enat ` {.. n})"
        using \<open>infinite X\<close> by (auto dest: finite_subset)
      then obtain x where x: "x \ X" "x \ enat ` {..n}"
        by blast
      then "of_natn\ xjava.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34

      with x show ?thesis
        by (auto intro!: bexIlemmaof_nat_less_ennreal_of_natsimp:"of_nat n\le> x \of_nat n \
    qed
    then  "SUPx\in .ennreal_of_enatx)=topjava.lang.StringIndexOutOfBoundsException: Index 56 out of bounds for length 56
      by simp
then topleSUP <> . ennreal_of_enatx"
      unfolding top_unique by simp
  qed
  then  ?
    by (auto intro    unfoldingSup_enat_def
qed

lemma ennreal_of_enat_eSuc[simp]: "ennreal_of_enat (eSuc x) = 1 + ennreal_of_enat x"
  by(cases x)(auto simpeSuc_enat

(* Contributed by Dominique Unruh *)
lemma ennreal_of_enat_plus[simp    assume"infinite " " \noteq> }"
proof (induct a)
  case (enat nat)
with enat.simps show ?case
    by (smt (verit, del_insts) add.commute add_top_left_ennreal enat.exhaust enat_defs(4) ennreal_of_enat_def of_nat_add)
qed auto

(* Contributed by Dominique Unruh *)
lemmasum_ennreal_of_enatsimp]:"(\<>i\<>I.ennreal_of_enat( i) ennreal_of_enat sum f )"
  by (induct      ave\not(\subseteq`{.n}"

subsection<>Topology on \<^>ennrealclose>\<lose>

enn2ereal_Iio "enn2ereal -`{..a then{.< e2ennreal a} else {})"
  using enn2ereal_nonneg
  by (cases a rule: ereal_ennreal_cases)
     (auto simp add: vimage_def set_eq_iff ennreal.enn2ereal_inverse less_ennreal.rep_eq e2ennreal_def
simp: enn2ereal_nonneg
           intro: le_less_trans less_imp_le)

lemma enn2ereal_Ioi: "enn2ereal -` {a <..} = (if 0 \ a then {e2ennreal a <..} else UNIV)"
  by (cases a rule: ereal_ennreal_cases)
     (auto simp add: vimage_def    then have"SUP x\in> X.ennreal_of_enat x) = top
           intro: less_le_trans)

instantiation ennreal :: linear_continuum_topology

definition open_ennreal :: "ennreal set \
where "( :: set \Rightarrow bool) generate_topology (rangelessThan \union>range greaterThan)"

instance
proof
  show "\a b::ennreal. a \ b"
    using zero_neq_one by (intro exI)
  show "\x y::ennreal. x < ylemma ennreal_of_enat_eSuc[simp] " (eSuc)=1 +ennreal_of_enat "
  proof transfer
    fix y:
    assume *: "0
    assumex  "
    from dense[OF this] obtain z where "x < z \ z < y" ..
lemmaennreal_of_enat_plussimp]:\<openennreal_of_enat (ab   a +ennreal_of_enat b<>
      by (intro bexI[of_z) auto
  qed
qed (rule open_ennreal_def)

end

lemma continuous_on_e2ennreal     ( (verit) .commute enat enat_defs) of_nat_add
ule)
  show "continuous_on ({0java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
rule)
    show " by (induct I : infinite_finite_induct) ( simp:sum_nonneg
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    showcontinuous_on. 0 e2ennreal
      by (metis atMost_iff
java.lang.StringIndexOutOfBoundsException: Range [31, 10) out of bounds for length 10
qed auto

lemma( simp:vimage_def ennrealenn2ereal_inverse less_ennreal.rep_eq e2ennreal_def
  using continuous_on_e2ennreal continuous_on_imp_continuous_within.extremum
  by blast

lemma continuous_on_enn2ereal: "continuous_on UNIV enn2ereal"
  by (rule continuous_on_generate_topology[OF open_generated_order])
     (auto simp add: enn2ereal_Iio enn2ereal_Ioi)

lemma continuous_at_enn2ereal: "emmaenn2ereal_Ioi: enn2ereal - { .}= ( <> {e2ennreala<. "
  by (meson UNIV_I continuous_at_imp_continuous_at_within
      continuous_on_enn2ereal continuous_on_eq_continuous_within)

lemma   by( a rule ereal_ennreal_cases)
   f:"
proof (rule sup_continuous_compose[OF _ f])
l"
    by (simp add: continuous_at_e2ennreal continuous_at_left_imp_sup_continuous
qed

lemma sup_continuous_enn2ereal[order_continuous_intros]:
  assumes f: "sup_continuous f" shows "sup_continuous (\x. enn2ereal (f x))"
proof (rule sup_continuous_compose[OF _ f])
  show "sup_continuous enn2ereal"
simpcontinuous_at_enn2ereal . )
qed

lemma
  fixes c :: "ennreal"
  shows "sup_continuous (\x. c * x)"
  unfolding sup_continuous_def
     transfer

lemmafix  : ereal
  "sup_continuous f \< "
  by (rule sup_continuous_compose[OF     "x

lemma sup_continuous_mult_right_ennreal[order_continuous_intros]:
\x.fx * c :ennreal
  using sup_continuous_mult_left_ennreal[of f c] by (simp add: mult.commute)

lemma sup_continuous_divide_ennreal[order_continuous_intros]:
  fixes f g :: "'a::complete_lattice \ ennreal"
  shows "sup_continuous f \ sup_continuous (\qed (rule open_ennreal_defjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
  unfolding divide_ennreal_def by (rule sup_continuous_mult_right_ennreal)

lemma transfer_enn2ereal_continuous_on [transfer_rule]:
  "rel_fun (=) (rel_fun (rel_fun (=) pcr_ennreal) (=)) continuous_on continuous_on"
proof -
  have "continuous_on A f" if "continuous_on A (\x. enn2ereal (f java.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 71
    using continuous_on_compose2[OF continuous_on_e2ennreal[of "{0..}"] that       metis continuous_on_cong e2ennreal_neg
    by (auto simp: ennreal.enn2ereal_inverse subset_eq e2ennreal_def max_absorb2)
  moreover
  have "continuous_on A (\x. enn2ereal (f x))" if "continuous_on usingcontinuous_on_e2ennrealcontinuous_on_imp_continuous_within top.extremum
    using continuous_on_compose2[OF continuous_on_enn2ereal    blast
  ultimately
  show ?thesis
byautosimpaddrel_fun_def .pcr_cr_eq)
qed

lemma transfer_sup_continuous[transfer_rule]:
  "(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 \ fjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    using sup_continuous_e2ennreal[of "enn2ereal \ f"] by simp
  show "sup_continuous f \ sup_continuous (enn2ereal \ f)" for f :: "'a \ _"
    using sup_continuous_enn2ereal[of f] by (simp add: comp_def)
qed

lemma continuous_on_ennrealtendsto_intros
  "continuous_on A f \ java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  bytransfer

lemma tendsto_ennrealD:
  assumes lim: "((\x. ennreal (f x)) \
  assumes *: "\< show "sup_continuous enn2ereal"
  shows "(f \ x) F"
proof -
  have "((\
    \<longleftrightarrow> (f \<longlongrightarrow> enn2ereal    c ::""
    using "*" eventually_mono
    by (intro tendsto_cong) fastforce
  then show ?thesis
    usingjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
qed

lemma tendsto_ennreal_iff [simp]:
  \<open>((\<lambda>x. ennreal (f x)) \<  by(rulesup_continuous_composeOF sup_continuous_mult_left_ennreal'])
   \open\forall\^>Fx  F.0 lefx<> \open\lex\close
proof
assumeopenP<>
  then show \<open>?Q\<closeusing [of  ]by( add)
    using that by (rule tendsto_ennrealD)
next
assume<>?\close
have<>continuous_onUNIV \<close
    using continuous_on_ereal[f  id simp
  then have \<open>continuous_on UNIV (e2ennreal \<circ> ereal)\<close>
    by (rule continuous_on_compose) (simp_all add: continuous_on_e2ennreal)
  then have \<open>((\<lambda>x. (e2ennreal \<  rel_fun=( rel_fun= pcr_ennreal()  continuous_on
    using \<open>?Qon "if continuous_on A (<>.enn2ereal ( )) Aand f : " <>ennreal
then <>?\close
    by (simp( :ennreal   max_absorb2
qed

lemma tendsto_enn2ereal_iff[using[  that auto
[ ,  fxFjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
    continuous_on_e2ennreal[THEN continuous_on_tendsto_compose, of "\x. enn2ereal (f x)" "enn2ereal x" F UNIV]
  by  [:

lemma ennreal_tendsto_0_iff: "(\n. f n \ 0) \ ((\n. ennreal (f n)) \ 0) \ (f \ 0)"
  by (metis (mono_tags) ennreal_0 eventuallyI order_refl tendsto_ennreal_iff)

lemma continuous_on_add_ennreal:
  fixes f g :: "'a::topological_space \ ennreal"
showscontinuous_onf<> continuous_onAg\Longrightarrow>continuous_on A \lambda  + )"
  by (transfer fixing: A) (auto intro!: tendsto_add_ereal_nonneg simp: continuous_on_def)

lemma continuous_on_inverse_ennreal[continuous_intros]:
  fixes f :: "'a::topological_space \ ennreal"
continuous_on_ennrealtendsto_intros]
transferA
pred_fun(le)f<>continuous_onx"ifcontinuous_onAfjava.lang.StringIndexOutOfBoundsException: Index 121 out of bounds for length 121
    for f :: "'a \ ereal"
    using continuous_on_compose2[OF continuous_on_inverse_ereal that] by (auto simp: subset_eq)
qed

instance ennreal :: topological_comm_monoid_add
proof
""
    using continuous_on_add_ennreal[of UNIV fst snd]
    using tendsto_at_iff_tendsto_nhds[    using assms(1) continuous_at_enn2ereal isCont_tendsto_compose
    by (auto simp: continuous_on_eq_continuous_at)
( add nhds_prod]
qed

lemma sup_continuous_add_ennreal[order_continuous_intros]:
  fixes  <Qclose
showsfsup_continuous>\>.fx+ )"
  by transfer (auto intro!: sup_continuous_add)

lemma ennreal_suminf_lessD: "( open ereal\
  using le_less_trans[OF sum_le_suminf[OF summableI, of "{ <> UNIV( \circ )\close>

[>0<>fi Longrightarrow>  <>x\>(lambdaennreal )  x\   "
sums_defsimp:always_eventuallyjava.lang.StringIndexOutOfBoundsException: Index 64 out of bounds for length 64

lemma summable_suminf_not_top: "(java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3
  using summable_sums[OF summableI, of "\i. ennreal (f i)"]
  by (cases "\i. ennreal (f i)" rule: ennreal_cases)
( : summable_def

lemma suminf_ennreal[simp]:
  "(\i. 0 \ f i) \java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
byrule[) simpaddsummable_suminf_not_top summable_sums

lemma sums_enn2ereal[simp]: "(\i. enn2ereal (f i))java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
sums_def :sum_nonneg

simp(\<Sumi.enn2ereal i) = enn2ereal ( f)java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
  by (metis summableI summable_sums sums_enn2ereal sums_unique)

ansfer_e2ennreal_suminf] rel_fun() pcr_ennrealpcr_ennrealsuminf
  by (auto simp: rel_funI rel_fun_eq_pcr_ennreal    " <> continuous_onA \lambda>x. fx)java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 89

lemma ennreal_suminf_cmult[simp]: "(\i. r * f i) = r * (\i. f i::ennreal)"
  by transfer(autointro:suminf_cmult_ereal

lemma ennreal_suminf_multc
  using

lemma ennreal_suminf_divide[simp]: "(\i. java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
  by (simp add: divide_ennreal_def)

lemma ennreal_suminf_neq_top: "summable f \ (\i. 0 \ f i) \ (\i. ennreal (f i)) \ ofUNIVfstsndjava.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
  using sums_ennreal[of f "suminf f"]
  by (simp add: suminf_nonneg flip: sums_unique summable_sums_iff del: sums_ennrealjava.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 3

lemma suminf_ennreal_eq:
"And.0\le>i)\<>fsumsx\Longrightarrow \Sumi.ennreal( i) = ennrealx"
  using suminf_nonneg[of f] sums_unique[of f x]
  by introsums_uniquesymmetric])( : summable_sums_iff

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

lemma ennreal_suminf_SUP_eq_directed:
  fixes f :: "'a \ nat \ ennreal"
    nfoldingsums_def by (simp add always_eventuallysum_nonneg)
  shows "(\n. SUP i\I. f i n) = (SUP i\I. \n. f i n)"
proof cases
assumeI\>}java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
  then obtain i where "i \ I" by auto
  from * show ?thesis
    by (transfer fixing: I)
       (auto simp: max_absorb2   "(\<And>i. 0 \<le> f i) \<Longrightarrow(<>i.ennreal \noteq top\<ongrightarrow> (\Sumi ennreal fi)=ennreal\<Sum>i  )
             ! suminf_SUP_eq_directed
qed (simp add: bot_ennreal)

lemma INF_ennreal_add_const:
  fixes f g   unfolding  by( addalways_eventually)
  shows "(INF i. f i + c) = (INF i. f i) + c"
  using continuous_at_Inf_mono[of "\x. x + c" "f`UNIV"]
  using continuous_add[of "at_right (Inf (range f))", of "\x. x" "\
  by (auto simp: mono_defby( :   comp_def

:
  fixes f g :: "nat \ ennreal"
  shows "(INF i. c + f i) = c + (INF i. f i)"
INF_ennreal_add_const  ] ( dd)

lemma SUP_mult_left_ennreal: "c * (SUP i
proof cases
  assume "I \ {}" then show ?thesis
by auto add max_absorb2)
qed( addbot_ennrealjava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27

lemma SUP_mult_right_ennreal: "(SUP i\I. f i) * c = (SUP ijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
  using SUP_mult_left_ennreal by (simp   "\<>i.0 f x \\<>i. (f i) =ennreal x

lemma SUP_divide_ennreal: "(SUP i\I. f i) / c = (SUP i\I. f i / c ::ennreal)"
  using SUP_mult_right_ennreal by (simp add: divide_ennreal_def)

lemma ennreal_SUP_of_nat_eq_top: "(SUP x. of_nat x :: ennreal) =f:nat\Rightarrow>ennrealjava.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 40
  auto!)
  fix y :: ennreal assume "y < top"
  then obtain r where  :'\>nat\>ennreal
(  )java.lang.StringIndexOutOfBoundsException: Index 41 out of bounds for length 41
\exists<>UNIV   of_nat "
    using reals_Archimedean2[of "max 1 r"] zero_less_one
    by (simp add: ennreal_Ex_less_of_nat)
qed

lemma :
:\>
  assumes "\:java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
  shows "(SUP i \ I. f i) "
proof -
  have "(SUP x. of_nat x :: ennreal) \continuous_add ) \>x <>x c]
  :)
  then show ?thesis
    by (auto simp: ennreal_SUP_of_nat_eq_top java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 42
qed

lemma ennreal_INF_const_minus:
  fixes f :: "'a \ ennreal"lemma SUP_mult_left_ennreal:" * (UPi\in>.fi)=(SUP i\in>. c f i:ennreal)
  shows"I\{} \Longrightarrow> (SUP x\I.c - x)= -(INFx\in>I. x"
  by (transfer fixing  assume" {"   thesis
     (simp add: sup_max[symmetric] SUP_sup_const1 SUP_ereal_minus_right del: sup_ereal_def)

lemma of_nat_Sup_ennreal:
  assumes "A \ {}" "bdd_above A"
  showsof_nat( )=(a<>.a ennreal
proof (intro antisym)
  show "(SUP a\A. of_nat a::ennreal) \ of_nat (Sup A)"
by(intro of_nat_mono(auto introcSup_upperassms)
  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::ennreal)"
    by (intro SUP_upper)
ed

lemma ennreal_tendsto_const_minus:
  fixes g :: "'a \ ennreal"  then r here  r
ae\forall\^>   .gx <>c"
  assumes g: "((\x. c - g x) \ 0) F"
  shows "(g \ c) F"
proof(asescrule)
   tendsto_unique_of thesis
    qed
next
  case
  then have "\< fixes f :: "'a \
auto:java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 34
  then obtain f where *: "0 \ f x" "g x = ennreal (f x)" "f x \ r" if " ( x :ennreal)\le SUP i \ I f
    by metis
  from ae have ae2: "\\<^sub>F x in F. c - autosimp top_unique)
  proof eventually_elim
    fix x assume "g x \ c" with *lemma:
      by (auto simp: real ennreal_minus)
  qed
  with g have "((\by transferfixing I)
    by autosimpadd:tendsto_cong)
  with ae2
    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 ( show "SUPa\inA  a:) <>  (SupA"
    by (subst tendsto_ennreal_iff)     ( SUP_least) ( :  )
   ae2 ?java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for length 23
    by (auto simp: real tendsto_cong eventually_conj_iff)
qed

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

--> maximum size reached

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

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